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 Th38:
  X common_on_dom H implies X common_on_dom r(#)H
proof
  assume
A1: X common_on_dom H;
A2: now
    let n;
    dom (H.n) = dom(r(#)(H.n)) by VALUED_1:def 5
      .= dom ((r(#)H).n) by Def1;
    hence X c= dom ((r(#)H).n) by A1;
  end;
  X <> {} by A1;
  hence thesis by A2;
end;
