
theorem :: PROPOSITION 4.12.(ii)
  for L be lower-bounded sup-Semilattice for f be Function of L,
CompactSublatt InclPoset Ids L st for x be Element of L holds f.x = downarrow x
  holds f is isomorphic
proof
  let L be lower-bounded sup-Semilattice;
  let f be Function of L, CompactSublatt InclPoset Ids L;
  assume
A1: for x be Element of L holds f.x = downarrow x;
A2: for x,y be Element of L holds x <= y iff f.x <= f.y
  proof
    let x,y be Element of L;
    reconsider fx = f.x as Element of InclPoset Ids L by YELLOW_0:58;
    reconsider fy = f.y as Element of InclPoset Ids L by YELLOW_0:58;
    thus x <= y implies f.x <= f.y
    proof
      assume x <= y;
      then downarrow x c= downarrow y by WAYBEL_0:21;
      then f.x c= downarrow y by A1;
      then fx c= fy by A1;
      then fx <= fy by YELLOW_1:3;
      hence thesis by YELLOW_0:60;
    end;
    x <= x;
    then
A3: x in downarrow x by WAYBEL_0:17;
    thus f.x <= f.y implies x <= y
    proof
      assume f.x <= f.y;
      then fx <= fy by YELLOW_0:59;
      then fx c= fy by YELLOW_1:3;
      then f.x c= downarrow y by A1;
      then downarrow x c= downarrow y by A1;
      hence thesis by A3,WAYBEL_0:17;
    end;
  end;
  now
    let y be object;
    assume
A4: y in the carrier of CompactSublatt InclPoset Ids L;
    the carrier of CompactSublatt InclPoset Ids L c= the carrier of
    InclPoset Ids L by YELLOW_0:def 13;
    then reconsider y9 = y as Element of InclPoset Ids L by A4;
    y9 is compact by A4,WAYBEL_8:def 1;
    then consider x be Element of L such that
A5: y9 = downarrow x by Th12;
    reconsider x9 = x as object;
    take x9;
    thus x9 in the carrier of L;
    thus y = f.x9 by A1,A5;
  end;
  then
A6: rng f = the carrier of CompactSublatt InclPoset Ids L by FUNCT_2:10;
  now
    let x,y be Element of L;
    assume f.x = f.y;
    then f.x = downarrow y by A1;
    then downarrow x = downarrow y by A1;
    hence x = y by WAYBEL_0:19;
  end;
  then f is one-to-one by WAYBEL_1:def 1;
  hence thesis by A6,A2,WAYBEL_0:66;
end;
