
theorem Th4:
  for S, T being complete LATTICE, f being monotone Function of S,
  T holds for x being Element of S holds f.x = sup (f.:downarrow x)
proof
  let S, T be complete LATTICE, f be monotone Function of S, T;
  let x be Element of S;
A1: for b being Element of T st b is_>=_than f.:downarrow x holds f.x <= b
  proof
    let b be Element of T;
    x <= x;
    then dom f = the carrier of S & x in downarrow x by FUNCT_2:def 1
,WAYBEL_0:17;
    then
A2: f.x in f.:downarrow x by FUNCT_1:def 6;
    assume b is_>=_than f.:downarrow x;
    hence thesis by A2;
  end;
  ex_sup_of downarrow x, S & sup downarrow x = x by WAYBEL_0:34;
  then downarrow x is_<=_than x by YELLOW_0:30;
  then f.:downarrow x is_<=_than f.x by YELLOW_2:14;
  hence thesis by A1,YELLOW_0:30;
end;
