reserve a,b,c,d,e,f for Real,
        k,m for Nat,
        D for non empty set,
        V for non trivial RealLinearSpace,
        u,v,w for Element of V,
        p,q,r for Element of ProjectiveSpace(V);
reserve o,p,q,r,s,t for Point of TOP-REAL 3,
        M for Matrix of 3,F_Real;
reserve pf for FinSequence of D;
reserve PQR for Matrix of 3,F_Real;
reserve R for Ring;

theorem Th77:
  for pf,qf,rf being FinSequence of F_Real
  for pt,qt,rt being FinSequence of 1-tuples_on REAL
  st PQR = <*<*p`1,q`1,r`1*>,<*p`2,q`2,r`2*>,<*p`3,q`3,r`3*>*> &
  p = pf & q = qf & r = rf &
  pt = M * pf & qt = M * qf & rt = M * rf
  holds (M * PQR)@ = <* M2F pt,M2F qt, M2F rt *>
  proof
    let pf,qf,rf be FinSequence of F_Real;
    let pt,qt,rt be FinSequence of 1-tuples_on REAL;
    assume that
A1: PQR = <*<*p`1,q`1,r`1*>,<*p`2,q`2,r`2*>,<*p`3,q`3,r`3*>*> and
A2: p = pf and
A3: q = qf and
A4: r = rf and
A5: pt = M * pf and
A6: qt = M * qf and
A7: rt = M * rf;
A8: PQR@ = <*<*p`1,p`2,p`3*>,<*q`1,q`2,q`3*>,<*r`1,r`2,r`3*>*> by A1,Th19;
A9: len PQR = 3 by MATRIX_0:24;
A10: width PQR = 3 by MATRIX_0:23;
A11: Indices(M * PQR) = [: Seg 3,Seg 3:] by MATRIX_0:24;
    len (PQR@) = 3 by MATRIX_0:23; then
A12: dom (PQR@) = Seg 3 by FINSEQ_1:def 3; then
A13: Col((PQR@)@,1) = Line(PQR@,1) by FINSEQ_1:1,MATRIX_0:58
                    .= p by A8,Th60;

A14: Col((PQR@)@,2) = Line(PQR@,2) by A12,FINSEQ_1:1,MATRIX_0:58
                    .= q by A8,Th60;

A15: Col((PQR@)@,3) = Line(PQR@,3) by A12,FINSEQ_1:1,MATRIX_0:58
                    .= r by A8,Th60;
    pf in TOP-REAL 3 & qf in TOP-REAL 3 & rf in TOP-REAL 3 by A2,A3,A4; then
A16: pf in REAL 3 & qf in REAL 3 & rf in REAL 3 by EUCLID:22; then
A17: len pf = 3 & len qf = 3 & len rf = 3 by EUCLID_8:50;
A18: width M = len (<*pf*>@) & width M = len (<*qf*>@) &
       width M = len (<*rf*>@)
    proof
      width <*pf*> = 3 by A17,Th61;
      then len (<*pf*>@) = width <*pf*> by MATRIX_0:29
                        .= len pf by MATRIX_0:23;
      then len (<*pf*>@) = 3 by A16,EUCLID_8:50;
      hence width M = len (<*pf*>@) by MATRIX_0:23;
      width <*qf*> = 3 by A17,Th61;
      then len (<*qf*>@) = width <*qf*> by MATRIX_0:29
                        .= len qf by MATRIX_0:23;
      then len (<*qf*>@) = 3 by A16,EUCLID_8:50;
      hence width M = len (<*qf*>@) by MATRIX_0:23;
      width <*rf*> = 3 by A17,Th61;
      then len (<*rf*>@) = width <*rf*> by MATRIX_0:29
                        .= len rf by MATRIX_0:23;
      then len (<*rf*>@) = 3 by A16,EUCLID_8:50;
      hence width M = len (<*rf*>@) by MATRIX_0:23;
    end;
A19: len pt = 3 & len qt = 3 & len rt = 3
    proof
      width M = len (<*pf*>@) & len M = 3 by MATRIX_0:23,A18;
      then len (M * <*pf*>@) = 3 by MATRIX_3:def 4;
      hence len pt = 3 by A5,LAPLACE:def 9;
      width M = len (<*qf*>@) & len M = 3 by A18,MATRIX_0:23;
      then len (M * <*qf*>@) = 3 by MATRIX_3:def 4;
      hence len qt = 3 by A6,LAPLACE:def 9;
      width M = len (<*rf*>@) & len M = 3 by A18,MATRIX_0:23;
      then len (M * <*rf*>@) = 3 by MATRIX_3:def 4;
      hence len rt = 3 by A7,LAPLACE:def 9;
    end;
    set PQRM = <*<*pt.1,pt.2,pt.3*>,<*qt.1,qt.2,qt.3*>,<*rt.1,rt.2,rt.3*>*>;
A20: dom (M * PQR) = Seg len (M * PQR) by FINSEQ_1:def 3
                 .= Seg 3 by MATRIX_0:24;
A21: width (M * PQR) = 3 by MATRIX_0:23;
A22: len ((M * PQR)@) = 3 by MATRIX_0:23; then
A23: dom ((M * PQR)@) = Seg 3 by FINSEQ_1:def 3;
A24: 1 in Seg width(M * PQR) & 2 in Seg width (M * PQR) &
       3 in Seg width (M * PQR) by A21,FINSEQ_1:1;
    now
      len <* M2F pt,M2F qt,M2F rt *> = 3 by FINSEQ_1:45;
      hence dom ((M * PQR)@) = dom <* M2F pt,M2F qt,M2F rt *>
        by A23,FINSEQ_1:def 3;
      thus for x be object st x in dom ((M * PQR)@) holds
        (M * PQR)@.x = <* M2F pt,M2F qt,M2F rt *>.x
      proof
        let x be object;
        assume
A25:     x in dom ((M * PQR)@);
        then reconsider y = x as Nat;
        y in Seg 3 by A25,A22,FINSEQ_1:def 3;
        then y = 1 or ... or y = 3 by FINSEQ_1:91;
        then per cases;
        suppose
A26:      y = 1;
A27:      M * <*pf*>@ is Matrix of 3,1,F_Real by A16,EUCLID_8:50,Th74; then
A28:      Indices (M * <*pf*>@) =[:Seg 3,Seg 1:] by MATRIX_0:23;
A29:      now
            thus len pt = 3 by A19;
            <* (M * PQR)*(1,1) *> = pt.1
            proof
              1 in Seg 3 by FINSEQ_1:1; then
A30:          width M = len PQR & [1,1] in Indices (M * PQR)
                by A9,MATRIX_0:23,A11,ZFMISC_1:87;
A31:          (M * <*pf*>@).1 = <* Line(M,1) "*" Col(PQR,1) *>
              proof
A32:            1 in Seg 3 by FINSEQ_1:1;
                Line(M * (<*pf*>@),1) = <* Line(M,1) "*" Col(PQR,1) *>
                proof
                  1 in Seg 3 & 1 in Seg 1 by FINSEQ_1:1; then
A33:              [1,1] in Indices (M * <*pf*>@) by A28,ZFMISC_1:87;
                  len pf = 3 by A16,EUCLID_8:50; then
                  Line(M * (<*pf*>@),1) = <* (M * (<*pf*>@))*(1,1) *> by Th75
                                       .= <* Line(M,1) "*" Col(<*pf*>@,1) *>
                                         by A18,A33,MATRIX_3:def 4
                                       .= <* Line(M,1) "*" pf *> by Th76
                                       .= <* Line(M,1) "*" Col(PQR,1) *>
                                         by A13,A9,A10,MATRIX_0:57,A2;
                  hence thesis;
                end;
                hence thesis by A32,A27,MATRIX_0:52;
              end;
              pt.1 = (M * <*pf*>@).1 by A5,LAPLACE:def 9;
              hence thesis by A30,MATRIX_3:def 4,A31;
            end;
            hence <* Line(M * PQR,1).1 *> = pt.1 by A24,MATRIX_0:def 7;
            <* (M * PQR)*(2,1) *> = pt.2
            proof
              1 in Seg 3 & 2 in Seg 3 by FINSEQ_1:1; then
A34:          width M = len PQR & [2,1] in Indices (M * PQR)
                by A9,MATRIX_0:23,A11,ZFMISC_1:87;
A34BIS:       (M * <*pf*>@).2 = <* Line(M,2) "*" Col(PQR,1) *>
              proof
A35:            2 in Seg 3 by FINSEQ_1:1;
                Line(M * (<*pf*>@),2) = <* Line(M,2) "*" Col(PQR,1) *>
                proof
                  1 in Seg 1 & 2 in Seg 3 by FINSEQ_1:1; then
A36:              [2,1] in Indices (M * <*pf*>@) by A28,ZFMISC_1:87;
                  len pf = 3 by A16,EUCLID_8:50; then
                  Line(M * (<*pf*>@),2) = <* (M * (<*pf*>@))*(2,1) *> by Th75
                                       .= <* Line(M,2) "*" Col(<*pf*>@,1) *>
                    by A18,A36,MATRIX_3:def 4
                                       .= <* Line(M,2) "*" pf *> by Th76
                                       .= <* Line(M,2) "*" Col(PQR,1) *>
                    by A13,A9,A10,MATRIX_0:57,A2;
                  hence thesis;
                end;
                hence thesis by A35,A27,MATRIX_0:52;
              end;
              pt.2 = (M * <*pf*>@).2 by A5,LAPLACE:def 9;
              hence thesis by A34,MATRIX_3:def 4,A34BIS;
            end;
            hence <* Line(M * PQR,2).1 *> = pt.2 by A24,MATRIX_0:def 7;
            <* (M * PQR)*(3,1) *> = pt.3
            proof
              1 in Seg 3 & 3 in Seg 3 by FINSEQ_1:1; then
A37:          width M = len PQR & [3,1] in Indices (M * PQR)
                by A9,MATRIX_0:23,A11,ZFMISC_1:87;
A38:          (M * <*pf*>@).3 = <* Line(M,3) "*" Col(PQR,1) *>
              proof
A39:            3 in Seg 3 by FINSEQ_1:1;
                Line(M * (<*pf*>@),3) = <* Line(M,3) "*" Col(PQR,1) *>
                proof
                  1 in Seg 1 & 3 in Seg 3 by FINSEQ_1:1; then
A39BIS:           [3,1] in Indices (M * <*pf*>@) by A28,ZFMISC_1:87;
                  len pf = 3 by A16,EUCLID_8:50; then
                  Line(M * (<*pf*>@),3) = <* (M * (<*pf*>@))*(3,1) *> by Th75
                                       .= <* Line(M,3) "*" Col(<*pf*>@,1) *>
                                         by A18,A39BIS,MATRIX_3:def 4
                                       .= <* Line(M,3) "*" pf *> by Th76
                                       .= <* Line(M,3) "*" Col(PQR,1) *>
                                         by A13,A9,A10,MATRIX_0:57,A2;
                  hence thesis;
                end;
                hence thesis by A39,A27,MATRIX_0:52;
              end;
              pt.3 = (M * <*pf*>@).3 by A5,LAPLACE:def 9;
              hence thesis by A37,MATRIX_3:def 4,A38;
            end;
            hence <* Line(M * PQR,3).1 *> = pt.3 by A24,MATRIX_0:def 7;
          end;
A40:      Line(M * PQR,1).1 = (M * PQR)*(1,1) &
              Line(M * PQR,2).1 = (M * PQR)*(2,1) &
              Line(M * PQR,3).1 = (M * PQR)*(3,1) by A24,MATRIX_0:def 7;
          1 in Seg 3 & 2 in Seg 3 & 3 in Seg 3 by FINSEQ_1:1;
          then [1,1] in Indices (M * PQR) &
            [2,1] in Indices (M * PQR) &
            [3,1] in Indices (M * PQR) by A11,ZFMISC_1:87; then
A41:        (M * PQR)@ * (1,1) = (M * PQR)*(1,1) &
            (M * PQR)@ * (1,2) = (M * PQR)*(2,1) &
            (M * PQR)@ * (1,3) = (M * PQR)*(3,1) by MATRIX_0:def 6;
          width ((M * PQR)@) = len (M * PQR) by A21,MATRIX_0:29; then
          Seg width ((M * PQR)@) = dom (M * PQR) by FINSEQ_1:def 3; then
A43:      Line((M * PQR)@,1).1 = (M * PQR)@*(1,1) &
            Line((M * PQR)@,1).2 = (M * PQR)@*(1,2) &
            Line((M * PQR)@,1).3 = (M * PQR)@*(1,3)
            by A20,FINSEQ_1:1,MATRIX_0:def 7;
A44:      1 in Seg 3 by FINSEQ_1:1;
          reconsider FMPQR = Line((M * PQR)@,1) as FinSequence of REAL;
          width ((M * PQR)@) = 3 by MATRIX_0:23; then
A45:      len FMPQR = 3 by MATRIX_0:def 7;
A46:      <* <*Line((M * PQR)@,1).1*>, <*Line((M * PQR)@,1).2*>,
            <*Line((M * PQR)@,1).3*> *> = F2M FMPQR by A45,DEF1;
          FMPQR = M2F F2M FMPQR by A45,Th70
               .= M2F pt by A46,A43,A41,A29,FINSEQ_1:45,A40;
          then (M * PQR)@.1 = M2F pt by A44,MATRIX_0:52;
          hence thesis by A26;
        end;
        suppose
A47:      y = 2;
A48:      M * <*qf*>@ is Matrix of 3,1,F_Real by A16,EUCLID_8:50,Th74; then
A49:      Indices (M * <*qf*>@) =[:Seg 3,Seg 1:] by MATRIX_0:23;
A50:      now
            thus len qt = 3 by A19;
            <* (M * PQR)*(1,2) *> = qt.1
            proof
              1 in Seg 3 & 2 in Seg 3 by FINSEQ_1:1; then
A51:          width M = len PQR & [1,2] in Indices (M * PQR)
                by A9,MATRIX_0:23,A11,ZFMISC_1:87;
A52:          (M * <*qf*>@).1 = <* Line(M,1) "*" Col(PQR,2) *>
              proof
A53:            1 in Seg 3 by FINSEQ_1:1;
                Line(M * (<*qf*>@),1) = <* Line(M,1) "*" Col(PQR,2) *>
                proof
                  1 in Seg 3 & 1 in Seg 1 by FINSEQ_1:1; then
A54:              [1,1] in Indices (M * <*qf*>@) by A49,ZFMISC_1:87;
                  len qf = 3 by A16,EUCLID_8:50; then
                  Line(M * (<*qf*>@),1) = <* (M * (<*qf*>@))*(1,1) *> by Th75
                                       .= <* Line(M,1) "*" Col(<*qf*>@,1) *>
                                         by A18,A54,MATRIX_3:def 4
                                       .= <* Line(M,1) "*" qf *> by Th76
                                       .= <* Line(M,1) "*" Col(PQR,2) *>
                                         by A14,A9,A10,MATRIX_0:57,A3;
                  hence thesis;
                end;
                hence thesis by A53,A48,MATRIX_0:52;
              end;
              qt.1 = (M * <*qf*>@).1 by A6,LAPLACE:def 9;
              hence thesis by A51,MATRIX_3:def 4,A52;
            end;
            hence <* Line(M * PQR,1).2 *> = qt.1 by A24,MATRIX_0:def 7;
            <* (M * PQR)*(2,2) *> = qt.2
            proof
              1 in Seg 3 & 2 in Seg 3 by FINSEQ_1:1; then
A55:          width M = len PQR & [2,2] in Indices (M * PQR)
                by A9,MATRIX_0:23,A11,ZFMISC_1:87;
A56:          (M * <*qf*>@).2 = <* Line(M,2) "*" Col(PQR,2) *>
              proof
A57:            2 in Seg 3 by FINSEQ_1:1;
                Line(M * (<*qf*>@),2) = <* Line(M,2) "*" Col(PQR,2) *>
                proof
                  1 in Seg 1 & 2 in Seg 3 by FINSEQ_1:1; then
A58:              [2,1] in Indices (M * <*qf*>@) by A49,ZFMISC_1:87;
                  len qf = 3 by A16,EUCLID_8:50; then
                  Line(M * (<*qf*>@),2) = <* (M * (<*qf*>@))*(2,1) *> by Th75
                                       .= <* Line(M,2) "*" Col(<*qf*>@,1) *>
                                         by A18,A58,MATRIX_3:def 4
                                       .= <* Line(M,2) "*" qf *> by Th76
                                       .= <* Line(M,2) "*" Col(PQR,2) *>
                                         by A14,A9,A10,MATRIX_0:57,A3;
                  hence thesis;
                end;
                hence thesis by A57,A48,MATRIX_0:52;
              end;
              qt.2 = (M * <*qf*>@).2 by A6,LAPLACE:def 9;
              hence thesis by A55,MATRIX_3:def 4,A56;
            end;
            hence <* Line(M * PQR,2).2 *> = qt.2 by A24,MATRIX_0:def 7;
            <* (M * PQR)*(3,2) *> = qt.3
            proof
              2 in Seg 3 & 3 in Seg 3 by FINSEQ_1:1; then
A57:          width M = len PQR & [3,2] in Indices (M * PQR)
                by A9,MATRIX_0:23,A11,ZFMISC_1:87;
A58:          (M * <*qf*>@).3 = <* Line(M,3) "*" Col(PQR,2) *>
              proof
A59:            3 in Seg 3 by FINSEQ_1:1;
                Line(M * (<*qf*>@),3) = <* Line(M,3) "*" Col(PQR,2) *>
                proof
                  1 in Seg 1 & 3 in Seg 3 by FINSEQ_1:1; then
A59BIS:           [3,1] in Indices (M * <*qf*>@) by A49,ZFMISC_1:87;
                  len qf = 3 by A16,EUCLID_8:50; then
                  Line(M * (<*qf*>@),3) = <* (M * (<*qf*>@))*(3,1) *> by Th75
                                       .= <* Line(M,3) "*" Col(<*qf*>@,1) *>
                                         by A18,A59BIS,MATRIX_3:def 4
                                       .= <* Line(M,3) "*" qf *> by Th76
                                       .= <* Line(M,3) "*" Col(PQR,2) *>
                                         by A14,A9,A10,MATRIX_0:57,A3;
                  hence thesis;
                end;
                hence thesis by A59,A48,MATRIX_0:52;
              end;
              qt.3 = (M * <*qf*>@).3 by A6,LAPLACE:def 9;
              hence thesis by A57,MATRIX_3:def 4,A58;
            end;
            hence <* Line(M * PQR,3).2 *> = qt.3 by A24,MATRIX_0:def 7;
          end;
A60:      Line(M * PQR,1).2 = (M * PQR)*(1,2) &
          Line(M * PQR,2).2 = (M * PQR)*(2,2) &
          Line(M * PQR,3).2 = (M * PQR)*(3,2) by A24,MATRIX_0:def 7;
          1 in Seg 3 & 2 in Seg 3 & 3 in Seg 3 by FINSEQ_1:1;
          then [1,2] in Indices (M * PQR) & [2,2] in Indices (M * PQR) &
            [3,2] in Indices (M * PQR) by A11,ZFMISC_1:87; then
A61:      (M * PQR)@ * (2,1) = (M * PQR)*(1,2) &
          (M * PQR)@ * (2,2) = (M * PQR)*(2,2) &
          (M * PQR)@ * (2,3) = (M * PQR)*(3,2) by MATRIX_0:def 6;
          width ((M * PQR)@) = len (M * PQR) by A21,MATRIX_0:29; then
          Seg width ((M * PQR)@) = dom (M * PQR) by FINSEQ_1:def 3; then
A62:      Line((M * PQR)@,2).1 = (M * PQR)@*(2,1) &
            Line((M * PQR)@,2).2 = (M * PQR)@*(2,2) &
            Line((M * PQR)@,2).3 = (M * PQR)@*(2,3)
            by A20,FINSEQ_1:1,MATRIX_0:def 7;
A63:      2 in Seg 3 by FINSEQ_1:1;
          reconsider FMPQR = Line((M * PQR)@,2) as FinSequence of REAL;
          width ((M * PQR)@) = 3 by MATRIX_0:23; then
A64:      len FMPQR = 3 by MATRIX_0:def 7; then
A65:      <* <*Line((M * PQR)@,2).1*>, <*Line((M * PQR)@,2).2*>,
            <*Line((M * PQR)@,2).3*> *> = F2M FMPQR by DEF1;
          FMPQR = M2F F2M FMPQR by A64,Th70
               .= M2F qt by A65,A62,A61,A50,FINSEQ_1:45,A60;
          then (M * PQR)@.2 = M2F qt by A63,MATRIX_0:52;
          hence thesis by A47;
        end;
        suppose
A66:      y = 3;
A67:      M * <*rf*>@ is Matrix of 3,1,F_Real by A16,EUCLID_8:50,Th74; then
A68:      Indices (M * <*rf*>@) =[:Seg 3,Seg 1:] by MATRIX_0:23;
A69:      now
            thus len rt = 3 by A19;
            <* (M * PQR)*(1,3) *> = rt.1
            proof
              1 in Seg 3 & 3 in Seg 3 by FINSEQ_1:1; then
A70:          width M = len PQR & [1,3] in Indices (M * PQR)
                by A9,MATRIX_0:23,A11,ZFMISC_1:87;
A71:          (M * <*rf*>@).1 = <* Line(M,1) "*" Col(PQR,3) *>
              proof
A72:            1 in Seg 3 by FINSEQ_1:1;
                Line(M * (<*rf*>@),1) = <* Line(M,1) "*" Col(PQR,3) *>
                proof
                  1 in Seg 3 & 1 in Seg 1 by FINSEQ_1:1; then
A73:              [1,1] in Indices (M * <*rf*>@) by A68,ZFMISC_1:87;
                  len rf = 3 by A16,EUCLID_8:50; then
                  Line(M * (<*rf*>@),1) = <* (M * (<*rf*>@))*(1,1) *> by Th75
                                       .= <* Line(M,1) "*" Col(<*rf*>@,1) *>
                                         by A18,A73,MATRIX_3:def 4
                                       .= <* Line(M,1) "*" rf *> by Th76
                                       .= <* Line(M,1) "*" Col(PQR,3) *>
                                         by A15,A9,A10,MATRIX_0:57,A4;
                  hence thesis;
                end;
                hence thesis by A72,A67,MATRIX_0:52;
              end;
              rt.1 = (M * <*rf*>@).1 by A7,LAPLACE:def 9;
              hence thesis by A70,MATRIX_3:def 4,A71;
            end;
            hence <* Line(M * PQR,1).3 *> = rt.1 by A24,MATRIX_0:def 7;
            <* (M * PQR)*(2,3) *> = rt.2
            proof
              2 in Seg 3 & 3 in Seg 3 by FINSEQ_1:1; then
A74:          width M = len PQR & [2,3] in Indices (M * PQR)
                by A9,MATRIX_0:23,A11,ZFMISC_1:87;
A75:           (M * <*rf*>@).2 = <* Line(M,2) "*" Col(PQR,3) *>
              proof
A76:            2 in Seg 3 by FINSEQ_1:1;
                Line(M * (<*rf*>@),2) = <* Line(M,2) "*" Col(PQR,3) *>
                proof
                  1 in Seg 1 & 2 in Seg 3 by FINSEQ_1:1; then
A77:              [2,1] in Indices (M * <*rf*>@) by A68,ZFMISC_1:87;
                  len rf = 3 by A16,EUCLID_8:50; then
                  Line(M * (<*rf*>@),2) = <* (M * (<*rf*>@))*(2,1) *> by Th75
                                       .= <* Line(M,2) "*" Col(<*rf*>@,1) *>
                                         by A18,A77,MATRIX_3:def 4
                                       .= <* Line(M,2) "*" rf *> by Th76
                                       .= <* Line(M,2) "*" Col(PQR,3) *>
                                         by A15,A9,A10,MATRIX_0:57,A4;
                  hence thesis;
                end;
                hence thesis by A76,A67,MATRIX_0:52;
              end;
              rt.2 = (M * <*rf*>@).2 by A7,LAPLACE:def 9;
              hence thesis by A74,MATRIX_3:def 4,A75;
            end;
            hence <* Line(M * PQR,2).3 *> = rt.2 by A24,MATRIX_0:def 7;
            <* (M * PQR)*(3,3) *> = rt.3
            proof
              3 in Seg 3 by FINSEQ_1:1; then
A78:          width M = len PQR & [3,3] in Indices (M * PQR)
                by A9,MATRIX_0:23,A11,ZFMISC_1:87;
A79:          (M * <*rf*>@).3 = <* Line(M,3) "*" Col(PQR,3) *>
              proof
A80:            3 in Seg 3 by FINSEQ_1:1;
                Line(M * (<*rf*>@),3) = <* Line(M,3) "*" Col(PQR,3) *>
                proof
                  1 in Seg 1 & 3 in Seg 3 by FINSEQ_1:1; then
A81:              [3,1] in Indices (M * <*rf*>@) by A68,ZFMISC_1:87;
                    len rf = 3 by A16,EUCLID_8:50; then
                  Line(M * (<*rf*>@),3) = <* (M * (<*rf*>@))*(3,1) *> by Th75
                                       .= <* Line(M,3) "*" Col(<*rf*>@,1) *>
                                         by A18,A81,MATRIX_3:def 4
                                       .= <* Line(M,3) "*" rf *> by Th76
                                       .= <* Line(M,3) "*" Col(PQR,3) *>
                                         by A15,A9,A10,MATRIX_0:57,A4;
                  hence thesis;
                end;
                hence thesis by A80,A67,MATRIX_0:52;
              end;
              rt.3 = (M * <*rf*>@).3 by A7,LAPLACE:def 9;
              hence thesis by A78,MATRIX_3:def 4,A79;
            end;
            hence <* Line(M * PQR,3).3 *> = rt.3 by A24,MATRIX_0:def 7;
          end;
A82:      Line(M * PQR,1).3 = (M * PQR)*(1,3) &
            Line(M * PQR,2).3 = (M * PQR)*(2,3) &
            Line(M * PQR,3).3 = (M * PQR)*(3,3) by A24,MATRIX_0:def 7;
          1 in Seg 3 & 2 in Seg 3 & 3 in Seg 3 by FINSEQ_1:1;
          then [1,3] in Indices (M * PQR) &
            [2,3] in Indices (M * PQR) &
            [3,3] in Indices (M * PQR) by A11,ZFMISC_1:87; then
A83:      (M * PQR)@ * (3,1) = (M * PQR)*(1,3) &
            (M * PQR)@ * (3,2) = (M * PQR)*(2,3) &
            (M * PQR)@ * (3,3) = (M * PQR)*(3,3) by MATRIX_0:def 6;
          width ((M * PQR)@) = len (M * PQR) by A21,MATRIX_0:29; then
          Seg width ((M * PQR)@) = dom (M * PQR) by FINSEQ_1:def 3; then
A84:      Line((M * PQR)@,3).1 = (M * PQR)@*(3,1) &
            Line((M * PQR)@,3).2 = (M * PQR)@*(3,2) &
            Line((M * PQR)@,3).3 = (M * PQR)@*(3,3)
            by A20,FINSEQ_1:1,MATRIX_0:def 7;
          3 in Seg 3 by FINSEQ_1:1; then
A85:      (M * PQR)@.3 = Line((M*PQR)@,3) by MATRIX_0:52;
          reconsider FMPQR = Line((M * PQR)@,3) as FinSequence of REAL;
          width ((M * PQR)@) = 3 by MATRIX_0:23; then
A86:      len FMPQR = 3 by MATRIX_0:def 7; then
A87:      <* <*Line((M * PQR)@,3).1*>, <*Line((M * PQR)@,3).2*>,
            <*Line((M * PQR)@,3).3*> *> = F2M FMPQR by DEF1;
          FMPQR = M2F F2M FMPQR by A86,Th70
               .= M2F rt by A87,A84,A83,A69,FINSEQ_1:45,A82;
          hence thesis by A85,A66;
        end;
      end;
    end;
    hence thesis by FUNCT_1:def 11;
  end;
