
theorem Th12:
  for S be non empty compact TopSpace,
     T be non empty MetrSpace,
     f,g be Point of MetricSpace_of_ContinuousFunctions(S,T),
     f1,g1 be Function of S,T,
     e be Real st f = f1 & g = g1 &
  for t be Point of S holds dist(f1.t,g1.t) <= e
    holds dist(f,g) <= e
  proof
    let S be non empty compact TopSpace,
        T be non empty MetrSpace,
        a,b be Point of MetricSpace_of_ContinuousFunctions(S,T),
        f1,g1 be Function of S,T,
        e be Real;
     assume A1: a=f1 & b=g1 &
       for t be Point of S holds dist(f1.t,g1.t) <= e;
     set M = MetricSpace_of_ContinuousFunctions(S,T);
     set A = ContinuousFunctions (S,T);
     reconsider f = the distance of M as Function of [:A,A:],REAL;
     a in A; then
     consider x be Function of S,TopSpaceMetr(T) such that
 A3: a = x & x is continuous;
     b in A; then
     consider y be Function of S,TopSpaceMetr(T) such that
 A4: b = y & y is continuous;
     consider Dist1 be RealMap of S such that
 A5: ( for t be Point of S
        holds Dist1.t=dist (In(x.t,T),In(y.t,T)) )
      & f.(a,b) = upper_bound rng Dist1 by Def5,A3,A4;
A6:  rng Dist1 <> {}
      & rng Dist1 is bounded_above
      & rng Dist1 is bounded_below by A3,A4,A5,Th9;
     now let r be Real;
       assume r in rng Dist1; then
       consider t being Element of S such that
   A7: r = Dist1.t by FUNCT_2:113;
       Dist1.t = dist (In(x.t,T),In(y.t,T)) by A5;
       hence r <=e by A7,A1,A3,A4;
     end;
     hence dist(a,b) <= e by A6,SEQ_4:45,A5;
   end;
