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, 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,Th12;
  end;
