reserve D for non empty set,
  D1, D2, x, y, Z for set,
  n, k for Nat,
  p, x1, r for Real,
  f for Function,
  Y for RealNormSpace,
  G, H, H1, H2, J for Functional_Sequence of D,the carrier of Y;
reserve
  x for Element of D,
  X for set,
  S1, S2 for sequence of Y,
  f for PartFunc of D,the carrier of Y;
reserve x for Element of D;

theorem Th36:
  X common_on_dom H1 & X common_on_dom H2
  implies X common_on_dom H1 + H2 & X common_on_dom H1-H2 
  proof
    assume that
    A1: X common_on_dom H1 and
    A2: X common_on_dom H2;

    now
      let n;
      X c= dom (H1.n) & X c= dom (H2.n) by A1, A2;
      then X c= dom (H1.n) /\ dom (H2.n) by XBOOLE_1:19;
      then X c= dom (H1.n + H2.n) by VFUNCT_1:def 1;
      hence X c= dom ((H1+H2).n) by Def5;
    end;
    hence X common_on_dom H1+H2 by A1;

    now
      let n;
      X c= dom (H1.n) & X c= dom (H2.n) by A1, A2;
      then X c= dom (H1.n) /\ dom (H2.n) by XBOOLE_1:19;
      then X c= dom (H1.n - H2.n) by VFUNCT_1:def 2;
      hence X c= dom ((H1-H2).n) by Th3;
    end;
    hence X common_on_dom H1-H2 by A1;
  end;
