 reserve R for Ring;
 reserve x, y, y1 for set;
 reserve a, b for Element of R;
 reserve V for LeftMod of R;
 reserve v, w for Vector of V;
 reserve u,v,w for Vector of V;
 reserve F,G,H,I for FinSequence of V;
 reserve j,k,n for Nat;
 reserve f,f9,g for sequence of V;

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;
