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

theorem Th15:
  for D being non empty set, M being Matrix of D holds [i,j] in
  Indices M iff i in dom Col(M,j) & j in dom Line(M,i)
proof
  let D be non empty set, M be Matrix of D;
  hereby
    assume
A1: [i,j] in Indices M;
    then
A2: i in dom M by Th13;
    then i in Seg len M by FINSEQ_1:def 3;
    then i in Seg len Col(M,j) by MATRIX_0:def 8;
    hence i in dom Col(M,j) by FINSEQ_1:def 3;
    j in dom (M.i) by A1,Th13;
    hence j in dom Line(M,i) by A2,MATRIX_0:60;
  end;
  assume that
A3: i in dom Col(M,j) and
A4: j in dom Line(M,i);
  i in Seg len Col(M,j) by A3,FINSEQ_1:def 3;
  then i in Seg len M by MATRIX_0:def 8;
  then
A5: i in dom M by FINSEQ_1:def 3;
  then j in dom (M.i) by A4,MATRIX_0:60;
  hence thesis by A5,Th13;
end;
