reserve V for RealLinearSpace;

theorem Th1:
  for W1,W2 being Subspace of V holds the carrier of W1 c= the
  carrier of W1 + W2
proof
  let W1,W2 be Subspace of V;
 let x be object;
  assume
A1: x in the carrier of W1;
  the carrier of W1 c= the carrier of V by RLSUB_1:def 2;
  then reconsider w = x as VECTOR of V by A1;
A2: w + 0.V = w & 0.V in W2 by RLSUB_1:17;
  x in W1 by A1;
  then x in {v + u where u,v is VECTOR of V : v in W1 & u in W2} by A2;
  hence thesis by RLSUB_2:def 1;
end;
