
theorem Th16: :: PROPOSITION 4.12.(v)
  for L be algebraic lower-bounded sup-Semilattice for f be
Function of L,InclPoset Ids CompactSublatt L st for x be Element of L holds f.x
  = compactbelow x holds f is isomorphic
proof
  let L be algebraic lower-bounded sup-Semilattice;
  let f be Function of L,InclPoset Ids CompactSublatt L;
  assume
A1: for x be Element of L holds f.x = compactbelow x;
A2: now
    let x,y be Element of L;
    thus x <= y implies f.x <= f.y
    proof
      assume x <= y;
      then compactbelow x c= compactbelow y by Th1;
      then f.x c= compactbelow y by A1;
      then f.x c= f.y by A1;
      hence thesis by YELLOW_1:3;
    end;
    thus f.x <= f.y implies x <= y
    proof
      assume f.x <= f.y;
      then f.x c= f.y by YELLOW_1:3;
      then f.x c= compactbelow y by A1;
      then
A3:   compactbelow x c= compactbelow y by A1;
      ex_sup_of compactbelow x,L & ex_sup_of compactbelow y,L by YELLOW_0:17;
      then sup compactbelow x <= sup compactbelow y by A3,YELLOW_0:34;
      then sup compactbelow x <= y by WAYBEL_8:def 3;
      hence thesis by WAYBEL_8:def 3;
    end;
  end;
  now
    let y be object;
    assume y in the carrier of InclPoset Ids CompactSublatt L;
    then reconsider y9 = y as Ideal of CompactSublatt L by YELLOW_2:41;
    reconsider y1 = y9 as non empty Subset of L by Th3;
    reconsider y1 as non empty directed Subset of L by YELLOW_2:7;
    set x = sup y1;
    reconsider x9 = x as object;
    take x9;
    thus x9 in the carrier of L;
A4: compactbelow x c= y9
    proof
      let d be object;
      assume d in compactbelow x;
      then d in {v where v is Element of L: x >= v & v is compact} by
WAYBEL_8:def 2;
      then consider d1 be Element of L such that
A5:   d1 = d and
A6:   d1 <= x and
A7:   d1 is compact;
      reconsider d9 = d1 as Element of CompactSublatt L by A7,WAYBEL_8:def 1;
      d1 << d1 by A7,WAYBEL_3:def 2;
      then consider z be Element of L such that
A8:   z in y1 and
A9:   d1 <= z by A6,WAYBEL_3:def 1;
      reconsider z9 = z as Element of CompactSublatt L by A8;
      d9 <= z9 by A9,YELLOW_0:60;
      hence thesis by A5,A8,WAYBEL_0:def 19;
    end;
    y9 c= compactbelow x
    proof
      let d be object;
      assume
A10:  d in y9;
      then reconsider d1 = d as Element of CompactSublatt L;
      reconsider d9 = d1 as Element of L by YELLOW_0:58;
      y9 is_<=_than sup y1 by YELLOW_0:32;
      then d9 is compact & d9 <= x by A10,LATTICE3:def 9,WAYBEL_8:def 1;
      hence thesis by WAYBEL_8:4;
    end;
    hence y = compactbelow x by A4,XBOOLE_0:def 10
      .= f.x9 by A1;
  end;
  then
A11: rng f = the carrier of InclPoset Ids CompactSublatt L by FUNCT_2:10;
  now
    let x,y be Element of L;
    assume f.x = f.y;
    then f.x = compactbelow y by A1;
    then compactbelow x = compactbelow y by A1;
    then sup compactbelow x = y by WAYBEL_8:def 3;
    hence x = y by WAYBEL_8:def 3;
  end;
  then f is one-to-one by WAYBEL_1:def 1;
  hence thesis by A11,A2,WAYBEL_0:66;
end;
