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 Th26:
  for W1,W2 be Subspace of V1 st W1/\W2=(0).V1 for w1 be OrdBasis
  of W1,w2 be OrdBasis of W2 holds w1^w2 is OrdBasis of W1+W2
proof
  let W1,W2 be Subspace of V1 such that
A1: W1/\W2=(0).V1;
  let w1 be OrdBasis of W1,w2 be OrdBasis of W2;
  reconsider R1=rng w1 as Basis of W1 by MATRLIN:def 2;
  reconsider R2=rng w2 as Basis of W2 by MATRLIN:def 2;
A2: R1\/R2=rng (w1^w2) by FINSEQ_1:31;
A3: R1 misses R2
  proof
    assume R1 meets R2;
    then consider x being object such that
A4: x in R1 and
A5: x in R2 by XBOOLE_0:3;
    x in W1 & x in W2 by A4,A5;
    then x in W1/\W2 by VECTSP_5:3;
    then x in the carrier of (0).V1 by A1;
    then x in {0.V1} by VECTSP_4:def 3;
    then x = 0.V1 by TARSKI:def 1
      .= 0.W1 by VECTSP_4:11;
    then not R1 is linearly-independent by A4,VECTSP_7:2;
    hence thesis by VECTSP_7:def 3;
  end;
A6: R1\/R2 is Basis of W1+W2 by A1,Th3;
  then reconsider w12=w1^w2 as FinSequence of W1+W2 by A2,FINSEQ_1:def 4;
  w1 is one-to-one & w2 is one-to-one by MATRLIN:def 2;
  then w12 is one-to-one by A3,FINSEQ_3:91;
  hence thesis by A6,A2,MATRLIN:def 2;
end;
