reserve i,j,n,k for Nat,
  a for Element of COMPLEX,
  R1,R2 for Element of i-tuples_on COMPLEX;

theorem Th39:
  for M being Matrix of COMPLEX st i in Seg len M holds Line(M,i)=
  (Line(M*',i))*'
proof
  let M be Matrix of COMPLEX;
  assume
A1: i in Seg len M;
A2: len Line(M,i)=width M by MATRIX_0:def 7;
A3: len ((Line(M*',i))*')=len (Line(M*',i)) by COMPLSP2:def 1
    .= width (M*') by MATRIX_0:def 7
    .= width M by Def1;
  for j be Nat st 1 <= j & j <= len (Line(M,i)) holds (Line(M,i)).j=((Line
  (M*',i))*').j
  proof
A4: i <= len M by A1,FINSEQ_1:1;
A5: 1 <= i by A1,FINSEQ_1:1;
    let j be Nat;
    assume that
A6: 1 <= j and
A7: j <= len Line(M,i);
A8: j in Seg width M by A2,A6,A7,FINSEQ_1:1;
    then
A9: j in Seg width (M*') by Def1;
    j <= len ((Line(M*',i))*') by A3,A7,MATRIX_0:def 7;
    then
A10: j <= len Line(M*',i) by COMPLSP2:def 1;
    j <= width M by A7,MATRIX_0:def 7;
    then
A11: [i,j] in Indices M by A6,A5,A4,Th1;
    (Line(M,i)).j = ((M*(i,j))*')*' by A8,MATRIX_0:def 7
      .= ((M*')*(i,j))*' by A11,Def1
      .= ((Line(M*',i)).j)*' by A9,MATRIX_0:def 7
      .= ((Line(M*',i))*').j by A6,A10,COMPLSP2:def 1;
    hence thesis;
  end;
  hence thesis by A2,A3,FINSEQ_1:14;
end;
