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 Th25:
  X common_on_dom H implies for x st x in X holds {x} common_on_dom H
proof
  assume
A1: X common_on_dom H;
  let x;
  assume
A2: x in X;
  thus {x} <> {};
  now
    let n;
    X c= dom(H.n) by A1;
    hence {x} c= dom(H.n) by A2,ZFMISC_1:31;
  end;
  hence thesis;
end;
