reserve D for non empty set,
  i,j,k for Nat,
  n,m for Nat,
  r for Real,
  e for real-valued FinSequence;

theorem Th17:
  for D1,D2 being non empty set,M1 being (Matrix of D1),M2 being
Matrix of D2 st M1 = M2 holds for j st j in Seg width M1 holds Col(M1,j) = Col(
  M2,j)
proof
  let D1,D2 be non empty set, M1 be (Matrix of D1), M2 be Matrix of D2 such
  that
A1: M1 = M2;
  hereby
    let j such that
A2: j in Seg width M1;
A3: for k be Nat st k in dom Col(M1,j) holds (Col(M1,j)).k = (Col(M2,j)).k
    proof
      let k be Nat;
      assume k in dom Col(M1,j);
      then k in Seg len Col(M1,j) by FINSEQ_1:def 3;
      then
A4:   k in Seg len M1 by MATRIX_0:def 8;
      then
A5:   [k,j] in Indices M1 by A2,Th12;
A6:   k in dom M1 by A4,FINSEQ_1:def 3;
      hence (Col(M1,j)).k = M1*(k,j) by MATRIX_0:def 8
        .= M2*(k,j) by A1,A5,MATRIXR1:23
        .= (Col(M2,j)).k by A1,A6,MATRIX_0:def 8;
    end;
    dom Col(M1,j) = Seg len Col(M1,j) by FINSEQ_1:def 3
      .= Seg len M1 by MATRIX_0:def 8
      .= Seg len Col(M2,j) by A1,MATRIX_0:def 8
      .= dom Col(M2,j) by FINSEQ_1:def 3;
    hence Col(M1,j) = Col(M2,j) by A3,FINSEQ_1:13;
  end;
end;
