
theorem Th11:
  for S be non empty compact TopSpace,
      T be non empty MetrSpace,
      f,g be Point of MetricSpace_of_ContinuousFunctions(S,T)
  holds
  for t be Point of S
    holds dist(In(f.t,T),In(g.t,T)) <= dist(f,g)
  proof
    let S be non empty compact TopSpace,
        T be non empty MetrSpace,
        a,b be Point of MetricSpace_of_ContinuousFunctions(S,T);
    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
A2: a=x & x is continuous;
    b in A; then
    consider y be Function of S,TopSpaceMetr(T) such that
A3: b=y & y is continuous;
    consider Dist1 be RealMap of S such that
A4: ( 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,A2,A3;
A5: rng Dist1 <> {}
      & rng Dist1 is bounded_above
      & rng Dist1 is bounded_below by A2,A3,A4,Th9;
    let t be Point of S;
A6: Dist1.t=dist (In(x.t,T),In(y.t,T)) by A4;
    Dist1.t in rng Dist1 by FUNCT_2:112;
    hence thesis by A6,A2,A3,A4,A5,SEQ_4:def 1;
  end;
