reserve a,b,c,d,e,f for Real,
        g           for positive Real,
        x,y         for Complex,
        S,T         for Element of REAL 2,
        u,v,w       for Element of TOP-REAL 3;
reserve a,b,c for Element of F_Real,
          M,N for Matrix of 3,F_Real;
reserve D        for non empty set;
reserve d1,d2,d3 for Element of D;
reserve A        for Matrix of 1,3,D;
reserve B        for Matrix of 3,1,D;
reserve u,v for non zero Element of TOP-REAL 3;
reserve P,Q,R for POINT of IncProjSp_of real_projective_plane,
            L for LINE of IncProjSp_of real_projective_plane,
        p,q,r for Point of real_projective_plane;
reserve u,v,w for non zero Element of TOP-REAL 3;

theorem
  for ra being non zero Real
  for O,N,M being invertible Matrix of 3,F_Real st
  O = symmetric_3(1,1,-1,0,0,0) & M = symmetric_3(ra,ra,-ra,0,0,0) &
  M = N@ * O * N & homography(M).:absolute = absolute
  holds homography(N).:absolute = absolute
  proof
    let ra be non zero Real;
    let O,N,M be invertible Matrix of 3,F_Real;
    assume that
A1: O = symmetric_3(1,1,-1,0,0,0) and
A2: M = symmetric_3(ra,ra,-ra,0,0,0) and
A4: M = N@ * O * N and
A5: homography(M).:absolute = absolute;
    reconsider O1 = O as Matrix of 3,REAL;
A6: len O1 = 3 & width O1 = 3 by MATRIX_0:24;
A7: homography(N).:absolute c= absolute
    proof
      let x be object;assume x in homography(N).:absolute;
      then consider y be object such that
A8:   y in dom homography(N) and
A9:   y in absolute and
A10:  (homography(N)).y = x by FUNCT_1:def 6;
A11:  rng homography(N) c= the carrier of ProjectiveSpace TOP-REAL 3
        by RELAT_1:def 19;
      reconsider y9 = y as Element of ProjectiveSpace TOP-REAL 3 by A8;
      consider z be object such that
A12:  z in dom homography(M) and
A13:  z in absolute and
A14:  (homography(M)).z = y by A9,A5,FUNCT_1:def 6;
      reconsider z9 = z as Element of ProjectiveSpace TOP-REAL 3 by A12;
A15:  x = (homography(N)).(homography(M).z9) by A14,A10
       .= (homography(N*M)).z by ANPROJ_9:13;
      reconsider NM = N * M as invertible Matrix of 3,F_Real;
      reconsider NMR = NM as Matrix of 3,REAL;
      consider u,v be Element of TOP-REAL 3,
                uf be FinSequence of F_Real,
                 p be FinSequence of 1-tuples_on REAL such that
A16:  z9 = Dir u & u is not zero & u = uf & p = NM * uf & v = M2F p &
        v is not zero & (homography(NM)).z9 = Dir v by ANPROJ_8:def 4;
      reconsider u1 = u as FinSequence of REAL by EUCLID:24;
      u is Element of REAL 3 by EUCLID:22; then
A17:  len u1 = 3 by EUCLID_8:50;
      reconsider x9 = x as Element of ProjectiveSpace TOP-REAL 3
        by A10,A11,A8,FUNCT_1:3;
      consider u9 be Element of TOP-REAL 3 such that
A18:  u9 is not zero and
A19:  x9 = Dir u9 by ANPROJ_1:26;
      reconsider uf9 = u9 as FinSequence of REAL by EUCLID:24;
      u9 is Element of REAL 3 by EUCLID:22;
      then len uf9 = 3 by EUCLID_8:50; then
A20:  len uf9 = len O1 & len uf9 = width O1 & len uf9 > 0 by MATRIX_0:24;
      are_Prop u9,v by A19,A18,A16,ANPROJ_1:22,A15;
      then consider b be Real such that
      b <> 0 and
A21:  u9 = b * v by ANPROJ_1:1;
      reconsider vf = v as FinSequence of REAL by EUCLID:24;
A22:  v is Element of REAL 3 by EUCLID:22; then
A23:  len vf = 3 by EUCLID_8:50;
A24:  width O1 = len vf & len vf > 0 by A6,A22,EUCLID_8:50;
      |( uf9, O1 * uf9 )| = 0
      proof
        width O1 = len vf & len vf > 0 by A23,MATRIX_0:24; then
A25:    len (O1 * vf) = len O1 by MATRIXR1:61
                     .= 3 by MATRIX_0:24; then
A26:    len vf = len (b * (O1 * vf)) by A23,RVSUM_1:117;
A27:    len vf = len (O1 * vf) by A25,A22,EUCLID_8:50;
A28:    |( uf9, O1 * uf9 )| = |( b * vf, b * (O1 * vf) )|
                                   by A21,A24,MATRIXR1:59
                           .= b * |( vf, b * (O1 * vf) )| by A26,RVSUM_1:121
                           .= b * (b * |(O1 * vf, vf)| ) by A27,RVSUM_1:121;
        |( vf, O1 * vf )| = 0
        proof
A29:      len N > 0 & width N > 0 by MATRIX_0:24;
A30:      (N@ * O * N)@ = (N@ * (N@ * O)@) by MATRIX14:30
                       .= (N@ * (O@ * N@@)) by MATRIX14:30
                       .= (N@ * (O@ * N)) by A29,MATRIX_0:57;
A31:      len N = 3 & width N = 3 & len M = 3 & width M = 3 by MATRIX_0:24;
A32:      len (O * (N * M)) = 3 by MATRIX_0:24;
A33:      len O = 3 & width O = 3 & len N = 3 by MATRIX_0:24; then
A34:      width (N@) = len O & width O = len N by MATRIX_0:24;
A35:      width M = len (N@) & width (N@) = len (O * (N * M))
            by A31,A32,MATRIX_0:29;
A36:      width (N@) = len O & width O = len (N * M) by MATRIX_0:24,A33;
A37:      width (N@ * O) = len N & width N = len M by A31,MATRIX_0:24;
A38:      width M = 3 & len M = 3 by MATRIX_0:24;
          reconsider ra3 = ra * ra * ra as Element of F_Real by XREAL_0:def 1;
          reconsider ea = ra as Element of F_Real by XREAL_0:def 1;
A39:      M = symmetric_3(ea,ea,-ea,0,0,0) by A2;
          O1 * NMR = O * (N * M) by ANPROJ_8:17; then
A40:      NMR@ * (O1 * NMR) = (N * M)@ * (O * (N * M)) by ANPROJ_8:17
                           .= ((N@ * (O@ * N)) * N@) * (O * (N * (N@ * O * N)))
                             by A30,A4,A31,MATRIX_3:22
                           .= ((N@ * (O * N)) * N@) * (O * (N * (N@ * O * N)))
                             by A1,PASCAL:12,MATRIX_6:def 5
                           .= (M * N@) * (O * (N * (N@ * O * N)))
                              by A4,A34,MATRIX_3:33
                           .= (M * ((N@) * (O * (N * M))))
                              by A4,A35,MATRIX_3:33
                           .= M * ((N@ * O) * (N * M)) by A36,MATRIX_3:33
                           .= M * ((N@ * O * N) * M) by A37,MATRIX_3:33
                           .= M * M * M by A4,A38,MATRIX_3:33
                           .= (ea * ea * ea) * symmetric_3(1,1,-1,0,0,0)
                              by A39,Th49
                           .= ra3 * O1 by Th46,A1;
          reconsider ONMRUF9 = O1 * (NMR * u1) as FinSequence of REAL;
A41:      u is Element of REAL 3 by EUCLID:22; then
A42:      len u1 = 3 by EUCLID_8:50;
A43:      width (NMR@) = 3 by MATRIX_0:24;
A44:      width NMR = 3 & len NMR = 3 by MATRIX_0:24;
A45:      len u1 = width (O1 * NMR) & len (O1 * NMR) = width (NMR@) &
            len (O1 * NMR) > 0 & len u1 > 0 by A42,A43,MATRIX_0:24;
          len u1 = width NMR & width O1 = len NMR & len u1 > 0 &
            len NMR > 0 by A41,EUCLID_8:50,A44,MATRIX_0:24; then
A46:      (NMR@) * (O1 * (NMR * u1)) = (NMR@) * ((O1 * NMR) * u1)
                                            by MATRIXR2:59
                                    .= (ra3 * O1) * u1
                                            by A40,A45,MATRIXR2:59;
          width O1 = len u1 & len u1 > 0 by A42,MATRIX_0:24;
          then len (O1 * u1) = len O1 by MATRIXR1:61
                            .= 3 by MATRIX_0:24;
          then
A47:      len u1 = len (O1 * u1) by A41,EUCLID_8:50;
A48:      len O1 = 3 by MATRIX_0:24;
          width NMR = 3 by MATRIX_0:24; then
A49:       len (NMR * u1) = len NMR by A42,MATRIXR1:61
                        .= 3 by MATRIX_0:24; then
A50:      width O1 = len (NMR * u1) by MATRIX_0:24;
A51:      len ONMRUF9 = len NMR & len u1 = width NMR & len u1 > 0 &
            len ONMRUF9 > 0
              by A44,A41,EUCLID_8:50,A49,A48,A50,MATRIXR1:61;
          consider s1,s2,s3,s4,s5,s6,s7,s8,s9 be Element of F_Real such that
A52:      NM = <* <* s1,s2,s3 *>,
                  <* s4,s5,s6 *>,
                  <* s7,s8,s9 *> *> by PASCAL:3;
          consider t1,t2,t3 be Real such that
A53:      u = <* t1,t2,t3 *> by EUCLID_5:1;
          reconsider et1 = t1, et2 = t2, et3 = t3 as Element of F_Real
            by XREAL_0:def 1;
A54:      vf = <* s1 * et1 + s2 * et2 + s3 * et3,
                  s4 * et1 + s5 * et2 + s6 * et3,
                  s7 * et1 + s8 * et2 + s9 * et3 *>
                    by A16,A52,A53,PASCAL:8;
          reconsider rs1 = s1, rs2 = s2, rs3 = s3,
                     rs4 = s4, rs5 = s5, rs6 = s6,
                     rs7 = s7, rs8 = s8, rs9 = s9 as Element of REAL;
          reconsider rt1 = t1, rt2 = t2, rt3 = t3 as Element of REAL
            by XREAL_0:def 1;
          NMR * u1 = <* rs1 * rt1 + rs2 * rt2 + rs3 * rt3,
                        rs4 * rt1 + rs5 * rt2 + rs6 * rt3,
                        rs7 * rt1 + rs8 * rt2 + rs9 * rt3 *>
            by A53,A52,PASCAL:9; then
A55:      |( vf, O1 * vf )| = |( u1, NMR@ * ONMRUF9 )|
                               by A54,A51,MATRPROB:48
                           .= |( u1, ra3 * (O1 * u1) )|
                               by A17,A46,Th51
                           .= ra3 * |( u1, O1 * u1 )| by RVSUM_1:121,A47;
A56:      len u1 = len O1 & len u1 = width O1 & len u1 > 0
               by A42,MATRIX_0:24;
          0 = SumAll QuadraticForm(u1,O1,u1) by A16,A13,A1,Th66
           .= |( u1, O1 * u1)| by A56,MATRPROB:44;
          hence thesis by A55;
         end;
         hence thesis by A28;
       end;
       then SumAll QuadraticForm(uf9,O1,uf9) = 0 by A20,MATRPROB:44;
       hence thesis by A18,A19,A1,Th66;
     end;
     absolute c= homography(N).:absolute
     proof
       let x be object;
       assume
A57:   x in absolute;
       then x in {P where P is Point of ProjectiveSpace TOP-REAL 3:
       for u being Element of TOP-REAL 3 st u is non zero & P = Dir u holds
         qfconic(1,1,-1,0,0,0,u) = 0} by PASCAL:def 2;
       then consider Q being Point of ProjectiveSpace TOP-REAL 3 such that
A58:   x = Q and
       for u being Element of TOP-REAL 3 st u is non zero & Q = Dir u holds
         qfconic(1,1,-1,0,0,0,u) = 0;
       reconsider P = homography(N~).Q as
         Element of ProjectiveSpace TOP-REAL 3;
A59:   homography(N).P = x by A58,ANPROJ_9:15;
       consider u2,v2 be Element of TOP-REAL 3,
                  uf2 be FinSequence of F_Real,
                   p2 be FinSequence of 1-tuples_on REAL such that
A60:   Q = Dir u2 & u2 is not zero & u2 = uf2 & p2 = (N~) * uf2 &
       v2 = M2F p2 & v2 is not zero & (homography(N~)).Q = Dir v2
         by ANPROJ_8:def 4;
       reconsider vf2 = v2 as FinSequence of REAL by EUCLID:24;
       v2 in TOP-REAL 3;
       then v2 in REAL 3 by EUCLID:22;
       then len vf2 = 3 by EUCLID_8:50; then
A61:   len vf2 = len O1 & len vf2 = width O1 & len vf2 > 0 by MATRIX_0:24;
       |( vf2, O1 * vf2 )| = 0
       proof
         reconsider uf3 = uf2 as FinSequence of REAL;
A62:     len O1 = 3 & width O1 = 3 by MATRIX_0:24;
            u2 in TOP-REAL 3; then
A63:     u2 in REAL 3 by EUCLID:22; then
A64:     len uf2 = 3 by A60,EUCLID_8:50;
A65:     len uf3 = len O1 by A63,A62,A60,EUCLID_8:50;
A66:     SumAll QuadraticForm(uf3,O1,uf3) = 0
              by A1,A58,A57,A60,Th66;
A67:     len (O1 * uf3) = len uf3 by A62,A65,MATRIXR1:61;
         reconsider NR = N as Matrix of 3,REAL;
         reconsider NI = N~ as Matrix of 3,3,REAL;
A68:     N~ is_reverse_of N by MATRIX_6:def 4;
A69:     NI * NR = (N~) * N by ANPROJ_8:17
                .= 1.(F_Real,3) by A68,MATRIX_6:def 2
                .= MXF2MXR 1.(F_Real,3) by MATRIXR1:def 2
                .= 1_Rmatrix(3) by MATRIXR2:def 2;
A70:     NR * NI = N * (N~) by ANPROJ_8:17
                .= 1.(F_Real,3) by A68,MATRIX_6:def 2
                .= MXF2MXR 1.(F_Real,3) by MATRIXR1:def 2
                .= 1_Rmatrix(3) by MATRIXR2:def 2;
         then NR is invertible by A69,MATRIXR2:def 5; then
A71:     Inv(NR) = NI by A69,A70,MATRIXR2:def 6;
         reconsider ea = ra as Element of F_Real by XREAL_0:def 1;
         M = symmetric_3(ea,ea,-ea,0,0,0) by A2; then
A72:     M = ea * O by A1,Th48
          .= ea * MXR2MXF O1 by MATRIXR1:def 1
          .= MXF2MXR (ea * MXR2MXF O1) by MATRIXR1:def 2
          .= ra * O1 by MATRIXR1:def 7;
         N@ * O = NR@ * O1 by ANPROJ_8:17; then
A73:     M = NR@ * O1 * NR by A4,ANPROJ_8:17;
         width (NI@ * O1) = 3 by MATRIXR2:4; then
A75:     len uf3 = width NI & width (NI@ * O1) = len NI &
           len uf3 > 0 & len NI > 0 by A64,MATRIX_0:24;
         width NI = len uf3 & len uf3 > 0 & len NI = 3 by A64,MATRIX_0:24;
         then
A76:     len (NI * uf3) = 3 & width O1 = 3 & len O1 = 3 &
           width (NI@) = 3 by MATRIX_0:24,MATRIXR1:61;
         width NI = len uf3 & len uf3 > 0 by A64,MATRIX_0:24;
         then len (NI * uf3) = len NI by MATRIXR1:61
                            .= 3 by MATRIX_0:24;
         then width O1 = len (NI * uf3) & len (NI * uf3) > 0 by MATRIX_0:24;
         then len (O1 * (NI * uf3)) = len O1 by MATRIXR1:61
                                   .= 3 by MATRIX_0:24; then
A77:     len (O1 * (NI * uf3)) = len NI & len uf3 = width NI &
           len uf3 > 0 & len (O1 * (NI * uf3)) > 0 by A64,MATRIX_0:24;
         vf2 = NI * uf3
         proof
           consider s1,s2,s3,s4,s5,s6,s7,s8,s9 be Element of F_Real such that
A78:       N~ = <* <* s1,s2,s3 *>,
                   <* s4,s5,s6 *>,
                   <* s7,s8,s9 *> *> by PASCAL:3;
           consider t1,t2,t3 be Real such that
A79:       u2 = <* t1,t2,t3 *> by EUCLID_5:1;
           reconsider et1 = t1, et2 = t2, et3 = t3 as Element of F_Real
             by XREAL_0:def 1;
           vf2 = <* s1 * et1 + s2 * et2 + s3 * et3,
                    s4 * et1 + s5 * et2 + s6 * et3,
                    s7 * et1 + s8 * et2 + s9 * et3 *>
                    by A60,A78,A79,PASCAL:8;
           hence thesis by A78,A79,A60,PASCAL:9;
         end;
         then |( vf2, O1 * vf2 )| = |( uf3, NI@ * (O1 * (NI * uf3)) )|
                                     by A77,MATRPROB:48
                                 .= |( uf3, (NI@ * O1) * (NI * uf3) )|
                                     by A76,MATRIXR2:59
                                 .= |( uf3, (NI@ * O1 * NI) * uf3 )|
                                     by A75,MATRIXR2:59
                                 .= |( uf3, (1/ra * O1) * uf3 )|
                   by A73,A69,A70,MATRIXR2:def 5,A72,Th53,A71
                                 .= |( (1/ra) * (O1 * uf3), uf3 )|
                                     by A64,Th51
                                 .= 1/ra * |( O1 * uf3, uf3 )|
                                     by A67,RVSUM_1:121
                                 .= 1/ra * 0
                                     by A66,A62,A65,MATRPROB:44
                                 .= 0;
         hence thesis;
       end;
       then SumAll QuadraticForm(vf2,O1,vf2) = 0 by A61,MATRPROB:44;
       then
A80:   P in absolute by A1,A60,Th66;
       dom homography(N) = the carrier of ProjectiveSpace TOP-REAL 3
         by FUNCT_2:def 1;
       hence thesis by A59,A80,FUNCT_1:def 6;
     end;
     hence thesis by A7;
   end;
