
theorem Th26:
  for S, T being complete Scott TopLattice, f being Function of S, T holds
  (( for x being Element of S holds
  f.x = "\/"({ f.w where w is Element of S : w <= x & w is compact },T ) )
  implies for x being Element of S holds
  f.x = "\/"({ f.w where w is Element of S : w << x },T) )
proof
  let S, T be complete Scott TopLattice, f be Function of S, T;
  assume
A1: for x being Element of S holds
  f.x = "\/"({ f.w where w is Element of S : w <= x & w is compact },T );
  then
A2: f is monotone by Th25;
  let x be Element of S;
A3: f.x = "\/"
  ({ f.w where w is Element of S : w <= x & w is compact },T ) by A1;
  set FW = { f.w where w is Element of S : w <= x & w is compact };
  set FX = { f.w where w is Element of S : w << x };
  set fx = f.x;
A4: FW c= FX
  proof
    let a be object;
    assume a in { f.w where w is Element of S : w <= x & w is compact };
    then consider w be Element of S such that
A5: a = f.w and
A6: w <= x and
A7: w is compact;
    w << w by A7;
    then w << x by A6,WAYBEL_3:2;
    hence thesis by A5;
  end;
A8: fx is_>=_than FX
  proof
    let b be Element of T;
    assume b in FX;
    then consider ww be Element of S such that
A9: b = f.ww and
A10: ww << x;
    ww <= x by A10,WAYBEL_3:1;
    hence thesis by A2,A9;
  end;
  for b being Element of T st b is_>=_than FX holds fx <= b
  proof
    let b be Element of T;
    assume b is_>=_than FX;
    then b is_>=_than FW by A4;
    hence thesis by A3,YELLOW_0:32;
  end;
  hence thesis by A8,YELLOW_0:30;
end;
