reserve k,t,i,j,m,n for Nat,
  x,y,y1,y2 for object,
  D for non empty set;
reserve K for Field,
  V for VectSp of K,
  a for Element of K,
  W for Element of V;
reserve KL1,KL2,KL3 for Linear_Combination of V,
  X for Subset of V;
reserve s for FinSequence,
  V1,V2,V3 for finite-dimensional VectSp of K,
  f,f1,f2 for Function of V1,V2,
  g for Function of V2,V3,
  b1 for OrdBasis of V1,
  b2 for OrdBasis of V2,
  b3 for OrdBasis of V3,
  v1,v2 for Vector of V2,
  v,w for Element of V1;
reserve p2,F for FinSequence of V1,
  p1,d for FinSequence of K,
  KL for Linear_Combination of V1;

theorem Th30:
  for P1,P2 be FinSequence of V1 st len P1 = len P2 holds
  Sum(P1 + P2) = (Sum P1) + (Sum P2)
proof
  let P1,P2 be FinSequence of V1;
  assume len P1 = len P2;
  then reconsider
  R1 = P1, R2 = P2 as Element of (len P1)-tuples_on (the carrier of
  V1) by FINSEQ_2:92;
  thus Sum(P1+P2) = Sum (R1 + R2) .= Sum P1 + Sum P2 by FVSUM_1:76;
end;
