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 Th10:
  for a be Element of V1 for F being FinSequence of K for G being
FinSequence of V1 st len F = len G & for k st k in dom F holds G.k = (F/.k) * a
  holds Sum(G) = Sum(F) * a
proof
  let a be Element of V1;
  let F be FinSequence of K;
  let G be FinSequence of V1;
  assume that
A1: len F = len G and
A2: for k st k in dom F holds G.k = (F/.k) * a;
  now
    let k;
    let v be Element of K;
    assume that
A3: k in dom G and
A4: v = F.k;
A5: k in dom F by A1,A3,FINSEQ_3:29;
    then v = F/.k by A4,PARTFUN1:def 6;
    hence G.k = v * a by A2,A5;
  end;
  hence thesis by A1,Th9;
end;
