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;

theorem Th25:
  X common_on_dom H implies for x be set st x in X holds {x} common_on_dom H
  proof
    assume
    A1: X common_on_dom H;
    let x be set;
    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;
