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

theorem Th13:
  for D being non empty set, M being Matrix of D holds [i,j] in
  Indices M iff i in dom M & j in dom (M.i)
proof
  let D be non empty set, M be Matrix of D;
  hereby
    assume
A1: [i,j] in Indices M;
    then
A2: j in Seg width M by Th12;
A3: i in Seg len M by A1,Th12;
    then i in dom M by FINSEQ_1:def 3;
    then j in Seg len(M.i) by A2,MATRIX_0:36;
    hence i in dom M & j in dom (M.i) by A3,FINSEQ_1:def 3;
  end;
  assume i in dom M & j in dom (M.i);
  hence thesis by MATRIX_0:37;
end;
