reserve x for set,
  K for Ring,
  r for Scalar of K,
  V for LeftMod of K,
  a,b,a1,a2 for Vector of V,
  A,A1,A2 for Subset of V,
  l for Linear_Combination of A,
  W for Subspace of V,
  Li for FinSequence of Submodules(V);
reserve S1,S2 for Element of V..W;

theorem Th17:
  a.W + b.W = (a+b).W & 0.(V.W) = (0.V).W
proof
  thus a.W + b.W = a..W + b..W by Def17
    .= (a+b).W by Def15;
  thus thesis;
end;
