reserve x,y,y1,y2 for set,
  p for FinSequence,
  i,k,l,n for Nat,
  V for RealLinearSpace,
  u,v,v1,v2,v3,w for VECTOR of V,
  a,b for Real,
  F,G,H1,H2 for FinSequence of V,
  A,B for Subset of V,
  f for Function of the carrier of V, REAL;

theorem
  len F = len G & (for k st k in dom F holds G.k = a * F/.k) implies Sum
  (G) = a * Sum(F)
proof
  assume that
A1: len F = len G and
A2: for k st k in dom F holds G.k = a * F/.k;
A3: dom F = Seg len F & dom G = Seg len G by FINSEQ_1:def 3;
  now
    let k be Nat,v;
    assume that
A4: k in dom G and
A5: v = F.k;
    v = F/.k by A1,A3,A4,A5,PARTFUN1:def 6;
    hence G.k = a * v by A1,A2,A3,A4;
  end;
  hence thesis by A1,RLVECT_1:39;
end;
