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;
