formal/mizar/anproj11.miz

:: Principle of Duality in Real Projective Plane: a Proof of the Converse 
:: of {D}esargues' Theorem and a Proof of the Converse of {P}appus' 
:: licenses, or see http://www.gnu.org/licenses/gpl.html and
:: http://creativecommons.org/licenses/by-sa/3.0/.

environ

 vocabularies INCSP_1, ANPROJ11, REAL_1, XCMPLX_0, ANPROJ_1, ANPROJ_2,
      PENCIL_1, MCART_1, EUCLID_5, ARYTM_1, ARYTM_3, CARD_1, EUCLID, FUNCT_1,
      NUMBERS, PRE_TOPC, RELAT_1, SUBSET_1, SUPINF_2, ANPROJ_9, TARSKI,
      INCPROJ, RVSUM_1, BKMODEL1, CARD_FIL, PROJRED2, PBOOLE, RELAT_2, AFF_2,
      VECTSP_1, ANALOAF;
 notations TARSKI, SUBSET_1, XCMPLX_0, PRE_TOPC, RVSUM_1, COLLSP, INCPROJ,
      ANPROJ_9, XREAL_0, NUMBERS, FUNCT_1, FINSEQ_2, EUCLID, ANPROJ_1,
      BKMODEL1, STRUCT_0, RLVECT_1, EUCLID_5, INCSP_1, PROJRED2, ANPROJ_2;
 constructors MONOID_0, EUCLID_5, ANPROJ_9, BKMODEL1, EUCLID_8, PROJRED2;
 registrations BKMODEL3, ORDINAL1, ANPROJ_1, STRUCT_0, XREAL_0, MONOID_0,
      EUCLID, VALUED_0, ANPROJ_2, FUNCT_1, FINSEQ_1, XCMPLX_0, INCPROJ, PASCAL;
 requirements SUBSET, NUMERALS, ARITHM, BOOLE;
 equalities BKMODEL1, XCMPLX_0, COLLSP, INCPROJ, ANPROJ_9, EUCLID_5;
 expansions TARSKI, XBOOLE_0, STRUCT_0, PROJRED2;
 theorems EUCLID_8, EUCLID_5, ANPROJ_1, ANPROJ_2, EUCLID, XCMPLX_1, RVSUM_1,
      FINSEQ_1, ANPROJ_8, INCPROJ, ANPROJ_9, XBOOLE_0, COLLSP, BKMODEL1,
      EUCLID_4;

begin ::Preliminaries

theorem
  for a,b,c,d,e,f,g,h,i being Real holds
  |{ |[a,b,c]|,
     |[d,e,f]|,
     |[g,h,i]| }| = a * e * i + b * f * g + c * d * h
                    - g * e * c - h * f * a - i * d * b
  proof
    let a,b,c,d,e,f,g,h,i be Real;
    reconsider p = |[a,b,c]|, q = |[d,e,f]|, r = |[g,h,i]| as
      Element of TOP-REAL 3;
A1: p`1 = a & p`2 = b & p`3 = c &
    q`1 = d & q`2 = e & q`3 = f &
    r`1 = g & r`2 = h & r`3 = i by EUCLID_5:2;
    |{ |[a,b,c]|,
       |[d,e,f]|,
       |[g,h,i]| }| = p`1 * q`2 * r`3 - p`3*q`2*r`1 - p`1*q`3*r`2
                      + p`2*q`3*r`1 - p`2*q`1*r`3 + p`3*q`1*r`2 by ANPROJ_8:27;
    hence thesis by A1;
  end;

theorem Th2:
  for a,b,c,d,e being Real holds
  |{ |[a,1,0]|,
     |[b,0,1]|,
     |[c,d,e]| }| = c - a * d - e * b
  proof
    let a,b,c,d,e be Real;
    reconsider p = |[a,1,0]|, q = |[b,0,1]|, r = |[c,d,e]| as
      Element of TOP-REAL 3;
A1: p`1 = a & p`2 = 1 & p`3 = 0 &
    q`1 = b & q`2 = 0 & q`3 = 1 &
    r`1 = c & r`2 = d & r`3 = e by EUCLID_5:2;
    |{ |[a,1,0]|,
       |[b,0,1]|,
       |[c,d,e]| }| = p`1 * q`2 * r`3 - p`3*q`2*r`1 - p`1*q`3*r`2
                      + p`2*q`3*r`1 - p`2*q`1*r`3 + p`3*q`1*r`2 by ANPROJ_8:27;
    hence thesis by A1;
  end;

theorem Th3:
  for a,b,c,d,e being Real holds
  |{ |[1,a,0]|,
     |[0,b,1]|,
     |[c,d,e]| }| = b * e + a * c - d
  proof
    let a,b,c,d,e be Real;
    reconsider p = |[1,a,0]|, q = |[0,b,1]|, r = |[c,d,e]| as
      Element of TOP-REAL 3;
A1: p`1 = 1 & p`2 = a & p`3 = 0 &
    q`1 = 0 & q`2 = b & q`3 = 1 &
    r`1 = c & r`2 = d & r`3 = e by EUCLID_5:2;
    |{ |[1,a,0]|,
       |[0,b,1]|,
       |[c,d,e]| }| = p`1 * q`2 * r`3 - p`3*q`2*r`1 - p`1*q`3*r`2
                      + p`2*q`3*r`1 - p`2*q`1*r`3 + p`3*q`1*r`2 by ANPROJ_8:27;
    hence thesis by A1;
  end;

theorem Th4:
  for a,b,c,d,e being Real holds
  |{ |[1,0,a]|,
     |[0,1,b]|,
     |[c,d,e]| }| = e - c * a - d * b
  proof
    let a,b,c,d,e be Real;
    reconsider p = |[1,0,a]|, q = |[0,1,b]|, r = |[c,d,e]| as
      Element of TOP-REAL 3;
A1: p`1 = 1 & p`2 = 0 & p`3 = a &
    q`1 = 0 & q`2 = 1 & q`3 = b &
    r`1 = c & r`2 = d & r`3 = e by EUCLID_5:2;
    |{ |[1,0,a]|,
       |[0,1,b]|,
       |[c,d,e]| }| = p`1 * q`2 * r`3 - p`3*q`2*r`1 - p`1*q`3*r`2
                      + p`2*q`3*r`1 - p`2*q`1*r`3 + p`3*q`1*r`2 by ANPROJ_8:27;
    hence thesis by A1;
  end;

theorem Th5:
  for u being Element of TOP-REAL 3 holds u is zero iff |( u, u )| = 0
  proof
    let u be Element of TOP-REAL 3;
    reconsider un = u as Element of REAL 3 by EUCLID:22;
    hereby
      assume u is zero;
      then 0.REAL 3 = u by EUCLID:66;
      then |( un,un )| = 0 by EUCLID_4:17;
      hence |( u, u )| = 0;
    end;
    assume |( u, u )| = 0;
    then un = 0.REAL 3 by EUCLID_4:17;
    hence thesis by EUCLID:66;
  end;

theorem
  for u,v,w being non zero Element of TOP-REAL 3 st |{u,v,w}| = 0
  holds ex p being non zero Element of TOP-REAL 3 st
  |(p,u)| = 0 & |(p,v)| = 0 & |(p,w)| = 0
  proof
    let u,v,w be non zero Element of TOP-REAL 3;
    assume
A1: |{u,v,w}| = 0;
    reconsider p = |[u`1,v`1,w`1]|,
               q = |[u`2,v`2,w`2]|,
               r = |[u`3,v`3,w`3]| as Element of TOP-REAL 3;
A2: p`1 = u`1 & p`2 = v`1 & p`3 = w`1 & q`1 = u`2 & q`2 = v`2 & q`3 = w`2 &
      r`1 = u`3 & r`2 = v`3 & r`3 = w`3 by EUCLID_5:2;
A3: |{ p,q,r }| = p`1 * q`2 * r`3 - p`3*q`2*r`1 - p`1*q`3*r`2 + p`2*q`3*r`1 -
      p`2*q`1*r`3 + p`3*q`1*r`2 by ANPROJ_8:27;
    |{u,v,w}| = u`1 * v`2 * w`3 - u`3*v`2*w`1 - u`1*v`3*w`2 + u`2*v`3*w`1 -
      u`2*v`1*w`3 + u`3*v`1*w`2 by ANPROJ_8:27;
    then consider a,b,c be Real such that
A4: a * p + b * q + c * r = 0.TOP-REAL 3 and
A5: a <> 0 or b <> 0 or c <> 0 by A1,A2,A3,ANPROJ_8:42;
A6: |[0,0,0]|
        = |[a * p`1,a * p`2,a * p`3]| + b * q + c * r by A4,EUCLID_5:4,7
       .= |[a * p`1,a * p`2,a * p`3]| + |[b * q`1,b * q`2,b * q`3]| + c * r
         by EUCLID_5:7
       .= |[a * p`1,a * p`2,a * p`3]| + |[b * q`1,b * q`2,b * q`3]|
         + |[c * r`1,c * r`2,c * r`3]| by EUCLID_5:7
       .= |[a * p`1+b*q`1,a * p`2+b*q`2,a * p`3+b*q`3]|
         + |[c * r`1,c * r`2,c * r`3]| by EUCLID_5:6
       .= |[a * p`1+b*q`1+c*r`1,a * p`2+b*q`2+c*r`2,a * p`3+b*q`3+c*r`3]|
         by EUCLID_5:6;
    reconsider p = |[a,b,c]| as non zero Element of TOP-REAL 3 by A5;
    take p;
    thus |(p,u)| = p`1 * u`1 + p`2 * u`2 + p`3 * u`3 by EUCLID_5:29
                .= a * u`1+p`2*u`2+p`3*u`3 by EUCLID_5:2
                .= a * u`1+b*u`2+p`3*u`3 by EUCLID_5:2
                .= a * u`1+b*u`2+c*u`3 by EUCLID_5:2
                .= 0 by A6,A2,FINSEQ_1:78;
    thus |(p,v)| = p`1 * v`1 + p`2 * v`2 + p`3 * v`3 by EUCLID_5:29
                .= a * v`1+p`2*v`2+p`3*v`3 by EUCLID_5:2
                .= a * v`1+b*v`2+p`3*v`3 by EUCLID_5:2
                .= a * v`1+b*v`2+c*v`3 by EUCLID_5:2
                .= 0 by A6,A2,FINSEQ_1:78;
    thus |(p,w)| = p`1 * w`1 + p`2 * w`2 + p`3 * w`3 by EUCLID_5:29
                .= a * w`1+p`2*w`2+p`3*w`3 by EUCLID_5:2
                .= a * w`1+b*w`2+p`3*w`3 by EUCLID_5:2
                .= a * w`1+b*w`2+c*w`3 by EUCLID_5:2
                .= 0 by A6,A2,FINSEQ_1:78;
  end;

theorem Th7:
  for u,v,w being non zero Element of TOP-REAL 3 st
  |(u,v)| = 0 & are_Prop w,v holds |(u,w)| = 0
  proof
    let u,v,w be non zero Element of TOP-REAL 3;
    assume that
A1: |(u,v)| = 0 and
A2: are_Prop w,v;
    consider a be Real such that
    a <> 0 and
A3: w = a * v by A2,ANPROJ_1:1;
    reconsider un = u,vn = v as Element of REAL 3 by EUCLID:22;
    thus |(u,w)| = |(a * vn,un)| by A3
                .= a * |(v,u)| by EUCLID_8:68
                .= 0 by A1;
  end;

theorem Th8:
  for a,u,v being non zero Element of TOP-REAL 3 st not are_Prop u,v &
  |(a,u)| = 0 & |(a,v)| = 0 holds are_Prop a,u <X> v
  proof
    let a,u,v be non zero Element of TOP-REAL 3;
    assume that
A1: not are_Prop u,v and
A2: |(a,u)| = 0 and
A3: |(a,v)| = 0;
    u <X> v is non zero by A1,ANPROJ_8:51;
    then reconsider uv = u <X> v as non zero Element of TOP-REAL 3;
A4: a`1 * u`1 + a`2 * u`2 + a`3 * u`3 = 0 &
      a`1 * v`1 + a`2 * v`2 + a`3 * v`3 = 0 by A2,A3,EUCLID_5:29;
    per cases by EUCLID_5:3,4;
    suppose
A5:   a`1 <> 0;
      then
A6:   u`1 = -a`2/a`1 * u`2 - a`3/a`1 * u`3 &
        v`1 = -a`2/a`1 * v`2 - a`3/a`1 * v`3 by A4,ANPROJ_8:13;
      set p1 = u,p2 = v;
      now
        reconsider r = a`1 as Real;
        thus
A7:     u <X> v = |[ 1 *( p1`2 * p2`3 - p1`3 * p2`2),
                     a`2/a`1 *( p1`2 * p2`3 - p1`3 * p2`2),
                     (a`3/a`1) * (- p1`3*p2`2 + p1`2*p2`3) ]| by A6
         .= ( p1`2 * p2`3 - p1`3 * p2`2) * |[ 1 ,a`2/a`1, a`3/a`1 ]|
           by EUCLID_5:8
         .= ( p1`2 * p2`3 - p1`3 * p2`2) * |[ a`1 / r, a`2 / r, a`3 / r ]|
           by A5,XCMPLX_1:60
         .= ( p1`2 * p2`3 - p1`3 * p2`2) * ((1/a`1) * a) by EUCLID_5:7
         .= (( p1`2 * p2`3 - p1`3 * p2`2) * (1/a`1)) * a by RVSUM_1:49;
        p1`2 * p2`3 - p1`3 * p2`2 <> 0
        proof
          assume p1`2 * p2`3 - p1`3 * p2`2 = 0;
          then u <X> v = |[0 * a`1,0 * a`2,0 * a`3]| by A7,EUCLID_5:7
                      .= 0.TOP-REAL 3 by EUCLID_5:4;
          hence thesis by A1,ANPROJ_8:51;
        end;
        hence (p1`2 * p2`3 - p1`3 * p2`2) * (1/a`1) <> 0 by A5;
      end;
      hence thesis by ANPROJ_1:1;
    end;
    suppose
A8:   a`2 <> 0;
      then
A9:   u`2 = -a`1/a`2 * u`1 - a`3/a`2 * u`3 &
        v`2 = -a`1/a`2 * v`1 - a`3/a`2 * v`3 by A4,ANPROJ_8:13;
      set p1 = u, p2 = v;
      now
        reconsider r = a`2 as Real;
        thus
A10:     u <X> v = |[ (a`1/a`2) *( p1`3 * p2`1 - p1`1 * p2`3),
                      1 *( p1`3 * p2`1 - p1`1 * p2`3),
                      (a`3/a`2) * ( p1`3*p2`1 - p1`1*p2`3) ]| by A9
               .= (p1`3*p2`1-p1`1*p2`3) * |[a`1/a`2,1,a`3/a`2]| by EUCLID_5:8
               .= (p1`3*p2`1-p1`1*p2`3) * |[a`1/r,r/r,a`3/r]| by A8,XCMPLX_1:60
               .= (p1`3*p2`1-p1`1*p2`3) * ((1/a`2) * a) by EUCLID_5:7
               .= ((p1`3*p2`1-p1`1*p2`3) * (1/a`2)) * a by RVSUM_1:49;
        p1`3*p2`1-p1`1*p2`3 <> 0
        proof
          assume p1`3*p2`1-p1`1*p2`3 = 0;
          then u <X> v = |[0 * a`1,0 * a`2,0 * a`3]| by A10,EUCLID_5:7
                      .= 0.TOP-REAL 3 by EUCLID_5:4;
          hence thesis by A1,ANPROJ_8:51;
        end;
        hence (p1`3*p2`1-p1`1*p2`3) * (1/a`2) <> 0 by A8;
      end;
      hence thesis by ANPROJ_1:1;
    end;
    suppose
A11:  a`3 <> 0;
      a`3 * u`3 + a`1 * u`1 + a`2 * u`2 = 0 &
        a`3 * v`3 + a`1 * v`1 + a`2 * v`2 = 0 by A4;
      then
A12:   u`3 = -a`1/a`3 * u`1 - a`2/a`3 * u`2 &
        v`3 = -a`1/a`3 * v`1 - a`2/a`3 * v`2 by A11,ANPROJ_8:13;
      set p1 = u, p2 = v;
      now
        reconsider r = a`3 as Real;
        thus
A13:    u <X> v = |[ (a`1/a`3) * (p1`1 * p2`2 - p1`2 * p2`1),
                      a`2/a`3 * (p1`1 * p2`2 - p1`2 * p2`1),
                      1 * (p1`1 * p2`2 - p1`2 * p2`1) ]| by A12
               .= (p1`1*p2`2-p1`2*p2`1) * |[a`1/a`3,a`2/a`3,1]| by EUCLID_5:8
               .= (p1`1*p2`2-p1`2*p2`1) * |[a`1/r,a`2/r,r/r]|
                 by A11,XCMPLX_1:60
               .= (p1`1*p2`2-p1`2*p2`1) * ((1/a`3) * a) by EUCLID_5:7
               .= ((p1`1*p2`2-p1`2*p2`1) * (1/a`3)) * a by RVSUM_1:49;
        p1`1*p2`2-p1`2*p2`1 <> 0
        proof
          assume p1`1*p2`2-p1`2*p2`1 = 0;
          then u <X> v = |[0 * a`1,0 * a`2,0 * a`3]| by A13,EUCLID_5:7
                      .= 0.TOP-REAL 3 by EUCLID_5:4;
          hence thesis by A1,ANPROJ_8:51;
        end;
        hence (p1`1*p2`2-p1`2*p2`1) * (1/a`3) <> 0 by A11;
      end;
      hence thesis by ANPROJ_1:1;
    end;
  end;

theorem Th9:
  for u,v being non zero Element of TOP-REAL 3
  for r being Real st r <> 0 & are_Prop u,v holds are_Prop r * u,v
  proof
    let u,v be non zero Element of TOP-REAL 3;
    let r be Real;
    assume that
A1: r <> 0 and
A2: are_Prop u,v;
    consider a be Real such that
A3: a <> 0 and
A4: u = a * v by ANPROJ_1:1,A2;
    r * u = (r * a) * v by A4,RVSUM_1:49;
    hence thesis by A1,A3,ANPROJ_1:1;
  end;

begin :: Alignment of definitions

definition
  let P being Point of ProjectiveSpace TOP-REAL 3;
  attr P is zero_proj1 means
:Def1:
  for u being non zero Element of TOP-REAL 3 st P = Dir u holds u.1 = 0;
end;

registration
  cluster zero_proj1 for Point of ProjectiveSpace TOP-REAL 3;
  existence
  proof
    take Dir001;
    reconsider p = |[0,0,1]| as non zero Element of TOP-REAL 3;
    now
      let u be non zero Element of TOP-REAL 3;
      assume Dir001 = Dir u;
      then are_Prop u, p by ANPROJ_1:22;
      then consider a be Real such that
      a <> 0 and
A1:   u = a * p by ANPROJ_1:1;
A2:   |[0,0,1]| = |[p.1,p.2,p.3]| by EUCLID_8:1,def 3;
      thus u.1 = a * p.1 by A1,RVSUM_1:44
              .= a * 0 by A2,FINSEQ_1:78
              .= 0;
    end;
    hence thesis;
  end;
end;

registration
  cluster non zero_proj1 for Point of ProjectiveSpace TOP-REAL 3;
  existence
  proof
    set P = Dir100;
    take P;
    reconsider u = |[1,0,0]| as non zero Element of TOP-REAL 3;
    now
      thus P = Dir u;
      |[1,0,0]| = |[u.1,u.2,u.3]| by EUCLID_8:1,def 1;
      hence u.1 <> 0 by FINSEQ_1:78;
    end;
    hence thesis;
  end;
end;

theorem Th10:
  for P being non zero_proj1 Point of ProjectiveSpace TOP-REAL 3
  for u being non zero Element of TOP-REAL 3 st P = Dir u holds
  u.1 <> 0
  proof
    let P be non zero_proj1 Point of ProjectiveSpace TOP-REAL 3;
    let u be non zero Element of TOP-REAL 3;
    assume
A1: P = Dir u;
    consider u9 be non zero Element of TOP-REAL 3 such that
A2: P = Dir u9 and
A3: u9.1 <> 0 by Def1;
    are_Prop u,u9 by A1,A2,ANPROJ_1:22;
    then consider a be Real such that
A4: a <> 0 and
A5: u = a * u9 by ANPROJ_1:1;
    assume u.1 = 0;
    then a * u9.1 = 0 by A5,RVSUM_1:44;
    hence thesis by A3,A4;
  end;

definition
  let P being non zero_proj1 Point of ProjectiveSpace TOP-REAL 3;
  func normalize_proj1(P) -> non zero Element of TOP-REAL 3 means
:Def2:
  Dir it = P & it.1 = 1;
  existence
  proof
    consider u be non zero Element of TOP-REAL 3 such that
A1: P = Dir u and
A2: u.1 <> 0 by Def1;
    reconsider v = |[1, u`2/u.1,u`3/u.1]| as non zero Element of TOP-REAL 3;
    take v;
A3: v`1 = 1 by EUCLID_5:2;
    u.1 * v = |[u.1 * 1, u.1 * (u`2/u.1),u.1*(u`3/u.1)]| by EUCLID_5:8
           .= |[u.1, u`2, u.1*(u`3/u.1)]| by XCMPLX_1:87,A2
           .= |[u`1, u`2, u`3]| by A2,XCMPLX_1:87
           .= u by EUCLID_5:3;
    then are_Prop u,v by A2,ANPROJ_1:1;
    hence thesis by A1,A3,ANPROJ_1:22;
  end;
  uniqueness
  proof
    let u,v be non zero Element of TOP-REAL 3 such that
A4: P = Dir u & u.1 = 1 and
A5: P = Dir v & v.1 = 1;
    are_Prop u,v by A4,A5,ANPROJ_1:22;
    then consider a be Real such that
    a <> 0 and
A6: u = a * v by ANPROJ_1:1;
A7: 1 = a * v.1 by A4,A6,RVSUM_1:44
     .= a by A5;
    a * v = |[a * v`1, a * v`2, a * v`3]| by EUCLID_5:7
         .= v by A7,EUCLID_5:3;
    hence thesis by A6;
  end;
end;

theorem Th11:
  for P being non zero_proj1 Point of ProjectiveSpace TOP-REAL 3
  for u being non zero Element of TOP-REAL 3 st P = Dir u holds
  normalize_proj1 P = |[1, u.2/u.1,u.3/u.1]|
  proof
    let P be non zero_proj1 Point of ProjectiveSpace TOP-REAL 3;
    let u9 be non zero Element of TOP-REAL 3;
    assume P = Dir u9;
    then Dir u9 = Dir normalize_proj1 P by Def2;
    then are_Prop u9,normalize_proj1 P by ANPROJ_1:22;
    then consider a be Real such that
    a <> 0 and
A1: normalize_proj1 P = a * u9 by ANPROJ_1:1;
A2: normalize_proj1 P = |[a * u9`1,a * u9`2,a * u9`3 ]| by A1,EUCLID_5:7;
A3: 1 = (normalize_proj1 P)`1 by Def2
     .= a * u9`1 by A2,EUCLID_5:2;
    then
A4: u9`1 = 1 / a & a = 1 / u9`1 by XCMPLX_1:73;
    normalize_proj1 P = |[ 1,u9`2 / u9`1,(1 / u9`1) * u9`3]|
                         by A1,A3,A4,EUCLID_5:7
                     .= |[ 1,u9.2 / u9.1,u9.3/u9.1]|;
    hence thesis;
  end;

theorem
  for P being non zero_proj1 Point of ProjectiveSpace TOP-REAL 3
  for Q being Point of ProjectiveSpace TOP-REAL 3 st
  Q = Dir normalize_proj1(P) holds Q is non zero_proj1 by Def2;

definition
  let P being Point of ProjectiveSpace TOP-REAL 3;
  attr P is zero_proj2 means
:Def3:
  for u being non zero Element of TOP-REAL 3 st P = Dir u holds u.2 = 0;
end;

registration
  cluster zero_proj2 for Point of ProjectiveSpace TOP-REAL 3;
  existence
  proof
    take Dir001;
    reconsider p = |[0,0,1]| as non zero Element of TOP-REAL 3;
    now
      let u be non zero Element of TOP-REAL 3;
      assume Dir001 = Dir u;
      then are_Prop u, p by ANPROJ_1:22;
      then consider a be Real such that
      a <> 0 and
A1:   u = a * p by ANPROJ_1:1;
A2:   |[0,0,1]| = |[p.1,p.2,p.3]| by EUCLID_8:1,def 3;
      thus u.2 = a * p.2 by A1,RVSUM_1:44
              .= a * 0 by A2,FINSEQ_1:78
              .= 0;
    end;
    hence thesis;
  end;
end;

registration
  cluster non zero_proj2 for Point of ProjectiveSpace TOP-REAL 3;
  existence
  proof
    set P = Dir010;
    take P;
    reconsider u = |[0,1,0]| as non zero Element of TOP-REAL 3;
    now
      thus P = Dir u;
      |[0,1,0]| = |[u.1,u.2,u.3]| by EUCLID_8:1,def 2;
      hence u.2 <> 0 by FINSEQ_1:78;
    end;
    hence thesis;
  end;
end;

theorem Th13:
  for P being non zero_proj2 Point of ProjectiveSpace TOP-REAL 3
  for u being non zero Element of TOP-REAL 3 st P = Dir u holds
  u.2 <> 0
  proof
    let P be non zero_proj2 Point of ProjectiveSpace TOP-REAL 3;
    let u be non zero Element of TOP-REAL 3;
    assume
A1: P = Dir u;
    consider u9 be non zero Element of TOP-REAL 3 such that
A2: P = Dir u9 and
A3: u9.2 <> 0 by Def3;
    are_Prop u,u9 by A1,A2,ANPROJ_1:22;
    then consider a be Real such that
A4: a <> 0 and
A5: u = a * u9 by ANPROJ_1:1;
    assume u.2 = 0;
    then a * u9.2 = 0 by A5,RVSUM_1:44;
    hence thesis by A3,A4;
  end;

definition
  let P being non zero_proj2 Point of ProjectiveSpace TOP-REAL 3;
  func normalize_proj2(P) -> non zero Element of TOP-REAL 3 means
  :Def4:
  Dir it = P & it.2 = 1;
  existence
  proof
    consider u be non zero Element of TOP-REAL 3 such that
A1: P = Dir u and
A2: u.2 <> 0 by Def3;
    reconsider v = |[u`1/u.2, 1,u`3/u.2]| as non zero Element of TOP-REAL 3;
    take v;
A3: v`2 = 1 by EUCLID_5:2;
    u.2 * v = |[u.2 * (u`1/u.2), u.2 * 1,u.2*(u`3/u.2)]| by EUCLID_5:8
           .= |[u`1, u.2, u.2*(u`3/u.2)]| by XCMPLX_1:87,A2
           .= |[u`1, u`2, u`3]| by A2,XCMPLX_1:87
           .= u by EUCLID_5:3;
    then are_Prop u,v by A2,ANPROJ_1:1;
    hence thesis by A1,A3,ANPROJ_1:22;
  end;
  uniqueness
  proof
    let u,v being non zero Element of TOP-REAL 3 such that
A4: P = Dir u & u.2 = 1 and
A5: P = Dir v & v.2 = 1;
    are_Prop u,v by A4,A5,ANPROJ_1:22;
    then consider a be Real such that
    a <> 0 and
A6: u = a * v by ANPROJ_1:1;
A7: 1 = a * v.2 by A4,A6,RVSUM_1:44
     .= a by A5;
    a * v = |[a * v`1, a * v`2, a * v`3]| by EUCLID_5:7
         .= v by A7,EUCLID_5:3;
    hence thesis by A6;
  end;
end;

theorem Th14:
  for P being non zero_proj2 Point of ProjectiveSpace TOP-REAL 3
  for u being non zero Element of TOP-REAL 3 st P = Dir u holds
  normalize_proj2 P = |[u.1/u.2,1,u.3/u.2]|
  proof
    let P be non zero_proj2 Point of ProjectiveSpace TOP-REAL 3;
    let u9 be non zero Element of TOP-REAL 3;
    assume P = Dir u9;
    then Dir u9 = Dir normalize_proj2 P by Def4;
    then are_Prop u9,normalize_proj2 P by ANPROJ_1:22;
    then consider a be Real such that
    a <> 0 and
A1: normalize_proj2 P = a * u9 by ANPROJ_1:1;
A2: normalize_proj2 P = |[a * u9`1,a * u9`2,a * u9`3 ]| by A1,EUCLID_5:7;
A3: 1 = (normalize_proj2 P)`2 by Def4
     .= a * u9`2 by A2,EUCLID_5:2;
    then
A4: u9`2 = 1 / a & a = 1 / u9`2 by XCMPLX_1:73;
    normalize_proj2 P = |[ u9`1 / u9`2,1,(1 / u9`2) * u9`3]|
                         by A1,A3,A4,EUCLID_5:7
                     .= |[ u9.1 / u9.2,1,u9.3/u9.2]|;
    hence thesis;
  end;

theorem
  for P being non zero_proj2 Point of ProjectiveSpace TOP-REAL 3
  for Q being Point of ProjectiveSpace TOP-REAL 3 st
  Q = Dir normalize_proj2(P) holds Q is non zero_proj2 by Def4;

definition
  let P being Point of ProjectiveSpace TOP-REAL 3;
  attr P is zero_proj3 means
:Def5:
  for u being non zero Element of TOP-REAL 3 st P = Dir u holds u.3 = 0;
end;

registration
  cluster zero_proj3 for Point of ProjectiveSpace TOP-REAL 3;
  existence
  proof
    take Dir100;
    reconsider p = |[1,0,0]| as non zero Element of TOP-REAL 3;
    now
      let u be non zero Element of TOP-REAL 3;
      assume Dir100 = Dir u;
      then are_Prop u, p by ANPROJ_1:22;
      then consider a be Real such that
      a <> 0 and
A1:   u = a * p by ANPROJ_1:1;
A2:   |[1,0,0]| = |[p.1,p.2,p.3]| by EUCLID_8:1,def 1;
      thus u.3 = a * p.3 by A1,RVSUM_1:44
              .= a * 0 by A2,FINSEQ_1:78
              .= 0;
    end;
    hence thesis;
  end;
end;

registration
  cluster non zero_proj3 for Point of ProjectiveSpace TOP-REAL 3;
  existence
  proof
    set P = Dir001;
    take P;
    reconsider u = |[0,0,1]| as non zero Element of TOP-REAL 3;
    now
      thus P = Dir u;
      |[0,0,1]| = |[u.1,u.2,u.3]| by EUCLID_8:1,def 3;
      hence u.3 <> 0 by FINSEQ_1:78;
    end;
    hence thesis;
  end;
end;

theorem Th16:
  for P being non zero_proj3 Point of ProjectiveSpace TOP-REAL 3
  for u being non zero Element of TOP-REAL 3 st P = Dir u holds
  u.3 <> 0
  proof
    let P be non zero_proj3 Point of ProjectiveSpace TOP-REAL 3;
    let u be non zero Element of TOP-REAL 3;
    assume
A1: P = Dir u;
    consider u9 be non zero Element of TOP-REAL 3 such that
A2: P = Dir u9 and
A3: u9.3 <> 0 by Def5;
    are_Prop u,u9 by A1,A2,ANPROJ_1:22;
    then consider a be Real such that
A4: a <> 0 and
A5: u = a * u9 by ANPROJ_1:1;
    assume u.3 = 0;
    then a * u9.3 = 0 by A5,RVSUM_1:44;
    hence thesis by A3,A4;
  end;

definition
  let P being non zero_proj3 Point of ProjectiveSpace TOP-REAL 3;
  func normalize_proj3(P) -> non zero Element of TOP-REAL 3 means
:Def6:
  Dir it = P & it.3 = 1;
  existence
  proof
    consider u be non zero Element of TOP-REAL 3 such that
A1: P = Dir u and
A2: u.3 <> 0 by Def5;
    reconsider v = |[u`1/u.3, u`2/u.3,1]| as non zero Element of TOP-REAL 3;
    take v;
A3: v`3 = 1 by EUCLID_5:2;
    u.3 * v = |[u.3 * (u`1/u.3), u.3 * (u`2/u.3),u.3*1]| by EUCLID_5:8
           .= |[u`1, u.3 * (u`2/u.3), u.3 ]| by XCMPLX_1:87,A2
           .= |[u`1, u`2, u`3]| by A2,XCMPLX_1:87
           .= u by EUCLID_5:3;
    then are_Prop u,v by A2,ANPROJ_1:1;
    hence thesis by A1,A3,ANPROJ_1:22;
  end;
  uniqueness
  proof
    let u,v be non zero Element of TOP-REAL 3 such that
A4: P = Dir u & u.3 = 1 and
A5: P = Dir v & v.3 = 1;
    are_Prop u,v by A4,A5,ANPROJ_1:22;
    then consider a be Real such that
    a <> 0 and
A6: u = a * v by ANPROJ_1:1;
A7: 1 = a * v.3 by A4,A6,RVSUM_1:44
     .= a by A5;
    a * v = |[a * v`1, a * v`2, a * v`3]| by EUCLID_5:7
         .= v by A7,EUCLID_5:3;
    hence thesis by A6;
  end;
end;

theorem Th17:
  for P being non zero_proj3 Point of ProjectiveSpace TOP-REAL 3
  for u being non zero Element of TOP-REAL 3 st P = Dir u holds
  normalize_proj3 P = |[u.1/u.3,u.2/u.3,1]|
  proof
    let P be non zero_proj3 Point of ProjectiveSpace TOP-REAL 3;
    let u9 be non zero Element of TOP-REAL 3;
    assume P = Dir u9;
    then Dir u9 = Dir normalize_proj3 P by Def6;
    then are_Prop u9,normalize_proj3 P by ANPROJ_1:22;
    then consider a be Real such that
    a <> 0 and
A1: normalize_proj3 P = a * u9 by ANPROJ_1:1;
A2: normalize_proj3 P = |[a * u9`1,a * u9`2,a * u9`3 ]| by A1,EUCLID_5:7;
A3: 1 = (normalize_proj3 P)`3 by Def6
     .= a * u9`3 by A2,EUCLID_5:2;
    then
A4: u9`3 = 1 / a & a = 1 / u9`3 by XCMPLX_1:73;
    normalize_proj3 P = |[ u9`1 / u9`3,(1 / u9`3) * u9`2,1]|
                        by A1,A3,A4,EUCLID_5:7
                     .= |[ u9.1 / u9.3,u9.2/u9.3,1]|;
    hence thesis;
  end;

theorem
  for P being non zero_proj3 Point of ProjectiveSpace TOP-REAL 3
  for Q being Point of ProjectiveSpace TOP-REAL 3 st
  Q = Dir normalize_proj3(P) holds Q is non zero_proj3 by Def6;

registration
  cluster non zero_proj1 non zero_proj2 for Point of
    ProjectiveSpace TOP-REAL 3;
  existence
  proof
    reconsider u = |[1,1,0]| as non zero Element of TOP-REAL 3;
    reconsider P = Dir u as Point of ProjectiveSpace TOP-REAL 3 by ANPROJ_1:26;
    take P;
    reconsider un = u as Element of REAL 3 by EUCLID:22;
    |[1,1,0]| = |[un.1,un.2,un.3]| by EUCLID_8:1;
    then u.1 <> 0 & u.2 <> 0 by FINSEQ_1:78;
    hence thesis;
  end;
end;
registration
  cluster non zero_proj1 non zero_proj3 for Point of
    ProjectiveSpace TOP-REAL 3;
  existence
  proof
    reconsider u = |[1,0,1]| as non zero Element of TOP-REAL 3;
    reconsider P = Dir u as Point of ProjectiveSpace TOP-REAL 3 by ANPROJ_1:26;
    take P;
    reconsider un = u as Element of REAL 3 by EUCLID:22;
    |[1,0,1]| = |[un.1,un.2,un.3]| by EUCLID_8:1;
    then u.1 <> 0 & u.3 <> 0 by FINSEQ_1:78;
    hence thesis;
  end;
end;
registration
  cluster non zero_proj2 non zero_proj3 for Point of
    ProjectiveSpace TOP-REAL 3;
  existence
  proof
    reconsider u = |[0,1,1]| as non zero Element of TOP-REAL 3;
    reconsider P = Dir u as Point of ProjectiveSpace TOP-REAL 3 by ANPROJ_1:26;
    take P;
    reconsider un = u as Element of REAL 3 by EUCLID:22;
    |[0,1,1]| = |[un.1,un.2,un.3]| by EUCLID_8:1;
    then u.2 <> 0 & u.3 <> 0 by FINSEQ_1:78;
    hence thesis;
  end;
end;

registration
  cluster non zero_proj1 non zero_proj2 non zero_proj3 for Point of
    ProjectiveSpace TOP-REAL 3;
  existence
  proof
    reconsider u = |[1,1,1]| as non zero Element of TOP-REAL 3;
    reconsider P = Dir u as Point of ProjectiveSpace TOP-REAL 3 by ANPROJ_1:26;
    take P;
    reconsider un = u as Element of REAL 3 by EUCLID:22;
    |[1,1,1]| = |[un.1,un.2,un.3]| by EUCLID_8:1;
    then u.1 <> 0 & u.2 <> 0 & u.3 <> 0 by FINSEQ_1:78;
    hence thesis;
  end;
end;

definition
  let P being non zero_proj1 Point of ProjectiveSpace TOP-REAL 3;
  func dir1a(P) -> non zero Element of TOP-REAL 3 equals
  |[- (normalize_proj1(P)).2,1,0]|;
  coherence;
end;
definition
  let P being non zero_proj1 Point of ProjectiveSpace TOP-REAL 3;
  func Pdir1a P -> Point of ProjectiveSpace TOP-REAL 3 equals
  Dir (dir1a P);
  coherence by ANPROJ_1:26;
end;

definition
  let P being non zero_proj1 Point of ProjectiveSpace TOP-REAL 3;
  func dir1b(P) -> non zero Element of TOP-REAL 3 equals
  |[- (normalize_proj1(P)).3,0,1]|;
  coherence;
end;

definition
  let P being non zero_proj1 Point of ProjectiveSpace TOP-REAL 3;
  func Pdir1b P -> Point of ProjectiveSpace TOP-REAL 3 equals
  Dir (dir1b P);
  coherence by ANPROJ_1:26;
end;

theorem
  for P being non zero_proj1 Point of ProjectiveSpace TOP-REAL 3 holds
  dir1a(P) <> dir1b(P) by FINSEQ_1:78;

theorem Th20:
  for P being non zero_proj1 Point of ProjectiveSpace TOP-REAL 3 holds
  Dir dir1a(P) <> Dir dir1b(P)
  proof
    let P being non zero_proj1 Point of ProjectiveSpace TOP-REAL 3;
    assume Dir dir1a(P) = Dir dir1b(P);
    then are_Prop dir1a(P),dir1b(P) by ANPROJ_1:22;
    then consider a be Real such that
A1: a <> 0 and
A2: dir1a(P) = a * dir1b(P) by ANPROJ_1:1;
    0 = (dir1a(P))`3 by EUCLID_5:2
     .= a * (dir1b(P))`3 by A2,RVSUM_1:44
     .= a * 1 by EUCLID_5:2;
    hence contradiction by A1;
  end;

theorem Th21:
  for P being non zero_proj1 Element of ProjectiveSpace TOP-REAL 3
  for u being non zero Element of TOP-REAL 3
  for v being Element of TOP-REAL 3 st u = normalize_proj1 P holds
  |{ dir1a P,dir1b P,v }| = |(u,v)|
  proof
    let P be non zero_proj1 Element of ProjectiveSpace TOP-REAL 3;
    let u be non zero Element of TOP-REAL 3;
    let v be Element of TOP-REAL 3;
    assume u = normalize_proj1 P;
    then
A1: u.1 = 1 & P = Dir u by Def2;
    then normalize_proj1 P = |[1, u.2/u.1,u.3/u.1]| by Th11;
    then (normalize_proj1(P))`2 = u.2/u.1 & (normalize_proj1(P))`3 = u.3/u.1
      by EUCLID_5:2;
    then |{ dir1a P,dir1b P,v }| = |{ |[ -u.2/u.1, 1  , 0   ]|,
                                      |[ -u.3/u.1, 0  , 1   ]|,
                                      |[ v`1     , v`2, v`3 ]| }|
                                    by EUCLID_5:3
      .= v`1 - (-u.2/u.1) * v`2 - v`3 * (-u.3/u.1) by Th2
      .= (1/u.1) * (u`1 * v`1 + u`2 * v`2 + v`3 * u`3) by A1
      .= (1/u.1) * |(u,v)| by EUCLID_5:29;
    hence thesis by A1;
  end;

theorem
  for P being non zero_proj1 Element of ProjectiveSpace TOP-REAL 3
  for u being non zero Element of TOP-REAL 3 st u = normalize_proj1 P holds
  |{ dir1a P,dir1b P,normalize_proj1 P }| = 1 + u.2 * u.2 + u.3 * u.3
  proof
    let P be non zero_proj1 Element of ProjectiveSpace TOP-REAL 3;
    let u be non zero Element of TOP-REAL 3;
    assume
A1: u = normalize_proj1 P;
    then
A2: u.1 = 1 by Def2;
    reconsider un = u as Element of REAL 3 by EUCLID:22;
    thus |{ dir1a P,dir1b P,normalize_proj1 P }| = |(un,un)| by A1,Th21
      .= u.1 * u.1 + u.2 * u.2 + u.3 * u.3 by EUCLID_8:63
      .= 1 + u.2 * u.2 + u.3 * u.3 by A2;
  end;

definition
  let P being non zero_proj2 Point of ProjectiveSpace TOP-REAL 3;
  func dir2a(P) -> non zero Element of TOP-REAL 3 equals
  |[1, - (normalize_proj2(P)).1,0]|;
  coherence;
end;
definition
  let P being non zero_proj2 Point of ProjectiveSpace TOP-REAL 3;
  func Pdir2a P -> Point of ProjectiveSpace TOP-REAL 3 equals
  Dir (dir2a P);
  coherence by ANPROJ_1:26;
end;

definition
  let P being non zero_proj2 Point of ProjectiveSpace TOP-REAL 3;
  func dir2b(P) -> non zero Element of TOP-REAL 3 equals
  |[0, - (normalize_proj2(P)).3,1]|;
  coherence;
end;

definition
  let P being non zero_proj2 Point of ProjectiveSpace TOP-REAL 3;
  func Pdir2b P -> Point of ProjectiveSpace TOP-REAL 3 equals
  Dir (dir2b P);
  coherence by ANPROJ_1:26;
end;

theorem
  for P being non zero_proj2 Point of ProjectiveSpace TOP-REAL 3 holds
  dir2a(P) <> dir2b(P) by FINSEQ_1:78;

theorem Th24:
  for P being non zero_proj2 Point of ProjectiveSpace TOP-REAL 3 holds
  Dir dir2a(P) <> Dir dir2b(P)
  proof
    let P being non zero_proj2 Point of ProjectiveSpace TOP-REAL 3;
    assume Dir dir2a(P) = Dir dir2b(P);
    then are_Prop dir2a(P),dir2b(P) by ANPROJ_1:22;
    then consider a be Real such that
A1: a <> 0 and
A2: dir2a(P) = a * dir2b(P) by ANPROJ_1:1;
    0 = (dir2a(P))`3 by EUCLID_5:2
     .= a * (dir2b(P))`3 by A2,RVSUM_1:44
     .= a * 1 by EUCLID_5:2;
    hence contradiction by A1;
  end;

theorem Th25:
  for P being non zero_proj2 Element of ProjectiveSpace TOP-REAL 3
  for u being non zero Element of TOP-REAL 3
  for v being Element of TOP-REAL 3 st u = normalize_proj2 P holds
  |{ dir2a P,dir2b P,v }| = - |(u,v)|
  proof
    let P be non zero_proj2 Element of ProjectiveSpace TOP-REAL 3;
    let u be non zero Element of TOP-REAL 3;
    let v be Element of TOP-REAL 3;
    assume u = normalize_proj2 P;
    then
A1: u.2 = 1 & P = Dir u by Def4;
    then normalize_proj2 P = |[u.1/u.2,1,u.3/u.2]| by Th14;
    then (normalize_proj2(P))`1 = u.1/u.2 & (normalize_proj2(P))`3 = u.3/u.2
      by EUCLID_5:2;
    then |{ dir2a P,dir2b P,v }| = |{ |[ 1,   -u.1/u.2, 0   ]|,
                                      |[ 0,   -u.3/u.2, 1   ]|,
                                      |[ v`1, v`2,      v`3 ]| }|
                                        by EUCLID_5:3
      .= (-u.3/u.2) * v`3 + (-u.1/u.2) * v`1 - v`2 by Th3
      .= -(1/u.2) * (u`1 * v`1 + u`2 * v`2 + u`3 * v`3) by A1
      .= -(1/u.2) * |(u,v)| by EUCLID_5:29;
    hence thesis by A1;
  end;

theorem
  for P being non zero_proj2 Element of ProjectiveSpace TOP-REAL 3
  for u being non zero Element of TOP-REAL 3 st u = normalize_proj2 P holds
  |{ dir2a P,dir2b P,normalize_proj2 P }| = - (u.1 * u.1 + 1 + u.3 * u.3)
  proof
    let P be non zero_proj2 Element of ProjectiveSpace TOP-REAL 3;
    let u be non zero Element of TOP-REAL 3;
    assume
A1: u = normalize_proj2 P;
    then
A2: u.2 = 1 by Def4;
    reconsider un = u as Element of REAL 3 by EUCLID:22;
    thus |{ dir2a P,dir2b P,normalize_proj2 P }| = - |(un,un)| by A1,Th25
      .= - (u.1 * u.1 + u.2 * u.2 + u.3 * u.3) by EUCLID_8:63
      .= - (u.1 * u.1 + 1 + u.3 * u.3) by A2;
  end;

definition
  let P being non zero_proj3 Point of ProjectiveSpace TOP-REAL 3;
  func dir3a(P) -> non zero Element of TOP-REAL 3 equals
  |[1,0,- (normalize_proj3(P)).1]|;
  coherence;
end;

definition
  let P being non zero_proj3 Point of ProjectiveSpace TOP-REAL 3;
  func Pdir3a P -> Point of ProjectiveSpace TOP-REAL 3 equals
  Dir (dir3a P);
  coherence by ANPROJ_1:26;
end;

definition
  let P being non zero_proj3 Point of ProjectiveSpace TOP-REAL 3;
  func dir3b(P) -> non zero Element of TOP-REAL 3 equals
  |[0,1,- (normalize_proj3(P)).2]|;
  coherence;
end;

definition
  let P being non zero_proj3 Point of ProjectiveSpace TOP-REAL 3;
  func Pdir3b P -> Point of ProjectiveSpace TOP-REAL 3 equals
  Dir (dir3b P);
  coherence by ANPROJ_1:26;
end;

theorem
  for P being non zero_proj3 Point of ProjectiveSpace TOP-REAL 3 holds
  dir3a(P) <> dir3b(P) by FINSEQ_1:78;

theorem Th28:
  for P being non zero_proj3 Point of ProjectiveSpace TOP-REAL 3 holds
  Dir dir3a(P) <> Dir dir3b(P)
  proof
    let P being non zero_proj3 Point of ProjectiveSpace TOP-REAL 3;
    assume Dir dir3a(P) = Dir dir3b(P);
    then are_Prop dir3a(P),dir3b(P) by ANPROJ_1:22;
    then consider a be Real such that
A1: a <> 0 and
A2: dir3a(P) = a * dir3b(P) by ANPROJ_1:1;
    0 = (dir3a(P))`2 by EUCLID_5:2
     .= a * (dir3b(P))`2 by A2,RVSUM_1:44
     .= a * 1 by EUCLID_5:2;
    hence contradiction by A1;
  end;

theorem Th29:
  for P being non zero_proj3 Element of ProjectiveSpace TOP-REAL 3
  for u being non zero Element of TOP-REAL 3
  for v being Element of TOP-REAL 3 st u = normalize_proj3 P holds
  |{ dir3a P,dir3b P,v }| = |(u,v)|
  proof
    let P be non zero_proj3 Element of ProjectiveSpace TOP-REAL 3;
    let u be non zero Element of TOP-REAL 3;
    let v be Element of TOP-REAL 3;
    assume u = normalize_proj3 P;
    then
A1: u.3 = 1 & P = Dir u by Def6;
    then normalize_proj3 P = |[u.1/u.3, u.2/u.3, 1]| by Th17;
    then (normalize_proj3(P))`1 = u.1/u.3 & (normalize_proj3(P))`2 = u.2/u.3
      by EUCLID_5:2;
    then |{ dir3a P,dir3b P,v }| = |{ |[ 1,   0,   -u.1/u.3 ]|,
                                      |[ 0,   1,   -u.2/u.3 ]|,
                                      |[ v`1, v`2, v`3 ]| }|  by EUCLID_5:3
      .= v`3 - v`1 * (-u.1/u.3) - v`2 * (-u.2/u.3) by Th4
      .= u`1 * v`1 + u`2 * v`2 + u`3 * v`3 by A1
      .= |(u,v)| by EUCLID_5:29;
    hence thesis;
  end;

theorem
  for P being non zero_proj3 Element of ProjectiveSpace TOP-REAL 3
  for u being non zero Element of TOP-REAL 3 st u = normalize_proj3 P holds
  |{ dir3a P,dir3b P,normalize_proj3 P }| = u.1 * u.1 + u.2 * u.2 + 1
  proof
    let P be non zero_proj3 Element of ProjectiveSpace TOP-REAL 3;
    let u be non zero Element of TOP-REAL 3;
    assume
A1: u = normalize_proj3 P;
    then
A2: u.3 = 1 by Def6;
    reconsider un = u as Element of REAL 3 by EUCLID:22;
    thus |{ dir3a P,dir3b P,normalize_proj3 P }| = |(un,un)| by A1,Th29
      .= u.1 * u.1 + u.2 * u.2 + u.3 * u.3 by EUCLID_8:63
      .= u.1 * u.1 + u.2 * u.2 + 1 by A2;
  end;

definition
  let P being non zero_proj1 Point of ProjectiveSpace TOP-REAL 3;
  func dual1 P -> Element of ProjectiveLines real_projective_plane equals
  Line(Pdir1a P,Pdir1b P);
  correctness
  proof
    reconsider P1 = Pdir1a P, P2 = Pdir1b P as Point of real_projective_plane;
    reconsider L = Line(P1,P2) as LINE of real_projective_plane
      by Th20,COLLSP:def 7;
    L in {B where B is Subset of real_projective_plane:
      B is LINE of real_projective_plane};
    hence thesis;
  end;
end;

definition
  let P being non zero_proj2 Point of ProjectiveSpace TOP-REAL 3;
  func dual2 P -> Element of ProjectiveLines real_projective_plane equals
  Line(Pdir2a P,Pdir2b P);
  correctness
  proof
    reconsider P1 = Pdir2a P, P2 = Pdir2b P as Point of real_projective_plane;
    reconsider L = Line(P1,P2) as LINE of real_projective_plane
      by Th24,COLLSP:def 7;
    L in {B where B is Subset of real_projective_plane:
      B is LINE of real_projective_plane};
    hence thesis;
  end;
end;

definition
  let P being non zero_proj3 Point of ProjectiveSpace TOP-REAL 3;
  func dual3 P -> Element of ProjectiveLines real_projective_plane equals
  Line(Pdir3a P,Pdir3b P);
  correctness
  proof
    reconsider P1 = Pdir3a P, P2 = Pdir3b P as Point of real_projective_plane;
    reconsider L = Line(P1,P2) as LINE of real_projective_plane
      by Th28,COLLSP:def 7;
    L in {B where B is Subset of real_projective_plane:
      B is LINE of real_projective_plane};
    hence thesis;
  end;
end;

theorem
  for P being non zero_proj1 non zero_proj2 Point of ProjectiveSpace TOP-REAL 3
  for u being non zero Element of TOP-REAL 3 st P = Dir u holds
  normalize_proj1 P = |[1, u.2/u.1, u.3/u.1]| &
  normalize_proj2 P = |[u.1/u.2, 1, u.3/u.2]| by Th11,Th14;

theorem
  for P being non zero_proj1 non zero_proj2 Point of ProjectiveSpace TOP-REAL 3
  for u being non zero Element of TOP-REAL 3 st P = Dir u holds
  normalize_proj1 P = u.2/u.1 * normalize_proj2 P &
  normalize_proj2 P = u.1/u.2 * normalize_proj1 P
  proof
    let P be non zero_proj1 non zero_proj2 Point of ProjectiveSpace TOP-REAL 3;
    let u be non zero Element of TOP-REAL 3;
    assume
A1: P = Dir u;
    set r = u.1 / u.2;
A2: u.1 <> 0 & u.2 <> 0 by A1,Th10,Th13;
A3: (u.1/u.2) * (u.2/u.1) = r * (1 / r) by XCMPLX_1:57
                         .= 1 by A2,XCMPLX_1:106;
    Dir normalize_proj1 P = P & Dir normalize_proj2 P = P by Def2,Def4;
    then are_Prop normalize_proj1 P,normalize_proj2 P by ANPROJ_1:22;
    then consider a be Real such that
    a <> 0 and
A4: normalize_proj1 P = a * normalize_proj2 P by ANPROJ_1:1;
    normalize_proj1 P = |[1, u.2/u.1, u.3/u.1]| &
      normalize_proj2 P = |[u.1/u.2, 1, u.3/u.2]| by A1,Th11,Th14;
    then
A5: |[1, u.2/u.1, u.3/u.1]| = |[ a * (u.1 / u.2),a*1,a * (u.3/u.2)]|
      by A4,EUCLID_5:8;
    hence normalize_proj1 P = (u.2/u.1) * normalize_proj2 P by A4,FINSEQ_1:78;
    (u.1/u.2) * normalize_proj1 P
      = (u.1/u.2) * ((u.2/u.1) * normalize_proj2 P) by A4,A5,FINSEQ_1:78
     .= ((u.1/u.2) * (u.2/u.1)) * normalize_proj2 P by RVSUM_1:49
     .= normalize_proj2 P by A3,RVSUM_1:52;
    hence normalize_proj2 P = (u.1/u.2) * normalize_proj1 P;
  end;

theorem Th33:
  for P being non zero_proj1 non zero_proj2 Point of ProjectiveSpace
  TOP-REAL 3 holds dual1 P = dual2 P
  proof
    let P be non zero_proj1 non zero_proj2 Point of ProjectiveSpace TOP-REAL 3;
    consider u be Element of TOP-REAL 3 such that
A1: u is not zero and
A2: P = Dir u by ANPROJ_1:26;
    reconsider u as non zero Element of TOP-REAL 3 by A1;
A3: normalize_proj1 P = |[1, u.2/u.1,u.3/u.1]| &
      normalize_proj2 P = |[u.1/u.2,1,u.3/u.2]| by A2,Th11,Th14;
    now
      now
        let x be object;
        assume x in Line(Pdir1a P,Pdir1b P);
        then consider P9 be Point of ProjectiveSpace TOP-REAL 3 such that
A4:     x = P9 and
A5:     Pdir1a P,Pdir1b P,P9 are_collinear;
        consider u9 be Element of TOP-REAL 3 such that
A6:     u9 is non zero and
A7:     P9 = Dir u9 by ANPROJ_1:26;
        set a2 = - (normalize_proj1(P)).2,
            a3 = - (normalize_proj1(P)).3,
            b1 = u9`1, b2 = u9`2, b3 = u9`3;
A8:     a2 = - (normalize_proj1(P))`2
          .= - u.2/u.1 by A3,EUCLID_5:2;
A9:     a3 = - (normalize_proj1(P))`3
          .= - u.3/u.1 by A3,EUCLID_5:2;
        0 = |{ dir1a P,dir1b P,u9 }| by A5,A6,A7,BKMODEL1:1
         .= |{ |[a2, 1 , 0]| ,
               |[a3, 0 , 1]|,
               |[b1, b2, b3]| }| by EUCLID_5:3
         .= b1 - a2 * b2 - a3 * b3 by Th2
         .= b1 + u.2/u.1 * b2 + u.3/u.1 * b3 by A8,A9;
        then
A10:    0 = u.1 * (b1 + u.2/u.1 * b2 + u.3/u.1 * b3)
         .= u.1 * b1 + u.1 * (u.2 / u.1) * b2 + u.1 * (u.3/u.1) * b3
         .= u.1 * b1 + u.2 * b2 + u.1 * (u.3/u.1) * b3
           by A2,Th10,XCMPLX_1:87
         .= u.1 * b1 + u.2 * b2 + u.3 * b3 by A2,Th10,XCMPLX_1:87;
        set c2 = - (normalize_proj2(P)).1,
            c3 = - (normalize_proj2(P)).3;
A11:    c2 = - (normalize_proj2(P))`1
          .= - u.1/u.2 by A3,EUCLID_5:2;
A12:    c3 = - (normalize_proj2(P))`3
          .= - u.3/u.2 by A3,EUCLID_5:2;
        |{ |[1,   c2,  0]|,
           |[0,   c3,  1]|,
           |[u9`1,u9`2,u9`3]| }| = (- u.1/u.2) *  b1 + (-u.3/u.2) * b3 - b2
             by A11,A12,Th3;
        then |{dir2a P,dir2b P,u9}|
          = (- u.1/u.2) *  b1 + (-u.3/u.2) * b3 + (-1) * b2 by EUCLID_5:3
         .= (- u.1/u.2) *  b1 + (-u.3/u.2) * b3 + (-u.2/u.2) * b2
           by XCMPLX_1:60,A2,Th13
         .= (u.1/(-u.2)) *  b1 + (-u.3/u.2) * b3 + (-u.2/u.2) * b2
           by XCMPLX_1:188
         .= (u.1/(-u.2)) *  b1 + (u.3/(-u.2)) * b3 + (-u.2/u.2) * b2
           by XCMPLX_1:188
         .= (u.1/(-u.2)) *  b1 + (u.3/(-u.2)) * b3 + (u.2/(-u.2)) * b2
           by XCMPLX_1:188
         .= (1 / -u.2) * (u.1 * b1 + u.2 * b2 + u.3 * b3)
         .= 0 by A10;
        then Pdir2a P,Pdir2b P,P9 are_collinear by A6,A7,BKMODEL1:1;
        hence x in Line(Pdir2a P,Pdir2b P) by A4;
      end;
      hence Line(Pdir1a P,Pdir1b P) c= Line(Pdir2a P,Pdir2b P);
      now
        let x be object;
        assume x in Line(Pdir2a P,Pdir2b P);
        then consider P9 be Point of ProjectiveSpace TOP-REAL 3 such that
A13:    x = P9 and
A14:    Pdir2a P,Pdir2b P,P9 are_collinear;
        consider u9 be Element of TOP-REAL 3 such that
A15:    u9 is non zero and
A16:    P9 = Dir u9 by ANPROJ_1:26;
        set a2 = - (normalize_proj1(P)).2,
            a3 = - (normalize_proj1(P)).3,
            b1 = u9`1, b2 = u9`2, b3 = u9`3;
        set c2 = - (normalize_proj2(P)).1,
            c3 = - (normalize_proj2(P)).3;
A17:    a2 = - (normalize_proj1(P))`2
          .= - u.2/u.1 by A3,EUCLID_5:2;
A18:    a3 = - (normalize_proj1(P))`3
          .= - u.3/u.1 by A3,EUCLID_5:2;
A19:    c2 = - (normalize_proj2(P))`1
          .= - u.1/u.2 by A3,EUCLID_5:2;
A20:    c3 = - (normalize_proj2(P))`3
          .= - u.3/u.2 by A3,EUCLID_5:2;
A21:    - u.2 <> 0 by A2,Th13;
A22:    0 = |{ dir2a P,dir2b P,u9 }| by A14,A15,A16,BKMODEL1:1
         .= |{ |[1, c2 , 0]| ,
               |[0, c3 , 1]|,
               |[b1, b2, b3]| }| by EUCLID_5:3
         .= c3 * b3 + c2 * b1 - b2 by Th3
         .= (- u.1/u.2) *  b1 + (-u.3/u.2) * b3 + (-1) * b2 by A19,A20
         .= (- u.1/u.2) *  b1 + (-u.3/u.2) * b3 + (-u.2/u.2) * b2
           by XCMPLX_1:60,A2,Th13
         .= (u.1/(-u.2)) *  b1 + (-u.3/u.2) * b3 + (-u.2/u.2) * b2
           by XCMPLX_1:188
         .= (u.1/(-u.2)) *  b1 + (u.3/(-u.2)) * b3 + (-u.2/u.2) * b2
           by XCMPLX_1:188
         .= (u.1/(-u.2)) *  b1 + (u.3/(-u.2)) * b3 + (u.2/(-u.2)) * b2
           by XCMPLX_1:188
         .= (1 / -u.2) * (u.1 * b1 + u.2 * b2 + u.3 * b3);
A23:    u.1/u.1 = 1 by XCMPLX_1:60,A2,Th10;
        |{dir1a P,dir1b P,u9}| = |{ |[a2, 1 , 0]| ,
                                    |[a3, 0 , 1]|,
                                    |[b1, b2, b3]| }| by EUCLID_5:3
         .= b1 - a2 * b2 - a3 * b3 by Th2
         .= (u.1/u.1) * b1 + u.2/u.1 * b2 + u.3/u.1 * b3 by A17,A18,A23
         .= (1/u.1) * (u.1 * b1 + u.2 * b2 + u.3 * b3)
         .= (1 / u.1) * 0 by A22,A21,XCMPLX_1:6
         .= 0;
        then Pdir1a P,Pdir1b P,P9 are_collinear by A15,A16,BKMODEL1:1;
        hence x in Line(Pdir1a P,Pdir1b P) by A13;
      end;
      hence Line(Pdir2a P,Pdir2b P) c= Line(Pdir1a P,Pdir1b P);
    end;
    hence thesis;
  end;

theorem Th34:
  for P being non zero_proj2 non zero_proj3 Point of ProjectiveSpace
  TOP-REAL 3 holds dual2 P = dual3 P
  proof
    let P be non zero_proj2 non zero_proj3 Point of ProjectiveSpace TOP-REAL 3;
    consider u be Element of TOP-REAL 3 such that
A1: u is not zero and
A2: P = Dir u by ANPROJ_1:26;
    reconsider u as non zero Element of TOP-REAL 3 by A1;
A3: normalize_proj3 P = |[u.1/u.3,u.2/u.3,1]| &
      normalize_proj2 P = |[u.1/u.2,1,u.3/u.2]| by A2,Th17,Th14;
    now
      now
        let x be object;
        assume x in Line(Pdir2a P,Pdir2b P);
        then consider P9 be Point of ProjectiveSpace TOP-REAL 3 such that
A4:     x = P9 and
A5:     Pdir2a P,Pdir2b P,P9 are_collinear;
        consider u9 be Element of TOP-REAL 3 such that
A6:     u9 is non zero and
A7:     P9 = Dir u9 by ANPROJ_1:26;
        set a2 = - (normalize_proj2(P)).1,
            a3 = - (normalize_proj2(P)).3,
            b1 = u9`1, b2 = u9`2, b3 = u9`3;
A8:     a2 = - (normalize_proj2(P))`1
          .= - u.1/u.2 by A3,EUCLID_5:2;
A9:     a3 = - (normalize_proj2(P))`3
          .= - u.3/u.2 by A3,EUCLID_5:2;
        0 = |{ dir2a P,dir2b P,u9 }| by A5,A6,A7,BKMODEL1:1
         .= |{ |[1, a2, 0]| ,
               |[0, a3, 1]|,
               |[b1, b2, b3]| }| by EUCLID_5:3
         .= a2 * b1 + a3 * b3 - b2 by Th3
         .= -(u.1/u.2 * b1 + b2 + u.3/u.2 * b3) by A8,A9;
        then
A10:    0 = u.2 * (u.1/u.2 *b1 + b2 + u.3/u.2 * b3)
         .= u.2 * b2 + u.2 * (u.1 / u.2) * b1 + u.2 * (u.3/u.2) * b3
         .= u.2 * b2 + u.1 * b1 + u.2 * (u.3/u.2) * b3
           by A2,Th13,XCMPLX_1:87
         .= u.2 * b2 + u.1 * b1 + u.3 * b3 by A2,Th13,XCMPLX_1:87;
        set c2 = - (normalize_proj3(P)).1,
            c3 = - (normalize_proj3(P)).2;
A11:    c2 = - (normalize_proj3(P))`1
          .= - u.1/u.3 by A3,EUCLID_5:2;
A12:    c3 = - (normalize_proj3(P))`2
          .= - u.2/u.3 by A3,EUCLID_5:2;
A13:    u.3 / u.3 = 1 by A2,Th16,XCMPLX_1:60;
        |{ |[1,   0,c2]|,
           |[0,   1,c3]|,
           |[u9`1,u9`2,u9`3]| }| = b3 - b1 * (-u.1/u.3) - b2 * (-u.2/u.3)
             by A11,A12,Th4
                                .= (u.1/u.3) * b1 + (u.2/u.3) * b2 + b3;
        then |{dir3a P,dir3b P,u9}|
          = (u.1 * (1/u.3)) *  b1 + (u.2/(u.3)) * b2 + (u.3/u.3) * b3
            by A13,EUCLID_5:3

         .= (1 /u.3) * (u.1 * b1 + u.2 * b2 + u.3 * b3)
         .= 0 by A10;
        then Pdir3a P,Pdir3b P,P9 are_collinear by A6,A7,BKMODEL1:1;
        hence x in Line(Pdir3a P,Pdir3b P) by A4;
      end;
      hence Line(Pdir2a P,Pdir2b P) c= Line(Pdir3a P,Pdir3b P);
      now
        let x be object;
        assume x in Line(Pdir3a P,Pdir3b P);
        then consider P9 be Point of ProjectiveSpace TOP-REAL 3 such that
A14:    x = P9 and
A15:    Pdir3a P,Pdir3b P,P9 are_collinear;
        consider u9 be Element of TOP-REAL 3 such that
A16:    u9 is non zero and
A17:    P9 = Dir u9 by ANPROJ_1:26;
        set a2 = - (normalize_proj3(P)).1,
            a3 = - (normalize_proj3(P)).2,
            b1 = u9`1, b2 = u9`2, b3 = u9`3;
        set c2 = - (normalize_proj2(P)).1,
            c3 = - (normalize_proj2(P)).3;
A18:    a2 = - (normalize_proj3(P))`1
          .= - u.1/u.3 by A3,EUCLID_5:2;
A19:    a3 = - (normalize_proj3(P))`2
          .= - u.2/u.3 by A3,EUCLID_5:2;
A20:    c2 = - (normalize_proj2(P))`1
          .= - u.1/u.2 by A3,EUCLID_5:2;
A21:    c3 = - (normalize_proj2(P))`3
          .= - u.3/u.2 by A3,EUCLID_5:2;
A22:    0 = |{ dir3a P,dir3b P,u9 }| by A15,A16,A17,BKMODEL1:1
         .= |{ |[1, 0, a2]| ,
               |[0, 1, a3]|,
               |[b1, b2, b3]| }| by EUCLID_5:3
         .= b3 - a2 * b1 - a3 * b2 by Th4
         .= (u.1/u.3) *  b1 + (u.2/u.3) * b2 + 1 * b3 by A18,A19
         .= (u.1/u.3) *  b1 + (u.2/u.3) * b2 + (u.3/u.3) * b3
           by XCMPLX_1:60,A2,Th16
         .= (1 /u.3) * (u.1 * b1 + u.2 * b2 + u.3 * b3);
A23:    u.3 <> 0 by A2,Th16;
        |{dir2a P,dir2b P,u9}| = |{ |[1 ,c2, 0]| ,
                                    |[0 ,c3, 1]|,
                                    |[b1, b2, b3]| }| by EUCLID_5:3
          .= c3 * b3 + c2 * b1 - b2 by Th3
          .= (-u.1/u.2) * b1 + (-1) * b2 + (-u.3/u.2) * b3 by A20,A21
          .= (-u.1/u.2) * b1 + (-u.2/u.2) * b2 + (-u.3/u.2) * b3
            by XCMPLX_1:60,A2,Th13
          .= (u.1/-u.2) * b1 + (-u.2/u.2) * b2 + (-u.3/u.2) * b3
            by XCMPLX_1:188
          .= (u.1/-u.2) * b1 + (u.2/-u.2) * b2 + (-u.3/u.2) * b3
            by XCMPLX_1:188
          .= (u.1/-u.2) * b1 + (u.2/-u.2) * b2 + (u.3/-u.2) * b3
            by XCMPLX_1:188
          .= (1/-u.2) * (u.1 * b1 + u.2 * b2 + u.3 * b3)
          .= (1/-u.2) * 0 by A23,XCMPLX_1:6,A22
          .= 0;
        then Pdir2a P,Pdir2b P,P9 are_collinear by A16,A17,BKMODEL1:1;
        hence x in Line(Pdir2a P,Pdir2b P) by A14;
      end;
      hence Line(Pdir3a P,Pdir3b P) c= Line(Pdir2a P,Pdir2b P);
    end;
    hence thesis;
  end;

theorem Th35:
  for P being non zero_proj1 non zero_proj3 Point of ProjectiveSpace
  TOP-REAL 3 holds dual1 P = dual3 P
  proof
    let P be non zero_proj1 non zero_proj3 Point of ProjectiveSpace TOP-REAL 3;
    consider u be Element of TOP-REAL 3 such that
A1: u is not zero and
A2: P = Dir u by ANPROJ_1:26;
    reconsider u as non zero Element of TOP-REAL 3 by A1;
A3: normalize_proj1 P = |[1, u.2/u.1,u.3/u.1]| &
      normalize_proj3 P = |[u.1/u.3,u.2/u.3,1]| by A2,Th11,Th17;
    now
      now
        let x be object;
        assume x in Line(Pdir1a P,Pdir1b P);
        then consider P9 be Point of ProjectiveSpace TOP-REAL 3 such that
A4:     x = P9 and
A5:     Pdir1a P,Pdir1b P,P9 are_collinear;
        consider u9 be Element of TOP-REAL 3 such that
A6:     u9 is non zero and
A7:     P9 = Dir u9 by ANPROJ_1:26;
        set a2 = - (normalize_proj1(P)).2,
            a3 = - (normalize_proj1(P)).3,
            b1 = u9`1, b2 = u9`2, b3 = u9`3;
A8:     a2 = - (normalize_proj1(P))`2
          .= - u.2/u.1 by A3,EUCLID_5:2;
A9:     a3 = - (normalize_proj1(P))`3
          .= - u.3/u.1 by A3,EUCLID_5:2;
        0 = |{ dir1a P,dir1b P,u9 }| by A5,A6,A7,BKMODEL1:1
         .= |{ |[a2, 1 , 0]| ,
               |[a3, 0 , 1]|,
               |[b1, b2, b3]| }| by EUCLID_5:3
         .= b1 - a2 * b2 - a3 * b3 by Th2
         .= b1 + u.2/u.1 * b2 + u.3/u.1 * b3 by A8,A9;
        then
A10:    0 = u.1 * (b1 + u.2/u.1 * b2 + u.3/u.1 * b3)
         .= u.1 * b1 + u.1 * (u.2 / u.1) * b2 + u.1 * (u.3/u.1) * b3
         .= u.1 * b1 + u.2 * b2 + u.1 * (u.3/u.1) * b3
           by A2,Th10,XCMPLX_1:87
         .= u.1 * b1 + u.2 * b2 + u.3 * b3 by A2,Th10,XCMPLX_1:87;
        set c2 = - (normalize_proj3(P)).1,
            c3 = - (normalize_proj3(P)).2;
A11:    c2 = - (normalize_proj3(P))`1
          .= - u.1/u.3 by A3,EUCLID_5:2;
A12:    c3 = - (normalize_proj3(P))`2
          .= - u.2/u.3 by A3,EUCLID_5:2;
        |{ |[1,   0, c2 ]|,
           |[0,   1, c3]|,
           |[u9`1,u9`2,u9`3]| }| = b3 - c2 * b1 - c3 * b2 by Th4;
        then |{dir3a P,dir3b P,u9}|
          = (u.1/u.3) * b1 + (u.2/u.3) * b2 + 1 * b3 by A11,A12,EUCLID_5:3
         .= (u.1/u.3) *  b1 + (u.2/u.3) * b2 + (u.3/u.3) * b3
           by XCMPLX_1:60,A2,Th16
         .= (1 / u.3) * (u.1 * b1 + u.2 * b2 + u.3 * b3)
         .= 0 by A10;
        then Pdir3a P,Pdir3b P,P9 are_collinear by A6,A7,BKMODEL1:1;
        hence x in Line(Pdir3a P,Pdir3b P) by A4;
      end;
      hence Line(Pdir1a P,Pdir1b P) c= Line(Pdir3a P,Pdir3b P);
      now
        let x be object;
        assume x in Line(Pdir3a P,Pdir3b P);
        then consider P9 be Point of ProjectiveSpace TOP-REAL 3 such that
A13:    x = P9 and
A14:    Pdir3a P,Pdir3b P,P9 are_collinear;
        consider u9 be Element of TOP-REAL 3 such that
A15:    u9 is non zero and
A16:    P9 = Dir u9 by ANPROJ_1:26;
        set a2 = - (normalize_proj1(P)).2,
            a3 = - (normalize_proj1(P)).3,
            b1 = u9`1, b2 = u9`2, b3 = u9`3;
        set c2 = - (normalize_proj3(P)).1,
            c3 = - (normalize_proj3(P)).2;
A17:    a2 = - (normalize_proj1(P))`2
          .= - u.2/u.1 by A3,EUCLID_5:2;
A18:    a3 = - (normalize_proj1(P))`3
          .= - u.3/u.1 by A3,EUCLID_5:2;
A19:    c2 = - (normalize_proj3(P))`1
          .= - u.1/u.3 by A3,EUCLID_5:2;
A20:    c3 = - (normalize_proj3(P))`2
          .= - u.2/u.3 by A3,EUCLID_5:2;
A21:    0 = |{ dir3a P,dir3b P,u9 }| by A14,A15,A16,BKMODEL1:1
         .= |{ |[1, 0,c2]| ,
               |[0, 1,c3]|,
               |[b1, b2, b3]| }| by EUCLID_5:3
         .= b3 - c2 * b1 - c3 * b2 by Th4
         .= (u.1/u.3) * b1 + (u.2/u.3) * b2 + 1 * b3 by A19,A20
         .= (u.1/u.3) * b1 + (u.2/u.3) * b2 + (u.3/u.3) * b3
           by XCMPLX_1:60,A2,Th16
         .= (1 / u.3) * (u.1 * b1 + u.2 * b2 + u.3 * b3);
A22:    u.3 <> 0 by A2,Th16;
A23:    u.1/u.1 = 1 by XCMPLX_1:60,A2,Th10;
        |{dir1a P,dir1b P,u9}| = |{ |[a2, 1 , 0]| ,
                                    |[a3, 0 , 1]|,
                                    |[b1, b2, b3]| }| by EUCLID_5:3
         .= b1 - a2 * b2 - a3 * b3 by Th2
         .= (u.1/u.1) * b1 + u.2/u.1 * b2 + u.3/u.1 * b3 by A17,A18,A23
         .= (1/u.1) * (u.1 * b1 + u.2 * b2 + u.3 * b3)
         .= (1 / u.1) * 0 by A22,A21,XCMPLX_1:6
         .= 0;
        then Pdir1a P,Pdir1b P,P9 are_collinear by A15,A16,BKMODEL1:1;
        hence x in Line(Pdir1a P,Pdir1b P) by A13;
      end;
      hence Line(Pdir3a P,Pdir3b P) c= Line(Pdir1a P,Pdir1b P);
    end;
    hence thesis;
  end;

theorem
  for P being non zero_proj1 non zero_proj2 non zero_proj3 Point of
  ProjectiveSpace TOP-REAL 3 holds dual1 P = dual2 P & dual1 P = dual3 P &
  dual2 P = dual3 P by Th33,Th34,Th35;

theorem Th37:
  for P being Element of ProjectiveSpace TOP-REAL 3 holds
  P is non zero_proj1 or P is non zero_proj2 or P is non zero_proj3
  proof
    let P be Element of ProjectiveSpace TOP-REAL 3;
    assume that
A1: P is zero_proj1 and
A2: P is zero_proj2 and
A3: P is zero_proj3;
    consider u be Element of TOP-REAL 3 such that
A4: u is not zero and
A5: Dir u = P by ANPROJ_1:26;
    reconsider u as non zero Element of TOP-REAL 3 by A4;
    u`1 = 0 & u`2 = 0 & u`3 = 0 by A1,A2,A3,A5;
    hence thesis by EUCLID_5:3,4;
  end;

definition
  let P being Point of ProjectiveSpace TOP-REAL 3;
  func dual P -> Element of ProjectiveLines real_projective_plane means
:Def22:
   ex P9 being non zero_proj1 Point of ProjectiveSpace TOP-REAL 3 st P9 = P &
   it = dual1 P9
if P is non zero_proj1,
  ex P9 being non zero_proj2 Point of ProjectiveSpace TOP-REAL 3 st P9 = P &
  it = dual2 P9
if (P is zero_proj1 & P is non zero_proj2),
  ex P9 being non zero_proj3 Point of ProjectiveSpace TOP-REAL 3 st P9 = P &
  it = dual3 P9
if (P is zero_proj1 & P is zero_proj2 & P is non zero_proj3);
  correctness
  proof
    per cases by Th37;
    suppose P is non zero_proj1;
      then reconsider P9 = P as
        non zero_proj1 Element of ProjectiveSpace TOP-REAL 3;
      dual1 P9 is Element of ProjectiveLines real_projective_plane;
      hence thesis;
    end;
    suppose
A1:   P is zero_proj1 & P is non zero_proj2;
      then reconsider P9 = P as
        non zero_proj2 Element of ProjectiveSpace TOP-REAL 3;
      dual2 P9 is Element of ProjectiveLines real_projective_plane;
      hence thesis by A1;
    end;
    suppose
A3:   P is zero_proj1 & P is zero_proj2 & P is non zero_proj3;
      then reconsider P9 = P as
        non zero_proj3 Element of ProjectiveSpace TOP-REAL 3;
      dual3 P9 is Element of ProjectiveLines real_projective_plane;
      hence thesis by A3;
    end;
  end;
end;

definition
  let P being Point of real_projective_plane;
  func # P -> Element of ProjectiveSpace TOP-REAL 3 equals P;
  coherence;
end;

definition
  let P being Point of real_projective_plane;
  func dual P -> Element of ProjectiveLines real_projective_plane equals
  dual #P;
  coherence;
end;

theorem Th38:
  for P being Element of real_projective_plane st #P is non zero_proj1 holds
  ex P9 being non zero_proj1 Point of ProjectiveSpace TOP-REAL 3 st
  P = P9 & dual P = dual1 P9
  proof
    let P be Element of real_projective_plane;
    assume
A1: #P is non zero_proj1;
    reconsider P1 = #P as non zero_proj1 Point of ProjectiveSpace TOP-REAL 3
      by A1;
    per cases;
    suppose P1 is non zero_proj2 & P1 is zero_proj3;
      hence thesis by Def22;
    end;
    suppose P1 is zero_proj2 & P1 is non zero_proj3;
      hence thesis by Def22;
    end;
    suppose P1 is zero_proj2 & P1 is zero_proj3;
      hence thesis by Def22;
    end;
    suppose P1 is non zero_proj2 & P1 is non zero_proj3;
      hence thesis by Def22;
    end;
  end;

theorem Th39:
  for P being Element of real_projective_plane st #P is non zero_proj2 holds
  ex P9 being non zero_proj2 Point of ProjectiveSpace TOP-REAL 3 st
  P = P9 & dual P = dual2 P9
  proof
    let P be Element of real_projective_plane;
    assume
A1: #P is non zero_proj2;
    reconsider P1 = #P as non zero_proj2 Point of ProjectiveSpace TOP-REAL 3
      by A1;
    per cases;
    suppose
      P1 is non zero_proj1 & P1 is zero_proj3;
      then reconsider P9 = P1 as
        non zero_proj1 non zero_proj2 Element of ProjectiveSpace TOP-REAL 3;
      dual P1 = dual1 P9 & dual1 P9 = dual2 P9 by Def22,Th33;
      hence thesis;
    end;
    suppose P1 is zero_proj1 & P1 is non zero_proj3;
      hence thesis by Def22;
    end;
    suppose P1 is zero_proj1 & P1 is zero_proj3;
      hence thesis by Def22;
    end;
    suppose P1 is non zero_proj1 & P1 is non zero_proj3;
      then reconsider P9 = P as non zero_proj1 non zero_proj2
        non zero_proj3 Element of ProjectiveSpace TOP-REAL 3;
      dual P1 = dual1 P9 & dual1 P9 = dual2 P9 by Def22,Th33;
      hence thesis;
    end;
  end;

theorem Th40:
  for P being Element of real_projective_plane st #P is non zero_proj3 holds
  ex P9 being non zero_proj3 Point of ProjectiveSpace TOP-REAL 3 st
  P = P9 & dual P = dual3 P9
  proof
    let P be Element of real_projective_plane;
    assume
A1: #P is non zero_proj3;
    reconsider P1 = #P as non zero_proj3 Point of ProjectiveSpace TOP-REAL 3
      by A1;
    per cases;
    suppose
A2:   P1 is non zero_proj2 & P1 is zero_proj1;
      then reconsider P9 = P as non zero_proj2 non zero_proj3
        Element of ProjectiveSpace TOP-REAL 3;
      dual P1 = dual2 P9 & dual3 P9 = dual2 P9 by A2,Def22,Th34;
      hence thesis;
    end;
    suppose
      P1 is zero_proj2 & P1 is non zero_proj1;
      then reconsider P9 = P as non zero_proj1 non zero_proj3
        Element of ProjectiveSpace TOP-REAL 3;
      dual P1 = dual1 P9 & dual1 P9 = dual3 P9 by Def22,Th35;
      hence thesis;
    end;
    suppose P1 is zero_proj2 & P1 is zero_proj1;
      hence thesis by Def22;
    end;
    suppose P1 is non zero_proj2 & P1 is non zero_proj1;
      then reconsider P9 = P as non zero_proj1 non zero_proj2
        non zero_proj3 Element of ProjectiveSpace TOP-REAL 3;
      dual P1 = dual1 P9 & dual1 P9 = dual3 P9 by Th35,Def22;
      hence thesis;
    end;
  end;

theorem Th41:
  for P being non zero_proj1 Element of ProjectiveSpace TOP-REAL 3 holds
  not P in Line(Pdir1a P,Pdir1b P)
  proof
    let P be non zero_proj1 Element of ProjectiveSpace TOP-REAL 3;
    assume P in Line(Pdir1a P,Pdir1b P); then
A1: Pdir1a P,Pdir1b P,P are_collinear by COLLSP:11;
    reconsider u = normalize_proj1 P as non zero Element of TOP-REAL 3;
A2: P = Dir u by Def2;
    |{ dir1a P,dir1b P,u }| = |( u, u )| by Th21;
    then |(u, u)| = 0 by A2,A1,BKMODEL1:1;
    hence thesis by Th5;
  end;

theorem Th42:
  for P being non zero_proj2 Element of ProjectiveSpace TOP-REAL 3 holds
  not P in Line(Pdir2a P,Pdir2b P)
  proof
    let P be non zero_proj2 Element of ProjectiveSpace TOP-REAL 3;
    assume P in Line(Pdir2a P,Pdir2b P); then
A1: Pdir2a P,Pdir2b P, P are_collinear by COLLSP:11;
    reconsider u = normalize_proj2 P as non zero Element of TOP-REAL 3;
A2: P = Dir u by Def4;
    |{ dir2a P,dir2b P,u }| = - |( u, u )| by Th25;
    then |(u, u)| = 0 by A2,A1,BKMODEL1:1;
    hence thesis by Th5;
  end;

theorem Th43:
  for P being non zero_proj3 Element of ProjectiveSpace TOP-REAL 3 holds
  not P in Line(Pdir3a P,Pdir3b P)
  proof
    let P be non zero_proj3 Element of ProjectiveSpace TOP-REAL 3;
    assume P in Line(Pdir3a P,Pdir3b P); then
A1: Pdir3a P,Pdir3b P, P are_collinear by COLLSP:11;
    reconsider u = normalize_proj3 P as non zero Element of TOP-REAL 3;
A2: P = Dir u by Def6;
    |{ dir3a P,dir3b P,u }| = |( u, u )| by Th29;
    then |(u, u)| = 0 by A2,A1,BKMODEL1:1;
    hence thesis by Th5;
  end;

theorem
  for P being Point of real_projective_plane holds not P in dual P
  proof
    let P be Point of real_projective_plane;
    reconsider P9 = P as Element of ProjectiveSpace TOP-REAL 3;
    per cases by Th37;
    suppose P9 is non zero_proj1;
      then reconsider P9 = P as
        non zero_proj1 Element of ProjectiveSpace TOP-REAL 3;
      #P = P9;
      then consider P99 be non zero_proj1 Point of ProjectiveSpace TOP-REAL 3
      such that
A1:   P = P99 and
A2:   dual P = dual1 P99 by Th38;
      assume P in dual P;
      hence contradiction by A1,A2,Th41;
    end;
    suppose P9 is non zero_proj2;
      then reconsider P9 = P as
        non zero_proj2 Element of ProjectiveSpace TOP-REAL 3;
      #P = P9;
      then consider P99 be non zero_proj2 Point of ProjectiveSpace TOP-REAL 3
      such that
A3:   P = P99 and
A4:   dual P = dual2 P99 by Th39;
      assume P in dual P;
      hence contradiction by Th42,A3,A4;
    end;
    suppose P9 is non zero_proj3;
      then reconsider P9 = P as
        non zero_proj3 Element of ProjectiveSpace TOP-REAL 3;
      #P = P9;
      then consider P99 be non zero_proj3 Point of ProjectiveSpace TOP-REAL 3
      such that
A5:   P = P99 and
A6:   dual P = dual3 P99 by Th40;
      assume P in dual P;
      hence contradiction by Th43,A5,A6;
    end;
  end;

definition
  let l being Element of ProjectiveLines real_projective_plane;
  func dual l -> Point of real_projective_plane means
:Def25:
  ex P,Q being Point of real_projective_plane st P <> Q &
  l = Line(P,Q) & it = L2P(P,Q);
  existence
  proof
    consider P,Q be Point of real_projective_plane such that
A1: P <> Q and
A2: l = Line(P,Q) by BKMODEL1:72;
    L2P(P,Q) is Point of real_projective_plane;
    hence thesis by A1,A2;
  end;
  uniqueness
  proof
    let P1,P2 be Point of real_projective_plane such that
A3: ex P,Q be Point of real_projective_plane st P <> Q &
      l = Line(P,Q) & P1 = L2P(P,Q) and
A4: ex P,Q being Point of real_projective_plane st P <> Q &
      l = Line(P,Q) & P2 = L2P(P,Q);
    consider P,Q be Point of real_projective_plane such that
A5: P <> Q and
A6: l = Line(P,Q) and
A7: P1 = L2P(P,Q) by A3;
    consider u,v be non zero Element of TOP-REAL 3 such that
A8: P = Dir u and
A9: Q = Dir v and
A10: L2P(P,Q) = Dir(u <X> v) by A5,BKMODEL1:def 5;
    consider P9,Q9 be Point of real_projective_plane such that
A11: P9 <> Q9 and
A12: l = Line(P9,Q9) and
A13: P2 = L2P(P9,Q9) by A4;
    consider u9,v9 be non zero Element of TOP-REAL 3 such that
A14: P9 = Dir u9 and
A15: Q9 = Dir v9 and
A16: L2P(P9,Q9) = Dir(u9 <X> v9) by A11,BKMODEL1:def 5;
    P,Q,P9 are_collinear & P,Q,Q9 are_collinear by A6,A12,COLLSP:10,11;
    then |{u,v,u9}| = 0 & |{u,v,v9}| = 0 by A8,A9,A14,A15,BKMODEL1:1;
    then
A17: |{u9,u,v}| = 0 & |{v9,u,v}| = 0 by EUCLID_5:33;
A18: now
      now
        assume u <X> v = 0.TOP-REAL 3;
        then are_Prop u,v by ANPROJ_8:51;
        hence contradiction by A8,A9,A5,ANPROJ_1:22;
      end;
      hence u <X> v is non zero;
      now
        assume u9 <X> v9 = 0.TOP-REAL 3;
        then are_Prop u9,v9 by ANPROJ_8:51;
        hence contradiction by A14,A15,A11,ANPROJ_1:22;
      end;
      hence u9 <X> v9 is non zero;
    end;
    then reconsider uv = u <X> v,
                    u9v9 = u9 <X> v9 as non zero Element of TOP-REAL 3;
    not are_Prop u9,v9 by A11,A14,A15,ANPROJ_1:22;
    then are_Prop uv,u9 <X> v9 by A17,Th8;
    hence thesis by A18,ANPROJ_1:22,A7,A13,A10,A16;
  end;
end;

theorem Th45:
  for P being Point of real_projective_plane holds dual dual P = P
  proof
    let P be Point of real_projective_plane;
    reconsider P9 = P as Point of ProjectiveSpace TOP-REAL 3;
    per cases by Th37;
    suppose P9 is non zero_proj1;
      then reconsider P9 = P as
        non zero_proj1 Point of ProjectiveSpace TOP-REAL 3;
      reconsider P = P9 as Point of real_projective_plane;
      consider P99 be non zero_proj1 Point of ProjectiveSpace TOP-REAL 3
      such that
A1:   P = P99 & dual P= dual1 P99 by Th38;
      reconsider l = Line(Pdir1a P9,Pdir1b P9) as
        Element of ProjectiveLines real_projective_plane by A1;
      consider P1,P2 be Point of real_projective_plane such that
A2:   P1 <> P2 and
A3:   l = Line(P1,P2) and
A4:   dual l = L2P(P1,P2) by Def25;
A5:   Line(P1,P2) = Line(Pdir1a P9,Pdir1b P9) & P1 in Line(P1,P2) &
        P2 in Line(P1,P2) by A3,COLLSP:10;
      then consider Q1 be Point of ProjectiveSpace TOP-REAL 3 such that
A6:   P1 = Q1 and
A7:   Pdir1a P9,Pdir1b P9,Q1 are_collinear;
      consider Q2 be Point of ProjectiveSpace TOP-REAL 3 such that
A8:   P2 = Q2 and
A9:   Pdir1a P9,Pdir1b P9,Q2 are_collinear by A5;
      consider u,v be non zero Element of TOP-REAL 3 such that
A10:  P1 = Dir u and
A11:  P2 = Dir v and
A12:  L2P(P1,P2) = Dir(u <X> v) by A2,BKMODEL1:def 5;
      consider w be Element of TOP-REAL 3 such that
A13:  w is not zero and
A14:  P9 = Dir w by ANPROJ_1:26;
      reconsider w as non zero Element of TOP-REAL 3 by A13;
      normalize_proj1 P9 = |[1, w.2/w.1,w.3/w.1]| by A14,Th11;
      then (normalize_proj1(P9))`2 = w.2/w.1 &
        (normalize_proj1(P9))`3 = w.3/w.1 by EUCLID_5:2;
      then
A15:  dir1a P9 <X> dir1b P9
        = |[ (1 * 1) - (0 * 0),
             (0 * (-w.3/w.1)) - ((-w.2/w.1) * 1),
             ((-w.2/w.1) * 0) - ((-w.3/w.1) * 1) ]| by EUCLID_5:15
       .= |[ w`1/w.1, w`2/w.1,w`3/w.1 ]| by A14,Th10,XCMPLX_1:60
       .= 1/w.1 * w by EUCLID_5:7;
      w.1 <> 0 by A14,Th10;
      then reconsider a = 1/w.1 * w as non zero Element of TOP-REAL 3
        by ANPROJ_9:3;
      now
        assume u <X> v = 0.TOP-REAL 3;
        then are_Prop u,v by ANPROJ_8:51;
        hence contradiction by A2,A10,A11,ANPROJ_1:22;
      end;
      then
A16:  u <X> v is non zero;
      now
        now
          thus not are_Prop u,v by A2,A10,A11,ANPROJ_1:22;
          thus 0 = |{ dir1a P9,dir1b P9,u }| by A10,A6,A7,BKMODEL1:1
                .= |{ u, dir1a P9,dir1b P9 }| by EUCLID_5:34
                .= |( a, u )| by A15;
          thus 0 = |{ dir1a P9,dir1b P9,v }| by A11,A8,A9,BKMODEL1:1
                .= |{ v, dir1a P9,dir1b P9 }| by EUCLID_5:34
                .= |( a, v )| by A15;
        end;
        then are_Prop 1/w.1 * w, u <X> v by Th8;
        hence are_Prop w.1 * a,u <X> v by A14,Th10,A16,Th9;
        thus w.1 * a = (w.1 * (1/w.1)) * w by RVSUM_1:49
                    .= 1 * w by A14,Th10,XCMPLX_1:106
                    .= w by RVSUM_1:52;
      end;
      hence thesis by A14,A1,A4,A12,A16,ANPROJ_1:22;
    end;
    suppose P9 is non zero_proj2;
      then reconsider P9 = P as
        non zero_proj2 Point of ProjectiveSpace TOP-REAL 3;
      reconsider P = P9 as Point of real_projective_plane;
      consider P99 be non zero_proj2 Point of ProjectiveSpace TOP-REAL 3
      such that
A17:  P = P99 & dual P= dual2 P99 by Th39;
      reconsider l = Line(Pdir2a P9,Pdir2b P9) as
        Element of ProjectiveLines real_projective_plane by A17;
      consider P1,P2 be Point of real_projective_plane such that
A18:  P1 <> P2 and
A19:  l = Line(P1,P2) and
A20:  dual l = L2P(P1,P2) by Def25;
A21:  Line(P1,P2) = Line(Pdir2a P9,Pdir2b P9) & P1 in Line(P1,P2) &
        P2 in Line(P1,P2) by A19,COLLSP:10;
      then consider Q1 be Point of ProjectiveSpace TOP-REAL 3 such that
A22:  P1 = Q1 and
A23:  Pdir2a P9,Pdir2b P9,Q1 are_collinear;

      consider Q2 be Point of ProjectiveSpace TOP-REAL 3 such that
A24:  P2 = Q2 and
A25:  Pdir2a P9,Pdir2b P9,Q2 are_collinear by A21;
      consider u,v be non zero Element of TOP-REAL 3 such that
A26:  P1 = Dir u and
A27:  P2 = Dir v and
A28:  L2P(P1,P2) = Dir(u <X> v) by A18,BKMODEL1:def 5;
      consider w be Element of TOP-REAL 3 such that
A29:  w is not zero and
A30:  P9 = Dir w by ANPROJ_1:26;
      reconsider w as non zero Element of TOP-REAL 3 by A29;
      normalize_proj2 P9 = |[w.1/w.2, 1, w.3/w.2]| by A30,Th14;
      then (normalize_proj2(P9))`1 = w.1/w.2 &
        (normalize_proj2(P9))`3 = w.3/w.2 by EUCLID_5:2;
      then
A31:  dir2a P9 <X> dir2b P9
        = |[ ((-w.1/w.2) * 1) - (0 * (-w.3/w.2)),
             (0 * 0) - (1 * 1),
             (1 * (-w.3/w.2)) - (0 * (-w.1/w.2)) ]| by EUCLID_5:15
       .= |[ -w.1/w.2, -w.2/w.2,-w.3/w.2 ]| by A30,Th13,XCMPLX_1:60
       .= |[ w.1/(-w.2), -w.2/w.2,-w.3/w.2 ]| by XCMPLX_1:188
       .= |[ w.1/(-w.2), w.2/(-w.2),-w.3/w.2 ]| by XCMPLX_1:188
       .= |[ w`1/(-w.2), w`2/(-w.2),w`3/(-w.2) ]| by XCMPLX_1:188
       .= 1/(-w.2) * w by EUCLID_5:7;
A32:  w.2 <> 0 by A30,Th13;
      then reconsider a = 1/(-w.2) * w as non zero Element of TOP-REAL 3
        by ANPROJ_9:3;
      now
        assume u <X> v = 0.TOP-REAL 3;
        then are_Prop u,v by ANPROJ_8:51;
        hence contradiction by A18,A26,A27,ANPROJ_1:22;
      end;
      then
A33:  u <X> v is non zero;
      now
        now
          thus not are_Prop u,v by A18,A26,A27,ANPROJ_1:22;
          thus 0 = |{ dir2a P9,dir2b P9,u }| by A26,A22,A23,BKMODEL1:1
                .= |{ u, dir2a P9,dir2b P9 }| by EUCLID_5:34
                .= |( a, u )| by A31;
          thus 0 = |{ dir2a P9,dir2b P9,v }| by A27,A24,A25,BKMODEL1:1
                .= |{ v, dir2a P9,dir2b P9 }| by EUCLID_5:34
                .= |( a, v )| by A31;
        end;
        then are_Prop 1/(-w.2) * w, u <X> v by Th8;
        hence are_Prop (-w.2) * a,u <X> v by A32,A33,Th9;
        thus (-w.2) * a = ((-w.2) * (1/(-w.2))) * w by RVSUM_1:49
                       .= 1 * w by A32,XCMPLX_1:106
                       .= w by RVSUM_1:52;
      end;
      hence thesis by A30,A17,A20,A28,A33,ANPROJ_1:22;
    end;
    suppose P9 is non zero_proj3;
      then reconsider P9 = P as
        non zero_proj3 Point of ProjectiveSpace TOP-REAL 3;
      reconsider P = P9 as Point of real_projective_plane;
      consider P99 be non zero_proj3 Point of ProjectiveSpace TOP-REAL 3
      such that
A34:  P = P99 & dual P= dual3 P99 by Th40;
      reconsider l = Line(Pdir3a P9,Pdir3b P9) as
        Element of ProjectiveLines real_projective_plane by A34;
      consider P1,P2 be Point of real_projective_plane such that
A35:  P1 <> P2 and
A36:  l = Line(P1,P2) and
A37:  dual l = L2P(P1,P2) by Def25;
A38:  Line(P1,P2) = Line(Pdir3a P9,Pdir3b P9) & P1 in Line(P1,P2) &
        P2 in Line(P1,P2) by A36,COLLSP:10;
      then consider Q1 be Point of ProjectiveSpace TOP-REAL 3 such that
A39:  P1 = Q1 and
A40:  Pdir3a P9,Pdir3b P9,Q1 are_collinear;
      consider Q2 be Point of ProjectiveSpace TOP-REAL 3 such that
A41:  P2 = Q2 and
A42:  Pdir3a P9,Pdir3b P9,Q2 are_collinear by A38;
      consider u,v be non zero Element of TOP-REAL 3 such that
A43:  P1 = Dir u and
A44:  P2 = Dir v and
A45:  L2P(P1,P2) = Dir(u <X> v) by A35,BKMODEL1:def 5;
      consider w be Element of TOP-REAL 3 such that
A46:  w is not zero and
A47:  P9 = Dir w by ANPROJ_1:26;
      reconsider w as non zero Element of TOP-REAL 3 by A46;
      normalize_proj3 P9 = |[w.1/w.3, w.2/w.3, 1]| by A47,Th17;
      then (normalize_proj3(P9))`1 = w.1/w.3 &
        (normalize_proj3(P9))`2 = w.2/w.3 by EUCLID_5:2;
      then
A48:  dir3a P9 <X> dir3b P9
        = |[ (0 * (-w.2/w.3)) - ((-w.1/w.3) * 1),
             ((-w.1/w.3) * 0) - (1 * (-w.2/w.3)),
             (1 * 1) - (0 * 0) ]| by EUCLID_5:15
       .= |[ w`1/w.3, w`2/w.3,w`3/w.3 ]| by A47,Th16,XCMPLX_1:60
       .= 1/(w.3) * w by EUCLID_5:7;
      w.3 <> 0 by A47,Th16;
      then reconsider a = 1/(w.3) * w as non zero Element of TOP-REAL 3
        by ANPROJ_9:3;
      now
        assume u <X> v = 0.TOP-REAL 3;
        then are_Prop u,v by ANPROJ_8:51;
        hence contradiction by A35,A43,A44,ANPROJ_1:22;
      end;
      then
A49:  u <X> v is non zero;
      now
        now
          thus not are_Prop u,v by A35,A43,A44,ANPROJ_1:22;
          thus 0 = |{ dir3a P9,dir3b P9,u }| by A43,A39,A40,BKMODEL1:1
                .= |{ u, dir3a P9,dir3b P9 }| by EUCLID_5:34
                .= |( a, u )| by A48;
          thus 0 = |{ dir3a P9,dir3b P9,v }| by A44,A41,A42,BKMODEL1:1
          .= |{ v, dir3a P9,dir3b P9 }| by EUCLID_5:34
          .= |( a, v )| by A48;
        end;
        then are_Prop 1/(w.3) * w, u <X> v by Th8;
        hence are_Prop (w.3) * a,u <X> v by A47,Th16,A49,Th9;
        thus w.3 * a = (w.3 * (1/w.3)) * w by RVSUM_1:49
                    .= 1 * w by A47,Th16,XCMPLX_1:106
                    .= w by RVSUM_1:52;
      end;
      hence thesis by A47,A34,A37,A45,A49,ANPROJ_1:22;
    end;
  end;

theorem Th46:
  for l being Element of ProjectiveLines real_projective_plane holds
  dual dual l = l
  proof
    let l be Element of ProjectiveLines real_projective_plane;
    consider P,Q be Point of real_projective_plane such that
A1: P <> Q and
A2: l = Line(P,Q) and
A3: dual l = L2P(P,Q) by Def25;
    reconsider P9 = P,Q9 = Q as Point of ProjectiveSpace TOP-REAL 3;
    consider u,v be non zero Element of TOP-REAL 3 such that
A4: P = Dir u and
A5: Q = Dir v and
A6: L2P(P,Q) = Dir(u <X> v) by A1,BKMODEL1:def 5;
    reconsider l2 = Line(P,Q) as LINE of real_projective_plane
      by A1,COLLSP:def 7;
    not are_Prop u,v by A1,A4,A5,ANPROJ_1:22;
    then u <X> v is non zero by ANPROJ_8:51;
    then reconsider uv = u <X> v as non zero Element of TOP-REAL 3;
    reconsider R = Dir uv as Point of ProjectiveSpace TOP-REAL 3
      by ANPROJ_1:26;
    reconsider R9 = R as Element of real_projective_plane;
A7: 0 = |( u <X> v,u )| by ANPROJ_8:44
     .= uv`1 * u`1 + uv`2 * u`2 + uv`3 * u`3 by EUCLID_5:29
     .= uv.1 * u`1 + uv.2 * u`2 + uv.3 * u`3;
A8: 0 = |( u <X> v,v )| by ANPROJ_8:45
     .= uv`1 * v`1 + uv`2 * v`2 + uv`3 * v`3 by EUCLID_5:29
     .= uv.1 * v`1 + uv.2 * v`2 + uv.3 * v`3;
    per cases by Th37;
    suppose
A9:   R is non zero_proj1;
      then reconsider R as non zero_proj1 Point of ProjectiveSpace TOP-REAL 3;
      #R9 = R;
      then consider P99 be non zero_proj1 Point of ProjectiveSpace TOP-REAL 3
      such that
A10:  R9 = P99 and
A11:  dual R9 = dual1 P99 by Th38;
A12:  uv.1 / uv.1 = 1 by A9,Th10,XCMPLX_1:60;
      normalize_proj1 P99 = |[1, uv.2/uv.1,uv.3/uv.1]| by A10,Th11;
      then
A13:  (normalize_proj1 P99)`2 = uv.2/uv.1 &
        (normalize_proj1 P99)`3 = uv.3/uv.1 by EUCLID_5:2;
      reconsider l1 = Line(Pdir1a P99,Pdir1b P99) as
        LINE of real_projective_plane by Th20,COLLSP:def 7;
      now
        |{ |[- uv.2/uv.1, 1,  0]|,
           |[- uv.3/uv.1, 0,  1]|,
           |[u`1,         u`2,u`3]| }|
          = u`1 - (-uv.2/uv.1) * u`2 - u`3 * (-uv.3/uv.1) by Th2
         .= (uv.1 / uv.1) * u`1  + (uv.2/uv.1) * u`2 + u`3 * (uv.3/uv.1)
           by A12
         .= (1/uv.1) * (uv.1 * u`1 + uv.2 * u`2 + uv.3 * u`3)
         .= 0 by A7;
        then|{ dir1a P99,dir1b P99,u }| = 0 by A13,EUCLID_5:3;
        then Pdir1a P99,Pdir1b P99, P9 are_collinear by A4,BKMODEL1:1;
        hence P in l1;
        |{ |[- uv.2/uv.1, 1,  0]|,
           |[- uv.3/uv.1, 0,  1]|,
           |[v`1,         v`2,v`3]| }|
          = (uv.1 / uv.1) * v`1 - (-uv.2/uv.1) * v`2 - v`3 * (-uv.3/uv.1)
           by A12,Th2
         .= (1/uv.1) * (uv.1 * v`1 + uv.2 * v`2 + uv.3 * v`3)
         .= 0 by A8;
        then|{ dir1a P99,dir1b P99,v }| = 0 by A13,EUCLID_5:3;
        then Pdir1a P99,Pdir1b P99, Q9 are_collinear by A5,BKMODEL1:1;
        hence Q in l1;
        thus P in l2 & Q in l2 by COLLSP:10;
      end;
      hence thesis by A1,A3,A6,A11,A2,COLLSP:20;
    end;
    suppose
A14:  R is non zero_proj2;
      then reconsider R as non zero_proj2 Point of ProjectiveSpace TOP-REAL 3;
      #R9 = R;
      then consider P99 be non zero_proj2 Point of ProjectiveSpace TOP-REAL 3
      such that
A15:  R9 = P99 and
A16:  dual R9 = dual2 P99 by Th39;
A17:  uv.2 / uv.2 = 1 by A14,Th13,XCMPLX_1:60;
      normalize_proj2 P99 = |[uv.1/uv.2,1,uv.3/uv.2]| by A15,Th14;
      then
A18:  (normalize_proj2 P99)`1 = uv.1/uv.2 &
        (normalize_proj2 P99)`3 = uv.3/uv.2 by EUCLID_5:2;
      reconsider l1 = Line(Pdir2a P99,Pdir2b P99) as
        LINE of real_projective_plane by Th24,COLLSP:def 7;
      now
        |{ |[1  , - uv.1/uv.2, 0]|,
           |[0  , - uv.3/uv.2, 1]|,
           |[u`1, u`2,         u`3]| }|
          = (-uv.3/uv.2) * u`3 + (-uv.1/uv.2) * u`1 - u`2 by Th3
         .= -((uv.3/uv.2) * u`3 + (uv.1/uv.2) * u`1 + (uv.2/uv.2) * u`2) by A17
         .= - ((1 / uv.2) * (uv.1 * u`1 + uv.2 * u`2 + uv.3 * u`3))
         .= 0 by A7;
        then|{ dir2a P99,dir2b P99,u }| = 0 by A18,EUCLID_5:3;
        then Pdir2a P99,Pdir2b P99, P9 are_collinear by A4,BKMODEL1:1;
        hence P in l1;
        |{ |[ 1 , - uv.1/uv.2, 0  ]|,
           |[ 0 , - uv.3/uv.2, 1  ]|,
           |[v`1, v`2,         v`3]| }|
          = (-uv.3/uv.2) * v`3 + (-uv.1/uv.2) * v`1 - v`2 by Th3
         .= -((uv.3/uv.2) * v`3 + (uv.1/uv.2) * v`1 + (uv.2/uv.2) * v`2) by A17
         .= - ((1 / uv.2) * (uv.1 * v`1 + uv.2 * v`2 + uv.3 * v`3))
         .= 0 by A8;
        then|{ dir2a P99,dir2b P99,v }| = 0 by A18,EUCLID_5:3;
        then Pdir2a P99,Pdir2b P99, Q9 are_collinear by A5,BKMODEL1:1;
        hence Q in l1;
        thus P in l2 & Q in l2 by COLLSP:10;
      end;
      hence thesis by A3,A6,A16,A2,A1,COLLSP:20;
    end;
    suppose
A19:  R is non zero_proj3;
      then reconsider R as non zero_proj3 Point of ProjectiveSpace TOP-REAL 3;
      #R9 = R;
      then consider P99 be non zero_proj3 Point of ProjectiveSpace TOP-REAL 3
      such that
A20:  R9 = P99 and
A21:  dual R9 = dual3 P99 by Th40;
A22:  uv.3 / uv.3 = 1 by A19,Th16,XCMPLX_1:60;
      normalize_proj3 P99 = |[uv.1/uv.3,uv.2/uv.3,1]| by A20,Th17;
      then
A23:  (normalize_proj3 P99)`1 = uv.1/uv.3 &
        (normalize_proj3 P99)`2 = uv.2/uv.3 by EUCLID_5:2;
      reconsider l1 = Line(Pdir3a P99,Pdir3b P99) as
        LINE of real_projective_plane by Th28,COLLSP:def 7;
      now
        |{ |[1  ,0  , - uv.1/uv.3]|,
           |[0  ,1  , - uv.2/uv.3]|,
           |[u`1,u`2, u`3        ]| }|
          = u`3 - u`1 * (-uv.1/uv.3) - u`2 * (-uv.2/uv.3) by Th4
         .= u`1 * (uv.1/uv.3) + u`2 * (uv.2/uv.3) + u`3 * (uv.3 / uv.3) by A22
         .= (1 / uv.3) * (uv.1 * u`1 + uv.2 * u`2 + uv.3 * u`3)
         .= 0 by A7;
        then|{ dir3a P99,dir3b P99,u }| = 0 by A23,EUCLID_5:3;
        then Pdir3a P99,Pdir3b P99, P9 are_collinear by A4,BKMODEL1:1;
        hence P in l1;
        |{ |[1  ,0  ,- uv.1/uv.3]|,
           |[0  ,1  ,- uv.2/uv.3]|,
           |[v`1,v`2,v`3]| }|
          = v`3 - v`1 * (-uv.1/uv.3) - v`2 * (-uv.2/uv.3) by Th4
         .= v`1 * (uv.1/uv.3) + v`2 * (uv.2/uv.3) + v`3 * (uv.3 / uv.3) by A22
         .= (1 / uv.3) * (uv.1 * v`1 + uv.2 * v`2 + uv.3 * v`3)
         .= 0 by A8;
        then|{ dir3a P99,dir3b P99,v }| = 0 by A23,EUCLID_5:3;
        then Pdir3a P99,Pdir3b P99, Q9 are_collinear by A5,BKMODEL1:1;
        hence Q in l1;
        thus P in l2 & Q in l2 by COLLSP:10;
      end;
      hence thesis by A3,A6,A21,A2,A1,COLLSP:20;
    end;
  end;

theorem
  for P,Q being Point of real_projective_plane holds
  (P <> Q iff dual P <> dual Q)
  proof
    let P,Q be Point of real_projective_plane;
    now
      assume
A1:   P <> Q;
      assume dual P = dual Q;
      then P = dual dual Q by Th45;
      hence contradiction by A1,Th45;
    end;
    hence thesis;
  end;

theorem Th48:
  for l,m being Element of ProjectiveLines real_projective_plane holds
  (l <> m iff dual l <> dual m)
  proof
    let l,m be Element of ProjectiveLines real_projective_plane;
    now
      assume
A1:   l <> m;
      assume dual l = dual m;
      then l = dual dual m by Th46;
      hence contradiction by A1,Th46;
    end;
    hence thesis;
  end;

begin

definition
  let l1,l2,l3 being Element of ProjectiveLines real_projective_plane;
  pred l1,l2,l3 are_concurrent means
  ex P being Point of real_projective_plane st P in l1 & P in l2 & P in l3;
end;

definition
  let l being Element of ProjectiveLines real_projective_plane;
  func #l -> LINE of IncProjSp_of real_projective_plane equals l;
  coherence;
end;

definition
  let l being LINE of IncProjSp_of real_projective_plane;
  func #l -> Element of ProjectiveLines real_projective_plane equals l;
  coherence;
end;

theorem Th49:
  for l1,l2,l3 being Element of ProjectiveLines real_projective_plane holds
  l1,l2,l3 are_concurrent iff #l1,#l2,#l3 are_concurrent
  proof
    let l1,l2,l3 be Element of ProjectiveLines real_projective_plane;
    now
      l1 in {B where B is Subset of real_projective_plane:
        B is LINE of real_projective_plane};
      hence ex B be Subset of real_projective_plane st l1 = B &
        B is LINE of real_projective_plane;
      l2 in {B where B is Subset of real_projective_plane:
        B is LINE of real_projective_plane};
      hence ex B be Subset of real_projective_plane st l2 = B &
        B is LINE of real_projective_plane;
      l3 in {B where B is Subset of real_projective_plane:
        B is LINE of real_projective_plane};
      hence ex B be Subset of real_projective_plane st l3 = B &
        B is LINE of real_projective_plane;
    end;
    then reconsider m1 = l1,m2 = l2,m3 = l3 as LINE of real_projective_plane;
    reconsider l91 = #l1, l92 = #l2, l93 = #l3 as
      LINE of IncProjSp_of real_projective_plane;
    hereby
      assume l1,l2,l3 are_concurrent;
      then consider P be Point of real_projective_plane such that
A1:   P in l1 and
A2:   P in l2 and
A3:   P in l3;
      reconsider P as Element of the Points of IncProjSp_of
        real_projective_plane;
      reconsider P9 = P as POINT of IncProjSp_of real_projective_plane;
      P in m1 & P in m2 & P in m3 by A1,A2,A3;
      then P9 on l91 & P9 on l92 & P9 on l93 by INCPROJ:5;
      hence #l1,#l2,#l3 are_concurrent;
    end;
    assume #l1,#l2,#l3 are_concurrent;
    then consider o be Element of the Points of IncProjSp_of
      real_projective_plane
    such that
A4: o on #l1 and
A5: o on #l2 and
A6: o on #l3;
    reconsider o9 = o as Point of real_projective_plane;
    o9 in m1 & o9 in m2 & o9 in m3 by A4,A5,A6,INCPROJ:5;
    hence l1,l2,l3 are_concurrent;
  end;

theorem
  for l1,l2,l3 being LINE of IncProjSp_of real_projective_plane holds
  l1,l2,l3 are_concurrent iff #l1,#l2,#l3 are_concurrent
  proof
    let l1,l2,l3 be LINE of IncProjSp_of real_projective_plane;
    reconsider l91 = #l1, l92 = #l2, l93 = #l3 as
      Element of ProjectiveLines real_projective_plane;
    hereby
      assume l1,l2,l3 are_concurrent;
      then #l91,#l92,#l93 are_concurrent;
      hence #l1,#l2,#l3 are_concurrent by Th49;
    end;
    assume #l1,#l2,#l3 are_concurrent;
    then ##l1,##l2,##l3 are_concurrent by Th49;
    hence l1,l2,l3 are_concurrent;
  end;

theorem
  for P,Q,R being Element of real_projective_plane st P,Q,R are_collinear holds
  Q,R,P are_collinear & R,P,Q are_collinear &
  P,R,Q are_collinear & R,Q,P are_collinear &
  Q,P,R are_collinear by ANPROJ_2:24;

theorem
  for l1,l2,l3 being Element of ProjectiveLines real_projective_plane st
  l1,l2,l3 are_concurrent holds l2,l1,l3 are_concurrent &
  l1,l3,l2 are_concurrent & l3,l2,l1 are_concurrent &
  l3,l2,l1 are_concurrent & l2,l3,l1 are_concurrent;

theorem
  for P,Q being Point of real_projective_plane
  for P9,Q9 being Element of ProjectiveSpace TOP-REAL 3 st
  P = P9 & Q = Q9 holds Line(P,Q) = Line(P9,Q9);

theorem Th54:
  for P being Point of real_projective_plane
  for l being Element of ProjectiveLines real_projective_plane st P in l holds
  dual l in dual P
  proof
    let P be Point of real_projective_plane;
    let l be Element of ProjectiveLines real_projective_plane;
    assume
A1: P in l;
    consider u be Element of TOP-REAL 3 such that
A2: u is not zero and
A3: P = Dir u by ANPROJ_1:26;
    reconsider u as non zero Element of TOP-REAL 3 by A2;
    reconsider P9 = P as Element of ProjectiveSpace TOP-REAL 3;
    reconsider dl = dual l as Point of ProjectiveSpace TOP-REAL 3;
    consider Pl,Ql be Point of real_projective_plane such that
A4: Pl <> Ql and
A5: l = Line(Pl,Ql) and
A6: dual l = L2P(Pl,Ql) by Def25;
    consider ul,vl be non zero Element of TOP-REAL 3 such that
A7: Pl = Dir ul and
A8: Ql = Dir vl and
A9: L2P(Pl,Ql) = Dir(ul <X> vl) by A4,BKMODEL1:def 5;
    reconsider ulvl = ul <X> vl as non zero Element of TOP-REAL 3
      by A4,A7,A8,BKMODEL1:78;
    consider S be Point of real_projective_plane such that
A10: P = S and
A11: Pl,Ql,S are_collinear by A1,A5;
    P,Pl,Ql are_collinear by A10,A11,ANPROJ_2:24;
    then
A12: |{u,ul,vl}| = 0 by A3,A7,A8,BKMODEL1:1;
    per cases by Th37;
    suppose P9 is non zero_proj1;
      then reconsider P9 as non zero_proj1 Point of ProjectiveSpace TOP-REAL 3;
      consider P99 be non zero_proj1 Point of ProjectiveSpace TOP-REAL 3
      such that
A13:  P9 = P99 and
A14:  dual P = dual1 P99 by Th38;
      consider S be Point of real_projective_plane such that
A15:  P = S and
A16:  Pl,Ql,S are_collinear by A1,A5;
      P,Pl,Ql are_collinear by A15,A16,ANPROJ_2:24;
      then
A17:  |{u,ul,vl}| = 0 by A3,A7,A8,BKMODEL1:1;
      Dir normalize_proj1 P9 = Dir u by A3,Def2;
      then
A18:  are_Prop normalize_proj1 P9,u by ANPROJ_1:22;
      |{dir1a P9,dir1b P9, ulvl }| = |(normalize_proj1 P9, ulvl)| by Th21
                                  .= 0 by A17,A18,Th7;
      then Pdir1a P9,Pdir1b P9, dl are_collinear by A6,A9,BKMODEL1:1;
      hence thesis by A13,A14;
    end;
    suppose P9 is non zero_proj2;
      then reconsider P9 as non zero_proj2 Point of ProjectiveSpace TOP-REAL 3;
      consider P99 be non zero_proj2 Point of ProjectiveSpace TOP-REAL 3
      such that
A19:  P9 = P99 and
A20:  dual P = dual2 P99 by Th39;
      Dir normalize_proj2 P9 = Dir u by A3,Def4;
      then
A21:  are_Prop normalize_proj2 P9,u by ANPROJ_1:22;
      |{dir2a P9,dir2b P9, ulvl }| = - |(normalize_proj2 P9, ulvl)| by Th25
                                  .= - 0 by A12,A21,Th7;
      then Pdir2a P9,Pdir2b P9, dl are_collinear by A6,A9,BKMODEL1:1;
      hence thesis by A19,A20;
    end;
    suppose P9 is non zero_proj3;
      then reconsider P9 as non zero_proj3 Point of ProjectiveSpace TOP-REAL 3;
      consider P99 be non zero_proj3 Point of ProjectiveSpace TOP-REAL 3
      such that
A22:  P9 = P99 and
A23:  dual P = dual3 P99 by Th40;
      Dir normalize_proj3 P9 = Dir u by A3,Def6;
      then
A24:  are_Prop normalize_proj3 P9,u by ANPROJ_1:22;
      |{dir3a P9,dir3b P9, ulvl }| = |(normalize_proj3 P9, ulvl)| by Th29
                                  .= 0 by A12,A24,Th7;
      then Pdir3a P9,Pdir3b P9, dl are_collinear by A6,A9,BKMODEL1:1;
      hence thesis by A22,A23;
    end;
  end;

theorem
  for P being Point of real_projective_plane
  for l being Element of ProjectiveLines real_projective_plane st
  dual l in dual P holds P in l
  proof
    let P be Point of real_projective_plane;
    let l be Element of ProjectiveLines real_projective_plane;
    assume dual l in dual P;
    then dual dual P in dual dual l by Th54;
    then P in dual dual l by Th45;
    hence thesis by Th46;
  end;

theorem Th56:
  for P,Q,R being Point of real_projective_plane st P,Q,R are_collinear holds
  dual P,dual Q, dual R are_concurrent
  proof
    let P,Q,R be Point of real_projective_plane;
    assume
A1: P,Q,R are_collinear;
    per cases;
    suppose
A2:  Q = R;
      reconsider lP = dual P,lQ = dual Q as LINE of real_projective_plane
        by INCPROJ:1;
      ex x be object st x in lP & x in lQ by BKMODEL1:76,XBOOLE_0:3;
      hence thesis by A2;
    end;
    suppose
A3:   Q <> R;
A4:  Q,R,P are_collinear by A1,ANPROJ_2:24;
      reconsider l = Line(Q,R) as LINE of real_projective_plane
        by A3,COLLSP:def 7;
      l in {B where B is Subset of real_projective_plane:
        B is LINE of real_projective_plane};
      then reconsider l = Line(Q,R) as Element of ProjectiveLines
        real_projective_plane;
      dual l in dual P & dual l in dual Q & dual l in dual R
        by A4,COLLSP:11,Th54,COLLSP:10;
      hence thesis;
    end;
  end;

theorem Th57:
  for l being Element of ProjectiveLines real_projective_plane
  for P,Q,R being Point of real_projective_plane st
  P in l & Q in l & R in l holds P,Q,R are_collinear
  proof
    let l be Element of ProjectiveLines real_projective_plane;
    let P,Q,R be Point of real_projective_plane;
    assume
A1: P in l & Q in l & R in l;
    l is LINE of real_projective_plane by INCPROJ:1;
    hence thesis by A1,COLLSP:16;
  end;

theorem Th58:
  for l1,l2,l3 being Element of ProjectiveLines real_projective_plane st
  l1,l2,l3 are_concurrent holds dual l1,dual l2,dual l3 are_collinear
  proof
    let l1,l2,l3 be Element of ProjectiveLines real_projective_plane;
    assume l1,l2,l3 are_concurrent;
    then consider P be Point of real_projective_plane such that
A1: P in l1 & P in l2 & P in l3;
    reconsider lP = dual P as Element of ProjectiveLines real_projective_plane;
    dual l1 in lP & dual l2 in lP & dual l3 in lP by A1,Th54;
    hence thesis by Th57;
  end;

theorem Th59:
  for P,Q,R being Point of real_projective_plane holds
  P,Q,R are_collinear iff dual P,dual Q,dual R are_concurrent
  proof
    let P,Q,R be Point of real_projective_plane;
    thus P,Q,R are_collinear implies dual P, dual Q, dual R are_concurrent
      by Th56;
    assume dual P, dual Q, dual R are_concurrent;
    then dual dual P, dual dual Q, dual dual R are_collinear
      by Th58;
    then P, dual dual Q, dual dual R are_collinear by Th45;
    then P, Q, dual dual R are_collinear by Th45;
    hence thesis by Th45;
  end;

theorem Th60:
  for l1,l2,l3 being Element of ProjectiveLines real_projective_plane holds
  l1,l2,l3 are_concurrent iff dual l1, dual l2, dual l3 are_collinear
  proof
    let l1,l2,l3 be Element of ProjectiveLines real_projective_plane;
    hereby
      assume l1,l2,l3 are_concurrent;
      then dual dual l1,l2,l3 are_concurrent by Th46;
      then dual dual l1,dual dual l2,l3 are_concurrent by Th46;
      then dual dual l1,dual dual l2,dual dual l3 are_concurrent by Th46;
      hence dual l1, dual l2, dual l3 are_collinear by Th59;
    end;
    assume dual l1, dual l2, dual l3 are_collinear;
    then dual dual l1, dual dual l2, dual dual l3 are_concurrent by Th59;
    then l1, dual dual l2, dual dual l3 are_concurrent by Th46;
    then l1, l2, dual dual l3 are_concurrent by Th46;
    hence thesis by Th46;
  end;

begin :: Some converse theorems

theorem
  real_projective_plane is reflexive &
  real_projective_plane is transitive &
  real_projective_plane is Vebleian &
  real_projective_plane is at_least_3rank &
  real_projective_plane is Fanoian &
  real_projective_plane is Desarguesian &
  real_projective_plane is Pappian &
  real_projective_plane is 2-dimensional;

::$N Converse reflexive
theorem
  for l,m,n being Element of ProjectiveLines real_projective_plane holds
  l,m,l are_concurrent & l,l,m are_concurrent & l,m,m are_concurrent
  proof
    let l1,l2,l3 be Element of ProjectiveLines real_projective_plane;
    dual l1,dual l2,dual l1 are_collinear &
      dual l1,dual l1,dual l2 are_collinear &
      dual l1,dual l2,dual l2 are_collinear by ANPROJ_2:def 7;
    then dual dual l1,dual dual l2,dual dual l1 are_concurrent &
      dual dual l1,dual dual l1,dual dual l2 are_concurrent &
      dual dual l1,dual dual l2,dual dual l2 are_concurrent by Th59;
    then l1,dual dual l2,dual dual l1 are_concurrent &
      l1,dual dual l1,dual dual l2 are_concurrent &
      l1,dual dual l2,dual dual l2 are_concurrent by Th46;
    then l1,l2,dual dual l1 are_concurrent &
      l1,l1,dual dual l2 are_concurrent &
      l1,l2,dual dual l2 are_concurrent by Th46;
    hence thesis;
  end;

::$N Converse transitive
theorem
  for l,m,n,n1,n2 being Element of ProjectiveLines real_projective_plane st
  l <> m & l,m,n are_concurrent & l,m,n1 are_concurrent &
  l,m,n2 are_concurrent holds n,n1,n2 are_concurrent
  proof
    let l,m,n,n1,n2 be Element of ProjectiveLines real_projective_plane;
    assume that
A1: l <> m and
A2: l,m,n are_concurrent and
A3: l,m,n1 are_concurrent and
A4: l,m,n2 are_concurrent;
    dual l <> dual m & dual l,dual m, dual n are_collinear &
      dual l,dual m, dual n1 are_collinear &
      dual l,dual m, dual n2 are_collinear by A1,A2,A3,A4,Th60,Th48;
    then dual dual n, dual dual n1, dual dual n2 are_concurrent
      by ANPROJ_2:def 8,Th59;
    then n, dual dual n1, dual dual n2 are_concurrent by Th46;
    then n, n1, dual dual n2 are_concurrent by Th46;
    hence thesis by Th46;
  end;

::$N Converse Vebliean
theorem
  for l,l1,l2,n,n1 being Element of ProjectiveLines real_projective_plane st
  l,l1,n are_concurrent & l1,l2,n1 are_concurrent
  ex n2 being Element of ProjectiveLines real_projective_plane st
  l,l2,n2 are_concurrent & n,n1,n2 are_concurrent
  proof
    let l,l1,l2,n,n1 be Element of ProjectiveLines real_projective_plane;
    assume that
A1: l,l1,n are_concurrent and
A2: l1,l2,n1 are_concurrent;
    dual l, dual l1, dual n are_collinear &
      dual l1,dual l2, dual n1 are_collinear by A1,A2,Th60;
    then consider P be Point of real_projective_plane such that
A3: dual l, dual l2, P are_collinear and
A4: dual n, dual n1, P are_collinear by ANPROJ_2:def 9;
    take dual P;
    dual dual l, dual dual l2, dual P are_concurrent &
      dual dual n, dual dual n1, dual P are_concurrent
      by A3,A4,Th59;
    then l, dual dual l2, dual P are_concurrent &
      n, dual dual n1, dual P are_concurrent by Th46;
    hence thesis by Th46;
  end;

::$N Converse at_least_3rank
theorem
  for l,m being Element of ProjectiveLines real_projective_plane holds
  ex n being Element of ProjectiveLines real_projective_plane st
  l <> n & m <> n & l,m,n are_concurrent
  proof
    let l,m be Element of ProjectiveLines real_projective_plane;
    consider r be Point of real_projective_plane such that
A1: dual l <> r and
A2: dual m <> r and
A3: dual l, dual m, r are_collinear by ANPROJ_2:def 10;
    now
      thus l <> dual r & m <> dual r by Th45,A1,A2;
      dual dual l, dual dual m, dual r are_concurrent by A3,Th59;
      then l, dual dual m, dual r are_concurrent by Th46;
      hence l, m, dual r are_concurrent by Th46;
    end;
    hence thesis;
  end;

::$N Converse Fanoian
theorem
  for l1,n2,m,n1,m1,l,n being Element of ProjectiveLines
  real_projective_plane holds
  (l1,n2,m are_concurrent & n1,m1,m are_concurrent & l1,n1,l are_concurrent &
  n2,m1,l are_concurrent & l1,m1,n are_concurrent & n2,n1,n are_concurrent &
  l,m,n are_concurrent implies
  (l1,n2,m1 are_concurrent or l1,n2,n1 are_concurrent or
  l1,n1,m1 are_concurrent or n2,n1,m1 are_concurrent))
  proof
    let l1,n2,m,n1,m1,l,n be Element of ProjectiveLines real_projective_plane;
    assume that
A1: l1,n2,m are_concurrent and
A2: n1,m1,m are_concurrent and
A3: l1,n1,l are_concurrent and
A4: n2,m1,l are_concurrent and
A5: l1,m1,n are_concurrent and
A6: n2,n1,n are_concurrent and
A7: l,m,n are_concurrent;
    dual l1,dual n2,dual m are_collinear &
      dual n1,dual m1,dual m are_collinear &
      dual l1,dual n1,dual l are_collinear &
      dual n2,dual m1,dual l are_collinear &
      dual l1,dual m1,dual n are_collinear &
      dual n2,dual n1,dual n are_collinear &
      dual l,dual m,dual n are_collinear by A1,A2,A3,A4,A5,A6,A7,Th60;
    then dual l1,dual n2,dual m1 are_collinear or
      dual l1,dual n2,dual n1 are_collinear or
      dual l1,dual n1,dual m1 are_collinear or
      dual n2,dual n1,dual m1 are_collinear by ANPROJ_2:def 11;
    hence thesis by Th60;
  end;

::$N Converse Desarguesian
theorem
  for k,l1,l2,l3,m1,m2,m3,n1,n2,n3 being Element of ProjectiveLines
  real_projective_plane st
  k <> m1 & l1 <> m1 & k <> m2 & l2 <> m2 & k <> m3 & l3 <> m3 &
  not k,l1,l2 are_concurrent & not k,l1,l3 are_concurrent &
  not k,l2,l3 are_concurrent &
  l1,l2,n3 are_concurrent & m1,m2,n3 are_concurrent &
  l2,l3,n1 are_concurrent & m2,m3,n1 are_concurrent &
  l1,l3,n2 are_concurrent & m1,m3,n2 are_concurrent &
  k,l1,m1 are_concurrent & k,l2,m2 are_concurrent &
  k,l3,m3 are_concurrent holds n1,n2,n3 are_concurrent
  proof
    let k,l1,l2,l3,m1,m2,m3,n1,n2,n3 be Element of ProjectiveLines
      real_projective_plane;
    assume that
A1: k <> m1 and
A2: l1 <> m1 and
A3: k <> m2 and
A4: l2 <> m2 and
A5: k <> m3 and
A6: l3 <> m3 and
A7: not k,l1,l2 are_concurrent and
A8: not k,l1,l3 are_concurrent and
A9: not k,l2,l3 are_concurrent and
A10: l1,l2,n3 are_concurrent and
A11: m1,m2,n3 are_concurrent and
A12: l2,l3,n1 are_concurrent and
A13: m2,m3,n1 are_concurrent and
A14: l1,l3,n2 are_concurrent and
A15: m1,m3,n2 are_concurrent and
A16: k,l1,m1 are_concurrent and
A17: k,l2,m2 are_concurrent and
A18: k,l3,m3 are_concurrent;
    dual k <> dual m1 & dual l1 <> dual m1 & dual k <> dual m2 &
      dual l2 <> dual m2 & dual k <> dual m3 & dual l3 <> dual m3 &
      not dual k,dual l1,dual l2 are_collinear &
      not dual k,dual l1,dual l3 are_collinear &
      not dual k,dual l2,dual l3 are_collinear &
      dual l1,dual l2,dual n3 are_collinear &
      dual m1,dual m2,dual n3 are_collinear &
      dual l2,dual l3,dual n1 are_collinear &
      dual m2,dual m3,dual n1 are_collinear &
      dual l1,dual l3,dual n2 are_collinear &
      dual m1,dual m3,dual n2 are_collinear &
      dual k,dual l1,dual m1 are_collinear &
      dual k,dual l2,dual m2 are_collinear &
      dual k,dual l3,dual m3 are_collinear
      by A1,A2,A3,A4,A5,A6,Th48,
         A7,A8,A9,A10,A11,A12,A13,A14,A15,A16,A17,A18,Th60;
    then dual n1,dual n2, dual n3 are_collinear by ANPROJ_2:def 12;
    hence thesis by Th60;
  end;

::$N Converse Pappian
theorem
  for k,l1,l2,l3,m1,m2,m3,n1,n2,n3 being Element of ProjectiveLines
    real_projective_plane st
  k <> l2 & k <> l3 & l2 <> l3 & l1 <> l2 & l1 <> l3 & k <> m2 &
  k <> m3 & m2 <> m3 & m1 <> m2 & m1 <> m3 &
  not k,l1,m1 are_concurrent & k,l1,l2 are_concurrent &
  k,l1,l3 are_concurrent & k,m1,m2 are_concurrent &
  k,m1,m3 are_concurrent & l1,m2,n3 are_concurrent &
  m1,l2,n3 are_concurrent & l1,m3,n2 are_concurrent &
  l3,m1,n2 are_concurrent & l2,m3,n1 are_concurrent &
  l3,m2,n1 are_concurrent holds n1,n2,n3 are_concurrent
  proof
    let k,l1,l2,l3,m1,m2,m3,n1,n2,n3 be Element of ProjectiveLines
      real_projective_plane;
    assume that
A1: k <> l2 and
A2: k <> l3 and
A3: l2 <> l3 and
A4: l1 <> l2 and
A5: l1 <> l3 and
A6: k <> m2 and
A7: k <> m3 and
A8: m2 <> m3 and
A9: m1 <> m2 and
A10: m1 <> m3 and
A11: not k,l1,m1 are_concurrent and
A12: k,l1,l2 are_concurrent and
A13: k,l1,l3 are_concurrent and
A14: k,m1,m2 are_concurrent and
A15: k,m1,m3 are_concurrent and
A16: l1,m2,n3 are_concurrent and
A17: m1,l2,n3 are_concurrent and
A18: l1,m3,n2 are_concurrent and
A19: l3,m1,n2 are_concurrent and
A20: l2,m3,n1 are_concurrent and
A21: l3,m2,n1 are_concurrent;
    now
      thus dual k <> dual l2 & dual k <> dual l3 & dual l2 <> dual l3
        by A1,A2,A3,Th48;
      thus dual l1 <> dual l2 & dual l1 <> dual l3 & dual k <> dual m2
        by A4,A5,A6,Th48;
      thus dual k <> dual m3 & dual m2 <> dual m3 & dual m1 <> dual m2 &
        dual m1 <> dual m3 by A7,A8,A9,A10,Th48;
      thus not dual k,dual l1,dual m1 are_collinear &
        dual k,dual l1,dual l2 are_collinear &
        dual k,dual l1,dual l3 are_collinear &
        dual k,dual m1,dual m2 are_collinear &
        dual k,dual m1,dual m3 are_collinear &
        dual l1,dual m2,dual n3 are_collinear &
        dual m1,dual l2,dual n3 are_collinear &
        dual l1,dual m3,dual n2 are_collinear &
        dual l3,dual m1,dual n2 are_collinear &
        dual l2,dual m3,dual n1 are_collinear &
        dual l3,dual m2,dual n1 are_collinear
        by A11,A12,A13,A14,A15,A16,A17,A18,A19,A20,A21,Th60;
    end;
    then dual n1,dual n2, dual n3 are_collinear by ANPROJ_2:def 13;
    hence thesis by Th60;
  end;

::$N Converse 2_dimensional
theorem
  for l,l1,m,m1 being Element of ProjectiveLines
  real_projective_plane holds ex n being Element of ProjectiveLines
  real_projective_plane st l,l1,n are_concurrent &
  m,m1,n are_concurrent
  proof
    let l,l1,m,m1 be Element of ProjectiveLines
      real_projective_plane;
    consider R be Point of real_projective_plane such that
A1: dual l, dual l1, R are_collinear and
A2: dual m, dual m1, R are_collinear by ANPROJ_2:def 14;
    dual dual l, dual dual l1, dual R are_concurrent &
      dual dual m, dual dual m1, dual R are_concurrent by A1,A2,Th59;
    then l, dual dual l1, dual R are_concurrent &
      m, dual dual m1, dual R are_concurrent by Th46;
    then l, l1, dual R are_concurrent &
      m, m1, dual R are_concurrent by Th46;
    hence thesis;
  end;