reserve T, T1 for non empty TopSpace;
reserve F,G,H for Subset-Family of T,
  A,B,C,D for Subset of T,
  O,U for open Subset of T,
  p,q for Point of T,
  x,y,X for set;
reserve Un for FamilySequence of T,
  r,r1,r2 for Real,
  n for Element of NAT;

theorem
  for F be Function of the carrier of T,Funcs(the carrier of T,the
carrier of T1) st for p be Point of T holds F.p is continuous Function of T,T1
  for S be Function of the carrier of T,bool the carrier of T st for p be Point
of T holds p in S.p & S.p is open & for p,q be Point of T st p in S.q holds F.p
  .p=F.q.p holds F Toler is continuous
proof
  let F be Function of the carrier of T,Funcs(the carrier of T,the carrier of
  T1) such that
A1: for p holds F.p is continuous Function of T,T1;
  let S be Function of the carrier of T,bool the carrier of T such that
A2: for p holds p in S.p & S.p is open & for p,q st p in S.q holds F.p.p
  =F.q.p;
  now
    let t be Point of T;
    for R being Subset of T1 st R is open & (F Toler).t in R ex H being
    Subset of T st H is open & t in H & (F Toler).:H c= R
    proof
      reconsider Ft=F.t as Function of T,T1 by A1;
      let R be Subset of T1 such that
A3:   R is open and
A4:   (F Toler).t in R;
      Ft is continuous by A1;
      then
A5:   Ft is_continuous_at t by TMAP_1:50;
      Ft.t in R by A4,Def8;
      then consider A being Subset of T such that
A6:   A is open & t in A and
A7:   Ft.:A c= R by A3,A5,TMAP_1:43;
      set H=A/\S.t;
A8:   (F Toler).:H c= R
      proof
        let FSh be object;
        assume FSh in (F Toler).:H;
        then consider h being object such that
A9:     h in dom (F Toler) and
A10:    h in H and
A11:    FSh=(F Toler).h by FUNCT_1:def 6;
        reconsider h9=h as Point of T by A9;
        h9 in S.t & FSh=F.h9.h9 by A10,A11,Def8,XBOOLE_0:def 4;
        then
A12:    FSh=Ft.h9 by A2;
A13:    the carrier of T =dom Ft by FUNCT_2:def 1;
        h9 in A by A10,XBOOLE_0:def 4;
        then FSh in Ft.:A by A12,A13,FUNCT_1:def 6;
        hence thesis by A7;
      end;
      take H;
      S.t is open & t in S.t by A2;
      hence thesis by A6,A8,XBOOLE_0:def 4;
    end;
    hence F Toler is_continuous_at t by TMAP_1:43;
  end;
  hence thesis by TMAP_1:50;
end;
