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);

theorem Th12:
  for x being Element of V..W ex a st x = a..W
proof
  let x be Element of V..W;
  consider a such that
A1: x = a + W by Def12;
  take a;
  thus thesis by A1;
end;
