The Mizar article:

Noetherian Lattices

by
Christoph Schwarzweller

Received June 9, 1999

Copyright (c) 1999 Association of Mizar Users

MML identifier: LATTICE6
[ MML identifier index ]


environ

 vocabulary FINSET_1, LATTICES, LATTICE3, BOOLE, FINSEQ_1, RELAT_1, FUNCT_1,
      ORDERS_1, FILTER_1, RELAT_2, BHSP_3, WELLORD1, ARYTM_3, WAYBEL_6,
      ZF_LANG, REALSET1, BINOP_1, TARSKI, LATTICE6;
 notation TARSKI, XBOOLE_0, ZFMISC_1, SUBSET_1, XREAL_0, RELAT_1, FUNCT_1,
      NAT_1, STRUCT_0, BINOP_1, LATTICES, LATTICE3, FINSET_1, WELLORD1,
      WAYBEL_6, GROUP_1, ORDERS_1, FINSEQ_1, WELLFND1, YELLOW_0, LATTICE2,
      REALSET1;
 constructors LATTICE2, WAYBEL_1, WAYBEL_6, NAT_1, GROUP_1, WELLFND1, WELLORD1,
      BINOP_1, REALSET1, MEMBERED;
 clusters SUBSET_1, STRUCT_0, LATTICE2, LATTICE3, YELLOW_1, FINSET_1, ORDERS_1,
      INT_1, FINSEQ_1, FINSEQ_6, WAYBEL_0, KNASTER, LATTICES;
 requirements NUMERALS, BOOLE, SUBSET, ARITHM;
 definitions WELLFND1, WELLORD1;
 theorems AXIOMS, TARSKI, FUNCT_1, LATTICES, LATTICE3, STRUCT_0, ORDERS_1,
      VECTSP_8, FINSEQ_1, WAYBEL_6, FINSET_1, GROUP_1, NAT_1, INT_1, REAL_1,
      RELAT_1, YELLOW_0, WELLORD1, WELLFND1, REALSET1, XBOOLE_0;
 schemes NAT_1, WELLFND1;

begin

definition
cluster finite Lattice;
existence
 proof
 set L = BooleLatt {};
 A1: the carrier of L = bool {} by LATTICE3:def 1;
   bool {} is finite by FINSET_1:24;
 then L is finite by A1,GROUP_1:def 13;
 hence thesis;
 end;
end;

Lm1:for L being finite Lattice
for X being Subset of L holds
X is empty or
ex a being Element of LattPOSet L
st a in X & for b being Element of LattPOSet L
           st b in X & b <> a holds not(b <= a)
proof
let L be finite Lattice;
let X be Subset of L;
defpred P[Nat] means
for L being finite Lattice
for X being Subset of LattPOSet L
for p being FinSequence st rng p = X & len p = $1 holds
X is empty or
ex a being Element of LattPOSet L
st a in X & for b being Element of LattPOSet L
           st b in X & b <> a holds not(b <= a);
A1: P[0]
    proof
    let L be finite Lattice;
    let X be Subset of LattPOSet L;
    let p be FinSequence;
    assume A2: rng p = X & len p = 0;
    then Seg 0 = dom p by FINSEQ_1:def 3;
    hence thesis by A2,FINSEQ_1:4,RELAT_1:65;
    end;
A3: for k being Nat st P[k] holds P[k + 1]
    proof
    let k be Nat;
    assume
A4:  P[k];
    per cases;
    suppose A5: k = 0;
        for L being finite Lattice
      for X being Subset of LattPOSet L
      for p being FinSequence st rng p = X & len p = 1 holds
      ex a being Element of LattPOSet L
      st a in X & for b being Element of LattPOSet L
                 st b in X & b <> a holds not(b <= a)
          proof
          let L be finite Lattice;
          let X be Subset of LattPOSet L;
          let p be FinSequence;
          assume A6: rng p = X & len p = 1;
          then A7:        Seg 1 = dom p by FINSEQ_1:def 3;
          consider a being set such that A8: p.1 = a;
            1 in dom p by A7,FINSEQ_1:4,TARSKI:def 1;
          then A9: a in rng p by A8,FUNCT_1:def 5;
          then reconsider a as Element of LattPOSet L by A6;
             rng p = { a } by A7,A8,FINSEQ_1:4,FUNCT_1:14;
          then for b being Element of LattPOSet L
          st b in X & b <> a holds not(b <= a) by A6,TARSKI:def 1;
          hence thesis by A6,A9;
      end;
      hence thesis by A5;
    suppose A10: k <> 0;
        for L being finite Lattice
      for X being Subset of LattPOSet L
      for p being FinSequence st rng p = X & len p = k + 1
      holds ex a being Element of LattPOSet L
      st a in X & for b being Element of LattPOSet L
                 st b in X & b <> a holds not(b <= a)
        proof
        let L be finite Lattice;
        let X be Subset of LattPOSet L;
        let p be FinSequence;
        assume A11: rng p = X & len p = k + 1;
        set q = p|(Seg k), X' = rng q;
        A12: k <= len p by A11,NAT_1:29;
        then A13: len q = k by FINSEQ_1:21;
        A14: rng q c= rng p by FUNCT_1:76;
          for x being set holds x in X' implies x in the carrier of LattPOSet L
          proof
          let x be set;
          assume x in X';
          then x in rng p by A14;
          hence thesis by A11;
          end;
        then A15: X' is Subset of LattPOSet L by TARSKI:def 3;
          q <> {} by A10,A13,FINSEQ_1:25;
        then X' <> {} by FINSEQ_1:27;
        then consider a being Element of LattPOSet L such that
           A16: a in X' & for b being Element of LattPOSet L
                        st b in X' & b <> a holds not(b <= a) by A4,A13,A15;
        consider b being set such that A17: p.(k+1) = b;
          Seg (k+1) = dom p by A11,FINSEQ_1:def 3;
        then k + 1 in dom p by FINSEQ_1:6;
        then A18: b in rng p by A17,FUNCT_1:def 5;
        then reconsider b as Element of LattPOSet L by A11;
        per cases;
        suppose ex c being Element of LattPOSet L
                st c in X & c <= b & c <> b;
           then consider c being Element of LattPOSet L
                such that A19: c in X & c <= b & c <> b;
             for u being Element of LattPOSet L
                st u in X & u <> a holds not(u <= a)
             proof
             let u be Element of LattPOSet L;
             assume A20: u in X & u <> a;
             per cases;
             suppose u in X';
               hence thesis by A16,A20;
             suppose A21: not(u in X');
               consider n being set such that
               A22: n in dom p & p.n = u by A11,A20,FUNCT_1:def 5;
               A23: Seg(k+1) = dom p by A11,FINSEQ_1:def 3;
               reconsider n as Nat by A22;
               A24: 1 <= n & n <= k+1 by A22,A23,FINSEQ_1:3;
A25:            n = k + 1
                 proof
                 assume n <> k + 1;
                 then n < k + 1 by A24,REAL_1:def 5;
                 then n <= k by NAT_1:38;
                 then n in Seg(k) by A24,FINSEQ_1:3;
                 then A26: n in dom q by A12,FINSEQ_1:21;
                 then q.n = u by A22,FUNCT_1:70;
                 hence thesis by A21,A26,FUNCT_1:def 5;
                 end;
               assume
A27:            u <= a;
               then A28: c <= a by A17,A19,A22,A25,ORDERS_1:26;
               A29: c in X'
                    proof
                    consider n being set such that
                    A30: n in dom p & p.n = c by A11,A19,FUNCT_1:def 5;
                    A31: Seg(k+1) = dom p by A11,FINSEQ_1:def 3;
                    reconsider n as Nat by A30;
                    A32: 1 <= n & n <= k+1 by A30,A31,FINSEQ_1:3;
                    then n < k + 1 by A17,A19,A30,REAL_1:def 5;
                    then n <= k by NAT_1:38;
                    then n in Seg(k) by A32,FINSEQ_1:3;
                    then A33: n in dom q by A12,FINSEQ_1:21;
                    then q.n = c by A30,FUNCT_1:70;
                    hence thesis by A33,FUNCT_1:def 5;
                    end;
                 c <> a by A17,A19,A22,A25,A27,ORDERS_1:25;
               hence thesis by A16,A28,A29;
             end;
           hence thesis by A11,A14,A16;
        suppose not(ex c being Element of LattPOSet L
                    st c in X & c <= b & c <> b);
         then for u being Element of LattPOSet L
                st u in X & u <> b holds not(u <= b);
           hence thesis by A11,A18;
        end;
      hence thesis;
    end;
A34: for k being Nat holds P[k] from Ind(A1,A3);
  the carrier of L is finite by GROUP_1:def 13;
then reconsider X as finite Subset of L
  by FINSET_1:13;
  LattPOSet L = RelStr(#the carrier of L, LattRel L#) by LATTICE3:def 2;
then reconsider X' = X as finite Subset of LattPOSet L;
consider p being FinSequence such that
A35: rng p = X' by FINSEQ_1:73;
   dom p = Seg len p by FINSEQ_1:def 3;
hence thesis by A34,A35;
end;

Lm2:for L being finite Lattice
for X being Subset of L holds
X is empty or
ex a being Element of LattPOSet L
st a in X & for b being Element of LattPOSet L
           st b in X & b <> a holds not(a <= b)
proof
let L be finite Lattice;
let X be Subset of L;
defpred P[Nat] means
for L being finite Lattice
for X being Subset of LattPOSet L
for p being FinSequence st rng p = X & len p = $1 holds
X is empty or
ex a being Element of LattPOSet L
st a in X & for b being Element of LattPOSet L
           st b in X & b <> a holds not(a <= b);
A1: P[0]
    proof
    let L be finite Lattice;
    let X be Subset of LattPOSet L;
    let p be FinSequence;
    assume A2: rng p = X & len p = 0;
    then Seg 0 = dom p by FINSEQ_1:def 3;
    hence thesis by A2,FINSEQ_1:4,RELAT_1:65;
    end;
A3: for k being Nat st P[k] holds P[k + 1]
    proof
    let k be Nat;
    assume
A4:  P[k];
    per cases;
    suppose A5: k = 0;
        for L being finite Lattice
      for X being Subset of LattPOSet L
      for p being FinSequence st rng p = X & len p = 1 holds
      ex a being Element of LattPOSet L
      st a in X & for b being Element of LattPOSet L
                 st b in X & b <> a holds not(a <= b)
          proof
          let L be finite Lattice;
          let X be Subset of LattPOSet L;
          let p be FinSequence;
          assume A6: rng p = X & len p = 1;
          then A7: dom p = { 1 } by FINSEQ_1:4,def 3;
          consider a being set such that A8: p.1 = a;
            1 in dom p by A7,TARSKI:def 1;
          then A9: a in rng p by A8,FUNCT_1:def 5;
          then reconsider a as Element of LattPOSet L by A6;
             rng p = { a } by A7,A8,FUNCT_1:14;
          then for b being Element of LattPOSet L
          st b in X & b <> a holds not(a <= b) by A6,TARSKI:def 1;
          hence thesis by A6,A9;
      end;
      hence thesis by A5;
    suppose A10: k <> 0;
        for L being finite Lattice
      for X being Subset of LattPOSet L
      for p being FinSequence st rng p = X & len p = k + 1
      holds ex a being Element of LattPOSet L
      st a in X & for b being Element of LattPOSet L
                 st b in X & b <> a holds not(a <= b)
        proof
        let L be finite Lattice;
        let X be Subset of LattPOSet L;
        let p be FinSequence;
        assume A11: rng p = X & len p = k + 1;
        set q = p|(Seg k), X' = rng q;
        A12: k <= len p by A11,NAT_1:29;
        then A13: len q = k by FINSEQ_1:21;
        A14: rng q c= rng p by FUNCT_1:76;
          for x being set holds x in X' implies x in the carrier of LattPOSet L
          proof
          let x be set;
          assume x in X';
          then x in rng p by A14;
          hence thesis by A11;
          end;
        then A15: X' is Subset of LattPOSet L by TARSKI:def 3;
          q <> {} by A10,A13,FINSEQ_1:25;
        then X' <> {} by FINSEQ_1:27;
        then consider a being Element of LattPOSet L such that
           A16: a in X' & for b being Element of LattPOSet L
                        st b in X' & b <> a holds not(a <= b) by A4,A13,A15;
        consider b being set such that A17: p.(k+1) = b;
          Seg (k+1) = dom p by A11,FINSEQ_1:def 3;
        then k + 1 in dom p by FINSEQ_1:6;
        then A18: b in rng p by A17,FUNCT_1:def 5;
        then reconsider b as Element of LattPOSet L by A11;
        per cases;
        suppose ex c being Element of LattPOSet L
                st c in X & b <= c & c <> b;
           then consider c being Element of LattPOSet L
                such that A19: c in X & b <= c & c <> b;
             for u being Element of LattPOSet L
                st u in X & u <> a holds not(a <= u)
             proof
             let u be Element of LattPOSet L;
             assume A20: u in X & u <> a;
               now per cases;
             case u in X';
               hence thesis by A16,A20;
             case A21: not(u in X');
               consider n being set such that
               A22: n in dom p & p.n = u by A11,A20,FUNCT_1:def 5;
               A23: Seg(k+1) = dom p by A11,FINSEQ_1:def 3;
               reconsider n as Nat by A22;
               A24: 1 <= n & n <= k+1 by A22,A23,FINSEQ_1:3;
A25:            n = k + 1
                 proof
                 assume n <> k + 1;
                 then n < k + 1 by A24,REAL_1:def 5;
                 then n <= k by NAT_1:38;
                 then n in Seg(k) by A24,FINSEQ_1:3;
                 then A26: n in dom q by A12,FINSEQ_1:21;
                 then q.n = u by A22,FUNCT_1:70;
                 hence thesis by A21,A26,FUNCT_1:def 5;
                 end;
               assume
A27:            a <= u;
               then A28: a <= c by A17,A19,A22,A25,ORDERS_1:26;
               A29: c in X'
                    proof
                    consider n being set such that
                    A30: n in dom p & p.n = c by A11,A19,FUNCT_1:def 5;
                    A31: Seg(k+1) = dom p by A11,FINSEQ_1:def 3;
                    reconsider n as Nat by A30;
                    A32: 1 <= n & n <= k+1 by A30,A31,FINSEQ_1:3;
                    then n < k + 1 by A17,A19,A30,REAL_1:def 5;
                    then n <= k by NAT_1:38;
                    then n in Seg(k) by A32,FINSEQ_1:3;
                    then A33: n in dom q by A12,FINSEQ_1:21;
                    then q.n = c by A30,FUNCT_1:70;
                    hence thesis by A33,FUNCT_1:def 5;
                    end;
                 c <> a by A17,A19,A22,A25,A27,ORDERS_1:25;
               hence thesis by A16,A28,A29;
             end;  :: cases
             hence thesis;
             end;
           hence thesis by A11,A14,A16;
        suppose not(ex c being Element of LattPOSet L
                    st c in X & b <= c & c <> b);
         then for u being Element of LattPOSet L
                st u in X & u <> b holds not(b <= u);
           hence thesis by A11,A18;
    end;
    hence thesis;
    end;
A34: for k being Nat holds P[k] from Ind(A1,A3);
  the carrier of L is finite by GROUP_1:def 13;
then reconsider X as finite Subset of L
  by FINSET_1:13;
  LattPOSet L = RelStr(#the carrier of L, LattRel L#) by LATTICE3:def 2;
then reconsider X' = X as finite Subset of LattPOSet L;
consider p being FinSequence such that
A35: rng p = X' by FINSEQ_1:73;
  dom p = Seg len p by FINSEQ_1:def 3;
hence thesis by A34,A35;
end;

Lm3:for L being finite Lattice
for X being Subset of L holds
X is empty or
ex a being Element of LattPOSet L
st for b being Element of LattPOSet L
   st b in X holds b <= a
proof
let L be finite Lattice;
let X be Subset of L;
defpred P[Nat] means
for L being finite Lattice
for X being Subset of LattPOSet L
for p being FinSequence st rng p = X & len p = $1 holds
X is empty or
ex a being Element of LattPOSet L
st for b being Element of LattPOSet L
   st b in X holds b <= a;
A1: P[0]
    proof
    let L be finite Lattice;
    let X be Subset of LattPOSet L;
    let p be FinSequence;
    assume A2: rng p = X & len p = 0;
    then dom p = {} by FINSEQ_1:4,def 3;
    hence thesis by A2,RELAT_1:65;
    end;
A3: for k being Nat st P[k] holds P[k + 1]
    proof
    let k be Nat;
    assume
A4:  P[k];
    per cases;
    suppose
A5:   k = 0;
        for L being finite Lattice
      for X being Subset of LattPOSet L
      for p being FinSequence st rng p = X & len p = 1 holds
      ex a being Element of LattPOSet L
      st for b being Element of LattPOSet L
         st b in X holds b <= a
          proof
          let L be finite Lattice;
          let X be Subset of LattPOSet L;
          let p be FinSequence;
          assume A6: rng p = X & len p = 1;
          then A7:        Seg 1 = dom p by FINSEQ_1:def 3;
          consider a being set such that A8: p.1 = a;
            1 in dom p by A7,FINSEQ_1:4,TARSKI:def 1;
          then a in rng p by A8,FUNCT_1:def 5;
          then reconsider a as Element of LattPOSet L by A6;
            rng p = { a } by A7,A8,FINSEQ_1:4,FUNCT_1:14;
          then for b being Element of LattPOSet L
          st b in X holds b <= a by A6,TARSKI:def 1;
          hence thesis;
      end;
      hence thesis by A5;
    suppose A9: k <> 0;
        for L being finite Lattice
      for X being Subset of LattPOSet L
      for p being FinSequence st rng p = X & len p = k + 1
      holds ex a being Element of LattPOSet L
      st for b being Element of LattPOSet L
        st b in X holds b <= a
        proof
        let L be finite Lattice;
        let X be Subset of LattPOSet L;
        let p be FinSequence;
        assume A10: rng p = X & len p = k + 1;
        set q = p|(Seg k), X' = rng q;
        A11: k <= k + 1 by NAT_1:29;
        then A12: len q = k by A10,FINSEQ_1:21;
          for x being set holds x in X' implies x in the carrier of LattPOSet L
          proof
          let x be set;
          assume
A13:         x in X';
            rng q c= rng p by FUNCT_1:76;
          then x in rng p by A13;
          hence thesis by A10;
          end;
        then A14: X' is Subset of LattPOSet L by TARSKI:def 3;
          q <> {} by A9,A12,FINSEQ_1:25;
        then X' <> {} by FINSEQ_1:27;
        then consider a being Element of LattPOSet L
             such that A15: for b being Element of LattPOSet L
                           st b in X' holds b <= a by A4,A12,A14;
        consider b being set such that A16: p.(k+1) = b;
          Seg (k+1) = dom p by A10,FINSEQ_1:def 3;
        then k + 1 in dom p by FINSEQ_1:6;
        then b in rng p by A16,FUNCT_1:def 5;
        then reconsider b as Element of LattPOSet L by A10;
        set a2 = %a "\/" %b;
        A17: %a "\/" a2 = (%a "\/" %a) "\/" %b by LATTICES:def 5
                    .= %a "\/" %b by LATTICES:17;
          %b "\/" a2 = (%b "\/" %b) "\/" %a by LATTICES:def 5
                .= %b "\/" %a by LATTICES:17;
        then A18:      %a [= a2 & %b [= a2 by A17,LATTICES:def 3;
        A19: (%a)% = %a by LATTICE3:def 3
             .= a by LATTICE3:def 4;
A20:    (%b)% = %b by LATTICE3:def 3
             .= b by LATTICE3:def 4;
          for c being Element of LattPOSet L
        st c in X holds c <= a2%
          proof
          let c be Element of LattPOSet L;
          assume A21: c in X;
          per cases;
          suppose
          c in X';
            then a <= a2% & c <= a by A15,A18,A19,LATTICE3:7;
            hence thesis by ORDERS_1:26;
          suppose A22: not(c in X');
            consider n being set such that
            A23: n in dom p & c = p.n by A10,A21,FUNCT_1:def 5;
            reconsider n as Nat by A23;
            A24: n in Seg(k + 1) by A10,A23,FINSEQ_1:def 3;
            then A25: 1 <= n & n <= k + 1 by FINSEQ_1:3;
              n >= k + 1
              proof
              assume not(n >= k + 1);
              then 1 <= n & n <= k by A24,FINSEQ_1:3,INT_1:20;
              then A26: n in Seg(k) by FINSEQ_1:3;
              then A27: n in dom q by A10,A11,FINSEQ_1:21;
                 dom(p|(Seg k))
                 = dom p /\ Seg(k) by RELAT_1:90
                .= Seg(k+1) /\ Seg(k) by A10,FINSEQ_1:def 3
                .= Seg(k) by A11,FINSEQ_1:9;
              then q.n = c by A23,A26,FUNCT_1:70;
              hence thesis by A22,A27,FUNCT_1:def 5;
              end;
            then c = b by A16,A23,A25,AXIOMS:21;
            hence thesis by A18,A20,LATTICE3:7;
          end;
        hence thesis;
    end;
    hence thesis;
    end;
A28: for k being Nat holds P[k] from Ind(A1,A3);
  the carrier of L is finite by GROUP_1:def 13;
then reconsider X as finite Subset of L
  by FINSET_1:13;
  LattPOSet L = RelStr(#the carrier of L, LattRel L#) by LATTICE3:def 2;
then reconsider X' = X as finite Subset of LattPOSet L;
consider p being FinSequence such that
A29: rng p = X' by FINSEQ_1:73;
   dom p = Seg len p by FINSEQ_1:def 3;
hence thesis by A28,A29;
end;
Lm4:for L being finite Lattice holds
ex a being Element of LattPOSet L
st for b being Element of LattPOSet L holds b <= a
proof
let L be finite Lattice;
  the carrier of L c= the carrier of L;
then reconsider L' = the carrier of L as Subset of L;
consider a being Element of LattPOSet L such that
A1: for b being Element of LattPOSet L
   st b in L' holds b <= a by Lm3;
  for b being Element of LattPOSet L holds b <= a
   proof
   let b be Element of LattPOSet L;
     LattPOSet L = RelStr(#the carrier of L, LattRel L#) by LATTICE3:def 2;
   hence thesis by A1;
   end;
hence thesis;
end;

Lm5:for L being finite Lattice holds L is complete
proof
let L be finite Lattice;
  for X being Subset of L
ex a being Element of L
st a is_less_than X &
   for b being Element of L
   st b is_less_than X holds b [= a
  proof
  let X be Subset of L;
  defpred P[Nat] means
  for X' being finite Subset of LattPOSet L
  for p being FinSequence st rng p = X' & len p = $1 holds
  ex a being Element of LattPOSet L st
  (for x being Element of LattPOSet L
   st x in X' holds a <= x) &
  (for x' being Element of LattPOSet L
   st for x being Element of LattPOSet L
      st x in X' holds x' <= x holds x' <= a);
  A1: P[0]
      proof
        for X' being finite Subset of LattPOSet L
      for p being FinSequence st rng p = X' & len p = 0 holds
      ex a being Element of LattPOSet L st
      (for x being Element of LattPOSet L
       st x in X' holds a <= x) &
      (for x' being Element of LattPOSet L
       st for x being Element of LattPOSet L
          st x in X' holds x' <= x holds x' <= a)
       proof
       let X' be finite Subset of LattPOSet L;
       let p be FinSequence;
       assume A2: rng p = X' & len p = 0;
       then A3:     dom p = {} by FINSEQ_1:4,def 3;
         ex a being Element of LattPOSet L st
       (for x being Element of LattPOSet L
        st x in X' holds a <= x) &
       (for x' being Element of LattPOSet L
        st for x being Element of LattPOSet L
           st x in X' holds x' <= x holds x' <= a)
         proof
         consider a being Element of LattPOSet L such that
         A4: for b being Element of LattPOSet L
            holds b <= a by Lm4;
         A5: for x being Element of LattPOSet L
            st x in X' holds a <= x by A2,A3,RELAT_1:65;
           for x' being Element of LattPOSet L
         st for x being Element of LattPOSet L
            st x in X' holds x' <= x holds x' <= a by A4;
         hence thesis by A5;
         end;
       hence thesis;
       end;
      hence thesis;
      end;
  A6: for k being Nat st P[k] holds P[k + 1]
    proof
    let k be Nat;
    assume
A7:  P[k];
      for X' being finite Subset of LattPOSet L
    for p being FinSequence st rng p = X' & len p = k + 1 holds
    ex a being Element of LattPOSet L st
    (for x being Element of LattPOSet L
     st x in X' holds a <= x) &
    (for x' being Element of LattPOSet L
     st for x being Element of LattPOSet L
        st x in X' holds x' <= x holds x' <= a)
      proof
      let X be finite Subset of LattPOSet L;
      let p be FinSequence;
      assume A8: rng p = X & len p = k + 1;
      set q = p|(Seg k), X' = rng q;
      A9: k <= k + 1 by NAT_1:29;
      then A10: len q = k by A8,FINSEQ_1:21;
        for x being set holds x in X' implies x in the carrier of LattPOSet L
        proof
        let x be set;
        assume
A11:       x in X';
          rng q c= rng p by FUNCT_1:76;
        then x in rng p by A11;
        hence thesis by A8;
        end;
      then X' is Subset of LattPOSet L by TARSKI:def 3;
      then consider a being Element of LattPOSet L such that
      A12: (for x being Element of LattPOSet L
           st x in X' holds a <= x) &
          (for x' being Element of LattPOSet L
           st for x being Element of LattPOSet L
              st x in X' holds x' <= x holds x' <= a) by A7,A10;
      consider b being set such that A13: p.(k+1) = b;
        Seg (k+1) = dom p by A8,FINSEQ_1:def 3;
      then k + 1 in dom p by FINSEQ_1:6;
      then A14: b in rng p by A13,FUNCT_1:def 5;
      then reconsider b as Element of LattPOSet L by A8;
      set a2 = %a "/\" %b;
      A15: a2 "\/" %a = %a by LATTICES:def 8;
        a2 "\/" %b = %b by LATTICES:def 8;
      then A16: a2 [= %a & a2 [= %b by A15,LATTICES:def 3;
      A17: (%a)% = %a by LATTICE3:def 3
           .= a by LATTICE3:def 4;
A18:  (%b)% = %b by LATTICE3:def 3
           .= b by LATTICE3:def 4;
      A19: for x being Element of LattPOSet L
           st x in X holds a2% <= x
          proof
          let c be Element of LattPOSet L;
          assume A20: c in X;
          per cases;
          suppose
          c in X';
            then a2% <= a & a <= c by A12,A16,A17,LATTICE3:7;
            hence thesis by ORDERS_1:26;
          suppose A21: not c in X';
            consider n being set such that
            A22: n in dom p & c = p.n by A8,A20,FUNCT_1:def 5;
            reconsider n as Nat by A22;
            A23: n in Seg(k + 1) by A8,A22,FINSEQ_1:def 3;
            then A24: 1 <= n & n <= k + 1 by FINSEQ_1:3;
              n >= k + 1
              proof
              assume not(n >= k + 1);
              then 1 <= n & n <= k by A23,FINSEQ_1:3,INT_1:20;
              then A25: n in Seg(k) by FINSEQ_1:3;
              then A26: n in dom q by A8,A9,FINSEQ_1:21;
                  dom(p|(Seg k))
                 = dom p /\ Seg(k) by RELAT_1:90
                .= Seg(k+1) /\ Seg(k) by A8,FINSEQ_1:def 3
                .= Seg(k) by A9,FINSEQ_1:9;
              then q.n = c by A22,A25,FUNCT_1:70;
              hence thesis by A21,A26,FUNCT_1:def 5;
              end;
            then c = b by A13,A22,A24,AXIOMS:21;
            hence thesis by A16,A18,LATTICE3:7;
          end;
        for x' being Element of LattPOSet L
      st for x being Element of LattPOSet L
         st x in X holds x' <= x holds x' <= a2%
        proof
        let x' be Element of LattPOSet L;
        assume A27: for x being Element of LattPOSet L
                   st x in X holds x' <= x;
          for x being Element of LattPOSet L
        st x in X' holds x' <= x
          proof
          let x be Element of LattPOSet L;
            rng q c= rng p by FUNCT_1:76;
          hence thesis by A8,A27;
          end;
        then A28: x' <= a by A12;
        A29: x' <= b by A8,A14,A27;
        A30: (%a)% = %a by LATTICE3:def 3
                 .= a by LATTICE3:def 4;
        A31: (%b)% = %b by LATTICE3:def 3
                 .= b by LATTICE3:def 4;
        A32: (%(x'))% = %(x') by LATTICE3:def 3
                    .= x' by LATTICE3:def 4;
        then A33:      %(x') [= %a & %(x') [= %b by A28,A29,A30,A31,LATTICE3:7;
          %(x') "/\" a2 = (%(x') "/\" %a) "/\" %b by LATTICES:def 7
                   .= %(x') "/\" %b by A33,LATTICES:21
                   .= %(x') by A33,LATTICES:21;
        then %(x') [= a2 by LATTICES:21;
        hence thesis by A32,LATTICE3:7;
        end;
      hence thesis by A19;
      end;
    hence thesis;
    end;
  A34: for k being Nat holds P[k] from Ind(A1,A6);
    the carrier of L is finite by GROUP_1:def 13;
  then reconsider X as finite Subset of L
    by FINSET_1:13;
    LattPOSet L = RelStr(#the carrier of L, LattRel L#) by LATTICE3:def 2;
  then reconsider X' = X as finite Subset of LattPOSet L;
  consider p being FinSequence such that
  A35: rng p = X' by FINSEQ_1:73;
    dom p = Seg len p by FINSEQ_1:def 3;
  then consider a being Element of LattPOSet L such that
  A36: (for x being Element of LattPOSet L
      st x in X' holds a <= x) &
     (for x' being Element of LattPOSet L
      st for x being Element of LattPOSet L
         st x in X' holds x' <= x holds x' <= a) by A34,A35;
    for x being Element of L st x in X holds %a [= x
    proof
    let x be Element of L;
    assume x in X;
    then consider x' being Element of LattPOSet L
    such that A37: x' = x & x' in X';
    A38: a <= x' by A36,A37;
    A39: (%a)% = %a by LATTICE3:def 3
         .= a by LATTICE3:def 4;
      x' = x% by A37,LATTICE3:def 3;
    hence thesis by A38,A39,LATTICE3:7;
    end;
  then A40: %a is_less_than X by LATTICE3:def 16;
    for b being Element of L
  st b is_less_than X holds b [= %a
    proof
    let b be Element of L;
    assume
A41:   b is_less_than X;
      for x being Element of LattPOSet L st x in X'
    holds b% <= x
      proof
      let x be Element of LattPOSet L;
      assume x in X';
      then consider x' being Element of L such that
      A42: x' = x & x' in X;
        b [= x' by A41,A42,LATTICE3:def 16;
      then b% <= (x')% by LATTICE3:7;
      hence thesis by A42,LATTICE3:def 3;
      end;
    then A43: b% <= a by A36;
      (%a)% = %a by LATTICE3:def 3
         .= a by LATTICE3:def 4;
    hence thesis by A43,LATTICE3:7;
    end;
  hence thesis by A40;
  end;
hence thesis by VECTSP_8:def 6;
end;

definition
cluster finite -> complete Lattice;
coherence by Lm5;
end;

definition
let L be Lattice;
let D be Subset of L;
func D% -> Subset of LattPOSet L equals :Def1:
 {d% where d is Element of L : d in D};
coherence
 proof
 set M' = {d% where d is Element of L : d in D};
   for x being set holds x in M' implies x in the carrier of LattPOSet L
   proof
   let x be set;
   assume x in M';
   then ex x' being Element of L st x = x'% & x' in D;
   hence thesis;
   end;
 then M' c= the carrier of (LattPOSet L) by TARSKI:def 3;
 hence thesis;
 end;
end;

definition
let L be Lattice;
let D be Subset of LattPOSet L;
func %D -> Subset of L equals :Def2:
 {%d where d is Element of LattPOSet L : d in D};
coherence
 proof
 set M' = {%d where d is Element of LattPOSet L : d in D};
   for x being set holds x in M' implies x in the carrier of L
   proof
   let x be set;
   assume x in M';
   then ex x' being Element of LattPOSet L st x = %x' & x' in D;
   hence thesis;
   end;
   then M' c= the carrier of L by TARSKI:def 3;
 hence thesis;
 end;
end;

definition let L be finite Lattice;
cluster LattPOSet L -> well_founded;
 coherence
  proof set R = LattPOSet L;
  let Y be set; assume A1: Y c= the carrier of R & Y <> {};
     LattPOSet L = RelStr(#the carrier of L, LattRel L#) by LATTICE3:def 2;
  then reconsider Y as Subset of L by A1;
  consider a being Element of LattPOSet L such that
  A2: a in Y & for b being Element of LattPOSet L
              st b in Y & b <> a holds not(b <= a) by A1,Lm1;
  A3: not(ex x being Element of R
          st x <> a & [x,a] in the InternalRel of R & x in Y)
      proof
      given x being Element of R such that
      A4: x <> a & [x,a] in the InternalRel of R & x in Y;
        x <= a by A4,ORDERS_1:def 9;
      hence thesis by A2,A4;
      end;
    (the InternalRel of R)-Seg(a) /\ Y = {}
     proof
     assume A5: (the InternalRel of R)-Seg(a) /\ Y <> {};
     consider z being Element of (the InternalRel of R)-Seg(a) /\ Y;
     A6: z in (the InternalRel of R)-Seg(a) & z in Y by A5,XBOOLE_0:def 3;
     then A7: z <> a & [z,a] in (the InternalRel of R) by WELLORD1:def 1;
       z is Element of R by A1,A6;
     hence thesis by A3,A6,A7;
     end;
   then (the InternalRel of R)-Seg(a) misses Y by XBOOLE_0:def 7;
   hence thesis by A2;
  end;
end;

Lm6:for L being finite Lattice holds (LattPOSet L)~ is well_founded
proof
let L be finite Lattice;
set R = LattPOSet L;
A1:(LattPOSet L)~ =
RelStr(#the carrier of R, (the InternalRel of R)~#) by LATTICE3:def 5;
  for Y being set st Y c= the carrier of R~ & Y <> {}
ex a being set st a in Y & (the InternalRel of R~)-Seg(a) misses Y
  proof
  let Y be set; assume A2: Y c= the carrier of R~ & Y <> {};
     LattPOSet L = RelStr(#the carrier of L, LattRel L#) by LATTICE3:def 2;
  then reconsider Y as Subset of L by A1,A2;
  consider a being Element of R such that
  A3: a in Y & for b being Element of LattPOSet L
              st b in Y & b <> a holds not(a <= b) by A2,Lm2;
  reconsider a as Element of R;
  reconsider a' = a as Element of R~ by A1;
  A4: for b being Element of (LattPOSet L)~
      st b in Y & b <> a holds not(b <= a')
      proof
      let b be Element of (LattPOSet L)~;
      assume A5: b in Y & b <> a;
      reconsider b' = b as Element of R by A1;
A6:      not(a <= b') by A3,A5;
        a~ = a' & (b')~ = b by LATTICE3:def 6;
      hence thesis by A6,LATTICE3:9;
      end;
  A7: not(ex x being Element of R~
          st x <> a & [x,a] in the InternalRel of R~ & x in Y)
      proof
      given x being Element of R~ such that
      A8: x <> a & [x,a] in the InternalRel of R~ & x in Y;
        x <= a' by A8,ORDERS_1:def 9;
      hence thesis by A4,A8;
      end;
    (the InternalRel of R~)-Seg(a) /\ Y = {}
     proof
     assume A9: (the InternalRel of R~)-Seg(a) /\ Y <> {};
     consider z being Element of (the InternalRel of R~)-Seg(a) /\ Y;
     A10: z in (the InternalRel of R~)-Seg(a) & z in Y by A9,XBOOLE_0:def 3;
     then A11: z <> a & [z,a] in (the InternalRel of R~) by WELLORD1:def 1;
       z is Element of R~ by A2,A10;
     hence thesis by A7,A10,A11;
     end;
     then (the InternalRel of R~)-Seg(a) misses Y by XBOOLE_0:def 7;
  hence thesis by A3;
  end;
then the InternalRel of R~ is_well_founded_in the carrier of R~ by WELLORD1:def
3;
hence thesis by WELLFND1:def 2;
end;

definition
let L be Lattice;
attr L is noetherian means :Def3:
  LattPOSet L is well_founded;
attr L is co-noetherian means :Def4:
  (LattPOSet L)~ is well_founded;
end;

definition
cluster noetherian upper-bounded lower-bounded complete Lattice;
existence
 proof
 consider L being finite Lattice;
 take L;
    LattPOSet L is well_founded;
 hence thesis by Def3;
 end;
end;

definition
cluster co-noetherian upper-bounded lower-bounded complete Lattice;
existence
 proof
 consider L being finite Lattice;
 take L;
    (LattPOSet L)~ is well_founded by Lm6;
 hence thesis by Def4;
 end;
end;

theorem
  for L being Lattice holds L is noetherian iff L.: is co-noetherian
proof
let L be Lattice;
set R = LattPOSet L;
set Ri = (LattPOSet L)~;
A1: now assume L is noetherian;
    then R is well_founded by Def3;
   then A2: (Ri)~ is well_founded by LATTICE3:8;
     (LattPOSet L)~ = LattPOSet (L.:) by LATTICE3:20;
   hence L.: is co-noetherian by A2,Def4;
   end;
  now assume A3: L.: is co-noetherian;
     (LattPOSet L)~ = LattPOSet (L.:) by LATTICE3:20;
   then (Ri)~ is well_founded by A3,Def4;
   then R is well_founded by LATTICE3:8;
   hence L is noetherian by Def3;
   end;
hence thesis by A1;
end;

Lm7:for L being finite Lattice holds
L is noetherian & L is co-noetherian
proof
let L be finite Lattice;
   LattPOSet L is well_founded & (LattPOSet L)~ is well_founded by Lm6;
hence thesis by Def3,Def4;
end;

definition
cluster finite -> noetherian Lattice;
coherence by Lm7;
cluster finite -> co-noetherian Lattice;
coherence by Lm7;
end;

definition
let L be Lattice;
let a,b be Element of L;
pred a is-upper-neighbour-of b means :Def5:
 a <> b & b [= a &
 for c being Element of L holds
 (b [= c & c [= a) implies (c = a or c = b);
synonym b is-lower-neighbour-of a;
end;

theorem
Th2:for L being Lattice
for a being Element of L
for b,c being Element of L st b <> c holds
((b is-upper-neighbour-of a & c is-upper-neighbour-of a)
 implies a = c "/\" b) &
((b is-lower-neighbour-of a & c is-lower-neighbour-of a)
 implies a = c "\/" b)
proof
let L be Lattice;
let a be Element of L;
let b,c be Element of L;
assume A1: b <> c;
A2: now assume A3: b is-upper-neighbour-of a & c is-upper-neighbour-of a;
   then A4: a [= b & a [= c by Def5;
     a = a "/\" a by LATTICES:18
    .= (b "/\" a) "/\" a by A4,LATTICES:21
    .= (b "/\" a) "/\" (c "/\" a) by A4,LATTICES:21
    .= b "/\" (a "/\" (c "/\" a)) by LATTICES:def 7
    .= b "/\" ((a "/\" a) "/\" c) by LATTICES:def 7
    .= b "/\" (a "/\" c) by LATTICES:18
    .= (b "/\" c) "/\" a by LATTICES:def 7;
   then A5: a [= b "/\" c by LATTICES:21;
A6: b "/\" c [= c by LATTICES:23;
     not(b "/\" c = c)
     proof
     assume b "/\" c = c;
     then c [= b by LATTICES:21;
     then c = a or c = b by A3,A4,Def5;
     hence contradiction by A1,A3,Def5;
     end;
   hence b "/\" c = a by A3,A5,A6,Def5;
   end;
  now assume A7: b is-lower-neighbour-of a & c is-lower-neighbour-of a;
   then A8: b [= a & c [= a by Def5;
     a = a "\/" a by LATTICES:17
    .= (b "\/" a) "\/" a by A8,LATTICES:def 3
    .= (b "\/" a) "\/" (c "\/" a) by A8,LATTICES:def 3
    .= b "\/" (a "\/" (a "\/" c)) by LATTICES:def 5
    .= b "\/" ((a "\/" a) "\/" c) by LATTICES:def 5
    .= b "\/" (a "\/" c) by LATTICES:17
    .= (b "\/" c) "\/" a by LATTICES:def 5;
   then A9: b "\/" c [= a by LATTICES:def 3;
A10:  c [= b "\/" c by LATTICES:22;
     not(b "\/" c = c)
     proof
     assume b "\/" c = c;
     then b [= c by LATTICES:def 3;
     then c = a or c = b by A7,A8,Def5;
     hence contradiction by A1,A7,Def5;
     end;
   hence b "\/" c = a by A7,A9,A10,Def5;
   end;
hence thesis by A2;
end;

theorem
Th3:for L being noetherian Lattice
for a being Element of L
for d being Element of L st a [= d & a <> d holds
ex c being Element of L
st c [= d & c is-upper-neighbour-of a
proof
let L be noetherian Lattice;
let a be Element of L;
let d be Element of L;
assume A1: a [= d & a <> d;
defpred P[Element of LattPOSet L] means
   a [= %($1) & a <> %($1) implies
   ex c being Element of L st
   c [= %($1) & c is-upper-neighbour-of a;
A2: LattPOSet L is well_founded by Def3;
A3: for x being Element of LattPOSet L
      st for y being Element of LattPOSet L
          st y <> x & [y,x] in the InternalRel of LattPOSet L holds P[y]
       holds P[x]
    proof
    let x be Element of LattPOSet L;
    assume A4: for y being Element of LattPOSet L st y <> x &
               [y,x] in the InternalRel of LattPOSet L holds P[y];
      a [= %x & a <> %x implies
    ex c being Element of L st
    c [= %x & c is-upper-neighbour-of a
      proof
      assume A5: a [= %x & a <> %x;
      A6: (%x)% = %x by LATTICE3:def 3 .= x by LATTICE3:def 4;
      per cases;
      suppose %x is-upper-neighbour-of a;
        hence thesis;
      suppose not(%x is-upper-neighbour-of a);
        then consider c being Element of L such that
        A7: a [= c & c [= %x & not(c = %x or c = a) by A5,Def5;
        A8: %(c%) = c% by LATTICE3:def 4 .= c by LATTICE3:def 3;
          c% <= x by A6,A7,LATTICE3:7;
        then [c%,x] in the InternalRel of LattPOSet L by ORDERS_1:def 9;
        then consider c' being Element of L such that
        A9: c' [= c & c' is-upper-neighbour-of a by A4,A7,A8;
          c' [= %x by A7,A9,LATTICES:25;
        hence thesis by A9;
      end;
    hence thesis;
    end;
A10: for x being Element of LattPOSet L holds P[x] from WFInduction(A3,A2);
  %(d%) = d% by LATTICE3:def 4 .= d by LATTICE3:def 3;
hence thesis by A1,A10;
end;

theorem
Th4:for L being co-noetherian Lattice
for a being Element of L
for d being Element of L st d [= a & a <> d holds
ex c being Element of L
st d [= c & c is-lower-neighbour-of a
proof
let L be co-noetherian Lattice;
let a be Element of L;
let d be Element of L;
assume A1: d [= a & a <> d;
defpred P[Element of (LattPOSet L)~] means
   %(~($1)) [= a & a <> %(~($1)) implies
   ex c being Element of L st
   %(~($1)) [= c & c is-lower-neighbour-of a;
A2: (LattPOSet L)~ is well_founded by Def4;
A3: for x being Element of (LattPOSet L)~
      st for y being Element of (LattPOSet L)~
          st y <> x & [y,x] in the InternalRel of (LattPOSet L)~ holds P[y]
       holds P[x]
    proof
    let x be Element of (LattPOSet L)~;
    assume A4: for y being Element of (LattPOSet L)~ st y <> x &
               [y,x] in the InternalRel of (LattPOSet L)~ holds P[y];
      %(~x) [= a & a <> %(~x) implies
    ex c being Element of L st
    %(~x) [= c & c is-lower-neighbour-of a
      proof
      assume A5: %(~x) [= a & a <> %(~x);
      A6: (%(~x))% = %(~x) by LATTICE3:def 3 .= ~x by LATTICE3:def 4;
      A7: (~x)~ = ~x by LATTICE3:def 6 .= x by LATTICE3:def 7;
      per cases;
      suppose %(~x) is-lower-neighbour-of a;
        hence thesis;
      suppose not(%(~x) is-lower-neighbour-of a);
        then consider c being Element of L such that
        A8: %(~x) [= c & c [= a & not(c = a or c = %(~x)) by A5,Def5;
        A9: %(~((c%)~)) = ~((c%)~) by LATTICE3:def 4
                        .= (c%)~ by LATTICE3:def 7
                        .= c% by LATTICE3:def 6
                        .= c by LATTICE3:def 3;
          ~x <= c% by A6,A8,LATTICE3:7;
        then (c%)~ <= x by A7,LATTICE3:9;
        then [(c%)~,x] in the InternalRel of (LattPOSet L)~ by ORDERS_1:def 9;
        then consider c' being Element of L such that
        A10: %(~((c%)~)) [= c' & c' is-lower-neighbour-of a by A4,A8,A9;
          %(~x) [= c' by A8,A9,A10,LATTICES:25;
        hence thesis by A10;
      end;
    hence thesis;
    end;
A11: for x being Element of (LattPOSet L)~ holds P[x] from WFInduction(A3,A2);
  %(~((d%)~)) = ~((d%)~) by LATTICE3:def 4 .= (d%)~ by LATTICE3:def 7
           .= d% by LATTICE3:def 6 .= d by LATTICE3:def 3;
hence thesis by A1,A11;
end;

theorem
Th5:for L being upper-bounded Lattice holds
not(ex b being Element of L st b is-upper-neighbour-of Top L)
proof
  let L be upper-bounded Lattice;
  given b being Element of L such that
  A1: b is-upper-neighbour-of Top L;
  A2: Top L [= b & Top L <> b by A1,Def5;
    b [= Top L by LATTICES:45;
  hence thesis by A2,LATTICES:26;
end;

theorem
Th6:for L being noetherian upper-bounded Lattice
for a being Element of L holds
a = Top L iff
not(ex b being Element of L st b is-upper-neighbour-of a)
proof
let L be noetherian upper-bounded Lattice;
let a be Element of L;
  now assume A1:
not(ex b being Element of L st b is-upper-neighbour-of a);
  for b being Element of L holds a "\/" b = a & b "\/" a = a
  proof
  let b be Element of L;
    consider c being Element of L such that
    A2: c = a "\/" b;
    A3: a [= c & b [= c by A2,LATTICES:22;
    per cases;
    suppose a <> c;
      then ex d being Element of L st
           d [= c & d is-upper-neighbour-of a by A3,Th3;
      hence thesis by A1;
    suppose a = c;
      hence thesis by A2;
  end;
hence a = Top L by LATTICES:def 17;
end;
hence thesis by Th5;
end;

theorem
Th7:for L being lower-bounded Lattice holds
not(ex b being Element of L st b is-lower-neighbour-of Bottom L)
proof
  let L be lower-bounded Lattice;
  given b being Element of L such that
  A1: b is-lower-neighbour-of Bottom L;
  A2: b [= Bottom L & b <> Bottom L by A1,Def5;
    Bottom L [= b by LATTICES:41;
  hence thesis by A2,LATTICES:26;
  end;

theorem
Th8:for L being co-noetherian lower-bounded Lattice
for a being Element of L holds
a = Bottom L iff
not(ex b being Element of L st b is-lower-neighbour-of a)
proof
let L be co-noetherian lower-bounded Lattice;
let a be Element of L;
  now assume A1:
not(ex b being Element of L st b is-lower-neighbour-of a);
  for b being Element of L holds a "/\" b = a & b "/\" a = a
  proof
  let b be Element of L;
    consider c being Element of L such that
    A2: c = a "/\" b;
    A3: c [= a & c [= b by A2,LATTICES:23;
    per cases;
    suppose a <> c;
      then ex d being Element of L st
           c [= d & d is-lower-neighbour-of a by A3,Th4;
      hence thesis by A1;
    suppose a = c;
      hence thesis by A2;
  end;
hence a = Bottom L by LATTICES:def 16;
end;
hence thesis by Th7;
end;

definition
let L be complete Lattice;
let a be Element of L;
func a*' -> Element of L equals :Def6:
  "/\"({d where d is Element of L : a [= d & d <> a},L);
correctness;
func *'a -> Element of L equals :Def7:
  "\/"({d where d is Element of L : d [= a & d <> a},L);
correctness;
end;

definition
let L be complete Lattice;
let a be Element of L;
attr a is completely-meet-irreducible means :Def8:
  a*' <> a;
attr a is completely-join-irreducible means :Def9:
  *'a <> a;
end;

theorem
Th9:for L being complete Lattice
for a being Element of L holds a [= a*' & *'a [= a
proof
let L be complete Lattice;
let a be Element of L;
set X = {d where d is Element of L : a [= d & d <> a};
A1: a is_less_than X
    proof
      for q being Element of L st q in X holds a [= q
      proof
      let q be Element of L;
      assume q in X;
      then ex q' being Element of L st
       q' = q & a [= q' & q' <> a;
      hence thesis;
      end;
    hence thesis by LATTICE3:def 16;
    end;
A2: a*' = "/\"(X,L) by Def6;
set X = {d where d is Element of L : d [= a & d <> a};
A3: X is_less_than a
    proof
      for q being Element of L st q in X holds q [= a
      proof
      let q be Element of L;
      assume q in X;
      then ex q' being Element of L st
       q' = q & q' [= a & q' <> a;
      hence thesis;
      end;
    hence thesis by LATTICE3:def 17;
    end;
  *'a = "\/"(X,L) by Def7;
hence thesis by A1,A2,A3,LATTICE3:34,def 21;
end;

theorem
  for L being complete Lattice holds
 Top L*' = Top L & (Top L)% is meet-irreducible
proof
let L be complete Lattice;
set X = {d where d is Element of L : Top L [= d & d <> Top L};
A1: X = {}
    proof
    assume X <> {};
    then reconsider X as non empty set;
    consider x being Element of X;
      x in X;
    then consider x' being Element of L such that
    A2: x' = x & Top L [= x' & x' <> Top L;
      x' [= Top L by LATTICES:45;
    hence thesis by A2,LATTICES:26;
    end;
A3: Top L is_less_than {}
    proof
      for q being Element of L st q in {} holds Top L [= q;
    hence thesis by LATTICE3:def 16;
    end;
A4: for b being Element of L st b is_less_than {} holds b [=
Top L
  by LATTICES:45;
then A5: "/\"({},L) = Top L by A3,LATTICE3:34;
     (Top L)% = Top (LattPOSet L)
    proof
       Top (LattPOSet L) = "/\"({},LattPOSet L) by YELLOW_0:def 12
                      .= "/\"({},L) by YELLOW_0:29
                      .= Top L by A3,A4,LATTICE3:34;
    hence thesis by LATTICE3:def 3;
    end;
hence thesis by A1,A5,Def6,WAYBEL_6:10;
end;

theorem
  for L being complete Lattice holds
*'Bottom L = Bottom L & (Bottom L)% is join-irreducible
proof
let L be complete Lattice;
set X = {d where d is Element of L : d [= Bottom L & d <>
Bottom L};
A1: X = {}
    proof
    assume X <> {};
    then reconsider X as non empty set;
    consider x being Element of X;
      x in X;
    then consider x' being Element of L such that
    A2: x' = x & x' [= Bottom L & x' <> Bottom L;
      Bottom L [= x' by LATTICES:41;
    hence thesis by A2,LATTICES:26;
    end;
A3: Bottom L is_great_than {}
    proof
      for q being Element of L st q in {} holds q [= Bottom L;
    hence thesis by LATTICE3:def 17;
    end;
A4: for b being Element of L st b is_great_than {} holds Bottom
L [= b
  by LATTICES:41;
then A5:  "\/"({},L) = Bottom L by A3,LATTICE3:def 21;
A6: (Bottom L)% = Bottom (LattPOSet L)
    proof
       Bottom (LattPOSet L) = "\/"({},LattPOSet L) by YELLOW_0:def 11
                      .= "\/"({},L) by YELLOW_0:29
                      .= Bottom L by A3,A4,LATTICE3:def 21;
    hence thesis by LATTICE3:def 3;
    end;
  Bottom (LattPOSet L) is join-irreducible
    proof
      for x,y being Element of LattPOSet L st Bottom (LattPOSet L) = x "\/" y
    holds x = Bottom (LattPOSet L) or y = Bottom (LattPOSet L)
      proof
      let x,y be Element of LattPOSet L;
      assume Bottom (LattPOSet L) = x "\/" y;
      then A7: Bottom (LattPOSet L) >= x & Bottom
(LattPOSet L) >= y by YELLOW_0:22;
      reconsider L' = LattPOSet L
          as lower-bounded antisymmetric non empty RelStr;
      reconsider x' = x as Element of L';
      reconsider y' = y as Element of L';
        x' >= Bottom (L') or y' >= Bottom (L') by YELLOW_0:44;
      hence x = Bottom (LattPOSet L) or y = Bottom
(LattPOSet L) by A7,ORDERS_1:25;
      end;
    hence thesis by WAYBEL_6:def 3;
    end;
hence thesis by A1,A5,A6,Def7;
end;

theorem
Th12:for L being complete Lattice
for a being Element of L st a is completely-meet-irreducible
holds a*' is-upper-neighbour-of a &
      for c being Element of L
      holds c is-upper-neighbour-of a implies c = a*'
proof
let L be complete Lattice;
let a be Element of L;
assume a is completely-meet-irreducible;
then A1: a*' <> a by Def8;
set X = { x where x is Element of L : a [= x & x <> a};
A2: a*' = "/\"(X,L) by Def6;
A3: a [= a*' by Th9;
  for c being Element of L st a [= c & c [= a*'
 holds c = a or c = a*'
  proof
  let c be Element of L;
  assume A4: a [= c & c [= a*';
  assume c <> a;
  then c in X by A4;
  then a*' [= c by A2,LATTICE3:38;
  hence thesis by A4,LATTICES:26;
  end;
then A5: for c being Element of L holds
     (a [= c & c [= a*') implies (c = a*' or c = a);
  for c being Element of L
holds c is-upper-neighbour-of a implies c = a*'
  proof
  let c be Element of L;
  assume
A6: c is-upper-neighbour-of a;
  then a <> c & a [= c &
       for d being Element of L holds
       (a [= d & d [= c) implies (d = c or d = a) by Def5;
  then c in X;
  then a [= a*' & a*' [= c by A2,Th9,LATTICE3:38;
  hence thesis by A1,A6,Def5;
  end;
hence thesis by A1,A3,A5,Def5;
end;

theorem
Th13:for L being complete Lattice
for a being Element of L st a is completely-join-irreducible
holds *'a is-lower-neighbour-of a &
      for c being Element of L
      holds c is-lower-neighbour-of a implies c = *'a
proof
let L be complete Lattice;
let a be Element of L;
assume a is completely-join-irreducible;
then A1: *'a <> a by Def9;
set X = { x where x is Element of L : x [= a & x <> a};
A2: *'a = "\/"(X,L) by Def7;
A3: *'a [= a by Th9;
A4: for c being Element of L st *'a [= c & c [= a
 holds c = a or c = *'a
  proof
  let c be Element of L;
  assume A5: *'a [= c & c [= a;
  assume c <> a;
  then c in X by A5;
  then c [= *'a by A2,LATTICE3:38;
  hence thesis by A5,LATTICES:26;
  end;
  for c being Element of L
holds c is-lower-neighbour-of a implies c = *'a
  proof
  let c be Element of L;
  assume
A6: c is-lower-neighbour-of a;
  then a <> c & c [= a &
       for d being Element of L holds
       (c [= d & d [= a) implies (d = c or d = a) by Def5;
  then c in X;
  then *'a [= a & c [= *'a by A2,Th9,LATTICE3:38;
  hence thesis by A1,A6,Def5;
  end;
hence thesis by A1,A3,A4,Def5;
end;

theorem
Th14:for L being noetherian complete Lattice
for a being Element of L holds
a is completely-meet-irreducible iff
ex b being Element of L
st b is-upper-neighbour-of a &
   for c being Element of L holds
   c is-upper-neighbour-of a implies c = b
proof
let L be noetherian complete Lattice;
let a be Element of L;
set X = { x where x is Element of L : a [= x & x <> a};
 hereby assume a is completely-meet-irreducible;
   then a*' is-upper-neighbour-of a &
        for c being Element of L
        holds c is-upper-neighbour-of a implies c = a*' by Th12;
   hence ex b being Element of L
         st b is-upper-neighbour-of a &
         for c being Element of L holds
         c is-upper-neighbour-of a implies c = b;
 end;
 given b being Element of L such that
    A1: b is-upper-neighbour-of a &
        for c being Element of L holds
        c is-upper-neighbour-of a implies c = b;
    A2: a [= b & a <> b by A1,Def5;
    then A3: b in X;
      for q being Element of L st q in X holds b [= q
      proof
      let q be Element of L;
      assume q in X;
      then ex q' being Element of L st
       q' = q & a [= q' & q' <> a;
      then ex c being Element of L st
       c [= q & c is-upper-neighbour-of a by Th3;
      hence thesis by A1;
      end;
    then b is_less_than X by LATTICE3:def 16;
    then b = "/\"(X,L) by A3,LATTICE3:42;
    hence a <> a*' by A2,Def6;
end;

theorem
Th15:for L being co-noetherian complete Lattice
for a being Element of L holds
a is completely-join-irreducible iff
ex b being Element of L
st b is-lower-neighbour-of a &
   for c being Element of L holds
   c is-lower-neighbour-of a implies c = b
proof
let L be co-noetherian complete Lattice;
let a be Element of L;
set X = { x where x is Element of L : x [= a & x <> a};
A1: now assume a is completely-join-irreducible;
   then *'a is-lower-neighbour-of a &
        for c being Element of L
        holds c is-lower-neighbour-of a implies c = *'a by Th13;
   hence ex b being Element of L
         st b is-lower-neighbour-of a &
         for c being Element of L holds
         c is-lower-neighbour-of a implies c = b;
   end;
  now
    given b being Element of L such that
    A2: b is-lower-neighbour-of a &
        for c being Element of L holds
        c is-lower-neighbour-of a implies c = b;
    A3: b [= a & a <> b by A2,Def5;
    then A4: b in X;
      for q being Element of L st q in X holds q [= b
      proof
      let q be Element of L;
      assume q in X;
      then ex q' being Element of L st
       q' = q & q' [= a & q' <> a;
      then ex c being Element of L st
       q [= c & c is-lower-neighbour-of a by Th4;
      hence thesis by A2;
      end;
    then b is_great_than X by LATTICE3:def 17;
    then b = "\/"(X,L) by A4,LATTICE3:41;
    then a <> *'a by A3,Def7;
    hence a is completely-join-irreducible by Def9;
    end;
hence thesis by A1;
end;

theorem
Th16:for L being complete Lattice
for a being Element of L holds
a is completely-meet-irreducible implies a% is meet-irreducible
proof
let L be complete Lattice;
let a be Element of L;
set X = {d where d is Element of L : a [= d & d <> a};
assume a is completely-meet-irreducible;
then A1: a <> a*' by Def8;
  for x,y being Element of LattPOSet L st a% = x "/\" y
holds x = a% or y = a%
    proof
    let x,y be Element of LattPOSet L;
    assume A2: a% = x "/\" y;
    then A3: a% <= x & a% <= y &
        for d being Element of LattPOSet L st d <= x & d <= y holds a% >= d
        by YELLOW_0:23;
    A4: a = a% by LATTICE3:def 3 .= %(a%) by LATTICE3:def 4;
    A5: y = %y by LATTICE3:def 4 .= (%y)% by LATTICE3:def 3;
    A6: x = %x by LATTICE3:def 4 .= (%x)% by LATTICE3:def 3;
    then A7: a [= %x & a [= %y by A3,A5,LATTICE3:7;
    A8: x = %x & y = %y & %(a%) = a% by LATTICE3:def 4;
    A9: a [= a*' by Th9;
    assume x <> a% & y <> a%;
    then A10: %x in X & %y in X by A4,A7,A8;
      a*' = "/\"(X,L) by Def6;
    then a*' is_less_than X &
    for b being Element of L st b is_less_than X holds b [= a*'
       by LATTICE3:34;
    then a*' [= %x & a*' [= %y by A10,LATTICE3:def 16;
    then (a*')% <= (%x)% & (a*')% <= (%y)% by LATTICE3:7;
    then A11: (a*')% <= a% by A2,A5,A6,YELLOW_0:23;
A12: a% <= (a*')% by A9,LATTICE3:7;
      a = a% & a*' = (a*')% by LATTICE3:def 3;
    hence contradiction by A1,A11,A12,ORDERS_1:25;
    end;
hence thesis by WAYBEL_6:def 2;
end;

Lm8:for L being Lattice
for a,b being Element of L holds
a "/\" b = a% "/\" b% & a "\/" b = a% "\/" b%
proof
let L be Lattice;
let a,b be Element of L;
consider c being Element of L such that
A1: c = a "/\" b;
  c [= a & c [= b by A1,LATTICES:23;
then A2: c% <= a% & c% <= b% by LATTICE3:7;
  for d being Element of LattPOSet L st d <= a% & d <= b% holds c% >= d
  proof
  let d be Element of LattPOSet L;
  assume A3: d <= a% & d <= b%;
  A4: (%d)% = %d by LATTICE3:def 3 .= d by LATTICE3:def 4;
  then A5: %d [= a & %d [= b by A3,LATTICE3:7;
    %d "/\" c = (%d "/\" a) "/\" b by A1,LATTICES:def 7
         .= %d "/\" b by A5,LATTICES:21
         .= %d by A5,LATTICES:21;
  then %d [= c by LATTICES:21;
  hence thesis by A4,LATTICE3:7;
  end;
then A6: c% = a% "/\" b% by A2,YELLOW_0:23;
consider c being Element of L such that
A7: c = a "\/" b;
  a [= c & b [= c by A7,LATTICES:22;
then A8: a% <= c% & b% <= c% by LATTICE3:7;
  for d being Element of LattPOSet L st a% <= d & b% <= d holds d >= c%
  proof
  let d be Element of LattPOSet L;
  assume A9: a% <= d & b% <= d;
  A10: (%d)% = %d by LATTICE3:def 3 .= d by LATTICE3:def 4;
  then A11: a [= %d & b [= %d by A9,LATTICE3:7;
    %d "\/" c = (%d "\/" a) "\/" b by A7,LATTICES:def 5
         .= %d "\/" b by A11,LATTICES:def 3
         .= %d by A11,LATTICES:def 3;
  then c [= %d by LATTICES:def 3;
  hence thesis by A10,LATTICE3:7;
  end;
then c% = a% "\/" b% by A8,YELLOW_0:22;
hence thesis by A1,A6,A7,LATTICE3:def 3;
end;

theorem
Th17:for L being complete noetherian Lattice
for a being Element of L st a <> Top L holds
a is completely-meet-irreducible iff a% is meet-irreducible
proof
let L be complete noetherian Lattice;
let a be Element of L;
assume a <> Top L;
then consider b being Element of L such that
     A1: b is-upper-neighbour-of a by Th6;
A2: b <> a & a [= b &
    for c being Element of L holds
    (a [= c & c [= b) implies (c = b or c = a) by A1,Def5;
  now assume
A3: a% is meet-irreducible;
      for d being Element of L holds
    d is-upper-neighbour-of a implies d = b
      proof
      let d be Element of L;
      assume A4: d is-upper-neighbour-of a;
      then A5: d <> a & a [= d &
               for c being Element of L holds
               (a [= c & c [= d) implies (c = d or c = a) by Def5;
      assume d <> b;
      then A6: a = d "/\" b by A1,A4,Th2;
      A7: a% = a & d% = d & b% = b by LATTICE3:def 3;
      then a% = d% "/\" b% by A6,Lm8;
      hence thesis by A2,A3,A5,A7,WAYBEL_6:def 2;
      end;
    hence a is completely-meet-irreducible by A1,Th14;
    end;
hence thesis by Th16;
end;

theorem
Th18:for L being complete Lattice
for a being Element of L holds
a is completely-join-irreducible implies a% is join-irreducible
proof
let L be complete Lattice;
let a be Element of L;
set X = {d where d is Element of L : d [= a & d <> a};
assume a is completely-join-irreducible;
then A1: a <> *'a by Def9;
  for x,y being Element of LattPOSet L st a% = x "\/" y
holds x = a% or y = a%
    proof
    let x,y be Element of LattPOSet L;
    assume A2: a% = x "\/" y;
    then A3: x <= a% & y <= a% &
        for d being Element of LattPOSet L st d >= x & d >= y holds a% <= d
        by YELLOW_0:22;
    A4: a = a% by LATTICE3:def 3 .= %(a%) by LATTICE3:def 4;
    A5: y = %y by LATTICE3:def 4 .= (%y)% by LATTICE3:def 3;
    A6: x = %x by LATTICE3:def 4 .= (%x)% by LATTICE3:def 3;
    then A7: %x [= a & %y [= a by A3,A5,LATTICE3:7;
    A8: x = %x & y = %y & %(a%) = a% by LATTICE3:def 4;
    A9: *'a [= a by Th9;
    assume x <> a% & y <> a%;
    then A10: %x in X & %y in X by A4,A7,A8;
      *'a = "\/"(X,L) by Def7;
    then X is_less_than *'a &
    for b being Element of L st X is_less_than b holds *'a [= b
       by LATTICE3:def 21;
    then %x [= *'a & %y [= *'a by A10,LATTICE3:def 17;
    then (*'a)% >= x & (*'a)% >= y by A5,A6,LATTICE3:7;
    then A11: (*'a)% >= a% by A2,YELLOW_0:22;
A12:  a% >= (*'a)% by A9,LATTICE3:7;
      a = a% & *'a = (*'a)% by LATTICE3:def 3;
    hence contradiction by A1,A11,A12,ORDERS_1:25;
    end;
hence thesis by WAYBEL_6:def 3;
end;

theorem
Th19:for L being complete co-noetherian Lattice
for a being Element of L st a <> Bottom L holds
a is completely-join-irreducible iff a% is join-irreducible
proof
let L be complete co-noetherian Lattice;
let a be Element of L;
assume a <> Bottom L;
then consider b being Element of L such that
     A1: b is-lower-neighbour-of a by Th8;
A2: a <> b & b [= a &
    for c being Element of L holds
    (b [= c & c [= a) implies (c = a or c = b) by A1,Def5;
  now assume
A3: a% is join-irreducible;
      for d being Element of L holds
    d is-lower-neighbour-of a implies d = b
      proof
      let d be Element of L;
      assume A4: d is-lower-neighbour-of a;
      then A5: a <> d & d [= a &
               for c being Element of L holds
               (d [= c & c [= a) implies (c = a or c = d) by Def5;
      assume d <> b;
      then A6: a = d "\/" b by A1,A4,Th2;
      A7: a% = a & d% = d & b% = b by LATTICE3:def 3;
      then a% = d% "\/" b% by A6,Lm8;
      hence thesis by A2,A3,A5,A7,WAYBEL_6:def 3;
      end;
    hence a is completely-join-irreducible by A1,Th15;
    end;
hence thesis by Th18;
end;

theorem
  for L being finite Lattice
for a being Element of L st a <> Bottom L & a <> Top L holds
(a is completely-meet-irreducible iff a% is meet-irreducible) &
(a is completely-join-irreducible iff a% is join-irreducible) by Th17,Th19;

definition
let L be Lattice;
let a be Element of L;
attr a is atomic means :Def10:
  a is-upper-neighbour-of Bottom L;
attr a is co-atomic means :Def11:
  a is-lower-neighbour-of Top L;
end;

theorem
  for L being complete Lattice
for a being Element of L st a is atomic holds
a is completely-join-irreducible
proof
let L be complete Lattice;
let a be Element of L;
assume a is atomic;
then A1: a is-upper-neighbour-of Bottom L by Def10;
then A2: a <> Bottom L & Bottom L [= a &
         for c being Element of L holds
         (Bottom L [= c & c [= a) implies (c = a or c = Bottom L) by Def5;
set X = { x where x is Element of L : x [= a & x <> a};
A3: for x being set holds x in X implies x in {Bottom L}
    proof
    let x be set;
    assume x in X;
    then A4:  ex x' being Element of L st x' = x & x' [= a & x'
<> a;
    then reconsider x as Element of L;
      Bottom L [= x by LATTICES:41;
    then x = Bottom L by A1,A4,Def5;
    hence thesis by TARSKI:def 1;
    end;
A5: for x being set holds x in {Bottom L} implies x in X
  proof
  let x be set; assume x in {Bottom L};
   then x = Bottom L by TARSKI:def 1;
  hence thesis by A2;
  end;
  Bottom L = "\/"({Bottom L},L) by LATTICE3:43
  .= "\/"(X,L) by A3,A5,TARSKI:2
  .= *'a by Def7;
hence thesis by A2,Def9;
end;

theorem
  for L being complete Lattice
for a being Element of L st a is co-atomic holds
a is completely-meet-irreducible
proof
let L be complete Lattice;
let a be Element of L;
assume a is co-atomic;
then A1: a is-lower-neighbour-of Top L by Def11;
then A2: a <> Top L & a [= Top L &
         for c being Element of L holds
         (a [= c & c [= Top L) implies (c = Top L or c = a) by Def5;
set X = { x where x is Element of L : a [= x & x <> a};
A3: for x being set holds x in X implies x in {Top L}
    proof
    let x be set;
    assume x in X;
    then A4:  ex x' being Element of L st x' = x & a [= x' & x'
<> a;
    then reconsider x as Element of L;
      x [= Top L by LATTICES:45;
    then x = Top L by A1,A4,Def5;
    hence thesis by TARSKI:def 1;
    end;
A5: for x being set holds x in {Top L} implies x in X
  proof
  let x be set; assume x in {Top L};
   then x = Top L by TARSKI:def 1;
  hence thesis by A2;
  end;
  Top L = "/\"({Top L},L) by LATTICE3:43
  .= "/\"(X,L) by A3,A5,TARSKI:2
  .= a*' by Def6;
hence thesis by A2,Def8;
end;

definition
let L be Lattice;
attr L is atomic means :Def12:
 for a being Element of L holds
 ex X being Subset of L st
 (for x being Element of L st x in X holds x is atomic) &
 a = "\/"(X,L);
end;

definition
  cluster strict non empty trivial Lattice;
  existence
  proof
 consider x being set;
 set X = {x};
 consider XJ,XM being BinOp of X;
 reconsider L = LattStr(#X,XJ,XM#) as non empty LattStr by STRUCT_0:def 1;
 A1: the carrier of L is trivial by REALSET1:def 12;
then A2: L is trivial by REALSET1:def 13;
   L is Lattice-like
   proof
   A3: for a,b being Element of L holds a"\/"b = b"\/"a
     by A2,REALSET1:def 20;
   A4: for a,b,c being Element of L
             holds a"\/"(b"\/"c) = (a"\/"b)"\/"c by A2,REALSET1:def 20;
   A5: for a,b being Element of L holds (a"/\"b)"\/"b = b
             by A2,REALSET1:def 20;
   A6: for a,b being Element of L holds a"/\"b = b"/\"a
             by A2,REALSET1:def 20;
   A7: for a,b,c being Element of L
                                   holds a"/\"(b"/\"c) = (a"/\"b)"/\"c
             by A2,REALSET1:def 20;
     for a,b being Element of L holds a"/\"(a"\/"b)=a
             by A2,REALSET1:def 20;
   then L is join-commutative join-associative meet-absorbing
        meet-commutative meet-associative join-absorbing
   by A3,A4,A5,A6,A7,LATTICES:def 4,def 5,def 6,def 7,def 8,def 9;
   hence thesis by LATTICES:def 10;
   end;
   then reconsider L as Lattice;
   take L;
   thus thesis by A1,REALSET1:def 13;
  end;
end;

Lm9:ex L being Lattice st L is atomic & L is complete
proof
 consider L being strict non empty trivial Lattice;
   the carrier of L is trivial non empty by REALSET1:def 13;
 then consider x being set such that
A1:  the carrier of L = {x} by REALSET1:def 12;
 reconsider x as Element of L by A1,TARSKI:def 1;
   for Y being set ex p being Element of L st Y is_less_than p &
 for r being Element of L st Y is_less_than r holds p [= r
    proof
    let Y be set;
      for q being Element of L st q in Y holds q [= x
      by A1,TARSKI:def 1;
    then A2: Y is_less_than x by LATTICE3:def 17;
      for r being Element of L st Y is_less_than r holds x [= r
      by A1,TARSKI:def 1;
    hence thesis by A2;
    end;
 then reconsider L as complete Lattice by LATTICE3:def 18;
   for a being Element of L holds
 ex X being Subset of L st
 (for u being Element of L st u in X holds u is atomic) &
 a = "\/"(X,L)
 proof
   let a be Element of L;
   A3: a = x by A1,TARSKI:def 1;
   A4: x = "\/"({},L) by A1,TARSKI:def 1;
   A5: for u being Element of L st u in {} holds u is atomic;
     for u being set holds u in {} implies u in the carrier of L;
   then {} is Subset of L by TARSKI:def 3;
   hence thesis by A3,A4,A5;
   end;
 then L is atomic by Def12;
 hence thesis;
end;

definition
cluster atomic complete Lattice;
existence by Lm9;
end;

definition
let L be complete Lattice;
let D be Subset of L;
attr D is supremum-dense means :Def13:
 for a being Element of L holds
 ex D' being Subset of D st a = "\/"(D',L);
attr D is infimum-dense means :Def14:
 for a being Element of L holds
 ex D' being Subset of D st a = "/\"(D',L);
end;

theorem
Th23:for L being complete Lattice
for D being Subset of L holds
D is supremum-dense iff
for a being Element of L holds
a = "\/"({d where d is Element of L: d in D & d [= a},L)
proof
let L be complete Lattice;
let D be Subset of L;
A1: now assume
 A2: D is supremum-dense;
   thus for a being Element of L holds
      a = "\/"({d where d is Element of L: d in D & d [= a},L)
     proof
     let a be Element of L;
     set X = {d where d is Element of L: d in D & d [= a};
     consider D' being Subset of D such that A3: a = "\/"(D',L) by A2,Def13;
       for x being set holds x in D' implies x in X
       proof
       let x be set; assume A4: x in D';
       then x in D;
       then reconsider x as Element of L;
         D' is_less_than a by A3,LATTICE3:def 21;
       then x [= a by A4,LATTICE3:def 17;
       hence thesis by A4;
       end;
     then D' c= X by TARSKI:def 3;
     then A5: a [= "\/"(X,L) by A3,LATTICE3:46;
       for q being Element of L st q in X holds q [= a
       proof
       let q be Element of L;
       assume q in X;
       then ex q' being Element of L st
        q' = q & q' in D & q' [= a;
       hence thesis;
       end;
     then X is_less_than a by LATTICE3:def 17;
     then "\/"(X,L) [= a by LATTICE3:def 21;
     hence thesis by A5,LATTICES:26;
     end;
   end;
  now assume A6:
    for a being Element of L holds
    a = "\/"({d where d is Element of L: d in D & d [= a},L);
      for a being Element of L holds
    ex D' being Subset of D st a = "\/"(D',L)
       proof
       let a be Element of L;
       set X = {d where d is Element of L: d in D & d [= a};
         for x being set holds x in X implies x in D
         proof
         let x be set; assume x in X;
         then ex x' being Element of L st x' = x & x' in D & x'
[= a;
         hence thesis;
         end;
       then A7: X is Subset of D by TARSKI:def 3;
         a = "\/"(X,L) by A6;
       hence thesis by A7;
       end;
    hence D is supremum-dense by Def13;
    end;
hence thesis by A1;
end;

theorem
Th24:for L being complete Lattice
for D being Subset of L holds
D is infimum-dense iff
for a being Element of L holds
a = "/\"({d where d is Element of L : d in D & a [= d},L)
proof
let L be complete Lattice;
let D be Subset of L;
A1: now assume
A2: D is infimum-dense;
   thus for a being Element of L holds
      a = "/\"({d where d is Element of L: d in D & a [= d},L)
     proof
     let a be Element of L;
     set X = {d where d is Element of L: d in D & a [= d};
     consider D' being Subset of D such that A3: a = "/\"(D',L) by A2,Def14;
       for x being set holds x in D' implies x in X
       proof
       let x be set; assume A4: x in D';
       then x in D;
       then reconsider x as Element of L;
         a is_less_than D' by A3,LATTICE3:34;
       then a [= x by A4,LATTICE3:def 16;
       hence thesis by A4;
       end;
     then D' c= X by TARSKI:def 3;
     then A5: "/\"(X,L) [= a by A3,LATTICE3:46;
       for q being Element of L st q in X holds a [= q
       proof
       let q be Element of L;
       assume q in X;
       then ex q' being Element of L st
        q' = q & q' in D & a [= q';
       hence thesis;
       end;
     then a is_less_than X by LATTICE3:def 16;
     then a [= "/\"(X,L) by LATTICE3:40;
     hence thesis by A5,LATTICES:26;
     end;
   end;
  now assume A6:
    for a being Element of L holds
    a = "/\"({d where d is Element of L: d in D & a [= d},L);
      for a being Element of L holds
    ex D' being Subset of D st a = "/\"(D',L)
       proof
       let a be Element of L;
       set X = {d where d is Element of L: d in D & a [= d};
         for x being set holds x in X implies x in D
         proof
         let x be set; assume x in X;
         then ex x' being Element of L st x' = x & x' in D & a
[= x';
         hence thesis;
         end;
       then A7: X is Subset of D by TARSKI:def 3;
         a = "/\"(X,L) by A6;
       hence thesis by A7;
       end;
    hence D is infimum-dense by Def14;
    end;
hence thesis by A1;
end;

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 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 D' being Subset of D such that
     A3: %a = "/\"(D',L) by A2,Def14;
     A4: %a is_less_than D' & for b being Element of L
          st b is_less_than D' holds b [= %a by A3,LATTICE3:34;
       for x being set holds x in D' implies x in the carrier of L
       proof
       let x be set; assume x in D';
       then x in D; hence thesis;
       end;
     then reconsider D' as Subset of L by TARSKI:def 3;
       for u being Element of LattPOSet L st u in (D')% holds a <= u
        proof
        let u be Element of LattPOSet L;
        assume u in (D')%;
        then u in {d% where d is Element of L : d in D'} by Def1
;
        then consider u' being Element of L such that
        A5: u = (u')% & u' in D';
        A6: %a [= u' by A4,A5,LATTICE3:def 16;
          (%a)% = %a by LATTICE3:def 3 .= a by LATTICE3:def 4;
        hence thesis by A5,A6,LATTICE3:7;
        end;
     then A7: (D')% is_>=_than a by LATTICE3:def 8;
       for b being Element of LattPOSet L st (D')% is_>=_than b holds b <= a
       proof
       let b be Element of LattPOSet L;
       assume
A8:     (D')% is_>=_than b;
       A9: b = %b by LATTICE3:def 4 .= (%b)% by LATTICE3:def 3;
       A10: a = %a by LATTICE3:def 4 .= (%a)% by LATTICE3:def 3;
         for u being Element of L st u in D' holds %b [= u
          proof
          let u be Element of L; assume u in D';
          then u% in {d% where d is Element of L : d in D'};
          then u% in (D')% by Def1;
          then b <= u% by A8,LATTICE3:def 8;
          hence thesis by A9,LATTICE3:7;
          end;
       then %b is_less_than D' by LATTICE3:def 16;
       then %b [= %a by A3,LATTICE3:34;
       hence thesis by A9,A10,LATTICE3:7;
       end;
     then A11: a = "/\"((D')%,LattPOSet L)by A7,YELLOW_0:33;
       for x being set holds x in (D')% implies x in D%
        proof
        let x be set; assume x in (D')%;
        then x in {d% where d is Element of L : d in D'} by Def1
;
        then ex x' being Element of L st x = (x')% & x' in D';
        then x in {d% where d is Element of L : d in D};
        hence thesis by Def1;
        end;
     then (D')% is Subset of D% by TARSKI:def 3;
     hence thesis by A11;
     end;
   hence D% is order-generating by WAYBEL_6:15;
   end;
  now assume
A12: D% is order-generating;
      for a being Element of L holds
    ex D' being Subset of D st a = "/\"(D',L)
       proof
       let a be Element of L;
       consider Y being Subset of D% such that
       A13: a% = "/\"(Y,LattPOSet L) by A12,WAYBEL_6:15;
       A14: a% is_<=_than Y & for b being Element of LattPOSet L
           st b is_<=_than Y holds a% >= b by A13,YELLOW_0:33;
         for x being set holds x in Y implies x in the carrier of LattPOSet L
          proof
          let x be set; assume x in Y;
          then x in D%; hence thesis;
          end;
       then reconsider Y as Subset of LattPOSet L
             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 q in {%d where d is Element of LattPOSet L : d in Y} by Def2;
          then consider q' being Element of LattPOSet L such that
          A15: q = %(q') & q' in Y;
          A16: a% <= q' by A14,A15,LATTICE3:def 8;
            q' = %(q') by LATTICE3:def 4 .= (%(q'))% by LATTICE3:def 3;
          hence thesis by A15,A16,LATTICE3:7;
          end;
       then A17: a is_less_than %Y by LATTICE3:def 16;
         for b being Element of L
       st b is_less_than %Y holds b [= a
          proof
          let b be Element of L;
          assume
A18:           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 %u in %Y by Def2;
            then A19:          b [= %u by A18,LATTICE3:def 16;
              (%u)% = %u by LATTICE3:def 3 .= u by LATTICE3:def 4;
            hence thesis by A19,LATTICE3:7;
            end;
          then b% is_<=_than Y by LATTICE3:def 8;
          then b% <= a% by A13,YELLOW_0:33;
          hence thesis by LATTICE3:7;
          end;
       then A20: a = "/\"(%Y,L) by A17,LATTICE3:34;
         for x being set holds x in %Y implies x in D
         proof
         let x be set; assume x in %Y;
         then x in {%d where d is Element of LattPOSet L : d in Y} by Def2;
         then consider x' being Element of LattPOSet L such that
         A21: x = %(x') & x' in Y;
           x' in D% by A21;
         then x' in {d% where d is Element of L : d in D} by
Def1;
         then consider y being Element of L such that
         A22: x' = y% & y in D;
           x = y% by A21,A22,LATTICE3:def 4
          .= y by LATTICE3:def 3;
         hence thesis by A22;
         end;
       then %Y is Subset of D by TARSKI:def 3;
       hence thesis by A20;
       end;
    hence D is infimum-dense by Def14;
    end;
hence thesis by A1;
end;

definition
let L be complete Lattice;
func MIRRS(L) -> Subset of L equals :Def15:
{a where a is Element of L : a is completely-meet-irreducible};
correctness
 proof
 set X = {a where a is Element of L :
                                     a is completely-meet-irreducible};
   for x being set holds x in X implies x in the carrier of L
   proof
   let x be set; assume x in X;
   then ex x' being Element of L st x' = x &
                          x' is completely-meet-irreducible;
   hence thesis;
   end;
  then X is Subset of L by TARSKI:def 3;
 hence thesis;
 end;
func JIRRS(L) -> Subset of L equals :Def16:
{a where a is Element of L : a is completely-join-irreducible };
correctness
 proof
 set X = {a where a is Element of L :
                                         a is completely-join-irreducible };
   for x being set holds x in X implies x in the carrier of L
   proof
   let x be set; assume x in X;
   then ex x' being Element of L st x' = x &
                                 x' is completely-join-irreducible;
   hence thesis;
   end;
  then X is Subset of L by TARSKI:def 3;
 hence thesis;
 end;
end;

theorem
  for L being complete Lattice
for D being Subset of L st D is supremum-dense holds JIRRS(L) c= D
proof
let L be complete Lattice;
let D be Subset of L;
assume A1: D is supremum-dense;
  for x being set holds x in JIRRS(L) implies x in D
  proof
  let x be set; assume A2: x in JIRRS(L); assume A3: not(x in D);
    JIRRS(L) = {a where a is Element of L :
                            a is completely-join-irreducible} by Def16;
  then consider x' being Element of L such that
  A4: x' = x & x' is completely-join-irreducible by A2;
  A5: x' = x & *'(x') <> x' by A4,Def9;
  reconsider x as Element of L by A4;
  set D' = {d where d is Element of L: d in D & d [= x};
  set X = {d where d is Element of L: d [= x & d <> x};
  A6: not(x in D')
      proof
      assume x in D';
      then ex x' being Element of L st
         x' = x & x' in D & x' [= x;
      hence thesis by A3;
      end;
    for u being set holds u in D' implies u in X
    proof
    let u be set; assume u in D';
    then consider u' being Element of L such that
    A7: u' = u & u' in D & u' [= x;
    reconsider u as Element of L by A7;
      u <> x by A6,A7;
    hence thesis by A7;
    end;
  then D' c= X by TARSKI:def 3;
  then "\/"(D',L) [= "\/"(X,L) by LATTICE3:46;
  then A8: x [= "\/"(X,L) by A1,Th23;
    for q being Element of L st q in X holds q [= x
    proof
    let q be Element of L;
    assume q in X;
    then ex q' being Element of L st
     q' = q & q' [= x & q' <> x;
    hence thesis;
    end;
  then X is_less_than x by LATTICE3:def 17;
  then "\/"(X,L) [= x by LATTICE3:def 21;
  then x = "\/"(X,L) by A8,LATTICES:26;
  hence thesis by A5,Def7;
  end;
hence thesis by TARSKI:def 3;
end;

theorem
  for L being complete Lattice
for D being Subset of L st D is infimum-dense holds MIRRS(L) c= D
proof
let L be complete Lattice;
let D be Subset of L;
assume A1: D is infimum-dense;
  for x being set holds x in MIRRS(L) implies x in D
  proof
  let x be set; assume A2: x in MIRRS(L); assume A3: not(x in D);
    MIRRS(L) = {a where a is Element of L :
                           a is completely-meet-irreducible} by Def15;
  then consider x' being Element of L such that
  A4: x' = x & x' is completely-meet-irreducible by A2;
  A5: x' = x & (x')*' <> x' by A4,Def8;
  reconsider x as Element of L by A4;
  set D' = {d where d is Element of L: d in D & x [= d};
  set X = {d where d is Element of L: x [= d & d <> x};
  A6: not(x in D')
      proof
      assume x in D';
      then ex x' being Element of L st
        x' = x & x' in D & x [= x';
      hence thesis by A3;
      end;
    for u being set holds u in D' implies u in X
    proof
    let u be set; assume u in D';
    then consider u' being Element of L such that
    A7: u' = u & u' in D & x [= u';
    reconsider u as Element of L by A7;
      u <> x by A6,A7;
    hence thesis by A7;
    end;
  then D' c= X by TARSKI:def 3;
  then "/\"(X,L) [= "/\"(D',L) by LATTICE3:46;
  then A8: "/\"(X,L) [= x by A1,Th24;
    for q being Element of L st q in X holds x [= q
    proof
    let q be Element of L;
    assume q in X;
    then ex q' being Element of L st
     q' = q & x [= q' & q' <> x;
    hence thesis;
    end;
  then x is_less_than X by LATTICE3:def 16;
  then x [= "/\"(X,L) by LATTICE3:40;
  then x = "/\"(X,L) by A8,LATTICES:26;
  hence thesis by A5,Def6;
  end;
hence thesis by TARSKI:def 3;
end;

Lm10:for L being co-noetherian complete Lattice holds MIRRS(L) is infimum-dense
proof
let L be co-noetherian complete Lattice;
defpred P[Element of (LattPOSet L)~] means
  ex D' being Subset of MIRRS(L) st $1 = ("/\"(D',L))%;
A1: (LattPOSet L)~ is well_founded by Def4;
A2: for x being Element of (LattPOSet L)~
    st for y being Element of (LattPOSet L)~
       st y <> x & [y,x] in the InternalRel of (LattPOSet L)~ holds P[y]
    holds P[x]
      proof
      let x be Element of (LattPOSet L)~;
      assume A3: for y being Element of (LattPOSet L)~
       st y <> x & [y,x] in the InternalRel of (LattPOSet L)~ holds P[y];
      set X =
        {d where d is Element of L: %(~x) [= d & d <> %(~x)};
      A4: for u being Element of L st u in X holds
          (u%)~ <> x & [(u%)~,x] in the InternalRel of (LattPOSet L)~
          proof
          let u be Element of L; assume u in X;
          then consider u' being Element of L such that
          A5: u' = u & %(~x) [= u' & %(~x) <> u';
A6:       (%(~x))% <= u% by A5,LATTICE3:7;
            ((%(~x))%)~ = (%(~x))% by LATTICE3:def 6
                          .= %(~x) by LATTICE3:def 3
                          .= ~x by LATTICE3:def 4 .= x by LATTICE3:def 7;
          then A7:       (u%)~ <= x by A6,LATTICE3:9;
          A8: %(~x) = ~x by LATTICE3:def 4 .= x by LATTICE3:def 7;
            (u%)~ = u% by LATTICE3:def 6 .= u by LATTICE3:def 3;
          hence thesis by A5,A7,A8,ORDERS_1:def 9;
          end;
      A9: for u being Element of L st u in X holds P[(u%)~]
         proof
         let u be Element of L; assume u in X;
         then (u%)~ <> x & [(u%)~,x] in
 the InternalRel of (LattPOSet L)~ by A4;
         hence thesis by A3;
         end;
      per cases;
      suppose %(~x) is completely-meet-irreducible;
        then %(~x) in {a where a is Element of L :
                                   a is completely-meet-irreducible };
        then %(~x) in MIRRS(L) by Def15;
        then for y being set holds y in {%(~x)} implies y in
 MIRRS(L) by TARSKI:def 1;
        then A10: {%(~x)} is Subset of MIRRS(L) by TARSKI:def 3;
          ("/\"({%(~x)},L))% = "/\"({%(~x)},L) by LATTICE3:def 3
                         .= %(~x) by LATTICE3:43
                         .= ~x by LATTICE3:def 4
                         .= x by LATTICE3:def 7;
        hence thesis by A10;
     suppose not( %(~x) is completely-meet-irreducible );
        then (%(~x))*' = %(~x) by Def8;
        then A11: %(~x) = "/\"(X,L) by Def6;
        set G = {H where H is Subset of MIRRS(L) :
                 ex u being Element of LattPOSet L st %u in X & %u = "/\"
(H,L)};
        set D' = union G;
          for v being set holds v in D' implies v in MIRRS(L)
            proof
            let v be set; assume v in D';
            then consider v' being set such that
            A12: v in v' & v' in G by TARSKI:def 4;
            consider H being Subset of MIRRS(L) such that A13: H = v' &
            ex u being Element of LattPOSet L st %u in X & %u = "/\"
(H,L) by A12;
            thus thesis by A12,A13;
            end;
        then A14: D' is Subset of MIRRS(L) by TARSKI:def 3;
          for q being Element of L st q in D' holds %(~x) [= q
            proof
            let q be Element of L;
            assume q in D';
            then consider h being set such that A15: q in h & h in
 G by TARSKI:def 4;
            consider h' being Subset of MIRRS(L) such that A16: h' = h &
            ex u being Element of LattPOSet L st %u in
 X & %u = "/\"(h',L) by A15;
            consider u being Element of LattPOSet L such that
            A17: %u in X & %u = "/\"(h,L) by A16;
              %u is_less_than h by A17,LATTICE3:34;
            then A18: %u [= q by A15,LATTICE3:def 16;
              %(~x) is_less_than X by A11,LATTICE3:34;
            then %(~x) [= %u by A17,LATTICE3:def 16;
            hence thesis by A18,LATTICES:25;
            end;
        then A19: %(~x) is_less_than D' by LATTICE3:def 16;
          for r being Element of L
        st r is_less_than D' holds r [= %(~x)
            proof
            let r be Element of L;
            assume
A20:          r is_less_than D';
              for q being Element of L st q in X holds r [= q
              proof
              let q be Element of L;
              assume A21: q in X;
              A22: q% = q by LATTICE3:def 3;
              consider D being Subset of MIRRS(L) such that
              A23: (q%)~ = ("/\"(D,L))% by A9,A21;
A24:           q = (q%)~ by A22,LATTICE3:def 6
                    .= "/\"(D,L) by A23,LATTICE3:def 3;
                for v being set holds v in D implies v in D'
                 proof
                 let v be set; assume A25: v in D;
                 A26: %(q%) in X by A21,A22,LATTICE3:def 4;
                   (q%)~ = q% by LATTICE3:def 6.= %(q%) by LATTICE3:def 4;
                 then %(q%) = "/\"(D,L) by A23,LATTICE3:def 3;
                 then D in G by A26;
                 hence thesis by A25,TARSKI:def 4;
                 end;
              then D c= D' by TARSKI:def 3;
              then for p being Element of L
                   st p in D holds r [= p by A20,LATTICE3:def 16;
              then r is_less_than D by LATTICE3:def 16;
              hence thesis by A24,LATTICE3:34;
              end;
            then r is_less_than X by LATTICE3:def 16;
            hence thesis by A11,LATTICE3:34;
            end;
        then A27: %(~x) = "/\"(D',L) by A19,LATTICE3:34;
        A28: %(~x) = ~x by LATTICE3:def 4 .= x by LATTICE3:def 7;
          ("/\"(D',L))% = "/\"(D',L) by LATTICE3:def 3;
        hence thesis by A14,A27,A28;
     end;
A29: for x being Element of (LattPOSet L)~ holds P[x] from WFInduction(A2,A1);
  for a being Element of L holds
     ex D' being Subset of MIRRS(L) st a = "/\"(D',L)
  proof
  let a be Element of L;
    (LattPOSet L)~ =
  RelStr(#the carrier of LattPOSet L,
          (the InternalRel of LattPOSet L)~#) by LATTICE3:def 5;
  then reconsider a' = a% as Element of (LattPOSet L)~;
  consider D' being Subset of MIRRS(L) such that A30: a' = ("/\"
(D',L))% by A29;
  A31: ("/\"(D',L))% = "/\"(D',L) by LATTICE3:def 3;
    a% = a by LATTICE3:def 3;
  hence thesis by A30,A31;
  end;
hence thesis by Def14;
end;

definition
let L be co-noetherian complete Lattice;
cluster MIRRS(L) -> infimum-dense;
coherence by Lm10;
end;

Lm11:for L being noetherian complete Lattice holds JIRRS(L) is supremum-dense
proof
let L be noetherian complete Lattice;
defpred P[Element of LattPOSet L] means
  ex D' being Subset of JIRRS(L) st %($1) = "\/"(D',L);
A1: LattPOSet L is well_founded by Def3;
A2: for x being Element of LattPOSet L
    st for y being Element of LattPOSet L
       st y <> x & [y,x] in the InternalRel of LattPOSet L holds P[y]
    holds P[x]
      proof
      let x be Element of LattPOSet L;
      assume A3: for y being Element of LattPOSet L
       st y <> x & [y,x] in the InternalRel of LattPOSet L holds P[y];
      set X = {d where d is Element of L: d [= %x & d <> %x};
      A4: for u being Element of L st u in X holds
          u% <> x & [u%,x] in the InternalRel of LattPOSet L
          proof
          let u be Element of L; assume u in X;
          then consider u' being Element of L such that
          A5: u' = u & u' [= %x & %x <> u';
          A6: %x = x by LATTICE3:def 4;
          then (%x)% = x by LATTICE3:def 3;
          then u% <= x by A5,LATTICE3:7;
          hence thesis by A5,A6,LATTICE3:def 3,ORDERS_1:def 9;
          end;
      A7: for u being Element of L st u in X holds P[u%]
         proof
         let u be Element of L; assume u in X;
         then u% <> x & [u%,x] in the InternalRel of LattPOSet L by A4;
         hence thesis by A3;
         end;
      per cases;
      suppose %x is completely-join-irreducible;
        then %x in {a where a is Element of L :
                                   a is completely-join-irreducible};
        then %x in JIRRS(L) by Def16;
        then for y being set holds y in {%x} implies y in
 JIRRS(L) by TARSKI:def 1;
        then A8: {%x} is Subset of JIRRS(L) by TARSKI:def 3;
          "\/"({%x},L) = %x by LATTICE3:43;
        hence thesis by A8;
     suppose not( %x is completely-join-irreducible );
        then *'(%x) = %x by Def9;
        then A9: %x = "\/"(X,L) by Def7;
        set G = {H where H is Subset of JIRRS(L) :
                 ex u being Element of LattPOSet L st %u in X & %u = "\/"
(H,L)};
        set D' = union G;
          for v being set holds v in D' implies v in JIRRS(L)
            proof
            let v be set; assume v in D';
            then consider v' being set such that
            A10: v in v' & v' in G by TARSKI:def 4;
            consider H being Subset of JIRRS(L) such that A11: H = v' &
            ex u being Element of LattPOSet L st %u in X & %u = "\/"
(H,L) by A10;
            thus thesis by A10,A11;
            end;
        then A12: D' is Subset of JIRRS(L) by TARSKI:def 3;
          for q being Element of L st q in D' holds q [= %x
            proof
            let q be Element of L;
            assume q in D';
            then consider h being set such that A13: q in h & h in
 G by TARSKI:def 4;
            consider h' being Subset of JIRRS(L) such that A14: h' = h &
            ex u being Element of LattPOSet L st %u in
 X & %u = "\/"(h',L) by A13;
            consider u being Element of LattPOSet L such that
            A15: %u in X & %u = "\/"(h,L) by A14;
              h is_less_than %u by A15,LATTICE3:def 21;
            then A16: q [= %u by A13,LATTICE3:def 17;
              X is_less_than %x by A9,LATTICE3:def 21;
            then %u [= %x by A15,LATTICE3:def 17;
            hence thesis by A16,LATTICES:25;
            end;
        then A17: D' is_less_than %x by LATTICE3:def 17;
          for r being Element of L
        st D' is_less_than r holds %x [= r
            proof
            let r be Element of L;
            assume
A18:          D' is_less_than r;
              for q being Element of L st q in X holds q [= r
              proof
              let q be Element of L;
              assume A19: q in X;
              A20: q% = q by LATTICE3:def 3;
              consider D being Subset of JIRRS(L) such that
              A21: %(q%) = "\/"(D,L) by A7,A19;
              A22: q = "\/"(D,L) by A20,A21,LATTICE3:def 4;
                for v being set holds v in D implies v in D'
                 proof
                 let v be set; assume A23: v in D;
                   D in G by A19,A21,A22;
                 hence thesis by A23,TARSKI:def 4;
                 end;
              then D c= D' by TARSKI:def 3;
              then for p being Element of L
                   st p in D holds p [= r by A18,LATTICE3:def 17;
              then D is_less_than r by LATTICE3:def 17;
              hence thesis by A22,LATTICE3:def 21;
              end;
            then X is_less_than r by LATTICE3:def 17;
            hence thesis by A9,LATTICE3:def 21;
            end;
        then %x = "\/"(D',L) by A17,LATTICE3:def 21;
        hence thesis by A12;
     end;
A24: for x being Element of LattPOSet L holds P[x] from WFInduction(A2,A1);
  for a being Element of L holds
     ex D' being Subset of JIRRS(L) st a = "\/"(D',L)
  proof
  let a be Element of L;
    %(a%) = a% by LATTICE3:def 4 .= a by LATTICE3:def 3;
  hence thesis by A24;
  end;
hence thesis by Def13;
end;

definition
let L be noetherian complete Lattice;
cluster JIRRS(L) -> supremum-dense;
coherence by Lm11;
end;

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