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;
reserve x for Element of D;
reserve 
  V, W for RealNormSpace, 
  H for Functional_Sequence of the carrier of V,the carrier of W;

theorem
  H is_unif_conv_on X & (for n holds (H.n)|X is_continuous_on X)
  implies lim(H, X) is_continuous_on X
  proof
    set l = lim(H, X);

    assume that
    A1: H is_unif_conv_on X and
    A2: for n holds (H.n)|X is_continuous_on X;

    A3: H is_point_conv_on X by A1, Th21;
    then
    A4: dom l = X by Def13;

    A5: X common_on_dom H by A1;

    for x0 be Point of V st x0 in X holds l|X is_continuous_in x0
    proof
      let x0 be Point of V;
      assume
      A6: x0 in X; then
      A60: x0 in dom(l|X) by RELAT_1:62, A4;

      for r be Real st 0 < r ex s be Real st 0 < s
      & for x1 be Point of V st x1 in dom (l|X)
      & ||.x1-x0.|| < s holds ||.(l|X)/.x1-(l|X)/.x0.|| < r
      proof
        let r be Real;
        assume
        A7: 0 < r;

        reconsider r as Element of REAL by XREAL_0:def 1;

        consider k such that
        A8: for n for x1 being Element of V st n>=k & x1 in X
        holds ||.(H.n)/.x1-l/.x1.||<r/3 by A1, A7, XREAL_1:222, Th42;

        0 < r/3 by A7, XREAL_1:222;
        then
        consider k1 be Nat such that
        A9: for n st n>=k1 holds ||.(H.n)/.x0 - l/.x0.|| < r/3 by A3, A6, Th20;
        reconsider m = max(k,k1) as Nat by TARSKI:1;    
        set h = H.m;
        A11: dom(h|X) = dom h /\ X by RELAT_1:61
        .= X by A5, XBOOLE_1:28;

        h|X is_continuous_on X by A2;
        then h is_continuous_on X by NFCONT_1:21;
        then h|X is_continuous_in x0 by A6, NFCONT_1:def 7;
        then consider s be Real such that
        A12:  0<s and
        A13:  for x1 be Point of V st x1 in dom (h|X) & ||. x1- x0 .||<s holds
        ||.( h|X)/.x1-(h|X)/.x0.|| < r/3 by A7, XREAL_1:222, NFCONT_1:7;
        take s;
        thus 0<s by A12;

        let x1 be Point of V;
        assume that
        A14: x1 in dom (l|X) and
        A15: ||. x1-x0 .||<s;

        A16: dom (l|X) = dom l /\ X by RELAT_1:61
        .= X by A4;
        then ||.(h|X)/.x1-(h|X)/.x0.||<r/3 by A11, A13, A14, A15;
        then ||.h/.x1-(h|X)/.x0.||<r/3 by A16, A11, A14, PARTFUN1:80;
        then
        A17: ||.h/.x1-h/.x0.||<r/3 by A11, A6, PARTFUN1:80;
        ||.h/.x0 - l/.x0.|| < r/3 by A9, XXREAL_0:25;
        then ||.(h/.x1-h/.x0)+(h/.x0-l/.x0).||
        <= ||.h/.x1-h/.x0.||+||.h/.x0-l/.x0.|| 
        & ||.h/.x1-h/.x0.|| +||.h/.x0-l/.x0.|| < r/3 + r/3
        by A17, NORMSP_1:def 1, XREAL_1:8;
        then
        A18: ||.(h/.x1-h/.x0)+(h/.x0-l/.x0).||<r/3+r/3 by XXREAL_0:2;
        ||.l/.x1 - l/.x0.|| = ||.l/.x1 -h/.x1 +h/.x1 - l/.x0.|| by RLVECT_4:1
        .= ||.l/.x1 -h/.x1 +(h/.x1 - l/.x0) .|| by RLVECT_1:28
        .= ||.l/.x1 -h/.x1 +(h/.x1 -h/.x0 + h/.x0 - l/.x0) .|| by RLVECT_4:1
        .= ||.l/.x1 -h/.x1 +((h/.x1 -h/.x0) + (h/.x0 - l/.x0)) .||
        by RLVECT_1:28; then
        A19: ||.l/.x1 - l/.x0.||
        <= ||.l/.x1-h/.x1.|| + ||.(h/.x1-h/.x0)+(h/.x0-l/.x0 ) .||
        by NORMSP_1:def 1;
        ||.h/.x1-l/.x1.||<r/3 by A8, A16, A14, XXREAL_0:25;
        then ||.-(l/.x1-h/.x1).||<r/3 by RLVECT_1:33;
        then ||.l/.x1-h/.x1.||<r/3 by NORMSP_1:2;
        then ||.l/.x1-h/.x1.|| + ||.(h/.x1-h/.x0)+(h/.x0-l/.x0).||
        < r/3 +(r/3+r/3) by A18, XREAL_1:8;
        then ||.l/.x1 - l/.x0.|| < r/3 +r/3+r/3 by A19, XXREAL_0:2;
        then ||.(l|X)/.x1 - l/.x0.|| < r by A14, PARTFUN1:80;
        hence thesis by A4,RELAT_1:68;
      end;
      hence thesis by NFCONT_1:7, A60;
    end;
    hence thesis by NFCONT_1:def 7, A4;
  end;
