reserve i, j, m, n, k for Nat,
  x, y for set,
  K for Field,
  a,a1 for Element of K;
reserve V1,V2,V3 for finite-dimensional VectSp of K,
  f for Function of V1,V2,

  b1,b19 for OrdBasis of V1,
  B1 for FinSequence of V1,
  b2 for OrdBasis of V2,
  B2 for FinSequence of V2,

  B3 for FinSequence of V3,
  v1,w1 for Element of V1,
  R,R1,R2 for FinSequence of V1,
  p,p1,p2 for FinSequence of K;

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 RELAT_1:70;
  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,RELAT_1:70,VECTSP_7:1;
  then reconsider A9=A as linearly-independent Subset of Lin A by A3,
VECTSP_9:12;
  b1 is one-to-one by MATRLIN:def 2;
  then
A4: b1|m is one-to-one by FUNCT_1:52;
  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;
