reserve
  j, k, l, n, m, t,i for Nat,
  K for comRing, 
  a for Element of K,
  M,M1,M2 for Matrix of n,m,K,
  pK,qK for FinSequence of K,
  A for Matrix of n,K;

theorem Th17:
  l in Seg width M & k in Seg width M & n>0 & m>0 & M1 = M@
  implies (RLineXS(M1,l,k,a))@ = RColXS(M,l,k,a)
proof
  assume that
A1: l in Seg width M and
A2: k in Seg width M and
A3: n>0 and
A4: m>0 and
A5: M1 = M@;
A6: width M = m by A3,MATRIX_0:23;
  then
A7: len M1 = m by A4,A5,MATRIX_0:54;
A8: width RLineXS(M1,l,k,a) = width M1 by Th1;
  len M = n by A3,MATRIX_0:23;
  then
A9: width M1 = n by A4,A5,A6,MATRIX_0:54;
  then
A10: len (RLineXS(M1,l,k,a)@) = n by A3,A8,MATRIX_0:54;
A11: dom M1 = Seg len M1 by FINSEQ_1:def 3
    .= Seg width M by A4,A5,A6,MATRIX_0:54;
  then len RLineXS(M1,l,k,a) = len M1 by A2,Def3;
  then width (RLineXS(M1,l,k,a)@) = m by A3,A7,A9,A8,MATRIX_0:54;
  then
A12: (RLineXS(M1,l,k,a)@) is Matrix of n,m,K by A3,A10,MATRIX_0:20;
  then consider M2 being Matrix of n,m,K such that
A13: M2 = (RLineXS(M1,l,k,a))@;
A14: for i,j st i in dom M & j in Seg width M holds (j = l implies M2*(i,j)
  = a*(M*(i,k))+(M*(i,l))) & (j <> l implies M2*(i,j) = M*(i,j))
  proof
    let i,j;
    assume that
A15: i in dom M and
A16: j in Seg width M;
A17: [i,j] in Indices M by A15,A16,ZFMISC_1:87;
    then
A18: [j,i] in Indices M1 by A5,MATRIX_0:def 6;
    then
A19: i in Seg width M1 by ZFMISC_1:87;
A20: len M1 = width M by A5,MATRIX_0:def 6;
    then
A21: k in dom M1 by A2,FINSEQ_1:def 3;
    dom (RLineXS(M1,l,k,a)) = Seg len (RLineXS(M1,l,k,a)) by FINSEQ_1:def 3
      .= Seg len M1 by A2,A11,Def3
      .= dom M1 by FINSEQ_1:def 3;
    then
A22: [j,i] in Indices (RLineXS(M1,l,k,a)) by A18,Th1;
A23: l in dom M1 by A1,A20,FINSEQ_1:def 3;
    thus j = l implies M2*(i,j) = a*(M*(i,k))+(M*(i,l))
    proof
A24:  [i,k] in Indices M by A2,A15,ZFMISC_1:87;
      assume
A25:  j = l;
      M2*(i,j) = (RLineXS(M1,l,k,a))*(j,i) by A13,A22,MATRIX_0:def 6
        .= a*(M1*(k,i))+(M1*(l,i)) by A23,A21,A19,A25,Def3
        .= a*(M*(i,k))+(M1*(l,i)) by A5,A24,MATRIX_0:def 6
        .= a*(M*(i,k))+(M*(i,l)) by A5,A17,A25,MATRIX_0:def 6;
      hence thesis;
    end;
A26: j in dom M1 by A18,ZFMISC_1:87;
    thus j <> l implies M2*(i,j) = M*(i,j)
    proof
      assume
A27:  j <> l;
      M2*(i,j) = (RLineXS(M1,l,k,a))*(j,i) by A13,A22,MATRIX_0:def 6
        .= M1*(j,i) by A21,A26,A19,A27,Def3
        .= M*(i,j) by A5,A17,MATRIX_0:def 6;
      hence thesis;
    end;
  end;
  for i,j st [i,j] in Indices RColXS(M,l,k,a) holds RColXS(M,l,k,a)*(i,j)
  = ((RLineXS(M1,l,k,a))@)*(i,j)
  proof
A28: Indices M = Indices RColXS(M,l,k,a) by MATRIX_0:26;
    let i,j;
    assume [i,j] in Indices RColXS(M,l,k,a);
    then
A29: i in dom M & j in Seg width M by A28,ZFMISC_1:87;
    then
A30: j = l implies ((RLineXS(M1,l,k,a))@)*(i,j) = a*(M*(i,k))+(M*(i,l)) by A13
,A14;
A31: j <> l implies ((RLineXS(M1,l,k,a))@)*(i,j) = M*(i,j) by A13,A14,A29;
    j = l implies RColXS(M,l,k,a)*(i,j) = a*(M*(i,k))+(M*(i,l)) by A2,A3,A4,A29
,Def6;
    hence thesis by A1,A2,A3,A4,A29,A30,A31,Def6;
  end;
  hence thesis by A12,MATRIX_0:27;
end;
