
theorem Th1:
  for V being RealUnitarySpace, W1,W2 being Subspace of V,
      x being object
 holds x in W1 + W2 iff ex v1,v2 being VECTOR of V st v1 in W1 & v2 in W2 &
  x = v1 + v2
proof
  let V be RealUnitarySpace;
  let W1,W2 be Subspace of V;
  let x be object;
  thus x in W1 + W2 implies ex v1,v2 being VECTOR of V st v1 in W1 & v2 in W2
  & x = v1 + v2
  proof
    assume x in W1 + W2;
    then x in the carrier of W1 + W2 by STRUCT_0:def 5;
    then x in {v + u where v,u is VECTOR of V : v in W1 & u in W2} by Def1;
    then consider v1,v2 being VECTOR of V such that
A1: x = v1 + v2 & v1 in W1 & v2 in W2;
    take v1,v2;
    thus thesis by A1;
  end;
  given v1,v2 be VECTOR of V such that
A2: v1 in W1 & v2 in W2 & x = v1 + v2;
  x in {v + u where v,u is VECTOR of V : v in W1 & u in W2} by A2;
  then x in the carrier of W1 + W2 by Def1;
  hence thesis by STRUCT_0:def 5;
end;
