
theorem Th11: :: PROPOSITION 4.12.(i)
  for S be lower-bounded sup-Semilattice for x be Element of
  InclPoset Ids S holds x is compact iff x is principal Ideal of S
proof
  let S be lower-bounded sup-Semilattice;
  reconsider InIdS = InclPoset Ids S as CLSubFrame of BoolePoset the carrier
  of S by Th8;
  let x be Element of InclPoset Ids S;
  reconsider x9 = x as Ideal of S by YELLOW_2:41;
  thus x is compact implies x is principal Ideal of S
  proof
    assume x is compact;
    then x in the carrier of CompactSublatt InIdS by WAYBEL_8:def 1;
    then consider F be Element of BoolePoset the carrier of S such that
A1: F is finite and
A2: x = meet { Y where Y is Element of InIdS : F c= Y } and
A3: F c= x by Th7;
A4: F c= the carrier of S by WAYBEL_8:26;
    ex y be Element of S st y in x9 & y is_>=_than x9
    proof
      reconsider F9 = F as Subset of S by WAYBEL_8:26;
      reconsider F9 as Subset of S;
      reconsider y = sup F9 as Element of S;
      take y;
      now
        per cases;
        suppose
          F9 <> {};
          hence y in x9 by A1,A3,WAYBEL_0:42;
        end;
        suppose
          F9 = {};
          then y = Bottom S by YELLOW_0:def 11;
          hence y in x9 by WAYBEL_4:21;
        end;
      end;
      hence y in x9;
      now
        now
          let u be object;
          assume
A5:       u in F;
          then reconsider u9 = u as Element of S by A4;
          ex_sup_of F9,S by A1,A5,YELLOW_0:54;
          then y is_>=_than F by YELLOW_0:30;
          then u9 <= y by A5,LATTICE3:def 9;
          hence u in downarrow y by WAYBEL_0:17;
        end;
        then
A6:     F c= downarrow y;
        let b be Element of S;
        assume
A7:     b in x9;
        downarrow y is Element of InIdS by YELLOW_2:41;
        then downarrow y in { Y where Y is Element of InIdS : F c= Y } by A6;
        then b in downarrow y by A2,A7,SETFAM_1:def 1;
        hence b <= y by WAYBEL_0:17;
      end;
      hence thesis by LATTICE3:def 9;
    end;
    hence thesis by WAYBEL_0:def 21;
  end;
  thus x is principal Ideal of S implies x is compact
  proof
    assume x is principal Ideal of S;
    then consider y be Element of S such that
A8: y in x9 and
A9: y is_>=_than x9 by WAYBEL_0:def 21;
    ex F be Element of BoolePoset the carrier of S st F is finite & F c=
    x & x = meet { Y where Y is Element of InIdS : F c= Y }
    proof
      reconsider F = {y} as Element of BoolePoset the carrier of S by
WAYBEL_8:26;
      take F;
      thus F is finite;
      for v be object st v in F holds v in x by A8,TARSKI:def 1;
      hence
A10:  F c= x;
A11:  now
        let z be object;
        thus z in x implies for Z be set holds Z in { Y where Y is Element of
InIdS:  F c= Y } implies z in Z
        proof
          assume
A12:      z in x;
          then reconsider z9 = z as Element of S by YELLOW_2:42;
A13:      z9 <= y by A9,A12,LATTICE3:def 9;
          let Z be set;
          assume Z in { Y where Y is Element of InIdS : F c= Y };
          then consider Z1 be Element of InIdS such that
A14:      Z1 = Z & F c= Z1;
          Z1 is Ideal of S & y in F by TARSKI:def 1,YELLOW_2:41;
          hence thesis by A14,A13,WAYBEL_0:def 19;
        end;
        thus ( for Z be set holds Z in { Y where Y is Element of InIdS : F c=
        Y } implies z in Z ) implies z in x
        proof
          assume
A15:      for Z be set holds Z in { Y where Y is Element of InIdS : F
          c= Y } implies z in Z;
          x in { Y where Y is Element of InIdS : F c= Y } by A10;
          hence thesis by A15;
        end;
      end;
      [#]S is Element of InIdS by YELLOW_2:41;
      then [#]S in { Y where Y is Element of InIdS : F c= Y };
      hence thesis by A11,SETFAM_1:def 1;
    end;
    then x in the carrier of CompactSublatt InIdS by Th7;
    hence thesis by WAYBEL_8:def 1;
  end;
end;
