 reserve
  S for non empty TopSpace,
  T for LinearTopSpace,
  X for non empty Subset of the carrier of S;
 reserve
    S,T for RealNormSpace,
    X for non empty Subset of the carrier of S;

theorem Th49:
  for S be non empty TopSpace,T be NormedLinearTopSpace,
      H be Functional_Sequence of the carrier of S,the carrier of T,
      LimH be Function of S,T
    st H is_unif_conv_on (the carrier of S)
      &
       ( for n be Nat holds
         ex Hn be Function of S,T st Hn = H.n & Hn is continuous )
     & LimH = lim(H,the carrier of S)
  holds LimH is continuous
  proof
    let S be non empty TopSpace,
        T be NormedLinearTopSpace,
        H be Functional_Sequence of the carrier of S,the carrier of T,
        LimH be Function of S,T;
    assume that
A1: H is_unif_conv_on the carrier of S and
A2: for n be Nat holds
      ex Hn be Function of S,T st Hn = H.n & Hn is continuous and
A3: LimH = lim(H,the carrier of S);
    set X = the carrier of S;
A4: H is_point_conv_on X by A1,SEQFUNC2:33;
    for x being Point of S holds LimH is_continuous_at x
    proof
      let x be Point of S;
      for G being a_neighborhood of LimH . x
        ex H being a_neighborhood of x st LimH .: H c= G
      proof
        let G be a_neighborhood of LimH . x;
        consider r being Real such that
A5:     r > 0
         & { y where y is Point of T : ||.y-LimH.x .|| < r } c= G by Th30;
A6:     0 < r/3 by XREAL_1:222,A5;
        reconsider r as Element of REAL by XREAL_0:def 1;
        consider k be Nat such that
A7:     for n be Nat for x1 being Element of S st n>=k & x1 in X holds
          ||.(H.n)/.x1-LimH/.x1.||<r/3 by A1,XREAL_1:222,A5,A3,SEQFUNC2:33;
        consider k1 be Nat such that
A8:     for n be Nat st n>=k1 holds ||.(H.n)/.x - LimH/.x.|| < r/3
          by A4,A6,A3,SEQFUNC2:11;
        reconsider m = max(k,k1) as Nat by XXREAL_0:def 10;
        consider h be Function of S,T such that
A9:     h = H.m & h is continuous by A2;
        set W = { y where y is Point of T : ||.y-h.x .|| < r/3 };
        now let z be object;
          assume z in W; then
          ex y be Point of T st z=y & ||.y-h.x .|| < r/3;
          hence z in the carrier of T;
        end; then
        reconsider W as Subset of T by TARSKI:def 3;
        W is a_neighborhood of h.x by XREAL_1:222,A5,Th30; then
        consider H being a_neighborhood of x such that
A10:    h.: H c= W by A9,TMAP_1:44,TMAP_1:def 2;
        take H;
        now let z be object;
          assume z in LimH .: H; then
          consider s be object such that
A11:      s in the carrier of S & s in H & z=LimH .s by FUNCT_2:64;
          reconsider s as Point of S by A11;
          h.s in h.:H by A11,FUNCT_2:35; then
          h.s in W by A10; then
A12:      ex y be Point of T st y =h.s & ||.y-h.x .|| < r/3;
          ||.h/.s-LimH/.s .|| < r/3 by A7,A9,XXREAL_0:25; then
A13:      ||.LimH/.s -h/.s.|| < r/3 by NORMSP_1:7;
A14:      ||.h/.x - LimH/.x.|| < r/3 by A8,A9,XXREAL_0:25;
          LimH.s-LimH.x = LimH.s - h/.s + h/.s -LimH.x by RLVECT_4:1
           .= LimH.s - h/.s + (h/.s -LimH.x) by RLVECT_1:28
           .= LimH.s - h/.s + (h/.s -h/.x + h/.x -LimH.x) by RLVECT_4:1
           .= LimH.s - h/.s + (h/.s -h/.x + (h/.x -LimH.x)) by RLVECT_1:28;
             then
A15:      ||.LimH.s-LimH.x.|| <= ||.LimH.s - h/.s.||
             + ||.h/.s -h/.x + (h/.x -LimH.x).|| by NORMSP_1:def 1;
A16:      ||.LimH.s - h/.s.|| + ||.h/.s -h/.x + (h/.x -LimH.x).||
             < r/3 + ||.h/.s -h/.x + (h/.x -LimH.x).|| by XREAL_1:8,A13;
A17:      ||.h/.s -h/.x + (h/.x -LimH.x).||
             <= ||.h/.s -h/.x.|| + ||.(h/.x -LimH.x).|| by NORMSP_1:def 1;
          ||.h/.s -h/.x.|| + ||.(h/.x -LimH.x).||
             < r/3 + r/3 by XREAL_1:8,A12,A14; then
          ||.h/.s -h/.x + (h/.x -LimH.x).|| < r/3 + r/3 by A17,XXREAL_0:2;
            then
          r/3 + ||.h/.s -h/.x + (h/.x -LimH.x).|| < r/3 + (r/3 + r/3)
            by XREAL_1:8; then
          ||.LimH.s - h/.s.|| + ||.h/.s -h/.x + (h/.x -LimH.x).||
            < r by A16,XXREAL_0:2; then
          ||.LimH.s-LimH.x .|| < r by A15,XXREAL_0:2;
          then LimH.s in { y where y is Point of T : ||.y-LimH.x .|| < r };
          hence z in G by A11,A5;
        end;
        hence LimH .: H c= G;
      end;
      hence thesis by TMAP_1:def 2;
    end;
    hence thesis by TMAP_1:44;
  end;
