theorem ThRank1:
  for V being finite-rank free Z_Module, b being OrdBasis of V
  holds len b = rank V
  proof
    let V be finite-rank free Z_Module, b be OrdBasis of V;
    reconsider R = rng b as Basis of V by defOrdBasis;
    A1: b is one-to-one by defOrdBasis;
    thus len b = card Seg len b by FINSEQ_1:57
    .= card dom b by FINSEQ_1:def 3
    .= card R by A1,CARD_1:70
    .= rank V by ZMODUL03:def 5;
  end;
