
theorem Th37:
  for s being Element of ProjectiveSpace TOP-REAL 3
  for p,q,r being Element of absolute st p,q,r are_mutually_distinct &
  s in tangent p /\ tangent q holds
  ex N being invertible Matrix of 3,F_Real st
  homography(N).:absolute = absolute &
  (homography(N)).Dir101 = p &
  (homography(N)).Dirm101 = q &
  (homography(N)).Dir011 = r &
  (homography(N)).Dir010 = s
  proof
    let s be Element of ProjectiveSpace TOP-REAL 3;
    let p,q,r be Element of absolute;
    assume that
A1: p,q,r are_mutually_distinct and
A2: s in tangent p /\ tangent q;
    reconsider P1 = p,P2 = q,P3 = r,P4 = s as Point of real_projective_plane;
    P4 in tangent p & P4 in tangent q by A2,XBOOLE_0:def 4;
    then not P1,P2,P3 are_collinear &
      not P1,P2,P4 are_collinear &
      not P1,P3,P4 are_collinear &
      not P2,P3,P4 are_collinear by A1,Th27;
    then consider N be invertible Matrix of 3,F_Real such that
A3: (homography(N)).Dir101 = P1 and
A4: (homography(N)).Dirm101 = P2 and
A5: (homography(N)).Dir011 = P3 and
A6: (homography(N)).Dir010 = P4
      by BKMODEL1:44,ANPROJ_9:31;
    consider na,nb,nc,nd,ne,nf,ng,nh,ni be Element of F_Real such that
A7: N = <* <* na,nb,nc *>,
               <* nd,ne,nf *>,
               <* ng,nh,ni *> *> by PASCAL:3;
    reconsider b = -1 as Element of F_Real by XREAL_0:def 1;
A8: b is non zero;
    reconsider a = 1 as Element of F_Real;
    a is non zero;
    then reconsider a = 1,b = -1 as non zero Element of F_Real by A8;
    reconsider N1 = <* <* a,0,0 *>,
                       <* 0,a,0 *>,
                       <* 0,0,b *> *> as invertible Matrix of 3,F_Real
                         by ANPROJ_9:9;
    reconsider M = (N@) * N1 * N as invertible Matrix of 3,F_Real;
A9: N1 = symmetric_3(a,a,b,0,0,0) by PASCAL:def 3;
    then
A10: M is symmetric by PASCAL:7,12;
    consider va,vb,vc,vd,ve,vf,vg,vh,vi be Element of F_Real such that
A11: M = <* <* va,vb,vc *>,
            <* vd,ve,vf *>,
            <* vg,vh,vi *> *> by PASCAL:3;
A12: vb = vd & vc = vg & vh = vf by A10,A11,PASCAL:6;
    reconsider ra = va,rb = vb, rc = vc, re = ve, rf = vf, ri = vi as Real;
A13: M = symmetric_3(ra,re,ri,rb,rc,rf) by A12,A11,PASCAL:def 3;
A14: p in conic(1,1,-1,0,0,0) & N~ is invertible;
    reconsider NR = MXF2MXR(N~) as Matrix of 3,REAL by MATRIXR1:def 2;
A15: N1 = symmetric_3(1,1,-1,0/2,0/2,0/2) by PASCAL:def 3;
    reconsider N2 = N1 as Matrix of 3,REAL;
A16: M = MXF2MXR((MXR2MXF(NR@))~) * N2 * MXF2MXR((MXR2MXF NR)~)
       by A15,BKMODEL1:53;
A17: (homography(N~)).p = Dir101 by A3,ANPROJ_9:15;
A18: not(ra = 0 & re = 0 & ri = 0 & rb = 0 & rc = 0 & rf = 0) &
      (homography(N~)).p in conic(ra,re,ri,2 * rb,2 * rc,2 * rf)
      by A13,A14,A15,A16,PASCAL:16;
    Dir101 in {P where P is Point of ProjectiveSpace TOP-REAL 3:
      for u be Element of TOP-REAL 3 st u is non zero & P = Dir u holds
      qfconic(ra,re,ri,2*rb,2*rc,2*rf,u) = 0} by A18,A17,PASCAL:def 2;
    then consider Q be Point of ProjectiveSpace TOP-REAL 3 such that
A19: Dir101 = Q and
A20: for u be Element of TOP-REAL 3 st u is non zero & Q = Dir u holds
      qfconic(ra,re,ri,2*rb,2*rc,2*rf,u) = 0;
A21: |[1,0,1]|.1 = 1 & |[1,0,1]|.2 = 0 & |[1,0,1]|.3 = 1;
    qfconic(ra,re,ri,2*rb,2*rc,2*rf, |[1,0,1]|) = 0 by A19,A20,BKMODEL1:41;
    then
A22: 0 = ra * 1 * 1 + re * 0 * 0 + ri * 1 * 1 + 2* rb * 1 * 0
               + 2* rc * 1 * 1 + 2 * rf * 0 * 1 by A21,PASCAL:def 1
          .= ra + ri + 2 * rc;
A23:(homography(N~)).q = Dirm101 by A4,ANPROJ_9:15;
       q in conic(1,1,-1,0,0,0) & N~ is invertible;
    then
A25: not(ra = 0 & re = 0 & ri = 0 & rb = 0 & rc = 0 & rf = 0) &
    (homography(N~)).q in conic(ra,re,ri,2 * rb,2 * rc,2 * rf)
      by A13,PASCAL:16,A15,A16;
    Dirm101 in {P where P is Point of ProjectiveSpace TOP-REAL 3:
      for u be Element of TOP-REAL 3 st u is non zero & P = Dir u holds
      qfconic(ra,re,ri,2*rb,2*rc,2*rf,u) = 0} by A23,A25,PASCAL:def 2;
    then consider Q be Point of ProjectiveSpace TOP-REAL 3 such that
A26: Dirm101 = Q and
A27: for u be Element of TOP-REAL 3 st u is non zero & Q = Dir u holds
      qfconic(ra,re,ri,2*rb,2*rc,2*rf,u) = 0;
A28: |[-1,0,1]|.1 = -1 & |[-1,0,1]|.2 = 0 & |[-1,0,1]|.3 = 1;
    qfconic(ra,re,ri,2*rb,2*rc,2*rf, |[-1,0,1]|) = 0
      by A26,A27,BKMODEL1:41;
    then
A29: 0 = ra * (-1) * (-1) + re * 0 * 0 + ri * 1 * 1
              + 2* rb * (-1) * 0 + 2* rc * (-1) * 1 + 2 * rf * 0 * 1
              by A28,PASCAL:def 1
          .= ra + ri - 2 * rc;
A30: (homography(N~)).r = Dir011 by A5,ANPROJ_9:15;
    r in conic(1,1,-1,0,0,0) & N~ is invertible;
    then
A31: for fa,fb,fc,fe,fi,ff be Real for N1,M being Matrix of 3,REAL
      for NR be Matrix of 3,REAL st N1 = symmetric_3(1,1,-1,0/2,0/2,0/2) &
      NR = MXF2MXR (N~) &
      M = MXF2MXR((MXR2MXF(NR@))~) * N1 * MXF2MXR((MXR2MXF NR)~) &
      M = symmetric_3(fa,fe,fi,fb,fc,ff) holds
      not(fa = 0 & fe = 0 & fi = 0 & fb = 0 & ff = 0 & fc = 0) &
      (homography(N~)).r in conic(fa,fe,fi,2 * fb,2 * fc,2 * ff) by PASCAL:16;
    not(ra = 0 & re = 0 & ri = 0 & rb = 0 & rc = 0 & rf = 0) &
      (homography(N~)).r in conic(ra,re,ri,2 * rb,2 * rc,2 * rf)
      by A9,A13,A31,A16;
    then Dir011 in {P where P is Point of ProjectiveSpace TOP-REAL 3:
      for u be Element of TOP-REAL 3 st u is non zero & P = Dir u holds
      qfconic(ra,re,ri,2*rb,2*rc,2*rf,u) = 0} by A30,PASCAL:def 2;
    then consider Q be Point of ProjectiveSpace TOP-REAL 3 such that
A32: Dir011 = Q and
A33: for u be Element of TOP-REAL 3 st u is non zero & Q = Dir u holds
      qfconic(ra,re,ri,2*rb,2*rc,2*rf,u) = 0;
A34: |[0,1,1]|.1 = 0 & |[0,1,1]|.2 = 1 & |[0,1,1]|.3 = 1;
    qfconic(ra,re,ri,2*rb,2*rc,2*rf, |[0,1,1]|) = 0 by A32,A33,BKMODEL1:41;
    then
A35: 0 = ra * 0 * 0 + re * 1 * 1 + ri * 1 * 1 + 2* rb * 0 * 1
              + 2 * rc * 0 * 1 + 2 * rf * 1 * 1 by A34,PASCAL:def 1
        .= re + ri + 2 * rf;
    rc = 0 & ra = - ri & rb = 0 & rf = 0 & ra = re
    proof
      thus rc = 0 by A22,A29;
      thus ra = - ri by A22,A29;
      consider k1 be Element of TOP-REAL 3 such that
A36:  k1 is non zero and
A37:  P1 = Dir k1 by ANPROJ_1:26;
      consider k1b be Element of TOP-REAL 3 such that
A38:  k1b is non zero and
A39:  P2 = Dir k1b by ANPROJ_1:26;
      consider k2 be Element of TOP-REAL 3 such that
A40:  k2 is non zero and
A41:  P4 = Dir k2 by ANPROJ_1:26;
      reconsider kf1 = k1,
                 kf1b = k1b,
                 kf2 =k2 as FinSequence of REAL by EUCLID:24;
A42: P4 in tangent p & N2 is Matrix of 3,REAL &
     p is Element of absolute & Q is Element of real_projective_plane &
     k1 is non zero Element of TOP-REAL 3 &
     k2 is non zero Element of TOP-REAL 3 & kf1 is FinSequence of REAL &
     kf2 is FinSequence of REAL &N2 = symmetric_3(1,1,-1,0,0,0) &
     p = Dir k1 & P4 = Dir k2 & k1 = kf1 & k2 = kf2
       by A2,XBOOLE_0:def 4,PASCAL:def 3,A36,A37,A40,A41;
A43: P4 in tangent q & N2 is Matrix of 3,REAL & p is Element of absolute &
     k1b is non zero Element of TOP-REAL 3 &
     k2 is non zero Element of TOP-REAL 3 & kf1b is FinSequence of REAL &
     kf2 is FinSequence of REAL & N2 = symmetric_3(1,1,-1,0,0,0) &
     q = Dir k1b & P4 = Dir k2 & k1b = kf1b & k2 = kf2
       by A2,A38,A40,PASCAL:def 3,A39,A41,XBOOLE_0:def 4;
     consider ua,va be Element of TOP-REAL 3,
                ufa be FinSequence of F_Real,
                 pa be FinSequence of 1-tuples_on REAL
     such that
A44: Dir101 = Dir ua & ua is not zero & ua = ufa & pa = N * ufa &
       va = M2F pa & va is not zero &
       homography(N).Dir101 = Dir va by ANPROJ_8:def 4;
     are_Prop k1,va by A3,A37,A44,ANPROJ_1:22,A36;
     then consider li be Real such that
A45: li <> 0 and
A46: k1 = li * va by ANPROJ_1:1;
A47: len (N * <*ufa*>@) = len N
     proof
       ufa in TOP-REAL 3 by A44; then
A48:   ufa in REAL 3 by EUCLID:22;
       then len ufa = 3 by EUCLID_8:50;
       then width <*ufa*> = 3 by ANPROJ_8:75; then
A50:    len (<*ufa*>@) = width <*ufa*> by MATRIX_0:29
                     .= len ufa by MATRIX_0:23;
       width N = 3 by MATRIX_0:24
              .= len (<*ufa*>@) by A50,A48,EUCLID_8:50;
       hence thesis by MATRIX_3:def 4;
     end;
A51: len pa = len N by A47,A44,LAPLACE:def 9
           .= 3 by MATRIX_0:23;
     then
A52: kf1 = M2F (li * pa) by A44,A46,ANPROJ_8:83;
     consider ub,vb be Element of TOP-REAL 3,
                ufb be FinSequence of F_Real,
                 pb be FinSequence of 1-tuples_on REAL
     such that
A53: Dir010 = Dir ub & ub is not zero & ub = ufb & pb = N * ufb &
       vb = M2F pb & vb is not zero & homography(N).Dir010 = Dir vb
       by ANPROJ_8:def 4;
     are_Prop ub,|[0,1,0]| by A53,ANPROJ_1:22,ANPROJ_9:def 6,10;
     then consider lub be Real such that
A54: lub <> 0 and
A55: ub = lub * |[0,1,0]| by ANPROJ_1:1;
A56: ufb = |[ lub * 0,lub * 1,lub *0]| by A53,A55,EUCLID_5:8
        .= |[0,lub,0]|;
     lub in REAL by XREAL_0:def 1;
     then reconsider MUFB = <* ufb *> as Matrix of 1,3,F_Real
       by A56,BKMODEL1:27;
A57: now
       len ufb = 3 by A56,FINSEQ_1:45;
       then dom ufb = {1,2,3} by FINSEQ_1:def 3,FINSEQ_3:1;
       then 1 in dom ufb & 2 in dom ufb & 3 in dom ufb by ENUMSET1:def 1;
       then MUFB*(1,1) = |[0,lub,0]|.1 & MUFB*(1,2) = |[0,lub,0]|.2 &
         MUFB*(1,3) = |[0,lub,0]|.3 by A56,ANPROJ_8:70;
       hence MUFB*(1,1) = 0 & MUFB*(1,2) = lub & MUFB*(1,3) = 0;
     end;
     are_Prop k2,vb by A41,A6,A53,ANPROJ_1:22,A40;
     then consider lj be Real such that
A58: lj <> 0 and
A59: k2 = lj * vb by ANPROJ_1:1;
A60: len (N * <*ufb*>@) = len N
     proof
       ufb in TOP-REAL 3 by A53;
       then
A61:   ufb in REAL 3 by EUCLID:22;
       then len ufb = 3 by EUCLID_8:50;
       then width <*ufb*> = 3 by ANPROJ_8:75;
       then
A61bis: len (<*ufb*>@) = width <*ufb*> by MATRIX_0:29
                     .= len ufb by MATRIX_0:23;
       width N = 3 by MATRIX_0:24
              .= len (<*ufb*>@) by A61,A61bis,EUCLID_8:50;
       hence thesis by MATRIX_3:def 4;
     end;
A62: len pb = len N by A60,A53,LAPLACE:def 9
           .= 3 by MATRIX_0:23;
     then
A63: M2F pb is Element of TOP-REAL 3 by ANPROJ_8:82; then
A64: M2F pb is Element of REAL 3 by EUCLID:22; then
A65: len M2F pb = 3 by EUCLID_8:50;
     M2F pa is Element of TOP-REAL 3 by A51,ANPROJ_8:82;
     then
A66: M2F pa is Element of REAL 3 by EUCLID:22; then
A67: len M2F pa = 3 by EUCLID_8:50;
A68: li * (N2 * (M2F pa )) = N2 * (li * M2F pa)
     proof
       width N2 = 3 by MATRIX_0:23;
       hence thesis by A67,MATRIXR1:59;
     end;
A69: len (li * (N2 * (M2F pa))) = 3 & len (N2 * (M2F pa)) = 3
     proof
       ColVec2Mx (M2F pa) = pa by A51,BKMODEL1:33;
       then reconsider Mpa = pa as Matrix of REAL;
A70:   len (N2 * (M2F pa))
         = len (Col(N2 * (ColVec2Mx (M2F pa)),1)) by MATRIXR1:def 11
        .= len (N2 * (ColVec2Mx (M2F pa))) by MATRIX_0:def 8
        .= len (N2 * Mpa) by A51,BKMODEL1:33;
       reconsider N2F = N2,MpaF = Mpa as Matrix of F_Real;
A71:   width N2F = len Mpa by A51,MATRIX_0:23;
       len(N2 * Mpa) = len(N2F * MpaF) by ANPROJ_8:17
                    .= len N2F by A71,MATRIX_3:def 4
                    .= 3 by MATRIX_0:23;
       hence thesis by A70,RVSUM_1:117;
     end;
     then
A72: len M2F pb = len (N2 * ((M2F(li * pa)))) by A51,ANPROJ_8:83,A68,A65;
A73: len M2F pb = len (N2 * (M2F(pa))) by A69,A64,EUCLID_8:50;
A74: kf2 = M2F (lj * pb) by A59,A53,A62,ANPROJ_8:83;
A75: len M2F(lj * pb) = len N2 & len M2F(li * pa) = width N2 &
       len M2F(li * pa) > 0
     proof
A76:   len N2 = 3 & width N2 = 3 by MATRIX_0:23;
       consider p1,p2,p3 be Real such that
A77:   p1 = (pb.1).1 & p2 = (pb.2).1 & p3 = (pb.3).1 &
         lj * pb = <* <* lj * p1 *>, <* lj * p2 *> , <* lj * p3 *> *>
         by A62,ANPROJ_8:def 3;
       len (lj * pb) = 3 by A77,FINSEQ_1:45; then
A78:   M2F (lj * pb) = <* ((lj * pb).1).1,
                          ((lj * pb).2).1,
                          ((lj * pb).3).1 *> by ANPROJ_8:def 2;
       consider p1,p2,p3 be Real such that
A79:   p1 = (pa.1).1 & p2 = (pa.2).1 & p3 = (pa.3).1 &
         li * pa = <* <* li * p1 *>, <* li * p2 *> , <* li * p3 *> *>
         by A51,ANPROJ_8:def 3;
       len (li * pa) = 3 by A79,FINSEQ_1:45;
       then M2F (li * pa) = <* ((li * pa).1).1,
                               ((li * pa).2).1,
                               ((li * pa).3).1 *> by ANPROJ_8:def 2;
       hence thesis by A76,A78,FINSEQ_1:45;
     end;
A80: 0 = SumAll QuadraticForm(M2F(lj * pb),N2,M2F(li * pa))
        by A42,Th26,A52,A74
      .= |( M2F(lj * pb), N2 * (M2F(li * pa)))| by A75,MATRPROB:44
      .= |( lj * M2F pb, N2 * (M2F(li * pa)))| by A62,ANPROJ_8:83
      .= lj * |( M2F pb, N2 * (M2F(li * pa)))| by A72,RVSUM_1:121
      .= lj * |( M2F pb, li * (N2 * (M2F pa )) )| by A68,A51,ANPROJ_8:83
      .= lj * (li * |( M2F pb, N2 * (M2F pa) )| ) by A73,RVSUM_1:121
      .= (lj * li) * |( M2F pb, N2 * (M2F pa) )|;
A81: nb * (- na + nc) + ne * (- nd + nf) = nh * (- ng + ni)
     proof
       consider ua2,va2 be Element of TOP-REAL 3,
                   ufa2 be FinSequence of F_Real,
                    pa2 be FinSequence of 1-tuples_on REAL
      such that
A82:  Dirm101 = Dir ua2 & ua2 is not zero & ua2 = ufa2 &
        pa2 = N * ufa2 & va2 = M2F pa2 &
        va2 is not zero & homography(N).Dirm101 = Dir va2 by ANPROJ_8:def 4;
      are_Prop k1b,va2 by A39,A4,A82,ANPROJ_1:22,A38;
      then consider li2 be Real such that
A83:  li2 <> 0 and
A84:  k1b = li2 * va2 by ANPROJ_1:1;
A85:  len (N * <*ufa2*>@) = len N
      proof
        ufa2 in TOP-REAL 3 by A82; then
A86:    ufa2 in REAL 3 by EUCLID:22;
A87:    len ufa2 = 3 by A86,EUCLID_8:50;
        width <*ufa2*> = 3 by A87,ANPROJ_8:75; then
A88:    len (<*ufa2*>@) = width <*ufa2*> by MATRIX_0:29
                       .= len ufa2 by MATRIX_0:23;
        width N = 3 by MATRIX_0:24
               .= len (<*ufa2*>@) by A88,A86,EUCLID_8:50;
        hence thesis by MATRIX_3:def 4;
      end;
A89:  len pa2 = len N by A85,A82,LAPLACE:def 9
             .= 3 by MATRIX_0:23;
A90:  kf1b = M2F (li2 * pa2) by A82,A84,A89,ANPROJ_8:83;
      M2F pa2 is Element of TOP-REAL 3 by A89,ANPROJ_8:82;
      then
A91:  M2F pa2 is Element of REAL 3 by EUCLID:22;
      then
A92:  len M2F pa2 = 3 by EUCLID_8:50;
A93:  li2 * (N2 * (M2F pa2 )) = N2 * (li2 * M2F pa2)
      proof
        width N2 = 3 by MATRIX_0:23;
        hence thesis by MATRIXR1:59,A92;
      end;
A94:  len (li2 * (N2 * (M2F pa2))) = 3 & len (N2 * (M2F pa2)) = 3
      proof
        ColVec2Mx (M2F pa2) = pa2 by A89,BKMODEL1:33;
        then reconsider Mpa2 = pa2 as Matrix of REAL;
A95:    len (N2 * (M2F pa2))
          = len (Col(N2 * (ColVec2Mx (M2F pa2)),1)) by MATRIXR1:def 11
         .= len (N2 * (ColVec2Mx (M2F pa2))) by MATRIX_0:def 8
         .= len (N2 * Mpa2) by A89,BKMODEL1:33;
        reconsider N2F2 = N2,MpaF2 = Mpa2 as Matrix of F_Real;
A96:    width N2F2 = len Mpa2 by A89,MATRIX_0:23;
        len(N2 * Mpa2) = len(N2F2 * MpaF2) by ANPROJ_8:17
                      .= len N2F2 by A96,MATRIX_3:def 4
                      .= 3 by MATRIX_0:23;
        hence thesis by A95,RVSUM_1:117;
      end;
      then
A97:  len M2F pb = len (N2 * ((M2F(li2 * pa2)))) by A89,ANPROJ_8:83,A93,A65;
A98:  len M2F pb = len (N2 * (M2F(pa2))) by A94,A64,EUCLID_8:50;
A99:  kf2 = M2F (lj * pb) by A59,A53,A62,ANPROJ_8:83;
A100: len M2F(lj * pb) = len N2 & len M2F(li2 * pa2) = width N2 &
      len M2F(li2 * pa2) > 0
      proof
A101:   len N2 = 3 & width N2 = 3 by MATRIX_0:23;
        consider p1,p2,p3 be Real such that
A102:   p1 = (pb.1).1 & p2 = (pb.2).1 & p3 = (pb.3).1 &
        lj * pb = <* <* lj * p1 *>, <* lj * p2 *> , <* lj * p3 *> *>
          by A62,ANPROJ_8:def 3;
        len (lj * pb) = 3 by A102,FINSEQ_1:45; then
A103:   M2F (lj * pb) = <* ((lj * pb).1).1,
                            ((lj * pb).2).1,
                            ((lj * pb).3).1 *> by ANPROJ_8:def 2;
        consider p1b,p2b,p3b be Real such that
A104:   p1b = (pa2.1).1 & p2b = (pa2.2).1 & p3b = (pa2.3).1 &
          li2 * pa2 = <* <* li2 * p1b *>,
                         <* li2 * p2b *> ,
                         <* li2 * p3b *> *> by A89,ANPROJ_8:def 3;
        len (li2 * pa2) = 3 by A104,FINSEQ_1:45;
        then M2F (li2 * pa2) = <* ((li2 * pa2).1).1,
                                  ((li2 * pa2).2).1,
                                  ((li2 * pa2).3).1 *> by ANPROJ_8:def 2;
        hence thesis by A101,A103,FINSEQ_1:45;
      end;
A105: 0 = SumAll QuadraticForm(M2F(lj * pb),N2,M2F(li2 * pa2))
        by A43,Th26,A90,A99
       .= |( M2F(lj * pb), N2 * (M2F(li2 * pa2)))| by A100,MATRPROB:44
       .= |( lj * M2F pb, N2 * (M2F(li2 * pa2)))| by A62,ANPROJ_8:83
       .= lj * |( M2F pb, N2 * (M2F(li2 * pa2)))| by A97,RVSUM_1:121
       .= lj * |( M2F pb, li2 * (N2 * (M2F pa2 )) )| by A93,A89,ANPROJ_8:83
       .= lj * (li2 * |( M2F pb, N2 * (M2F pa2) )| ) by A98,RVSUM_1:121
       .= (lj * li2) * |( M2F pb, N2 * (M2F pa2) )|;
A106: M2F pa2 = <* (pa2.1).1,(pa2.2).1,(pa2.3).1 *> by A89,ANPROJ_8:def 2;
      dom (M2F pa2) = Seg 3 by A91,EUCLID_8:50;
      then (M2F pa2).1 in REAL & (M2F pa2).2 in REAL & (M2F pa2).3 in REAL
        by FINSEQ_1:1,FINSEQ_2:11;
      then reconsider s1 = (pa2.1).1,s2 = (pa2.2).1,s3=(pa2.3).1
        as Element of REAL by A106;
A107: M2F pb = <* (pb.1).1,(pb.2).1,(pb.3).1 *> by A62,ANPROJ_8:def 2;
      dom (M2F pb) = Seg 3 by A64,EUCLID_8:50;
      then (M2F pb).1 in REAL & (M2F pb).2 in REAL & (M2F pb).3 in REAL
        by FINSEQ_1:1,FINSEQ_2:11;
      then reconsider t1 = (pb.1).1,t2 = (pb.2).1,t3 = (pb.3).1
        as Element of F_Real by A107;
      reconsider r1 = 1, r2 = 0, r3 = -1 as Element of F_Real by XREAL_0:def 1;
      nb * (- na + nc) + ne * (- nd + nf) = nh * (- ng + ni)
      proof
        reconsider r1 = 1, r2 = 0, r3 = -1 as Element of F_Real
          by XREAL_0:def 1;
A108:   M2F pa2 = <* (pa2.1).1,(pa2.2).1,(pa2.3).1 *> by A89,ANPROJ_8:def 2;
        dom (M2F pa2) = Seg 3 by A91,EUCLID_8:50;
        then (M2F pa2).1 in REAL & (M2F pa2).2 in REAL & (M2F pa2).3 in REAL
          by FINSEQ_1:1,FINSEQ_2:11;
        then reconsider s1 = (pa2.1).1,s2 = (pa2.2).1,s3=(pa2.3).1
          as Element of REAL by A108;
A109:   M2F pb = <* (pb.1).1,(pb.2).1,(pb.3).1 *> by A62,ANPROJ_8:def 2;
        dom (M2F pb) = Seg 3 by A64,EUCLID_8:50;
        then (M2F pb).1 in REAL & (M2F pb).2 in REAL & (M2F pb).3 in REAL
          by FINSEQ_1:1,FINSEQ_2:11;
        then reconsider t1 = (pb.1).1,t2 = (pb.2).1,t3 = (pb.3).1
          as Element of F_Real by A109;
        M2F pa2 = <* s1,s2,s3 *> by A89,ANPROJ_8:def 2; then
A110:   N2 * M2F pa2 = <* 1 * s1 + 0 * s2 + 0 * s3,
                          0 * s1 + 1 * s2 + 0 * s3,
                          0 * s1 + 0 * s2 + (-1) * s3 *> by PASCAL:9
                    .= <* s1,s2,-s3 *>;
        M2F pb = <* t1,t2,t3 *> by A62,ANPROJ_8:def 2; then
A111:   (M2F pb).1 = t1 &
          (M2F pb).2 = t2 &
          (M2F pb).3 = t3 &
          <* s1,s2,-s3 *>.1 = s1 &
          <* s1,s2,-s3 *>.2 = s2 &
          <* s1,s2,-s3 *>.3 = -s3;
A112:   M2F pb is Element of REAL 3 by A63,EUCLID:22;
A113:   |[ s1,s2,-s3 ]| is Element of REAL 3 by EUCLID:22;
        0 = |( M2F pb  , <* s1,s2,-s3 *> )| by A110,A105,A83,A58
         .= t1 * s1 + t2 * s2 + t3 * (-s3) by A112,A113,EUCLID_8:63,A111;
        then
A114:   t1 * s1 + t2 * s2 = t3 * s3;
        |[-1,0,1]| is non zero by EUCLID_5:4,FINSEQ_1:78;
        then are_Prop ua2,|[-1,0,1]| by A82,ANPROJ_1:22;
        then consider lua2 be Real such that
A115:   lua2 <> 0 and
A116:   ua2 = lua2 * |[-1,0,1]| by ANPROJ_1:1;
A117:   ua2 = |[ lua2 * (-1), lua2 * 0, lua2 * 1]| by A116,EUCLID_5:8
           .= |[ - lua2,0,lua2]|;
        reconsider za1 = - lua2,za2 = 0,za3 = lua2 as Element of F_Real
          by XREAL_0:def 1;
        lua2 in REAL & -lua2 in REAL by XREAL_0:def 1;
        then reconsider MUFA = <* ufa2 *> as Matrix of 1,3,F_Real
          by A117,A82,BKMODEL1:27;
        now
          len ufa2 = 3 by A117,A82,FINSEQ_1:45;
          then dom ufa2 = {1,2,3} by FINSEQ_1:def 3,FINSEQ_3:1;
          then 1 in dom ufa2 & 2 in dom ufa2 & 3 in dom ufa2 by ENUMSET1:def 1;
          then MUFA*(1,1) = |[- lua2,0,lua2]|.1 &
               MUFA*(1,2) = |[ - lua2,0,lua2]|.2 &
               MUFA*(1,3) = |[ - lua2,0,lua2]|.3 by A117,A82,ANPROJ_8:70;
          hence MUFA*(1,1) = - lua2 &
                MUFA*(1,2) = 0 &
                MUFA*(1,3) = lua2;
        end;
        then
A119:   (<* ufa2 *>)@ = <* <* - lua2 *>, <* 0 *>, <* lua2 *> *>
          by BKMODEL1:31;
        reconsider nlua2 = - lua2 as Element of F_Real by XREAL_0:def 1;
        0 is Element of F_Real & lua2 is Element of F_Real by XREAL_0:def 1;
        then reconsider MUFAT = <* <* nlua2 *>, <* 0 *>, <* lua2 *> *>
          as Matrix of 3,1,F_Real by BKMODEL1:28;
A120:   N * MUFAT is Matrix of 3,1,F_Real by BKMODEL1:24;
A121:   N * ufa2 = N * MUFAT by A119,LAPLACE:def 9;
        then N * ufa2 = <* <* (N * ufa2)*(1,1) *>,
        <* (N * ufa2)*(2,1) *> ,
                           <* (N * ufa2)*(3,1) *> *> by A120,BKMODEL1:30;
        then
A122:   pa2.1 = <* (N * ufa2)*(1,1) *> & pa2.2 = <* (N * ufa2)*(2,1) *> &
          pa2.3 = <* (N * ufa2)*(3,1) *> by A82;
        N * MUFAT is Matrix of 3,1,F_Real by BKMODEL1:24;
        then
A123:   Indices (N * MUFAT) = [: Seg 3,Seg 1:] by MATRIX_0:23;
        width N = 3 by MATRIX_0:24;
        then
A124:   width N = len MUFAT by MATRIX_0:23;
A125:    Col(MUFAT,1) = <* za1,za2,za3 *> by ANPROJ_8:5;
A126:    Line(N,1) = <* na,nb,nc *> &
          Line(N,2) = <* nd,ne,nf *> &
          Line(N,3) = <* ng,nh,ni *> by A7,ANPROJ_9:4;
        (N * MUFAT)*(1,1) = Line(N,1) "*" Col(MUFAT,1) &
          (N * MUFAT)*(2,1) = Line(N,2) "*" Col(MUFAT,1) &
          (N * MUFAT)*(3,1) = Line(N,3) "*" Col(MUFAT,1)
          by A123,A124,MATRIX_3:def 4,ANPROJ_8:2;
        then (N * MUFAT)*(1,1) = na * za1 + nb * za2 + nc * za3 &
          (N * MUFAT)*(2,1) = nd * za1 + ne * za2 + nf * za3 &
          (N * MUFAT)*(3,1) = ng * za1 + nh * za2 + ni * za3
          by A125,A126,ANPROJ_8:7;
        then
A127:   (pa2.1).1 = na * nlua2 + nc * lua2 &
          (pa2.2).1 = nd * nlua2 + nf * lua2 &
          (pa2.3).1 = ng * nlua2 + ni * lua2 by A121,A122;
        reconsider z1 = 0,z2 = lub,z3 = 0 as Element of F_Real
          by XREAL_0:def 1;
        0 is Element of F_Real & lub is Element of F_Real by XREAL_0:def 1;
        then
        reconsider MUFBT = <* <* 0 *>, <* lub *>, <* 0 *> *>
          as Matrix of 3,1,F_Real by BKMODEL1:28;
A128:   N * MUFBT is Matrix of 3,1,F_Real by BKMODEL1:24;
A129:   N * ufb = N * (<*ufb*>@) by LAPLACE:def 9
               .= N * MUFBT by A57,BKMODEL1:31;
        then N * ufb = <* <* (N * ufb)*(1,1) *>,
                          <* (N * ufb)*(2,1) *> ,
                          <* (N * ufb)*(3,1) *> *> by A128,BKMODEL1:30;
        then
A130:   pb.1 = <* (N * ufb)*(1,1) *> & pb.2 = <* (N * ufb)*(2,1) *> &
          pb.3 = <* (N * ufb)*(3,1) *> by A53;
        N * MUFBT is Matrix of 3,1,F_Real by BKMODEL1:24;
        then
A131:   Indices (N * MUFBT) = [: Seg 3,Seg 1:] by MATRIX_0:23;
        width N = 3 by MATRIX_0:24; then
A132:   width N = len MUFBT by MATRIX_0:23;
        reconsider z1 = 0, z2 = lub, z3 = 0 as Element of F_Real
          by XREAL_0:def 1;
A133:   Col(MUFBT,1) = <* z1,z2,z3 *> by ANPROJ_8:5;
A134:   Line(N,1) = <* na,nb,nc *> &
          Line(N,2) = <* nd,ne,nf *> &
          Line(N,3) = <* ng,nh,ni *> by ANPROJ_9:4,A7;
        (N * MUFBT)*(1,1) = Line(N,1) "*" Col(MUFBT,1) &
          (N * MUFBT)*(2,1) = Line(N,2) "*" Col(MUFBT,1) &
          (N * MUFBT)*(3,1) = Line(N,3) "*" Col(MUFBT,1)
          by A132,MATRIX_3:def 4,A131,ANPROJ_8:2;
        then (N * MUFBT)*(1,1) = na * z1 + nb * z2 + nc * z3 &
          (N * MUFBT)*(2,1) = nd * z1 + ne * z2 + nf * z3 &
          (N * MUFBT)*(3,1) = ng * z1 + nh * z2 + ni * z3
           by A133,A134,ANPROJ_8:7;
        then (pb.1).1 = nb * lub & (pb.2).1 = ne * lub &
          (pb.3).1 = nh * lub by A129,A130;
        then (lua2 * lub) * (nb * (- na + nc)) +
          (lua2 * lub) * (ne * (- nd + nf))
          = (lua2 * lub ) * (nh * (- ng + ni))
          by A127,A114;
        then (lua2 * lub) * (nb * (- na + nc))
          + (lua2 * lub) * (ne * (- nd + nf))
          - (lua2 * lub ) * (nh * (- ng + ni)) = 0;
        then (lua2 * lub) * ((nb * (- na + nc))
          + (ne * (- nd + nf)) - (nh * (- ng + ni))) = 0;
        then (nb * (- na + nc)) + (ne * (- nd + nf)) - (nh * (- ng + ni)) = 0
          by A54,A115;
        hence thesis;
      end;
      hence thesis;
    end;
A136: nb * (na + nc) + ne * (nd + nf) = nh * (ng + ni)
    proof
      reconsider r1 = 1, r2 = 0, r3 = -1 as Element of F_Real by XREAL_0:def 1;
A137: M2F pa = <* (pa.1).1,(pa.2).1,(pa.3).1 *> by A51,ANPROJ_8:def 2;
      dom (M2F pa) = Seg 3 by A66,EUCLID_8:50;
      then (M2F pa).1 in REAL & (M2F pa).2 in REAL & (M2F pa).3 in REAL
        by FINSEQ_1:1,FINSEQ_2:11;
      then reconsider s1 = (pa.1).1,s2 = (pa.2).1,s3=(pa.3).1
        as Element of REAL by A137;
A138: M2F pb = <* (pb.1).1,(pb.2).1,(pb.3).1 *> by A62,ANPROJ_8:def 2;
      dom (M2F pb) = Seg 3 by A64,EUCLID_8:50;
      then (M2F pb).1 in REAL & (M2F pb).2 in REAL & (M2F pb).3 in REAL
        by FINSEQ_1:1,FINSEQ_2:11;
      then reconsider t1 = (pb.1).1,t2 = (pb.2).1,t3 = (pb.3).1
        as Element of F_Real by A138;
      M2F pa = <* s1,s2,s3 *> by A51,ANPROJ_8:def 2;
      then
A139: N2 * M2F pa
        = <* 1 * s1 + 0 * s2 + 0 * s3,
             0 * s1 + 1 * s2 + 0 * s3,
             0 * s1 + 0 * s2 + (-1) * s3 *> by PASCAL:9
       .= <* s1,s2,-s3 *>;
      M2F pb = <* t1,t2,t3 *> by A62,ANPROJ_8:def 2; then
A140: (M2F pb).1 = t1 &
        (M2F pb).2 = t2 &
        (M2F pb).3 = t3 &
        <* s1,s2,-s3 *>.1 = s1 &
        <* s1,s2,-s3 *>.2 = s2 &
        <* s1,s2,-s3 *>.3 = -s3;
A141: M2F pb is Element of REAL 3 by A63,EUCLID:22;
A142: |[ s1,s2,-s3 ]| is Element of REAL 3 by EUCLID:22;
      0 = |( M2F pb  , <* s1,s2,-s3 *> )| by A139,A80,A45,A58
       .= t1 * s1 + t2 * s2 + t3 * (-s3)
         by A141,A142,EUCLID_8:63,A140;
      then
A143: t1 * s1 + t2 * s2 = t3 * s3;
      |[1,0,1]| is non zero by EUCLID_5:4,FINSEQ_1:78;
      then are_Prop ua,|[1,0,1]| by A44,ANPROJ_1:22;
      then consider lua be Real such that
A145: lua <> 0 and
A146: ua = lua * |[1,0,1]| by ANPROJ_1:1;
A147: ua = |[ lua * 1, lua * 0, lua * 1]| by A146,EUCLID_5:8
        .= |[lua,0,lua]|;
      reconsider za1 = lua,za2 = 0,za3 = lua as Element of F_Real
        by XREAL_0:def 1;
      lua in REAL by XREAL_0:def 1;
      then reconsider MUFA = <* ufa *> as Matrix of 1,3,F_Real
        by A147,A44,BKMODEL1:27;
      now
        len ufa = 3 by A147,A44,FINSEQ_1:45;
        then dom ufa = {1,2,3} by FINSEQ_1:def 3,FINSEQ_3:1;
        then 1 in dom ufa & 2 in dom ufa & 3 in dom ufa by ENUMSET1:def 1;
        then MUFA*(1,1) = |[lua,0,lua]|.1 &
             MUFA*(1,2) = |[lua,0,lua]|.2 &
             MUFA*(1,3) = |[lua,0,lua]|.3 by A147,A44,ANPROJ_8:70;
        hence MUFA*(1,1) = lua & MUFA*(1,2) = 0 & MUFA*(1,3) = lua;
      end;
      then
A148: (<* ufa *>)@ = <* <* lua *>, <* 0 *>, <* lua *> *>
        by BKMODEL1:31;
      0 is Element of F_Real & lua is Element of F_Real by XREAL_0:def 1;
      then reconsider MUFAT = <* <* lua *>, <* 0 *>, <* lua *> *>
        as Matrix of 3,1,F_Real by BKMODEL1:28;
A149: N * MUFAT is Matrix of 3,1,F_Real by BKMODEL1:24;
A150: N * ufa = N * MUFAT by A148,LAPLACE:def 9;
      N * ufa = <* <* (N * ufa)*(1,1) *>,
                   <* (N * ufa)*(2,1) *> ,
                   <* (N * ufa)*(3,1) *> *> by A149,A150,BKMODEL1:30;
      then pa.1 = <* (N * ufa)*(1,1) *> & pa.2 = <* (N * ufa)*(2,1) *> &
        pa.3 = <* (N * ufa)*(3,1) *> by A44;
      then
A152: (pa.1).1 = (N * MUFAT)*(1,1) & (pa.2).1 = (N * MUFAT)*(2,1) &
        (pa.3).1 = (N * MUFAT)*(3,1) by A150;
      N * MUFAT is Matrix of 3,1,F_Real by BKMODEL1:24;
      then
A153: Indices (N * MUFAT) = [: Seg 3,Seg 1:] by MATRIX_0:23;
      width N = 3 by MATRIX_0:24;
      then
A154: width N = len MUFAT by MATRIX_0:23;
A155: Col(MUFAT,1) = <* za1,za2,za3 *> by ANPROJ_8:5;
A156: Line(N,1) = <* na,nb,nc *> &
        Line(N,2) = <* nd,ne,nf *> &
        Line(N,3) = <* ng,nh,ni *> by ANPROJ_9:4,A7;
      (N * MUFAT)*(1,1) = Line(N,1) "*" Col(MUFAT,1) &
        (N * MUFAT)*(2,1) = Line(N,2) "*" Col(MUFAT,1) &
        (N * MUFAT)*(3,1) = Line(N,3) "*" Col(MUFAT,1)
        by A154,MATRIX_3:def 4,A153,ANPROJ_8:2;
      then
A157: (pa.1).1 = na * za1 + nb * za2 + nc * za3 &
        (pa.2).1 = nd * za1 + ne * za2 + nf * za3 &
        (pa.3).1 = ng * za1 + nh * za2 + ni * za3 by A152,A155,A156,ANPROJ_8:7;
      reconsider z1 = 0,z2 = lub,z3 = 0 as Element of F_Real by XREAL_0:def 1;
      0 is Element of F_Real & lub is Element of F_Real by XREAL_0:def 1;
      then reconsider MUFBT = <* <* 0 *>, <* lub *>, <* 0 *> *>
        as Matrix of 3,1,F_Real by BKMODEL1:28;
A158: N * MUFBT is Matrix of 3,1,F_Real by BKMODEL1:24;
A159: N * ufb = N * (<*ufb*>@) by LAPLACE:def 9
             .= N * MUFBT by A57,BKMODEL1:31;
      N * ufb = <* <* (N * ufb)*(1,1) *>,
                   <* (N * ufb)*(2,1) *>,
                   <* (N * ufb)*(3,1) *> *> by A158,A159,BKMODEL1:30;
      then
A160: pb.1 = <* (N * ufb)*(1,1) *> & pb.2 = <* (N * ufb)*(2,1) *> &
        pb.3 = <* (N * ufb)*(3,1) *> by A53;
      N * MUFBT is Matrix of 3,1,F_Real by BKMODEL1:24;
      then
A161: Indices (N * MUFBT) = [: Seg 3,Seg 1:] by MATRIX_0:23;
      width N = 3 by MATRIX_0:24;
      then
A162: width N = len MUFBT by MATRIX_0:23;
      reconsider z1 = 0, z2 = lub, z3 = 0 as Element of F_Real
        by XREAL_0:def 1;
A163: Col(MUFBT,1) = <* z1,z2,z3 *> by ANPROJ_8:5;
A164: Line(N,1) = <* na,nb,nc *> &
        Line(N,2) = <* nd,ne,nf *> &
        Line(N,3) = <* ng,nh,ni *> by ANPROJ_9:4,A7;
      (N * MUFBT)*(1,1) = Line(N,1) "*" Col(MUFBT,1) &
        (N * MUFBT)*(2,1) = Line(N,2) "*" Col(MUFBT,1) &
        (N * MUFBT)*(3,1) = Line(N,3) "*" Col(MUFBT,1)
        by A162,MATRIX_3:def 4,A161,ANPROJ_8:2;
      then (N * MUFBT)*(1,1) = na * z1 + nb * z2 + nc * z3 &
        (N * MUFBT)*(2,1) = nd * z1 + ne * z2 + nf * z3 &
        (N * MUFBT)*(3,1) = ng * z1 + nh * z2 + ni * z3
        by A163,A164,ANPROJ_8:7;
      then (pb.1).1 = nb * lub & (pb.2).1 = ne * lub &
        (pb.3).1 = nh * lub by A160,A159;
      then (lua * lub) * (nb * (na + nc)) +
        (lua * lub) * (ne * (nd + nf)) = (lua * lub ) * (nh * (ng + ni))
        by A157,A143;
      then (lua * lub) * (nb * (na + nc))
        + (lua * lub) * (ne * (nd + nf))
        - (lua * lub ) * (nh * (ng + ni)) = 0;
      then (lua * lub) * ((nb * (na + nc))
        + (ne * (nd + nf)) - (nh * (ng + ni))) = 0;
      then (nb * (na + nc)) + (ne * (nd + nf)) - (nh * (ng + ni)) = 0
        by A54,A145;
      hence thesis;
    end;
    <* <* na,nb,nc *>,
       <* nd,ne,nf *>,
       <* ng,nh,ni *> *> = <* <* N*(1,1),N*(1,2),N*(1,3)*>,
                              <* N*(2,1),N*(2,2),N*(2,3)*>,
                              <* N*(3,1),N*(3,2),N*(3,3)*> *>
                              by A7,MATRIXR2:37;
    then
A166: na = N*(1,1) & nb = N*(1,2) & nc = N*(1,3) &
       nd = N*(2,1) & ne = N*(2,2) & nf = N*(2,3) &
       ng = N*(3,1) & nh = N*(3,2) & ni = N*(3,3) by PASCAL:2;
    width N > 0 by MATRIX_0:23;
    then len N = 3 & len N1 = 3 & width N1 = 3 &
      width (N@) = len N by MATRIX_0:29,23;
    then
A167: <* <* ra,rb,rc *>,
         <* rb,re,rf *>,
         <* rc,rf,ri *> *> = N@ * (N1 * N) by A12,A11,MATRIX_3:33
                          .= <* <* (N@ * (N1 * N))*(1,1),
                                   (N@ * (N1 * N))*(1,2),
                                   (N@ * (N1 * N))*(1,3) *>,
                                <* (N@ * (N1 * N))*(2,1),
                                   (N@ * (N1 * N))*(2,2),
                                   (N@ * (N1 * N))*(2,3) *>,
                                <* (N@ * (N1 * N))*(3,1),
                                   (N@ * (N1 * N))*(3,2),
                                   (N@ * (N1 * N))*(3,3) *> *>
                                     by MATRIXR2:37;
A168: (N@ * (N1 * N))*(1,1) = a * (N*(1,1)) * (N*(1,1))
       + a * (N*(2,1)) * (N*(2,1)) + b * (N*(3,1)) * (N*(3,1)) &
    (N@ * (N1 * N))*(1,2) = a * (N*(1,1)) * (N*(1,2))
       + a * (N*(2,1)) * (N*(2,2)) + b * (N*(3,1)) * (N*(3,2)) &
    (N@ * (N1 * N))*(1,3) = a * (N*(1,1)) * (N*(1,3))
       + a * (N*(2,1)) * (N*(2,3)) + b * (N*(3,1)) * (N*(3,3)) &
    (N@ * (N1 * N))*(2,1) = a * (N*(1,2)) * (N*(1,1))
       + a * (N*(2,2)) * (N*(2,1)) + b * (N*(3,2)) * (N*(3,1)) &
    (N@ * (N1 * N))*(2,2) = a * (N*(1,2)) * (N*(1,2))
       + a * (N*(2,2)) * (N*(2,2)) + b * (N*(3,2)) * (N*(3,2)) &
    (N@ * (N1 * N))*(2,3) = a * (N*(1,2)) * (N*(1,3))
       + a * (N*(2,2)) * (N*(2,3)) + b * (N*(3,2)) * (N*(3,3)) &
    (N@ * (N1 * N))*(3,1) = a * (N*(1,3)) * (N*(1,1))
       + a * (N*(2,3)) * (N*(2,1)) + b * (N*(3,3)) * (N*(3,1)) &
    (N@ * (N1 * N))*(3,2) = a * (N*(1,3)) * (N*(1,2))
       + a * (N*(2,3)) * (N*(2,2)) + b * (N*(3,3)) * (N*(3,2)) &
    (N@ * (N1 * N))*(3,3) = a * (N*(1,3)) * (N*(1,3))
       + a * (N*(2,3)) * (N*(2,3)) + b * (N*(3,3)) * (N*(3,3))
       by BKMODEL1:23;
A169: ra = na * na + nd * nd - ng * ng &
     rb = na * nb + nd * ne - ng * nh &
     rc = na * nc + nd * nf - ng * ni &
     rb = na * nb + nd * ne - ng * nh &
     re = nb * nb + ne * ne - nh * nh &
     rf = nb * nc + ne * nf - nh * ni &
     rc = na * nc + nd * nf - ng * ni &
     rf = nb * nc + ne * nf - nh * ni &
     ri = nc * nc + nf * nf - ni * ni by A166,A167,PASCAL:2,A168;
A170: nb * na + nb * nc + ne * nd + ne * nf = nh * ng + nh * ni by A136;
    - nb * na + nb * nc + -ne * nd + ne * nf = -nh * ng + nh * ni by A81;
    hence thesis by A170,A169,A22,A29,A35;
  end;
  then
A170: M = symmetric_3(ra,ra,-ra,0,0,0) by A12,A11,PASCAL:def 3;
A171: ra <> 0
    proof
      assume ra = 0;
      then Det M = 0.F_Real by A170,BKMODEL1:22;
      hence contradiction by LAPLACE:34;
    end;
    then
A172: homography(M).:absolute = absolute by A170,Th29;
    take N;
    thus thesis by A9,A171,A172,A170,A3,A4,A5,A6,BKMODEL1:93;
  end;
