theorem
  for B1 be FinSequence of V1,W1 be Subspace of V1, B2 be FinSequence of
  W1 st B1 = B2 holds Sum B1 = Sum B2
proof
  let B1 be FinSequence of V1,W1 be Subspace of V1, B2 be FinSequence of W1
  such that
A1: B1 = B2;
  defpred P[Nat] means for B1 be FinSequence of V1,W1 be Subspace of V1, B2 be
  FinSequence of W1 st B1 = B2 & len B1=$1 holds Sum B1 = Sum B2;
A2: for n st P[n] holds P[n+1]
  proof
    let n such that
A3: P[n];
    set n1=n+1;
    let B1 be FinSequence of V1,W1 be Subspace of V1, B2 be FinSequence of W1
    such that
A4: B1 = B2 and
A5: len B1=n1;
A6: len (B1|n)=n by A5,FINSEQ_1:59,NAT_1:11;
    then
A7: Sum (B1|n)=Sum (B2|n) by A3,A4;
    1<=n1 by NAT_1:11;
    then
A8: n1 in dom B1 by A5,FINSEQ_3:25;
    then
A9: B2.n1=B2/.n1 by A4,PARTFUN1:def 6;
A10: B1.n1=B1/.n1 by A8,PARTFUN1:def 6;
A11: dom (B1|n)=Seg n by A6,FINSEQ_1:def 3;
    hence Sum B1 = Sum (B1|n)+B1/.n1 by A5,A10,A6,RLVECT_1:38
      .= Sum (B2|n)+B2/.n1 by A4,A10,A9,A7,VECTSP_4:13
      .=Sum B2 by A4,A5,A9,A6,A11,RLVECT_1:38;
  end;
A12: P[0]
  proof
    let B1 be FinSequence of V1,W1 be Subspace of V1, B2 be FinSequence of W1;
    assume B1 = B2 & len B1=0;
    then Sum B1=0.V1 & Sum B2=0.W1 by RLVECT_1:75;
    hence thesis by VECTSP_4:11;
  end;
  for n holds P[n] from NAT_1:sch 2(A12,A2);
  then P[len B1];
  hence thesis by A1;
end;
