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
  Z c= X & Z <> {} & H is_unif_conv_on X implies H is_unif_conv_on Z
  proof
    assume that
    A1: Z c= X and
    A2: Z <> {} and
    A3: H is_unif_conv_on X;

    consider f such that
    A4: dom f = X and
    A5: for p st p>0 ex k st for n,x st n>=k & x in X
    holds ||.(H.n)/.x - f/.x.|| < p by A3;

    now
      take g = f|Z;
      thus
      A7: dom g = dom f /\ Z by RELAT_1:61
      .= Z by A1, A4, XBOOLE_1:28;
      let p;
      assume p>0;
      then consider k such that
      A8: for n,x st n>=k & x in X holds ||.(H.n)/.x - f/.x.|| < p by A5;
      take k;

      let n, x;
      assume that
      A9: n >= k and
      A10: x in Z;

      x in Z & x in dom f by A7, A10, RELAT_1:57;
      then f/.x = f.x by PARTFUN1:def 6
      .= g.x by A7, A10, FUNCT_1:47
      .= g/.x by A10, A7, PARTFUN1:def 6; 
      hence ||.(H.n)/.x -g/.x.|| < p by A1, A8, A9, A10;
    end;
    hence thesis by A1, A2, A3, Th22;
  end;
