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
  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;
