
theorem Th48:
  for L being complete Semilattice st for x being Element of L, J
  being set, f being Function of J,the carrier of L holds x "/\" Sup f = sup (x
  "/\" FinSups f) holds for x being Element of L, N being prenet of L st N is
  eventually-directed holds x "/\" sup N = sup ({x} "/\" rng netmap (N,L))
proof
  let L be complete Semilattice such that
A1: for x being Element of L, J being set for f being Function of J,the
  carrier of L holds x "/\" Sup f = sup (x "/\" FinSups f);
  let x be Element of L, N be prenet of L such that
A2: N is eventually-directed;
  reconsider R = rng netmap (N,L) as non empty directed Subset of L by A2,Th18;
  reconsider xx = {x} as non empty directed Subset of L by WAYBEL_0:5;
  set f = the mapping of N;
  set h = the mapping of FinSups f;
A3: ex_sup_of xx "/\" R,L by WAYBEL_0:75;
A4: rng the mapping of x "/\" FinSups f is_<=_than sup ({x} "/\" rng netmap
  (N,L))
  proof
    let a be Element of L;
A5: {x} "/\" rng h = {x "/\" y where y is Element of L: y in rng h} by
YELLOW_4:42;
    assume a in rng the mapping of x "/\" FinSups f;
    then a in {x} "/\" rng h by Th23;
    then consider y being Element of L such that
A6: a = x "/\" y and
A7: y in rng h by A5;
    for x being set holds ex_sup_of f.:x,L by YELLOW_0:17;
    then rng netmap(FinSups f,L) c= finsups rng f by Th24;
    then y in finsups rng f by A7;
    then consider Y being finite Subset of rng f such that
A8: y = "\/"(Y,L) and
A9: ex_sup_of Y,L;
    rng netmap (N,L) is directed by A2,Th18;
    then consider z being Element of L such that
A10: z in rng f and
A11: z is_>=_than Y by WAYBEL_0:1;
A12: x <= x;
    "\/"(Y,L) <= z by A9,A11,YELLOW_0:30;
    then
A13: x "/\" y <= x "/\" z by A8,A12,YELLOW_3:2;
    x in {x} by TARSKI:def 1;
    then x "/\" z <= sup (xx "/\" rng f) by A3,A10,YELLOW_4:1,37;
    hence a <= sup ({x} "/\" rng netmap (N,L)) by A6,A13,YELLOW_0:def 2;
  end;
  x "/\" FinSups f is eventually-directed by Th27;
  then rng netmap(x "/\" FinSups f,L) is directed by Th18;
  then ex_sup_of rng the mapping of x "/\" FinSups f,L by WAYBEL_0:75;
  then sup (x "/\" FinSups f) = "\/"(rng the mapping of x "/\" FinSups f,L) &
"\/"( rng the mapping of x "/\" FinSups f,L) <= sup ({x} "/\" rng netmap (N,L))
  by A4,YELLOW_0:def 9,YELLOW_2:def 5;
  then
A14: x "/\" Sup netmap (N,L) <= sup ({x} "/\" rng netmap (N,L)) by A1;
  ex_sup_of R,L & Sup netmap (N,L) = "\/"(rng netmap(N,L),L) by WAYBEL_0:75
,YELLOW_2:def 5;
  then sup ({x} "/\" rng netmap (N,L)) <= x "/\" Sup netmap (N,L) by A3,
YELLOW_4:53;
  hence thesis by A14,ORDERS_2:2;
end;
