theorem
  rng(b1/^m) is linearly-independent Subset of V1 & for A be Subset of
  V1 st A = rng(b1/^m) holds b1/^m is OrdBasis of Lin A
proof
  reconsider RNG=rng b1 as Basis of V1 by MATRLIN:def 2;
A1: RNG is linearly-independent by VECTSP_7:def 3;
  rng (b1/^m) c= RNG by FINSEQ_5:33;
  hence rng(b1/^m) is linearly-independent Subset of V1 by A1,VECTSP_7:1
,XBOOLE_1:1;
  let A be Subset of V1 such that
A2: A = rng(b1 /^m);
A3: A c= the carrier of Lin (A)
  proof
    let x be object;
    assume x in A;
    then x in Lin A by VECTSP_7:8;
    hence thesis;
  end;
  A is linearly-independent by A1,A2,FINSEQ_5:33,VECTSP_7:1;
  then reconsider A9=A as linearly-independent Subset of Lin A by A3,
VECTSP_9:12;
  b1 is one-to-one & b1=(b1|m)^(b1/^m) by MATRLIN:def 2,RFINSEQ:8;
  then
A4: b1/^m is one-to-one by FINSEQ_3:91;
  Lin A9= the ModuleStr of Lin A by VECTSP_9:17;
  then rng(b1/^m) is Basis of Lin A & b1/^m is FinSequence of Lin A by A2,
FINSEQ_1:def 4,VECTSP_7:def 3;
  hence thesis by A4,MATRLIN:def 2;
end;
