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;
reserve L,L1,L2,L3 for Linear_Combination of V;
reserve l for Linear_Combination of A;

theorem
  Carrier(L) = {v1,v2} & v1 <> v2 implies Sum(L) = v1 * L.v1 + v2 * L.v2
proof
  assume that
A1: Carrier(L) = {v1,v2} and
A2: v1 <> v2;
  L is Linear_Combination of {v1,v2} by A1,Def5;
  hence thesis by A2,Th33;
end;
