
theorem
  for L being complete Lattice for D being Subset of L holds D is
  infimum-dense iff D% is order-generating
proof
  let L be complete Lattice;
  let D be Subset of L;
A1: now
    assume
A2: D% is order-generating;
    for a being Element of L holds ex D9 being Subset of D st a = "/\"(D9 ,L)
    proof
      let a be Element of L;
      consider Y being Subset of D% such that
A3:   a% = "/\"(Y,LattPOSet L) by A2,WAYBEL_6:15;
A4:   for x being object
holds x in Y implies x in the carrier of LattPOSet L
      proof
        let x be object;
        assume x in Y;
        then x in D%;
        hence thesis;
      end;
A5:   a% is_<=_than Y by A3,YELLOW_0:33;
      reconsider Y as Subset of LattPOSet L by A4,TARSKI:def 3;
A6:   for b being Element of L st b is_less_than %Y holds b [= a
      proof
        let b be Element of L;
        assume
A7:     b is_less_than %Y;
        for u being Element of LattPOSet L st u in Y holds b% <= u
        proof
          let u be Element of LattPOSet L;
          assume u in Y;
          then %u in {%d where d is Element of LattPOSet L : d in Y};
          then
A8:       b [= %u by A7,LATTICE3:def 16;
          (%u)% = %u by LATTICE3:def 3
            .= u by LATTICE3:def 4;
          hence thesis by A8,LATTICE3:7;
        end;
        then b% is_<=_than Y by LATTICE3:def 8;
        then b% <= a% by A3,YELLOW_0:33;
        hence thesis by LATTICE3:7;
      end;
      for x being object holds x in %Y implies x in D
      proof
        let x be object;
        assume x in %Y;
        then consider x9 being Element of LattPOSet L such that
A9:     x = %(x9) and
A10:    x9 in Y;
        x9 in D% by A10;
        then consider y being Element of L such that
A11:    x9 = y% and
A12:    y in D;
        x = y% by A9,A11,LATTICE3:def 4
          .= y by LATTICE3:def 3;
        hence thesis by A12;
      end;
      then
A13:  %Y is Subset of D by TARSKI:def 3;
      for q being Element of L st q in %Y holds a [= q
      proof
        let q be Element of L;
        assume q in %Y;
        then consider q9 being Element of LattPOSet L such that
A14:    q = %(q9) and
A15:    q9 in Y;
A16:    q9 = %(q9) by LATTICE3:def 4
          .= (%(q9))% by LATTICE3:def 3;
        a% <= q9 by A5,A15,LATTICE3:def 8;
        hence thesis by A14,A16,LATTICE3:7;
      end;
      then a is_less_than %Y by LATTICE3:def 16;
      then a = "/\"(%Y,L) by A6,LATTICE3:34;
      hence thesis by A13;
    end;
    hence D is infimum-dense;
  end;
  now
    assume
A17: D is infimum-dense;
    for a being Element of LattPOSet L ex Y being Subset of D% st a = "/\"
    (Y,LattPOSet L)
    proof
      let a be Element of LattPOSet L;
      consider D9 being Subset of D such that
A18:  %a = "/\"(D9,L) by A17;
A19:  for x being object holds x in D9 implies x in the carrier of L
      proof
        let x be object;
        assume x in D9;
        then x in D;
        hence thesis;
      end;
A20:  %a is_less_than D9 by A18,LATTICE3:34;
      reconsider D9 as Subset of L by A19,TARSKI:def 3;
A21:  for b being Element of LattPOSet L st (D9)% is_>=_than b holds b <= a
      proof
        let b be Element of LattPOSet L;
A22:    b = %b by LATTICE3:def 4
          .= (%b)% by LATTICE3:def 3;
        assume
A23:    (D9)% is_>=_than b;
        for u being Element of L st u in D9 holds %b [= u
        proof
          let u be Element of L;
          assume u in D9;
          then u% in {d% where d is Element of L : d in D9};
          then b <= u% by A23,LATTICE3:def 8;
          hence thesis by A22,LATTICE3:7;
        end;
        then %b is_less_than D9 by LATTICE3:def 16;
        then
A24:    %b [= %a by A18,LATTICE3:34;
        a = %a by LATTICE3:def 4
          .= (%a)% by LATTICE3:def 3;
        hence thesis by A22,A24,LATTICE3:7;
      end;
      for x being object holds x in (D9)% implies x in D%
      proof
        let x be object;
        assume x in (D9)%;
        then ex x9 being Element of L st x = (x9)% & x9 in D9;
        hence thesis;
      end;
      then
A25:  (D9)% is Subset of D% by TARSKI:def 3;
      for u being Element of LattPOSet L st u in (D9)% holds a <= u
      proof
        let u be Element of LattPOSet L;
A26:    (%a)% = %a by LATTICE3:def 3
          .= a by LATTICE3:def 4;
        assume u in (D9)%;
        then consider u9 being Element of L such that
A27:    u = (u9)% and
A28:    u9 in D9;
        %a [= u9 by A20,A28,LATTICE3:def 16;
        hence thesis by A27,A26,LATTICE3:7;
      end;
      then (D9)% is_>=_than a by LATTICE3:def 8;
      then a = "/\"((D9)%,LattPOSet L)by A21,YELLOW_0:33;
      hence thesis by A25;
    end;
    hence D% is order-generating by WAYBEL_6:15;
  end;
  hence thesis by A1;
end;
