
theorem Th41:
  for X being non empty TopSpace, Y being monotone-convergence
  T_0-TopSpace, N being net of ContMaps(X,Omega Y), x being Point of X, f being
Function of X, Omega Y st (for a being Point of X holds commute(N,a,Omega Y) is
eventually-directed) & f = "\/"(rng the mapping of N, (Omega Y) |^ the carrier
  of X) holds f.x = sup commute(N,x,Omega Y)
proof
  let X be non empty TopSpace, Y be monotone-convergence T_0-TopSpace, N be
  net of ContMaps(X,Omega Y), x be Point of X, f be Function of X, Omega Y such
  that
A1: for a being Point of X holds commute(N,a,Omega Y) is
  eventually-directed and
A2: f = "\/"(rng the mapping of N, (Omega Y) |^ the carrier of X);
  set n = the mapping of N, m = the mapping of commute(N,x,Omega Y), L = (
  Omega Y) |^ the carrier of X;
A3: for x being Point of X holds ex_sup_of commute(N,x,Omega Y)
  proof
    let x be Point of X;
    commute(N,x,Omega Y) is eventually-directed by A1;
    hence thesis by Th31;
  end;
  then
A4: ex_sup_of rng n,L by Th26;
A5: dom n = the carrier of N by FUNCT_2:def 1;
  then
A6: dom m = the carrier of N by Lm6;
A7: for a being Element of Omega Y st rng m is_<=_than a holds f.x <= a
  proof
    let a be Element of Omega Y;
    defpred P[set,set] means ($1 = x implies $2 = a) & ($1 <> x implies ex u
    being Element of X st $1 = u & $2 = sup commute(N,u,Omega Y));
A8: Omega Y = ((the carrier of X) --> Omega Y).x;
A9: for e being Element of X ex u being Element of Omega Y st P[e,u]
    proof
      let e be Element of X;
      per cases;
      suppose
        e = x;
        hence thesis;
      end;
      suppose
A10:    e <> x;
        reconsider e as Element of X;
        take sup commute(N,e,Omega Y);
        thus thesis by A10;
      end;
    end;
    consider t being Function of the carrier of X, the carrier of Omega Y such
    that
A11: for u being Element of X holds P[u,t.u] from FUNCT_2:sch 3(A9);
    reconsider t as Function of X, Omega Y;
    reconsider tt = t as Element of L by WAYBEL24:19;
    reconsider p = "\/"(rng n,L), q = tt as Element of product ((the carrier
    of X) --> Omega Y) by YELLOW_1:def 5;
    assume
A12: for e being Element of Omega Y st e in rng m holds e <= a;
    tt is_>=_than rng n
    proof
      let s be Element of L;
      reconsider ss = s as Function of X, Omega Y by WAYBEL24:19;
      reconsider s9 = s, t9 = tt as Element of product ((the carrier of X) -->
      Omega Y) by YELLOW_1:def 5;
      assume s in rng n;
      then consider i being object such that
A13:  i in dom n and
A14:  s = n.i by FUNCT_1:def 3;
      reconsider i as Point of N by A13;
A15:  for j being Element of X holds s9.j <= t9.j
      proof
        let j be Element of X;
A17:    ss.j = (the mapping of commute(N,j,Omega Y)).i by A14,Th24;
        per cases;
        suppose
          j <> x;
          then ex u being Element of X st j = u & t.j = sup commute(N,u,Omega
          Y) by A11;
          then
A18:      t9.j = Sup the mapping of commute(N,j,Omega Y) by WAYBEL_2:def 1
            .= sup rng the mapping of commute(N,j,Omega Y);
          commute(N,j,Omega Y) is eventually-directed by A1;
          then ex_sup_of commute(N,j,Omega Y) by Th31;
          then
A19:      ex_sup_of rng the mapping of commute(N,j,Omega Y), Omega Y;
          i in dom the mapping of commute(N,j,Omega Y) by A13,Lm6;
          then ss.j in rng the mapping of commute(N,j,Omega Y) by A17,
FUNCT_1:def 3;
          hence thesis by A18,A19,YELLOW_4:1;
        end;
        suppose
A20:      j = x;
A21:      m.i in rng m by A6,FUNCT_1:def 3;
          ss.x = m.i by A14,Th24;
          then ss.x <= a by A12,A21;
          hence thesis by A11,A20;
        end;
      end;
      L = product ((the carrier of X) --> Omega Y) by YELLOW_1:def 5;
      hence s <= tt by A15,WAYBEL_3:28;
    end;
    then "\/"(rng n,L) <= tt by A4,YELLOW_0:30;
    then
A22: p <= q by YELLOW_1:def 5;
    tt.x = a by A11;
    hence thesis by A2,A8,A22,WAYBEL_3:28;
  end;
  rng m is_<=_than f.x
  proof
    let w be Element of Omega Y;
    assume w in rng m;
    then consider i being object such that
A23: i in dom m and
A24: m.i = w by FUNCT_1:def 3;
    reconsider i as Point of N by A5,A23,Lm6;
    reconsider g = n.i as Function of X, Omega Y by Lm5;
    g in rng n by A5,FUNCT_1:def 3;
    then g <= f by A2,A3,Th26,Th40;
    then ex a, b being Element of Omega Y st a = g.x & b = f.x & a <= b;
    hence w <= f.x by A24,Th24;
  end;
  hence f.x = Sup the mapping of commute(N,x,Omega Y) by A7,YELLOW_0:30
    .= sup commute(N,x,Omega Y) by WAYBEL_2:def 1;
end;
