
theorem
  for n being Nat, M being Matrix of n, F_Rat
  st M is without_repeated_line
  holds Det M <> 0.F_Rat iff lines M is linearly-independent
  proof
    let n be Nat, M being Matrix of n, F_Rat;
    assume AS: M is without_repeated_line;
    hereby assume Det M <> 0.F_Rat;
      then the_rank_of M = n by MATRIX13:83;
      hence lines M is linearly-independent by MATRIX13:110;
    end;
    assume lines M is linearly-independent;
    then the_rank_of M = n by AS,MATRIX13:121;
    hence Det M <> 0.F_Rat  by MATRIX13:83;
  end;
