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;
reserve S for Element of V.W;

theorem Th22:
  a/W + b/W = (a+b)/W & r*(a/W) = (r*a)/W
proof
  thus a/W + b/W = a.W + b.W .= (a+b)/W by Th17;
  thus r*(a/W) = (LMULT W).(r,a.W) by VECTSP_1:def 12
    .= r*(a.W qua Element of V.W) by Def21
    .= (r*a)/W by Def20;
end;
