reserve V for RealLinearSpace;

theorem Th9:
  for V1,V2,W being Subspace of V, W1,W2 being Subspace of W st W1
  = V1 & W2 = V2 holds W1 + W2 = V1 + V2
proof
  let V1,V2,W be Subspace of V, W1,W2 be Subspace of W such that
A1: W1 = V1 & W2 = V2;
  reconsider W3 = W1 + W2 as Subspace of V by RLSUB_1:27;
  now
    let v be VECTOR of V;
A2: the carrier of W1 c= the carrier of W & the carrier of W2 c= the
    carrier of W by RLSUB_1:def 2;
    hereby
      assume
A3:   v in W3;
      then reconsider w = v as VECTOR of W by Th8;
      consider w1,w2 being VECTOR of W such that
A4:   w1 in W1 & w2 in W2 and
A5:   w = w1 + w2 by A3,RLSUB_2:1;
      reconsider v1 = w1, v2 = w2 as VECTOR of V by RLSUB_1:10;
      v = v1 + v2 by A5,RLSUB_1:13;
      hence v in V1 + V2 by A1,A4,RLSUB_2:1;
    end;
    assume v in V1 + V2;
    then consider v1,v2 being VECTOR of V such that
A6: v1 in V1 & v2 in V2 and
A7: v = v1 + v2 by RLSUB_2:1;
    v1 in the carrier of W1 & v2 in the carrier of W2 by A1,A6;
    then reconsider w1 = v1, w2 = v2 as VECTOR of W by A2;
    v = w1 + w2 by A7,RLSUB_1:13;
    hence v in W3 by A1,A6,RLSUB_2:1;
  end;
  hence thesis by RLSUB_1:31;
end;
