 reserve i,n for Nat;
 reserve r for Real;
 reserve ra for Element of F_Real;
 reserve a,b,c for non zero Element of F_Real;
 reserve u,v for Element of TOP-REAL 3;
 reserve p1 for FinSequence of (1-tuples_on REAL);
 reserve pf,uf for FinSequence of F_Real;
 reserve N for Matrix of 3,F_Real;
 reserve K for Field;
 reserve k for Element of K;
 reserve N,N1,N2 for invertible Matrix of 3,F_Real;
 reserve P,P1,P2,P3 for Point of ProjectiveSpace TOP-REAL 3;

theorem Th14:
  (homography(N1)).((homography(N2)).P) = (homography(N1 * N2)).P
  proof
    consider u12,v12 being Element of TOP-REAL 3,
    u12f being FinSequence of F_Real,
    p12 being FinSequence of (1-tuples_on REAL) such that
A1: P = Dir u12 and
A2: u12 is not zero and
A3: u12 = u12f and
A4: p12 = (N1 * N2) * u12f and
A5: v12 = M2F p12 and
A6: v12 is not zero and
A7: (homography(N1 * N2)).P = Dir v12 by ANPROJ_8:def 4;
    consider u2,v2 being Element of TOP-REAL 3,
    u2f being FinSequence of F_Real,
    p2 being FinSequence of (1-tuples_on REAL) such that
A8: P = Dir u2 and
A9: u2 is not zero and
A10: u2 = u2f and
A11: p2 = N2 * u2f and
A12: v2 = M2F p2 and
A13: v2 is not zero and
A14: (homography(N2)).P = Dir v2 by ANPROJ_8:def 4;
    consider u1,v1 being Element of TOP-REAL 3,
    u1f being FinSequence of F_Real,
    p1 being FinSequence of (1-tuples_on REAL) such that
A15: (homography(N2)).P = Dir u1 and
A16: u1 is not zero and
A17: u1 = u1f and
A18: p1 = N1 * u1f and
A19: v1 = M2F p1 and
A20: v1 is not zero and
A21: (homography(N1)).((homography(N2)).P) = Dir v1 by ANPROJ_8:def 4;
    are_Prop u12,u2 by A1,A8,A2,A9,ANPROJ_1:22;
    then consider a be Real such that
A22: a <> 0 and
A23: u2 = a * u12 by ANPROJ_1:1;
    are_Prop v2,u1 by A14,A15,A16,A13,ANPROJ_1:22;
    then consider b be Real such that
A24: b <> 0 and
A25: u1 = b * v2 by ANPROJ_1:1;
    u1 in TOP-REAL 3; then
    F2: u1 in REAL 3 by EUCLID:22;
    width(<*u1f*>) = len u1 by A17,ANPROJ_8:75
                  .= 3 by F2,EUCLID_8:50;
    then len ( <*u1f*>@ ) = width <*u1f*> by MATRIX_0:29
                         .= len u1 by ANPROJ_8:75,A17
                         .= 3 by F2,EUCLID_8:50; then
A26: width N1 = len (<*u1f*>@) by MATRIX_0:24;
A27: len (N1 * u1f) = len (N1 * (<*u1f*>@)) by LAPLACE:def 9
                    .= len N1 by A26,MATRIX_3:def 4
                    .= 3 by MATRIX_0:24;
A28: v1 = <* (N1 * <*u1f*>@)*(1,1),
             (N1 * <*u1f*>@)*(2,1),
             (N1 * <*u1f*>@)*(3,1) *>
    proof
      p1.1 = <* (N1 * <*u1f*>@)*(1,1) *> &
        p1.2 = <* (N1 * <*u1f*>@)*(2,1) *> & p1.3 = <* (N1 * <*u1f*>@)*(3,1) *>
        by A17,A18,FINSEQ_1:1,Lem02;
      then (p1.1).1 = (N1 * <*u1f*>@)*(1,1) &
        (p1.2).1 = (N1 * <*u1f*>@)*(2,1) & (p1.3).1 = (N1 * <*u1f*>@)*(3,1);
      hence thesis by A19,A27,A18,ANPROJ_8:def 2;
    end;
    u2 in TOP-REAL 3; then
A29: u2 in REAL 3 by EUCLID:22;
    width(<*u2f*>) = len u2 by A10,ANPROJ_8:75
                  .= 3 by A29,EUCLID_8:50;
    then
    len ( <*u2f*>@ ) = width <*u2f*> by MATRIX_0:29
                    .= len u2 by ANPROJ_8:75,A10
                    .= 3 by A29,EUCLID_8:50; then
A30: width N2 = len (<*u2f*>@) by MATRIX_0:24;
A31: len (N2 * u2f) = len (N2 * (<*u2f*>@)) by LAPLACE:def 9
                    .= len N2 by A30,MATRIX_3:def 4
                    .= 3 by MATRIX_0:24;
A32: v2 = <* (N2 * <*u2f*>@)*(1,1),
             (N2 * <*u2f*>@)*(2,1),
             (N2 * <*u2f*>@)*(3,1) *>
    proof
      p2.1 = <* (N2 * <*u2f*>@)*(1,1) *> &
        p2.2 = <* (N2 * <*u2f*>@)*(2,1) *> & p2.3 = <* (N2 * <*u2f*>@)*(3,1) *>
        by A10,A11,FINSEQ_1:1,Lem02;
      then (p2.1).1 = (N2 * <*u2f*>@)*(1,1) &
        (p2.2).1 = (N2 * <*u2f*>@)*(2,1) & (p2.3).1 = (N2 * <*u2f*>@)*(3,1);
      hence thesis by A12,A31,A11,ANPROJ_8:def 2;
    end;
    u12 in TOP-REAL 3; then
A33: u12 in REAL 3 by EUCLID:22;
    width(<*u12f*>) = len u12 by A3,ANPROJ_8:75
                   .= 3 by A33,EUCLID_8:50;
    then
A34: len ( <*u12f*>@ ) = width <*u12f*> by MATRIX_0:29
                      .= len u12 by ANPROJ_8:75,A3
                      .= 3 by A33,EUCLID_8:50;
A35: width (N1 * N2) = len (<*u12f*>@) by MATRIX_0:24,A34;
A36: len ((N1 * N2) * u12f) = len ((N1 * N2) * (<*u12f*>@)) by LAPLACE:def 9
                            .= len (N1 * N2) by A35,MATRIX_3:def 4
                            .= 3 by MATRIX_0:24;
A37: v12 = <* ((N1*N2) * <*u12f*>@)*(1,1),
              ((N1*N2) * <*u12f*>@)*(2,1),
              ((N1*N2) * <*u12f*>@)*(3,1) *>
    proof
      p12.1 = <* ((N1 * N2) * <*u12f*>@)*(1,1) *> &
        p12.2 = <* ((N1 * N2) * <*u12f*>@)*(2,1) *> &
        p12.3 = <* ((N1 * N2) * <*u12f*>@)*(3,1) *>
        by A3,A4,FINSEQ_1:1,Lem02;
      then (p12.1).1 = ((N1 * N2) * <*u12f*>@)*(1,1) &
        (p12.2).1 = ((N1 * N2) * <*u12f*>@)*(2,1) &
        (p12.3).1 = ((N1 * N2) * <*u12f*>@)*(3,1);
      hence thesis by A5,A36,A4,ANPROJ_8:def 2;
    end;
    reconsider v2f = v2 as FinSequence of F_Real by RVSUM_1:145;
    reconsider invb = 1/b as Real;
A38: v2f = invb * u1
    proof
      invb * u1 = (invb * b) * v2f by A25,RVSUM_1:49
               .= 1 * v2f by A24,XCMPLX_1:106
               .= v2f by RVSUM_1:52;
      hence thesis;
    end;
    1 in Seg 3 & 2 in Seg 3 & 3 in Seg 3 by FINSEQ_1:1;
    then
A39: (N1 * <*u1f*>@)*(1,1) = (1/invb) * (Line(N1,1) "*" v2f) &
       (N1 * <*u1f*>@)*(2,1) = (1/invb) * (Line(N1,2) "*" v2f) &
       (N1 * <*u1f*>@)*(3,1) = (1/invb) * (Line(N1,3) "*" v2f)
       by A24,XCMPLX_1:50,A38,A17,Lem03;
A40: len Line(N1,1) = width N1 & len Line(N1,2) = width N1 &
       len Line(N1,3) = width N1 by MATRIX_0:def 7;
    width N1 = 3 by MATRIX_0:24;
    then 1 in Seg width N1 & 2 in Seg width N1 &
      3 in Seg width N1 by FINSEQ_1:1; then
A41: Line(N1,1).1 = N1*(1,1) & Line(N1,1).2 = N1*(1,2) &
      Line(N1,1).3 = N1*(1,3) & Line(N1,2).1 = N1*(2,1) &
      Line(N1,2).2 = N1*(2,2) & Line(N1,2).3 = N1*(2,3) &
      Line(N1,3).1 = N1*(3,1) & Line(N1,3).2 = N1*(3,2) &
      Line(N1,3).3 = N1*(3,3) by MATRIX_0:def 7; then
A42: Line(N1,1) = <* N1*(1,1),N1*(1,2),N1*(1,3) *> &
      Line(N1,2) = <* N1*(2,1),N1*(2,2),N1*(2,3) *> &
      Line(N1,3) = <* N1*(3,1),N1*(3,2),N1*(3,3) *>
      by A40,MATRIX_0:24,FINSEQ_1:45;
A43: len Line(N1 * N2,1) = width (N1 * N2) &
       len Line(N1 * N2,2) = width (N1 * N2) &
       len Line(N1 * N2,3) = width (N1 * N2) by MATRIX_0:def 7;
    width (N1 * N2) = 3 by MATRIX_0:24;
    then 1 in Seg width (N1 * N2) & 2 in Seg width (N1 * N2) &
    3 in Seg width (N1 * N2) by FINSEQ_1:1;
    then
A44: Line(N1*N2,1).1 = (N1*N2)*(1,1) & Line(N1*N2,1).2 = (N1*N2)*(1,2) &
      Line(N1*N2,1).3 = (N1*N2)*(1,3) & Line(N1*N2,2).1 = (N1*N2)*(2,1) &
      Line(N1*N2,2).2 = (N1*N2)*(2,2) & Line(N1*N2,2).3 = (N1*N2)*(2,3) &
      Line(N1*N2,3).1 = (N1*N2)*(3,1) & Line(N1*N2,3).2 = (N1*N2)*(3,2) &
      Line(N1*N2,3).3 = (N1*N2)*(3,3) by MATRIX_0:def 7;
    reconsider fa = a as Element of F_Real by XREAL_0:def 1;
A45: v1.1 = (N1 * <*u1f*>@)*(1,1) & v1.2 = (N1 * <*u1f*>@)*(2,1) &
      v1.3 = (N1 * <*u1f*>@)*(3,1) by A28;
    reconsider t1 = v2f.1, t2 = v2f.2, t3 = v2f.3 as Element of F_Real
      by A32;
    len v2f = 3 by A32,FINSEQ_1:45;
    then
A46: v2f = <* t1,t2,t3 *> by FINSEQ_1:45;
    N2 * <*u2f*>@ is Matrix of 3,1,F_Real by A10,FINSEQ_3:153,ANPROJ_8:91;
    then
A47: [1,1] in Indices(N2 * <*u2f*>@) & [2,1] in Indices(N2 * <*u2f*>@) &
     [3,1] in Indices(N2 * <*u2f*>@) by ANPROJ_8:2,MATRIX_0:23;
A48: t1 = (N2 * <*u2f*>@)*(1,1) by A32
       .= Line(N2,1) "*" Col(<*u2f*>@,1) by A30,A47,MATRIX_3:def 4
       .= Line(N2,1) "*" u2f by ANPROJ_8:93;
A49: t2 = (N2 * <*u2f*>@)*(2,1) by A32
       .= Line(N2,2) "*" Col(<*u2f*>@,1) by A30,A47,MATRIX_3:def 4
       .= Line(N2,2) "*" u2f by ANPROJ_8:93;
A50: t3 = (N2 * <*u2f*>@)*(3,1) by A32
       .= Line(N2,3) "*" Col(<*u2f*>@,1) by A30,A47,MATRIX_3:def 4
       .= Line(N2,3) "*" u2f by ANPROJ_8:93;
    now
      width N2 = 3 by MATRIX_0:23;
      then 1 in Seg width N2 & 2 in Seg width N2 & 3 in Seg width N2
        by FINSEQ_1:1;
      hence Line(N2,1).1 = N2*(1,1) & Line(N2,1).2 = N2*(1,2) &
        Line(N2,1).3 = N2*(1,3) & Line(N2,2).1 = N2*(2,1) &
        Line(N2,2).2 = N2*(2,2) & Line(N2,2).3 = N2*(2,3)  &
        Line(N2,3).1 = N2*(3,1) & Line(N2,3).2 = N2*(3,2) &
        Line(N2,3).3 = N2*(3,3) by MATRIX_0:def 7;
      thus [1,1] in Indices N2 & [1,2] in Indices N2 &
        [1,3] in Indices N2 & [2,1] in Indices N2 & [2,2] in Indices N2 &
        [2,3] in Indices N2 & [3,1] in Indices N2 & [3,2] in Indices N2 &
        [3,3] in Indices N2 by ANPROJ_8:1,MATRIX_0:23;
    end;
    then reconsider l11 = Line(N2,1).1,l12 = Line(N2,1).2,
      l13 = Line(N2,1).3, l21 = Line(N2,2).1,l22 = Line(N2,2).2,
      l23 = Line(N2,2).3, l31 = Line(N2,3).1,l32 = Line(N2,3).2,
      l33 = Line(N2,3).3 as Element of F_Real;
    reconsider ru121 = u12f.1,ru122 = u12f.2,
      ru123 = u12f.3 as Element of F_Real by XREAL_0:def 1;
    len Line(N2,1) = width N2 & len Line(N2,2) = width N2 &
      len Line(N2,3) = width N2 by MATRIX_0:def 7;
    then
A51: Line(N2,1) = <* l11,l12,l13 *> & Line(N2,2) = <* l21,l22,l23 *> &
      Line(N2,3) = <* l31,l32,l33 *> by MATRIX_0:23,FINSEQ_1:45;
    reconsider ru21 = u2.1,ru22 = u2.2,ru23 = u2.3 as Element of F_Real
      by XREAL_0:def 1;
A52: ru21 = a * u12f.1 & ru22 = a * u12f.2 & ru23 = a * u12f.3
      by A3,A23,RVSUM_1:44;
    len u2 = 3 by A29,EUCLID_8:50;
    then
A53: u2f = <* ru21,ru22,ru23 *> by A10,FINSEQ_1:45;
    reconsider t121 = v12.1, t122 = v12.2, t123 = v12.3 as Element of F_Real
      by A37;
    reconsider v12f = v12 as FinSequence of F_Real by RVSUM_1:145;
    width N1 = 3 by MATRIX_0:24; then
A54: width N1 = len N2 by MATRIX_0:24;
A55: [1,1] in Indices(N1*N2) & [1,2] in Indices(N1*N2) &
        [1,3] in Indices(N1*N2) & [2,1] in Indices(N1*N2) &
        [2,2] in Indices(N1*N2) & [2,3] in Indices(N1*N2) &
        [3,1] in Indices(N1*N2) & [3,2] in Indices(N1*N2) &
        [3,3] in Indices(N1*N2) by ANPROJ_8:1,MATRIX_0:23;
A56: width(N1*N2)=len(<*u12f*>@) by A34,MATRIX_0:24;
    u12 in TOP-REAL 3;
    then u12f in REAL 3 by EUCLID:22,A3; then
A57: len u12f = 3 by EUCLID_8:50;
    (N1 * N2) * <*u12f*>@ is Matrix of 3,1,F_Real
      by A3,FINSEQ_3:153,ANPROJ_8:91; then
A58: [1,1] in Indices((N1*N2) * <*u12f*>@) &
      [2,1] in Indices((N1*N2) * <*u12f*>@) &
      [3,1] in Indices((N1*N2) * <*u12f*>@) by MATRIX_0:23,ANPROJ_8:2;
A59: u12f = <* ru121,ru122,ru123 *> by A57,FINSEQ_1:45;
A60: Col(N2,1) = <* l11,l21,l31 *> & Col(N2,2) = <* l12,l22,l32 *> &
       Col(N2,3) = <* l13,l23,l33 *>
    proof
      now
        len N2 = 3 by MATRIX_0:24;
        hence len Col(N2,1) = 3 & len Col(N2,2) = 3 & len Col(N2,3) = 3
          by MATRIX_0:def 8;
        len N2 = 3 & width N2 = 3 by MATRIX_0:24;
        then dom N2 = Seg 3 & Seg width N2 = Seg 3 by FINSEQ_1:def 3;
        then 1 in dom N2 & 2 in dom N2 & 3 in dom N2 &
          1 in Seg width N2 & 2 in Seg width N2 & 3 in Seg width N2
          by FINSEQ_1:1;
        hence  Col(N2,1).1 = l11 & Col(N2,2).1 = l12 & Col(N2,3).1 = l13 &
          Col(N2,1).2 = l21 & Col(N2,2).2 = l22 & Col(N2,3).2 = l23 &
          Col(N2,1).3 = l31 & Col(N2,2).3 = l32 & Col(N2,3).3 = l33
          by MATRIX_0:42;
      end;
      hence thesis by FINSEQ_1:45;
    end;
    Dir v1 = Dir v12
    proof
      ex c be Real st c <> 0 & v1 = c * v12
      proof
        set c = a * b;
        take c;
        thus c <> 0 by A22,A24,XCMPLX_1:6;
A61:
        now
          thus (N1 * <*u1f*>@)*(1,1) = c * (((N1*N2) * <*u12f*>@)*(1,1))
          proof
            v1.1 = c * v12.1
            proof
              reconsider s1 = N1*(1,1),s2 = N1*(1,2),
                         s3 = N1*(1,3) as Element of F_Real;
A62:          Line(N1,1) "*" v2f =
                s1 * (<* l11,l12,l13 *> "*" <* ru21, ru22, ru23 *>) +
                s2 * (<* l21,l22,l23 *> "*" <* ru21,ru22,ru23 *>) +
                s3 * (<* l31,l32,l33 *> "*" <* ru21,ru22,ru23 *>)
                by A51,A53,A48,A49,A50,A42,A46,ANPROJ_8:7
                                .= s1 * (l11 * ru21 + l12 * ru22
                                                    + l13 * ru23) +
                                  s2 * (Line(N2,2) "*" u2f)
                                  + s3 * (Line(N2,3) "*" u2f)
                                  by A51,A53,ANPROJ_8:7
                                .= s1 * (l11 * ru21 + l12 * ru22
                                                    + l13 * ru23) +
                                  s2 * (l21 * ru21 + l22 * ru22 + l23 * ru23) +
                                  s3 * (Line(N2,3) "*" u2f)
                                  by A51,A53,ANPROJ_8:7
                                .= s1 * (l11 * ru21 + l12 * ru22
                                                    + l13 * ru23) +
                                  s2 * (l21 * ru21 + l22 * ru22 + l23 * ru23) +
                                  s3 * (l31 * ru21 + l32 * ru22 + l33 * ru23)
                                  by A51,A53,ANPROJ_8:7
                                .= a * ((s1 * l11 + s2 * l21 + s3 * l31)
                                     * ru121 +
                                  (s1 * l12 + s2 * l22 + s3 * l32) * ru122  +
                                  (s1 * l13 + s2 * l23 + s3 * l33) * ru123)
                                  by A52;
              reconsider s121 = (N1*N2)*(1,1),s122 = (N1*N2)*(1,2),
                s123 = (N1*N2)*(1,3) as Element of F_Real;
A63:          t121 = ((N1*N2) * <*u12f*>@)*(1,1) by A37
                  .= Line(N1*N2,1) "*" Col(<*u12f*>@,1)
                    by A56,A58,MATRIX_3:def 4
                  .= Line(N1*N2,1) "*" u12f by ANPROJ_8:93
                  .= <* s121,s122,s123 *> "*" <* ru121,ru122,ru123 *>
                    by A59,A44,A43,MATRIX_0:24,FINSEQ_1:45
                  .= s121 * ru121 + s122 * ru122 + s123 * ru123 by ANPROJ_8:7;
              now
                thus s121 = Line(N1,1) "*" Col(N2,1) by A54,A55,MATRIX_3:def 4
                         .= <* s1,s2,s3 *>  "*" Col(N2,1)
                           by A41,A40,MATRIX_0:24,FINSEQ_1:45
                         .= (s1 * l11 + s2 * l21 + s3 * l31) by A60,ANPROJ_8:7;
                thus s122 = Line(N1,1) "*" Col(N2,2) by A54,A55,MATRIX_3:def 4
                         .= <* s1,s2,s3 *>  "*" Col(N2,2)
                           by A41,A40,MATRIX_0:24,FINSEQ_1:45
                         .= (s1 * l12 + s2 * l22 + s3 * l32) by A60,ANPROJ_8:7;
                thus s123 = Line(N1,1) "*" Col(N2,3) by A54,A55,MATRIX_3:def 4
                         .= <* s1,s2,s3 *>  "*" Col(N2,3)
                           by A41,A40,MATRIX_0:24,FINSEQ_1:45
                         .= (s1 * l13 + s2 * l23 + s3 * l33) by A60,ANPROJ_8:7;
              end;
              then a * b * t121 = b * (Line(N1,1) "*" v2f) by A63,A62
                               .= (N1 * <*u1f*>@)*(1,1) by A39,XCMPLX_1:52
                               .= v1.1 by A28;
              hence thesis;
            end;
            hence thesis by A45,A37;
          end;
          thus (N1 * <*u1f*>@)*(2,1) = c * (((N1*N2) * <*u12f*>@)*(2,1))
          proof
            v1.2 = c * v12.2
            proof
              reconsider s1 = N1*(2,1),s2 = N1*(2,2),
                s3 = N1*(2,3) as Element of F_Real;
A64:          Line(N1,2) "*" v2f =
                s1 * (<* l11,l12,l13 *> "*" <* ru21, ru22, ru23 *>) +
                s2 * (<* l21,l22,l23 *> "*" <* ru21,ru22,ru23 *>) +
                s3 * (<* l31,l32,l33 *> "*" <* ru21,ru22,ru23 *>)
                by A51,A53,A48,A49,A50,A42,A46,ANPROJ_8:7
                                .= s1 * (l11 * ru21 + l12 * ru22
                                                    + l13 * ru23) +
                                  s2 * (Line(N2,2) "*" u2f) +
                                  s3 * (Line(N2,3) "*" u2f)
                                  by A51,A53,ANPROJ_8:7
                                .= s1 * (l11 * ru21 + l12 * ru22
                                                    + l13 * ru23) +
                                  s2 * (l21 * ru21 + l22 * ru22 + l23 * ru23) +
                                  s3 * (Line(N2,3) "*" u2f)
                                  by A51,A53,ANPROJ_8:7
                                .= s1 * (l11 * ru21 + l12 * ru22
                                                    + l13 * ru23) +
                                  s2 * (l21 * ru21 + l22 * ru22 + l23 * ru23) +
                                  s3 * (l31 * ru21 + l32 * ru22 + l33 * ru23)
                                  by A51,A53,ANPROJ_8:7
                                  .= a * ((s1 * l11 + s2 * l21 + s3 * l31)
                                       * ru121 +
                                    (s1 * l12 + s2 * l22 + s3 * l32) * ru122  +
                                    (s1 * l13 + s2 * l23 + s3 * l33) * ru123)
                                    by A52;
              reconsider s121 = (N1*N2)*(2,1),s122 = (N1*N2)*(2,2),
                s123 = (N1*N2)*(2,3) as Element of F_Real;
A65:          t122 = ((N1*N2) * <*u12f*>@)*(2,1) by A37
                  .= Line(N1*N2,2) "*" Col(<*u12f*>@,1)
                    by A56,A58,MATRIX_3:def 4
                  .= Line(N1*N2,2) "*" u12f by ANPROJ_8:93
                  .= <* s121,s122,s123 *> "*" <* ru121,ru122,ru123 *>
                    by A59,A44,A43,MATRIX_0:24,FINSEQ_1:45
                  .= s121 * ru121 + s122 * ru122 + s123 * ru123
                    by ANPROJ_8:7;
              now
                thus s121 = Line(N1,2) "*" Col(N2,1) by A54,A55,MATRIX_3:def 4
                         .= <* s1,s2,s3 *> "*" Col(N2,1)
                           by A41,A40,MATRIX_0:24,FINSEQ_1:45
                         .= (s1 * l11 + s2 * l21 + s3 * l31) by A60,ANPROJ_8:7;
                thus s122 = Line(N1,2) "*" Col(N2,2) by A54,A55,MATRIX_3:def 4
                         .= <* s1,s2,s3 *>  "*" Col(N2,2)
                           by A41,A40,MATRIX_0:24,FINSEQ_1:45
                         .= (s1 * l12 + s2 * l22 + s3 * l32) by A60,ANPROJ_8:7;
                thus s123 = Line(N1,2) "*" Col(N2,3) by A54,A55,MATRIX_3:def 4
                         .= <* s1,s2,s3 *>  "*" Col(N2,3)
                           by A41,A40,MATRIX_0:24,FINSEQ_1:45
                         .= (s1 * l13 + s2 * l23 + s3 * l33) by A60,ANPROJ_8:7;
              end;
              then a * b * t122 = b * (Line(N1,2) "*" v2f) by A65,A64
                               .= (N1 * <*u1f*>@)*(2,1) by A39,XCMPLX_1:52
                               .= v1.2 by A28;
              hence thesis;
            end;
            hence thesis by A45,A37;
          end;
          thus (N1 * <*u1f*>@)*(3,1) = c * (((N1*N2) * <*u12f*>@)*(3,1))
          proof
            v1.3 = c * v12.3
            proof
              reconsider s1 = N1*(3,1),s2 = N1*(3,2),
                s3 = N1*(3,3) as Element of F_Real;
A66:          Line(N1,3) "*" v2f =
                s1 * (<* l11,l12,l13 *> "*" <* ru21, ru22, ru23 *>) +
                s2 * (<* l21,l22,l23 *> "*" <* ru21,ru22,ru23 *>) +
                s3 * (<* l31,l32,l33 *> "*" <* ru21,ru22,ru23 *>)
                by A51,A53,A48,A49,A50,A42,A46,ANPROJ_8:7
                                .= s1 * (l11 * ru21 + l12 * ru22
                                                    + l13 * ru23) +
                                  s2 * (Line(N2,2) "*" u2f) +
                                  s3 * (Line(N2,3) "*" u2f)
                                  by A51,A53,ANPROJ_8:7
                                .= s1 * (l11 * ru21 + l12 * ru22
                                                    + l13 * ru23) +
                                  s2 * (l21 * ru21 + l22 * ru22 + l23 * ru23) +
                                  s3 * (Line(N2,3) "*" u2f)
                                  by A51,A53,ANPROJ_8:7
                                .= s1 * (l11 * ru21 + l12 * ru22
                                                    + l13 * ru23) +
                                  s2 * (l21 * ru21 + l22 * ru22 + l23 * ru23) +
                                  s3 * (l31 * ru21 + l32 * ru22 + l33 * ru23)
                                  by A51,A53,ANPROJ_8:7
                                  .= a * ((s1 * l11 + s2 * l21 + s3 * l31)
                                       * ru121 +
                                  (s1 * l12 + s2 * l22 + s3 * l32) * ru122 +
                                  (s1 * l13 + s2 * l23 + s3 * l33) * ru123)
                                  by A52;
              reconsider s121 = (N1*N2)*(3,1),s122 = (N1*N2)*(3,2),
                s123 = (N1*N2)*(3,3) as Element of F_Real;
A67:          t123 = ((N1*N2) * <*u12f*>@)*(3,1) by A37
                  .= Line(N1*N2,3) "*" Col(<*u12f*>@,1)
                    by A56,A58,MATRIX_3:def 4
                  .= Line(N1*N2,3) "*" u12f by ANPROJ_8:93
                  .= <* s121,s122,s123 *> "*" <* ru121,ru122,ru123 *>
                    by A59,A44,A43,MATRIX_0:24,FINSEQ_1:45
                  .= s121 * ru121 + s122 * ru122 + s123 * ru123 by ANPROJ_8:7;
              now
                thus s121 = Line(N1,3) "*" Col(N2,1) by A54,A55,MATRIX_3:def 4
                         .= <* s1,s2,s3 *>  "*" Col(N2,1)
                           by A41,A40,MATRIX_0:24,FINSEQ_1:45
                         .= (s1 * l11 + s2 * l21 + s3 * l31) by A60,ANPROJ_8:7;
                thus s122 = Line(N1,3) "*" Col(N2,2) by A54,A55,MATRIX_3:def 4
                         .= <* s1,s2,s3 *>  "*" Col(N2,2)
                           by A41,A40,MATRIX_0:24,FINSEQ_1:45
                         .= (s1 * l12 + s2 * l22 + s3 * l32) by A60,ANPROJ_8:7;
                thus s123 = Line(N1,3) "*" Col(N2,3) by A54,A55,MATRIX_3:def 4
                         .= <* s1,s2,s3 *>  "*" Col(N2,3)
                           by A41,A40,MATRIX_0:24,FINSEQ_1:45
                         .= (s1 * l13 + s2 * l23 + s3 * l33) by A60,ANPROJ_8:7;
              end;
              then a * b * t123 = b * (Line(N1,3) "*" v2f) by A67,A66
                               .= (N1 * <*u1f*>@)*(3,1) by A39,XCMPLX_1:52
                               .= v1.3 by A28;
              hence thesis;
            end;
            hence thesis by A45,A37;
          end;
        end;
        <* c * (((N1*N2) * <*u12f*>@)*(1,1)),
           c * (((N1*N2) * <*u12f*>@)*(2,1)),
           c * (((N1*N2) * <*u12f*>@)*(3,1)) *>
          = c * |[ (((N1*N2) * <*u12f*>@)*(1,1)),
                   (((N1*N2) * <*u12f*>@)*(2,1)),
                   (((N1*N2) * <*u12f*>@)*(3,1)) ]| by EUCLID_5:8
         .= c * <* ((N1*N2) * <*u12f*>@)*(1,1),
                   ((N1*N2) * <*u12f*>@)*(2,1),
                   ((N1*N2) * <*u12f*>@)*(3,1) *>;
        hence thesis by A28,A37,A61;
      end;
      then are_Prop v1,v12 by ANPROJ_1:1;
      hence thesis by A6,A20,ANPROJ_1:22;
    end;
    hence thesis by A7,A21;
  end;
