theorem Th83:
  for M be Matrix of n,K holds the_rank_of M = n iff Det M <> 0.K
proof
  let M be Matrix of n,K;
A1: [:Seg n,Seg n:] c= Indices M by MATRIX_0:24;
A2: len M=n by MATRIX_0:24;
  then
A3: the_rank_of M <= n by Th74;
A4: width M=n by MATRIX_0:24;
  then
A5: M = Segm(M,Seg n,Seg n) by A2,Th46
    .= EqSegm(M,Seg n,Seg n) by Def3;
A6: card Seg n=n by FINSEQ_1:57;
  thus the_rank_of M = n implies Det M <> 0.K
  proof
    assume the_rank_of M = n;
    then consider P,Q such that
A7: [:P,Q:] c= Indices M and
A8: card P = card Q and
A9: card P = n and
A10: Det EqSegm(M,P,Q)<>0.K by Def4;
    P c= Seg n by A2,A7,A8,Th67;
    then
A11: P=Seg n by A6,A9,CARD_2:102;
    Q c= Seg n by A4,A7,A8,Th67;
    then Q=Seg n by A6,A8,A9,CARD_2:102;
    then M = Segm(M,P,Q) by A2,A4,A11,Th46
      .= EqSegm(M,P,Q) by A8,Def3;
    hence thesis by A9,A10;
  end;
  assume Det M <> 0.K;
  then the_rank_of M >= n by A6,A5,A1,Def4;
  hence thesis by A3,XXREAL_0:1;
end;
