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 Th25:
  for W1 be Subspace of V1 st W1 = (Omega).V1 for w be Vector of
W1, v be Vector of V1,w1 be OrdBasis of W1 st v = w & b1 = w1 holds v|--b1 = w
  |-- w1
proof
  let W1 be Subspace of V1 such that
A1: W1 = (Omega).V1;
  let w be Vector of W1,v be Vector of V1,w1 be OrdBasis of W1 such that
A2: v = w and
A3: b1 = w1;
  consider KL be Linear_Combination of W1 such that
A4: w = Sum(KL) & Carrier KL c= rng w1 and
A5: for k st 1<=k & k<=len (w|--w1) holds (w|--w1)/.k=KL.(w1/.k) by
MATRLIN:def 7;
  consider K1 be Linear_Combination of V1 such that
A6: Carrier K1=Carrier KL & Sum K1=Sum KL and
A7: K1|the carrier of W1=KL by Lm4;
A8: len w1 = len (w|-- w1) by MATRLIN:def 7;
  now
    let k such that
A9: 1<=k & k<=len (w|--w1);
A10: k in dom w1 by A8,A9,FINSEQ_3:25;
    dom K1 = the carrier of W1 by A1,FUNCT_2:def 1;
    then KL=K1 by A7;
    hence (w|--w1)/.k = K1.(w1/.k) by A5,A9
      .= K1.(w1.k) by A10,PARTFUN1:def 6
      .= K1.(b1/.k) by A3,A10,PARTFUN1:def 6;
  end;
  hence thesis by A2,A3,A4,A6,A8,MATRLIN:def 7;
end;
