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 Th21:
  len b1 = dim V1
proof
  reconsider R=rng b1 as Basis of V1 by MATRLIN:def 2;
A1: b1 is one-to-one by MATRLIN:def 2;
  thus len b1 = card Seg len b1 by FINSEQ_1:57
    .= card dom b1 by FINSEQ_1:def 3
    .= card R by A1,CARD_1:70
    .= dim V1 by VECTSP_9:def 1;
end;
