
theorem Th2:
  for V being RealUnitarySpace, W1,W2 being Subspace of V, v being
  VECTOR of V st v in W1 or v in W2 holds v in W1 + W2
proof
  let V be RealUnitarySpace;
  let W1,W2 be Subspace of V;
  let v be VECTOR of V;
  assume
A1: v in W1 or v in W2;
  now
    per cases by A1;
    suppose
A2:   v in W1;
      v = v + 0.V & 0.V in W2 by RLVECT_1:4,RUSUB_1:11;
      hence thesis by A2,Th1;
    end;
    suppose
A3:   v in W2;
      v = 0.V + v & 0.V in W1 by RLVECT_1:4,RUSUB_1:11;
      hence thesis by A3,Th1;
    end;
  end;
  hence thesis;
end;
