theorem Th31:
  for M being Matrix of k,F_Real holds
  the_rank_of M < k iff
  lines M is linearly-dependent or not M is without_repeated_line
  proof
    let M be Matrix of k,F_Real;
A1: the_rank_of M = k iff lines M is linearly-independent &
    M is without_repeated_line by MATRIX13:121;
A2: the_rank_of M <= len M by MATRIX13:74;
    len M = k by MATRIX_0:def 2;
    hence thesis by A1,A2,XXREAL_0:1;
  end;
