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
  len R1 = len R2 implies Sum (R1+R2)=Sum R1 + Sum R2
proof
  assume len R1=len R2;
  then reconsider
  r1=R1,r2=R2 as Element of (len R1)-tuples_on the carrier of V1 by FINSEQ_2:92
;
  thus Sum (R1+R2) = Sum (r1+r2) .= Sum R1+Sum R2 by FVSUM_1:76;
end;
