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 Th15:
  l in Seg width M & k in Seg width M & n>0 & m>0 & M1 = M@
  implies (ILine(M1,l,k))@ = ICol(M,l,k)
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;
A7: width ILine(M1,l,k) = width M1 by Th1;
  len M = n by A3,MATRIX_0:23;
  then
A8: width (M1) = n by A4,A5,A6,MATRIX_0:54;
  then
A9: len (ILine(M1,l,k)@) = n by A3,A7,MATRIX_0:54;
A10: len ILine(M1,l,k) = len M1 by Def1;
  len (M1) = m by A4,A5,A6,MATRIX_0:54;
  then width (ILine(M1,l,k)@) = m by A3,A8,A10,A7,MATRIX_0:54;
  then
A11: (ILine(M1,l,k)@) is Matrix of n,m,K by A3,A9,MATRIX_0:20;
  then consider M2 being Matrix of n,m,K such that
A12: M2 = (ILine(M1,l,k))@;
A13: for i,j st i in dom M & j in Seg width M holds (j = l implies M2*(i,j)
= M*(i,k)) & (j = k implies M2*(i,j) = M*(i,l)) & (j <> l & j <> k implies M2*(
  i,j) = M*(i,j))
  proof
    let i,j;
    assume that
A14: i in dom M and
A15: j in Seg width M;
A16: [i,j] in Indices M by A14,A15,ZFMISC_1:87;
    then
A17: [j,i] in Indices M1 by A5,MATRIX_0:def 6;
    then
A18: j in dom M1 & i in Seg width M1 by ZFMISC_1:87;
    dom (ILine(M1,l,k)) = Seg len (ILine(M1,l,k)) by FINSEQ_1:def 3
      .= Seg len M1 by Def1
      .= dom M1 by FINSEQ_1:def 3;
    then
A19: [j,i] in Indices (ILine(M1,l,k)) by A17,Th1;
    thus j = l implies M2*(i,j) = M*(i,k)
    proof
A20:  [i,k] in Indices M by A2,A14,ZFMISC_1:87;
      assume
A21:  j = l;
      M2*(i,j) = (ILine(M1,l,k))*(j,i) by A12,A19,MATRIX_0:def 6
        .= M1*(k,i) by A18,A21,Def1
        .= M*(i,k) by A5,A20,MATRIX_0:def 6;
      hence thesis;
    end;
    thus j = k implies M2*(i,j) = M*(i,l)
    proof
A22:  [i,l] in Indices M by A1,A14,ZFMISC_1:87;
      assume
A23:  j = k;
      M2*(i,j) = (ILine(M1,l,k))*(j,i) by A12,A19,MATRIX_0:def 6
        .= M1*(l,i) by A18,A23,Def1
        .= M*(i,l) by A5,A22,MATRIX_0:def 6;
      hence thesis;
    end;
    assume
A24: j <> l & j <> k;
    M2*(i,j) = (ILine(M1,l,k))*(j,i) by A12,A19,MATRIX_0:def 6
      .= M1*(j,i) by A18,A24,Def1
      .= M*(i,j) by A5,A16,MATRIX_0:def 6;
    hence thesis;
  end;
  for i,j st [i,j] in Indices ICol(M,l,k) holds ICol(M,l,k)*(i,j) = ((
  ILine(M1,l,k))@)*(i,j)
  proof
A25: Indices M = Indices ICol(M,l,k) by MATRIX_0:26;
    let i,j;
    assume [i,j] in Indices ICol(M,l,k);
    then
A26: i in dom M & j in Seg width M by A25,ZFMISC_1:87;
    then
A27: j = l implies ((ILine(M1,l,k))@)*(i,j) = M*(i,k) by A12,A13;
A28: j = k implies ICol(M,l,k)*(i,j) = M*(i,l) by A1,A3,A4,A26,Def4;
A29: j = k implies ((ILine(M1,l,k))@)*(i,j) = M*(i,l) by A12,A13,A26;
A30: j <> l & j <> k implies ((ILine(M1,l,k))@)*(i,j) = M*(i,j) by A12,A13,A26;
    j = l implies ICol(M,l,k)*(i,j) = M*(i,k) by A2,A3,A4,A26,Def4;
    hence thesis by A1,A2,A3,A4,A26,A27,A29,A30,A28,Def4;
  end;
  hence thesis by A11,MATRIX_0:27;
end;
