reserve r for Real,
  X for set,
  f, g, h for real-valued Function;
reserve T for non empty TopSpace,
  A for closed Subset of T;

theorem Th20:
  for F being Functional_Sequence of the carrier of T, REAL st F
  is_unif_conv_on the carrier of T & for i being Nat holds F.i is
  continuous Function of T, R^1 holds lim(F, the carrier of T) is continuous
  Function of T, R^1
proof
  let F be Functional_Sequence of the carrier of T, REAL such that
A1: F is_unif_conv_on the carrier of T and
A2: for i being Nat holds F.i is continuous Function of T, R^1;
  F is_point_conv_on the carrier of T by A1,SEQFUNC:43;
  then dom lim(F, the carrier of T) = the carrier of T by SEQFUNC:def 13;
  then reconsider l=lim(F, the carrier of T) as Function of T,R^1 by
FUNCT_2:def 1,TOPMETR:17;
  now
    let p be Point of T;
    now
      let e be Real such that
A3:   e>0;
      reconsider e3=e/3 as Real;
A4:   e3>0 by A3,XREAL_1:139;
      then consider k being Nat such that
A5:   for n being Nat for x being Point of T st n>=k & x in
      the carrier of T holds |.(F.n).x-(lim(F,the carrier of T)).x.|<e3 by A1,
SEQFUNC:43;
      reconsider Fk=F.k as continuous Function of T, R^1 by A2;
A6:   |.Fk.p-l.p.|<e3 by A5;
      Fk is_continuous_at p by TMAP_1:44;
      then consider H being Subset of T such that
A7:   H is open & p in H and
A8:   for y being Point of T st y in H holds |.Fk.y-Fk.p.|<e3 by A4,Th19;
      take H;
      thus H is open & p in H by A7;
      let y be Point of T such that
A9:   y in H;
      |.Fk.y-l.y.|<e3 by A5;
      then |.-(Fk.y-l.y).|<e3 by COMPLEX1:52;
      then
      |.(Fk.p-l.p)+(-Fk.y+l.y).| <= |.Fk.p-l.p.|+|.-(Fk.y-l.y).| &
      |.Fk.p-l.p.|+|.-(Fk.y-l.y).| < e3+e3 by A6,COMPLEX1:56,XREAL_1:8;
      then |.(Fk.p-l.p)+(-Fk.y+l.y).| < 2*e3 by XXREAL_0:2;
      then
      |.Fk.y-Fk.p+((Fk.p-l.p)+(-Fk.y+l.y)).| <= |.Fk.y-Fk.p.|+
      |.(Fk.p-l.p)+(-Fk .y+l.y).| &
       |.Fk.y-Fk.p.|+|.(Fk.p-l.p)+(-Fk.y+l.y).| < e3+2*e3 by
A8,A9,COMPLEX1:56,XREAL_1:8;
      hence |.l.y-l.p.|<e by XXREAL_0:2;
    end;
    hence l is_continuous_at p by Th19;
  end;
  hence thesis by TMAP_1:44;
end;
