reserve D for non empty set,
  D1,D2,x,y for set,
  n,k for Nat,
  p,x1 ,r for Real,
  f for Function;
reserve F for Functional_Sequence of D1,D2;
reserve G,H,H1,H2,J for Functional_Sequence of D,REAL;
reserve x for Element of D,
  X,Y for set,
  S1,S2 for Real_Sequence,
  f for PartFunc of D,REAL;
reserve H for Functional_Sequence of REAL,REAL;

theorem
  H is_unif_conv_on X & (for n holds (H.n)|X is continuous)
    implies lim(H,X)|X is continuous
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;
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 Real st x0 in dom(l|X) holds l|X is_continuous_in x0
  proof
    let x0 be Real;
    assume x0 in dom(l|X);
    then
A6: x0 in X by RELAT_1:57;
    reconsider x0 as Element of REAL by XREAL_0:def 1;
    for r be Real st 0<r ex s be Real st 0<s & for x1 be
Real st x1 in dom (l|X) & |.x1-x0.|<s holds |.(l|X).x1-(l|X).x0.|<r
    proof
      let r be Real;
      assume 0<r;
      then
A7:   0 < r/3 by XREAL_1:222;
      reconsider r as Element of REAL by XREAL_0:def 1;
      consider k such that
A8:   for n for x1 being Element of REAL st n>=k & x1 in X
       holds |.(H.n).x1-l.x1.|<r/3 by A1,A7,Th42;
      consider k1 be Nat such that
A9:   for n st n>=k1 holds |.(H.n).x0 - l.x0.| < r/3 by A3,A6,A7,Th20;
      reconsider m = max(k,k1) as Nat by TARSKI:1;    
      set h = H.m;
A10:  X c= dom h by A5;
A11:  dom(h|X) = dom h /\ X by RELAT_1:61
        .= X by A10,XBOOLE_1:28;
      h|X is continuous by A2;
      then h|X is_continuous_in x0 by A6,A11,FCONT_1:def 2;
      then consider s be Real such that
A12:  0<s and
A13:  for x1 be Real st x1 in dom (h|X) & |.x1-x0.|<s holds
      |.( h|X).x1-(h|X).x0.|<r/3 by A7,FCONT_1:3;
      reconsider s as Real;
      take s;
      thus 0<s by A12;
      let x1 be Real;
      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,FUNCT_1:47;
      then
A17:  |.h.x1-h.x0.|<r/3 by A6,FUNCT_1:49;
      |.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,COMPLEX1:56,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-h.x0)+(h.x0-l.x0)).|;
      then
A19:  |.l.x1 - l.x0.| <= |.l.x1-h.x1.|+ |.(h.x1-h.x0)+(h.x0-l.x0 ) .|
      by COMPLEX1:56;
      |.h.x1-l.x1.|<r/3 by A8,A16,A14,XXREAL_0:25;
      then |.-(l.x1-h.x1).|<r/3;
      then |.l.x1-h.x1.|<r/3 by COMPLEX1:52;
      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,FUNCT_1:47;
      hence thesis by A4,RELAT_1:68;
    end;
    hence thesis by FCONT_1:3;
  end;
  hence thesis by FCONT_1:def 2;
end;
