reserve R for Ring,
  V for RightMod of R,
  a,b for Scalar of R,
  x,y for set,
  p,q ,r for FinSequence,
  i,k for Nat,
  u,v,v1,v2,v3,w for Vector of V,
  F,G,H for FinSequence of V,
  A,B for Subset of V,
  f for Function of V, R,
  S,T for finite Subset of V;

theorem
  v1 <> v2 & v2 <> v3 & v1 <> v3 implies Sum{v1,v2,v3} = v1 + v2 + v3
proof
  assume v1 <> v2 & v2 <> v3 & v1 <> v3;
  then
A1: <* v1,v2,v3 *> is one-to-one by FINSEQ_3:95;
  rng<* v1,v2,v3 *> = {v1,v2,v3} & Sum<* v1,v2,v3 *> = v1 + v2 + v3 by
FINSEQ_2:128,RLVECT_1:46;
  hence thesis by A1,Def1;
end;
