
theorem Th26:
  for X being non empty TopSpace, Y being T_0-TopSpace, N being
net of ContMaps(X,Omega Y) st for x being Point of X holds ex_sup_of commute(N,
x,Omega Y) holds ex_sup_of rng the mapping of N, (Omega Y) |^ the carrier of X
proof
  let X be non empty TopSpace, Y be T_0-TopSpace, N be net of ContMaps(X,Omega
  Y) such that
A1: for x being Point of X holds ex_sup_of commute(N,x,Omega Y);
  deffunc F(Element of X) = sup commute(N,$1,Omega Y);
  set n = the mapping of N, L = (Omega Y) |^ the carrier of X;
  consider f being Function of the carrier of X, the carrier of Omega Y such
  that
A2: for u being Element of X holds f.u = F(u) from FUNCT_2:sch 4;
  reconsider a = f as Element of L by WAYBEL24:19;
  ex a being Element of L st rng n is_<=_than a & for b being Element of L
  st rng n is_<=_than b holds a <= b
  proof
    take a;
A3: dom n = the carrier of N by FUNCT_2:def 1;
A4: L = product ((the carrier of X) --> Omega Y) by YELLOW_1:def 5;
    thus rng n is_<=_than a
    proof
      let k be Element of L;
      reconsider k9 = k, a9 = a as Element of product ((the carrier of X) -->
      Omega Y) by YELLOW_1:def 5;
      assume k in rng n;
      then consider i being object such that
A5:   i in dom n and
A6:   k = n.i by FUNCT_1:def 3;
      reconsider i as Point of N by A5;
      for u being Element of X holds k9.u <= a9.u
      proof
        let u be Element of X;
        ex_sup_of commute(N,u,Omega Y) by A1;
        then
A8:     ex_sup_of
 rng the mapping of commute(N,u,Omega Y), Omega Y;
A9:     k9.u = (the mapping of commute(N,u,Omega Y)).i by A6,Th24;
        dom the mapping of commute(N,u,Omega Y) = the carrier of N by A3,Lm6;
        then
A10:    k9.u in rng the mapping of commute(N,u,Omega Y) by A9,FUNCT_1:def 3;
        a9.u = sup commute(N,u,Omega Y) by A2
          .= Sup the mapping of commute(N,u,Omega Y) by WAYBEL_2:def 1
          .= sup rng the mapping of commute(N,u,Omega Y);
        hence thesis by A8,A10,YELLOW_4:1;
      end;
      hence k <= a by A4,WAYBEL_3:28;
    end;
    let b be Element of L such that
A11: for k being Element of L st k in rng n holds k <= b;
    reconsider a9 = a, b9 = b as Element of product ((the carrier of X) -->
    Omega Y) by YELLOW_1:def 5;
    for u being Element of X holds a9.u <= b9.u
    proof
      let u be Element of X;
      ex_sup_of commute(N,u,Omega Y) by A1;
      then
A12:  ex_sup_of rng the mapping of commute(N,u,Omega Y), Omega Y;
A13:  Omega Y = ((the carrier of X) --> Omega Y).u;
A14:  rng the mapping of commute(N,u,Omega Y) is_<=_than b.u
      proof
        let s be Element of Omega Y;
        assume s in rng the mapping of commute(N,u,Omega Y);
        then consider i being object such that
A15:    i in dom the mapping of commute(N,u,Omega Y) and
A16:    (the mapping of commute(N,u,Omega Y)).i = s by FUNCT_1:def 3;
        reconsider i as Point of N by A3,A15,Lm6;
        n.i is Function of X, Omega Y by WAYBEL24:21;
        then reconsider k = n.i as Element of L by WAYBEL24:19;
        reconsider k9 = k as Element of product ((the carrier of X) --> Omega
        Y) by YELLOW_1:def 5;
        n.i in rng n by A3,FUNCT_1:def 3;
        then k <= b by A11;
        then
A17:    k9 <= b9 by YELLOW_1:def 5;
        s = n.i.u by A16,Th24;
        hence s <= b.u by A13,A17,WAYBEL_3:28;
      end;
      a9.u = sup commute(N,u,Omega Y) by A2
        .= Sup the mapping of commute(N,u,Omega Y) by WAYBEL_2:def 1
        .= sup rng the mapping of commute(N,u,Omega Y);
      hence thesis by A12,A14,YELLOW_0:30;
    end;
    hence thesis by A4,WAYBEL_3:28;
  end;
  hence thesis by YELLOW_0:15;
end;
