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 Th22:
  Y c= X & Y<>{} & X common_on_dom H implies Y common_on_dom H
proof
  assume that
A1: Y c= X and
A2: Y<>{} and
A3: X common_on_dom H;
  now
    let n;
    X c= dom (H.n) by A3;
    hence Y c= dom (H.n) by A1;
  end;
  hence thesis by A2;
end;
