
theorem Th12:
  for K be add-associative right_zeroed right_complementable
associative Abelian well-unital distributive non empty doubleLoopStr for V be
  VectSp of K for V1,V2,W be Subspace of V, W1,W2 be Subspace of W st W1 = V1 &
  W2 = V2 holds W1 + W2 = V1 + V2
proof
  let K be add-associative right_zeroed right_complementable associative
  Abelian well-unital distributive non empty doubleLoopStr, V be VectSp of K;
  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 VECTSP_4:26;
  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 VECTSP_4:def 2;
    hereby
      assume
A3:   v in W3;
      then reconsider w = v as Vector of W by Th11;
      consider w1,w2 be Vector of W such that
A4:   w1 in W1 & w2 in W2 and
A5:   w = w1 + w2 by A3,VECTSP_5:1;
      reconsider v1 = w1, v2 = w2 as Vector of V by VECTSP_4:10;
      v = v1 + v2 by A5,VECTSP_4:13;
      hence v in V1 + V2 by A1,A4,VECTSP_5:1;
    end;
    assume v in V1 + V2;
    then consider v1,v2 be Vector of V such that
A6: v1 in V1 & v2 in V2 and
A7: v = v1 + v2 by VECTSP_5:1;
    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,VECTSP_4:13;
    hence v in W3 by A1,A6,VECTSP_5:1;
  end;
  hence thesis by VECTSP_4:30;
end;
