reserve X for set,
        n,m,k for Nat,
        K for Field,
        f for n-element real-valued FinSequence,
        M for Matrix of n,m,F_Real;

theorem
  the_rank_of M = n implies M is OrdBasis of Lin lines M
proof
  A1: lines M c=[#]Lin lines M by Lm4;
  then reconsider L=lines M as Subset of Lin lines M;
  reconsider B=M as FinSequence of Lin lines M by A1,FINSEQ_1:def 4;
  assume that
   A2: the_rank_of M=n;
  A3: M is one-to-one by A2,MATRIX13:121;
  lines M is linearly-independent by A2,MATRIX13:121;
  then A4: L is linearly-independent by VECTSP_9:12;
  Lin L=Lin lines M by VECTSP_9:17;
  then L is Basis of Lin lines M by A4,VECTSP_7:def 3;
  then B is OrdBasis of Lin lines M by A3,MATRLIN:def 2;
  hence thesis;
end;
