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;
