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

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 & X common_on_dom H1(#)H2
proof
  assume that
A1: X common_on_dom H1 and
A2: X common_on_dom H2;
A3: X <> {} 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 VALUED_1:def 1;
    hence X c= dom ((H1+H2).n) by Def5;
  end;
  hence X common_on_dom H1+H2 by A3;
  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 VALUED_1:12;
    hence X c= dom ((H1-H2).n) by Th3;
  end;
  hence X common_on_dom H1-H2 by A3;
  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 VALUED_1:def 4;
    hence X c= dom ((H1(#)H2).n) by Def7;
  end;
  hence thesis by A3;
end;
