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 Th19:
  H is_point_conv_on X iff X common_on_dom H & for x st x in X
  holds (H#x) is convergent
  proof
    defpred X[set] means $1 in X;
    deffunc U(Element of D) = In(lim(H#$1),the carrier of Y);
    consider f such that
    A1: for x holds x in dom f iff X[x] and
    A2: for x st x in dom f holds f.x = U(x) from SEQ_1:sch 3;
    thus H is_point_conv_on X implies X common_on_dom H & for x st x in X
    holds (H#x) is convergent
    proof
      assume
      A3: H is_point_conv_on X;
      hence X common_on_dom H;
      let x;
      assume
      A4: x in X;
      ex f st X = dom f & for x st x in X holds (H#x) is convergent
      & lim(H#x) = f.x by A3, Th18;
      hence thesis by A4;
    end;

    assume that
    A5: X common_on_dom H and
    A6: for x st x in X holds (H#x) is convergent;
    now
      take f;
      thus
      A7: X = dom f
      proof
        thus X c= dom f
        proof
          let x be object such that
          A8: x in X;
          X c= dom (H.0) by A5;
          then X c= D by XBOOLE_1:1;
          then reconsider x9 = x as Element of D by A8;
          x9 in dom f by A1, A8;
          hence thesis;
        end;
        let x be object;
        assume x in dom f;
        hence thesis by A1;
      end;

      let x;
      assume
      A9: x in X;
      hence (H#x) is convergent by A6;
      In(lim(H#x),the carrier of Y) = lim(H#x) by SUBSET_1:def 8;
      hence f.x = lim(H#x) by A2, A7, A9;
    end;
    hence thesis by A5, Th18;
  end;
