### The Mizar article:

### Auxiliary and Approximating Relations

**by****Adam Grabowski**

- Received October 21, 1996
Copyright (c) 1996 Association of Mizar Users

- MML identifier: WAYBEL_4
- [ MML identifier index ]

environ vocabulary RELAT_2, ORDERS_1, RELAT_1, WAYBEL_3, LATTICES, LATTICE3, WAYBEL_0, BOOLE, YELLOW_1, SETFAM_1, BHSP_3, YELLOW_0, QUANTAL1, PRE_TOPC, FUNCT_1, FILTER_2, SEQM_3, PARTFUN1, CAT_1, GROUP_1, FUNCOP_1, FUNCT_2, CARD_3, RLVECT_2, WELLORD1, YELLOW_2, FILTER_0, WELLORD2, ORDINAL2, WAYBEL_2, SUBSET_1, ORDERS_2, WAYBEL_4; notation TARSKI, XBOOLE_0, ZFMISC_1, SUBSET_1, SETFAM_1, STRUCT_0, ORDERS_1, PRE_TOPC, PARTFUN1, LATTICE3, RELAT_1, RELAT_2, RELSET_1, FUNCT_1, FUNCT_2, WELLORD1, YELLOW_0, PRE_CIRC, CARD_3, PRALG_1, YELLOW_1, ALG_1, YELLOW_2, YELLOW_4, WAYBEL_0, WAYBEL_1, WAYBEL_2, WAYBEL_3; constructors ORDERS_3, PRE_CIRC, TOLER_1, WAYBEL_1, WAYBEL_2, WAYBEL_3, YELLOW_4, ALG_1; clusters LATTICE3, RELSET_1, STRUCT_0, WAYBEL_0, WAYBEL_2, WAYBEL_3, YELLOW_1, YELLOW_2, SUBSET_1, PARTFUN1, XBOOLE_0; requirements SUBSET, BOOLE; definitions LATTICE3, XBOOLE_0, TARSKI, YELLOW_0, WAYBEL_0, RELAT_1, YELLOW_2, ORDERS_3; theorems FUNCOP_1, FUNCT_1, FUNCT_2, LATTICE3, ORDERS_1, ORDERS_3, PARTFUN1, PRE_TOPC, RELAT_1, RELAT_2, RELSET_1, SETFAM_1, STRUCT_0, TARSKI, WAYBEL_0, WAYBEL_3, WAYBEL_2, WELLORD1, WELLORD2, YELLOW_0, YELLOW_1, YELLOW_2, YELLOW_4, ZFMISC_1, XBOOLE_0, XBOOLE_1; schemes FUNCT_2, PARTFUN1, RELSET_1, YELLOW_2, XBOOLE_0; begin :: Auxiliary Relations definition let L be non empty reflexive RelStr; canceled; func L-waybelow -> Relation of L means :Def2: for x,y being Element of L holds [x,y] in it iff x << y; existence proof defpred P[set,set] means ex x', y' be Element of L st x' = $1 & y' = $2 & x' << y'; consider R being Relation of the carrier of L, the carrier of L such that A1: for x,y being set holds [x,y] in R iff x in the carrier of L & y in the carrier of L & P[x,y] from Rel_On_Set_Ex; reconsider R as Relation of L ; take R; let x,y be Element of L; hereby assume [x,y] in R; then consider x', y' being Element of L such that A2: x' = x & y' = y & x' << y' by A1; thus x << y by A2; end; assume x << y; hence [x,y] in R by A1; end; uniqueness proof let A1,A2 be Relation of L; assume A3: for x,y being Element of L holds [x,y] in A1 iff x << y; assume A4: for x,y being Element of L holds [x,y] in A2 iff x << y; thus A1 = A2 proof let a,b be set; hereby assume A5: [a,b] in A1; then a in the carrier of L & b in the carrier of L by ZFMISC_1:106; then reconsider a' = a, b' = b as Element of L; a' << b' by A3,A5; hence [a,b] in A2 by A4; end; assume A6: [a,b] in A2; then a in the carrier of L & b in the carrier of L by ZFMISC_1:106; then reconsider a' = a, b' = b as Element of L; a' << b' by A4,A6; hence [a,b] in A1 by A3; end; end; end; definition let L be RelStr; func IntRel L -> Relation of L equals :Def3: the InternalRel of L; coherence ; correctness; end; Lm1: for L being RelStr, x, y being Element of L holds [x, y] in IntRel L iff x <= y proof let L be RelStr, x, y be Element of L; hereby assume [x, y] in IntRel L; then [x, y] in the InternalRel of L by Def3; hence x <= y by ORDERS_1:def 9; end; assume x <= y; then [x, y] in the InternalRel of L by ORDERS_1:def 9; hence [x, y] in IntRel L by Def3; end; definition let L be RelStr, R be Relation of L; attr R is auxiliary(i) means :Def4: for x, y being Element of L holds [x,y] in R implies x <= y; attr R is auxiliary(ii) means :Def5: for x, y, z, u being Element of L holds ( u <= x & [x,y] in R & y <= z implies [u,z] in R ); end; definition let L be non empty RelStr, R be Relation of L; attr R is auxiliary(iii) means :Def6: for x, y, z being Element of L holds ( [x,z] in R & [y,z] in R implies [(x "\/" y),z] in R ); attr R is auxiliary(iv) means :Def7: for x being Element of L holds [Bottom L,x] in R; end; :: Definition 1.9 p.43 definition let L be non empty RelStr, R be Relation of L; attr R is auxiliary means :Def8: R is auxiliary(i) auxiliary(ii) auxiliary(iii) auxiliary(iv); end; definition let L be non empty RelStr; cluster auxiliary -> auxiliary(i) auxiliary(ii) auxiliary(iii) auxiliary(iv) Relation of L; coherence by Def8; cluster auxiliary(i) auxiliary(ii) auxiliary(iii) auxiliary(iv) -> auxiliary Relation of L; coherence by Def8; end; definition let L be lower-bounded with_suprema transitive antisymmetric RelStr; cluster auxiliary Relation of L; existence proof set A = IntRel L; take A; thus A is auxiliary(i) proof let x, y be Element of L; assume [x,y] in A; then [x,y] in the InternalRel of L by Def3; hence thesis by ORDERS_1:def 9; end; thus A is auxiliary(ii) proof let x, y, z, u be Element of L; assume that A1: u <= x and A2: [x,y] in A and A3: y <= z; [x,y] in the InternalRel of L by A2,Def3; then x <= y by ORDERS_1:def 9; then u <= y by A1,ORDERS_1:26; then u <= z by A3,ORDERS_1:26; then [u,z] in the InternalRel of L by ORDERS_1:def 9; hence thesis by Def3; end; thus A is auxiliary(iii) proof let x, y, z be Element of L; assume that A4: [x,z] in A and A5: [y,z] in A; [x,z] in the InternalRel of L by A4,Def3; then A6: x <= z by ORDERS_1:def 9; [y,z] in the InternalRel of L by A5,Def3; then A7: y <= z by ORDERS_1:def 9; ex q being Element of L st x <= q & y <= q & for c being Element of L st x <= c & y <= c holds q <= c by LATTICE3:def 10; then (x "\/" y) <= z by A6,A7,LATTICE3:def 13; then [x "\/" y, z] in the InternalRel of L by ORDERS_1:def 9; hence thesis by Def3; end; thus A is auxiliary(iv) proof let x be Element of L; Bottom L <= x by YELLOW_0:44; then [Bottom L,x] in the InternalRel of L by ORDERS_1:def 9; hence thesis by Def3; end; end; end; theorem Th1: for L being lower-bounded sup-Semilattice for AR being auxiliary(ii) auxiliary(iii) Relation of L for x, y, z, u being Element of L holds [x, z] in AR & [y, u] in AR implies [x "\/" y, z "\/" u] in AR proof let L be lower-bounded sup-Semilattice; let AR be auxiliary(ii) auxiliary(iii) Relation of L; let x,y,z,u be Element of L; assume A1: [x, z] in AR & [y, u] in AR; A2: x <= x & y <= y; z <= z "\/" u & u <= z "\/" u by YELLOW_0:22; then [x, z "\/" u] in AR & [y, z "\/" u] in AR by A1,A2,Def5; hence [x "\/" y, z "\/" u] in AR by Def6; end; Lm2: for L be lower-bounded sup-Semilattice for AR be auxiliary(i) auxiliary(ii) Relation of L holds AR is transitive proof let L be lower-bounded sup-Semilattice; let AR be auxiliary(i) auxiliary(ii) Relation of L; A1: field AR c= (the carrier of L) \/ (the carrier of L) by RELSET_1:19; now let x,y,z be set; assume A2: x in field AR & y in field AR & z in field AR & [x,y] in AR & [y,z] in AR; then reconsider x'=x, y'=y, z'=z as Element of L by A1; reconsider x', y', z' as Element of L; A3: x' <= y' by A2,Def4; z' <= z'; hence [x,z] in AR by A2,A3,Def5; end; then AR is_transitive_in field AR by RELAT_2:def 8; hence thesis by RELAT_2:def 16; end; definition let L be lower-bounded sup-Semilattice; cluster auxiliary(i) auxiliary(ii) -> transitive Relation of L; coherence by Lm2; end; definition let L be RelStr; cluster IntRel L -> auxiliary(i); coherence proof let x, y be Element of L; assume [x, y] in IntRel L; hence x <= y by Lm1; end; end; definition let L be transitive RelStr; cluster IntRel L -> auxiliary(ii); coherence proof set A = IntRel L; let x,y,z,u be Element of L; assume that A1: u <= x and A2: [x,y] in A and A3: y <= z; x <= y by A2,Lm1; then u <= y by A1,ORDERS_1:26; then u <= z by A3,ORDERS_1:26; hence [u,z] in A by Lm1; end; end; definition let L be with_suprema antisymmetric RelStr; cluster IntRel L -> auxiliary(iii); coherence proof set A = IntRel L; let x,y,z be Element of L; assume that A1: [x,z] in A and A2: [y,z] in A; A3: x <= z by A1,Lm1; A4: y <= z by A2,Lm1; ex q being Element of L st x <= q & y <= q & for c being Element of L st x <= c & y <= c holds q <= c by LATTICE3:def 10; then x "\/" y <= z by A3,A4,LATTICE3:def 13; hence thesis by Lm1; end; end; definition let L be lower-bounded antisymmetric non empty RelStr; cluster IntRel L -> auxiliary(iv); coherence proof set A = IntRel L; for x be Element of L holds [Bottom L,x] in A proof let x be Element of L; Bottom L <= x by YELLOW_0:44; hence thesis by Lm1; end; hence A is auxiliary(iv) by Def7; end; end; reserve a for set; definition let L be lower-bounded sup-Semilattice; func Aux L means :Def9: a in it iff a is auxiliary Relation of L; existence proof defpred P[set] means $1 is auxiliary Relation of L; consider X be set such that A1: for a holds a in X iff a in bool [:the carrier of L,the carrier of L:] & P[a] from Separation; take X; thus thesis by A1; end; uniqueness proof let A1,A2 be set such that A2: a in A1 iff a is auxiliary Relation of L and A3: a in A2 iff a is auxiliary Relation of L; thus A1 = A2 proof thus A1 c= A2 proof let a; assume a in A1; then a is auxiliary Relation of L by A2; hence thesis by A3; end; let a; assume a in A2; then a is auxiliary Relation of L by A3; hence thesis by A2; end; end; end; definition let L be lower-bounded sup-Semilattice; cluster Aux L -> non empty; coherence proof IntRel L is auxiliary Relation of L; hence thesis by Def9; end; end; Lm3: for L being lower-bounded sup-Semilattice for AR being auxiliary(i) Relation of L for a,b being set st [a,b] in AR holds [a,b] in IntRel L proof let L be lower-bounded sup-Semilattice; let AR be auxiliary(i) Relation of L; let a,b be set; assume A1: [a,b] in AR; then a in the carrier of L & b in the carrier of L by ZFMISC_1:106; then reconsider a' = a, b' = b as Element of L; a' <= b' by A1,Def4; hence thesis by Lm1; end; theorem Th2: for L being lower-bounded sup-Semilattice for AR being auxiliary(i) Relation of L holds AR c= IntRel L proof let L be lower-bounded sup-Semilattice; let AR be auxiliary(i) Relation of L; let a,b be set; assume [a,b] in AR; hence thesis by Lm3; end; theorem for L being lower-bounded sup-Semilattice holds Top InclPoset Aux L = IntRel L proof let L be lower-bounded sup-Semilattice; IntRel L = "/\"({},InclPoset Aux L) proof set P = InclPoset Aux L; set I = IntRel L; I in Aux L by Def9; then I in the carrier of P by YELLOW_1:1; then reconsider I as Element of P; A1: I is_<=_than {} by YELLOW_0:6; for b being Element of P st b is_<=_than {} holds I >= b proof let b be Element of P; b in the carrier of InclPoset Aux L; then b in Aux L by YELLOW_1:1; then reconsider b' = b as auxiliary Relation of L by Def9; assume b is_<=_than {}; b' c= I by Th2; hence thesis by YELLOW_1:3; end; hence thesis by A1,YELLOW_0:31; end; hence thesis by YELLOW_0:def 12; end; definition let L be lower-bounded sup-Semilattice; cluster InclPoset Aux L -> upper-bounded; coherence proof set I = IntRel L; I in Aux L by Def9; then I in the carrier of InclPoset Aux L by YELLOW_1:1; then reconsider I as Element of InclPoset Aux L; take I; I is_>=_than the carrier of InclPoset Aux L proof let b be Element of InclPoset Aux L; assume b in the carrier of InclPoset Aux L; then b in Aux L by YELLOW_1:1; then reconsider b' = b as auxiliary Relation of L by Def9; b' c= I by Th2; hence thesis by YELLOW_1:3; end; hence thesis; end; end; definition let L be non empty RelStr; func AuxBottom L -> Relation of L means :Def10: for x,y be Element of L holds [x,y] in it iff x = Bottom L; existence proof defpred P[set,set] means $1 = Bottom L; consider R being Relation of the carrier of L, the carrier of L such that A1: for a,b be set holds [a,b] in R iff a in the carrier of L & b in the carrier of L & P[a,b] from Rel_On_Set_Ex; reconsider R as Relation of L ; take R; let x,y be Element of L; thus [x,y] in R implies x = Bottom L by A1; assume x = Bottom L; hence [x,y] in R by A1; end; uniqueness proof let A1,A2 be Relation of L; assume A2: for x,y be Element of L holds [x,y] in A1 iff x = Bottom L; assume A3: for x,y be Element of L holds [x,y] in A2 iff x = Bottom L; thus A1 = A2 proof let a,b be set; hereby assume A4: [a,b] in A1; then a in the carrier of L & b in the carrier of L by ZFMISC_1:106; then reconsider a' = a, b' = b as Element of L; [a',b'] in A1 by A4; then a' = Bottom L by A2; then [a',b'] in A2 by A3; hence [a,b] in A2; end; assume A5: [a,b] in A2; then a in the carrier of L & b in the carrier of L by ZFMISC_1:106; then reconsider a' = a, b' = b as Element of L; [a',b'] in A2 by A5; then a' = Bottom L by A3; then [a',b'] in A1 by A2; hence [a,b] in A1; end; end; end; definition let L be lower-bounded sup-Semilattice; cluster AuxBottom L -> auxiliary; coherence proof set A = AuxBottom L; A1:A is auxiliary(i) proof let x, y be Element of L; assume [x,y] in A; then x = Bottom L by Def10; hence thesis by YELLOW_0:44; end; A2:A is auxiliary(ii) proof let x, y, z, u be Element of L; assume A3: u <= x & [x,y] in A & y <= z; then A4: x = Bottom L by Def10; Bottom L <= u by YELLOW_0:44; then u = Bottom L by A3,A4,ORDERS_1:25; hence thesis by Def10; end; A5:A is auxiliary(iii) proof let x, y, z be Element of L; assume [x,z] in A & [y,z] in A; then A6: x = Bottom L & y = Bottom L by Def10; then x <= y; then x "\/" y = Bottom L by A6,YELLOW_0:24; hence thesis by Def10; end; for x be Element of L holds [Bottom L,x] in A by Def10; then A is auxiliary(iv) by Def7; hence thesis by A1,A2,A5,Def8; end; end; theorem Th4: for L being lower-bounded sup-Semilattice for AR being auxiliary(iv) Relation of L holds AuxBottom L c= AR proof let L be with_suprema lower-bounded Poset; let AR be auxiliary(iv) Relation of L; let a,b be set; assume A1: [a,b] in AuxBottom L; then a in the carrier of L & b in the carrier of L by ZFMISC_1:106; then reconsider a' = a, b' = b as Element of L; [a',b'] in AuxBottom L by A1; then a' = Bottom L by Def10; then [a',b'] in AR by Def7; hence [a,b] in AR; end; theorem for L being lower-bounded sup-Semilattice for AR being auxiliary(iv) Relation of L holds Bottom InclPoset Aux L = AuxBottom L proof let L be with_suprema lower-bounded Poset; let AR be auxiliary(iv) Relation of L; AuxBottom L in Aux L by Def9; then AuxBottom L in the carrier of InclPoset Aux L by YELLOW_1:1; then reconsider N = AuxBottom L as Element of InclPoset Aux L; AuxBottom L = "\/"({},InclPoset Aux L) proof A1: N is_>=_than {} by YELLOW_0:6; for b being Element of InclPoset Aux L st b is_>=_than {} holds N <= b proof let b be Element of InclPoset Aux L; assume b is_>=_than {}; b in the carrier of InclPoset Aux L; then b in Aux L by YELLOW_1:1; then reconsider b' = b as auxiliary Relation of L by Def9; N c= b' by Th4; hence thesis by YELLOW_1:3; end; hence thesis by A1,YELLOW_0:30; end; hence thesis by YELLOW_0:def 11; end; definition let L be lower-bounded sup-Semilattice; cluster InclPoset Aux L -> lower-bounded; coherence proof set I = AuxBottom L; I in Aux L by Def9; then I in the carrier of InclPoset Aux L by YELLOW_1:1; then reconsider I as Element of InclPoset Aux L; take I; I is_<=_than the carrier of InclPoset Aux L proof let b be Element of InclPoset Aux L; assume b in the carrier of InclPoset Aux L; then b in Aux L by YELLOW_1:1; then reconsider b' = b as auxiliary Relation of L by Def9; I c= b' by Th4; hence thesis by YELLOW_1:3; end; hence thesis; end; end; theorem Th6: for L being lower-bounded sup-Semilattice for a,b being auxiliary(i) Relation of L holds a /\ b is auxiliary(i) Relation of L proof let L be with_suprema lower-bounded Poset; let a,b be auxiliary(i) Relation of L; reconsider ab = a /\ b as Relation of L ; for x, y be Element of L holds [x,y] in ab implies x <= y proof let x, y be Element of L; assume [x,y] in ab; then [x,y] in a & [x,y] in b by XBOOLE_0:def 3; hence x <= y by Def4; end; hence thesis by Def4; end; theorem Th7: for L being lower-bounded sup-Semilattice for a,b being auxiliary(ii) Relation of L holds a /\ b is auxiliary(ii) Relation of L proof let L be with_suprema lower-bounded Poset; let a,b be auxiliary(ii) Relation of L; reconsider ab = a /\ b as Relation of L ; for x, y, z, u be Element of L holds u <= x & [x,y] in ab & y <= z implies [u,z] in ab proof let x, y, z, u be Element of L; assume A1: u <= x & [x,y] in ab & y <= z; then [x,y] in a & [x,y] in b by XBOOLE_0:def 3; then [u,z] in a & [u,z] in b by A1,Def5; hence thesis by XBOOLE_0:def 3; end; hence thesis by Def5; end; theorem Th8: for L being lower-bounded sup-Semilattice for a,b being auxiliary(iii) Relation of L holds a /\ b is auxiliary(iii) Relation of L proof let L be with_suprema lower-bounded Poset; let a,b be auxiliary(iii) Relation of L; reconsider ab = a /\ b as Relation of L ; for x, y, z be Element of L holds [x,z] in ab & [y,z] in ab implies [(x "\/" y),z] in ab proof let x, y, z be Element of L; assume [x,z] in ab & [y,z] in ab; then [x,z] in a & [x,z] in b & [y,z] in a & [y,z] in b by XBOOLE_0:def 3; then [(x "\/" y),z] in a & [(x "\/" y),z] in b by Def6; hence thesis by XBOOLE_0:def 3; end; hence thesis by Def6; end; theorem Th9: for L being lower-bounded sup-Semilattice for a,b being auxiliary(iv) Relation of L holds a /\ b is auxiliary(iv) Relation of L proof let L be with_suprema lower-bounded Poset; let a,b be auxiliary(iv) Relation of L; reconsider ab = a /\ b as Relation of L ; for x be Element of L holds [Bottom L,x] in ab proof let x be Element of L; [Bottom L,x] in a & [Bottom L,x] in b by Def7; hence thesis by XBOOLE_0:def 3; end; hence thesis by Def7; end; theorem Th10: for L being lower-bounded sup-Semilattice for a,b being auxiliary Relation of L holds a /\ b is auxiliary Relation of L proof let L be with_suprema lower-bounded Poset; let a,b be auxiliary Relation of L; reconsider ab = a /\ b as Relation of L ; ab is auxiliary(i) auxiliary(ii) auxiliary(iii) auxiliary(iv) by Th6,Th7,Th8,Th9; hence thesis by Def8; end; theorem Th11: for L being lower-bounded sup-Semilattice for X being non empty Subset of InclPoset Aux L holds meet X is auxiliary Relation of L proof let L be with_suprema lower-bounded Poset; let X be non empty Subset of InclPoset Aux L; X c= the carrier of InclPoset Aux L; then A1: X c= Aux L by YELLOW_1:1; consider a be Element of X; a in X; then a is auxiliary Relation of L by A1,Def9; then meet X c= [:the carrier of L,the carrier of L:] by SETFAM_1:8; then meet X is Relation of the carrier of L by RELSET_1:def 1; then reconsider ab = meet X as Relation of L ; now thus ab is auxiliary(i) proof let x, y be Element of L; assume A2: [x,y] in ab; consider V be Element of X; A3: V in X; A4: [x,y] in V by A2,SETFAM_1:def 1; V is auxiliary Relation of L by A1,A3,Def9; hence x <= y by A4,Def4; end; thus ab is auxiliary(ii) proof let x, y, z, u be Element of L; assume A5: u <= x & [x,y] in ab & y <= z; for Y be set st Y in X holds [u,z] in Y proof let Y be set; assume A6: Y in X; then A7: Y is auxiliary Relation of L by A1,Def9; [x,y] in Y by A5,A6,SETFAM_1:def 1; hence thesis by A5,A7,Def5; end; hence thesis by SETFAM_1:def 1; end; thus ab is auxiliary(iii) proof let x, y, z be Element of L; assume A8: [x,z] in ab & [y,z] in ab; for Y be set st Y in X holds [(x "\/" y),z] in Y proof let Y be set; assume A9: Y in X; then A10: Y is auxiliary Relation of L by A1,Def9; [x,z] in Y & [y,z] in Y by A8,A9,SETFAM_1:def 1; hence thesis by A10,Def6; end; hence thesis by SETFAM_1:def 1; end; thus ab is auxiliary(iv) proof let x be Element of L; for Y be set st Y in X holds [Bottom L,x] in Y proof let Y be set; assume Y in X; then Y is auxiliary Relation of L by A1,Def9; hence thesis by Def7; end; hence thesis by SETFAM_1:def 1; end; end; hence thesis by Def8; end; definition let L be lower-bounded sup-Semilattice; cluster InclPoset Aux L -> with_infima; coherence proof for x, y be set st (x in Aux L & y in Aux L) holds x /\ y in Aux L proof let x, y be set; assume x in Aux L & y in Aux L; then x is auxiliary Relation of L & y is auxiliary Relation of L by Def9; then x /\ y is auxiliary Relation of L by Th10; hence x /\ y in Aux L by Def9; end; hence InclPoset Aux L is with_infima by YELLOW_1:12; end; end; definition let L be lower-bounded sup-Semilattice; cluster InclPoset Aux L -> complete; coherence proof for X be Subset of InclPoset Aux L holds ex_inf_of X, InclPoset Aux L proof let X be Subset of InclPoset Aux L; set N = meet X; per cases; suppose A1: X <> {}; then N is auxiliary Relation of L by Th11; then N in Aux L by Def9; then N in the carrier of InclPoset Aux L by YELLOW_1:1; then reconsider N as Element of InclPoset Aux L; A2: X is_>=_than N proof let b be Element of InclPoset Aux L; assume b in X; then N c= b by SETFAM_1:4; hence N <= b by YELLOW_1:3; end; for b being Element of InclPoset Aux L st X is_>=_than b holds N >= b proof let b be Element of InclPoset Aux L; assume A3: X is_>=_than b; for Z1 be set st Z1 in X holds b c= Z1 proof let Z1 be set; assume A4: Z1 in X; then reconsider Z1' = Z1 as Element of InclPoset Aux L; b <= Z1' by A3,A4,LATTICE3:def 8; hence b c= Z1 by YELLOW_1:3; end; then b c= meet X by A1,SETFAM_1:6; hence thesis by YELLOW_1:3; end; hence thesis by A2,YELLOW_0:16; suppose X = {}; hence thesis by YELLOW_0:43; end; hence thesis by YELLOW_2:30; end; end; definition let L be non empty RelStr, x be Element of L; let AR be Relation of the carrier of L; A1: {y where y is Element of L : [y,x] in AR} c= the carrier of L proof let z be set; assume z in {y where y is Element of L: [y,x] in AR}; then ex y being Element of L st z = y & [y,x] in AR; hence thesis; end; func AR-below x -> Subset of L equals :Def11: {y where y is Element of L: [y,x] in AR}; correctness by A1; A2: {y where y is Element of L: [x,y] in AR} c= the carrier of L proof let z be set; assume z in {y where y is Element of L: [x,y] in AR}; then ex y being Element of L st z = y & [x,y] in AR; hence thesis; end; func AR-above x -> Subset of L equals :Def12: {y where y is Element of L: [x,y] in AR}; correctness by A2; end; theorem Th12: for L being lower-bounded sup-Semilattice, x being Element of L for AR being auxiliary(i) Relation of L holds AR-below x c= downarrow x proof let L be lower-bounded sup-Semilattice, x be Element of L, AR be auxiliary(i) Relation of L; let a; A1:AR-below x = {y where y is Element of L: [y,x] in AR} by Def11; assume a in AR-below x; then consider y1 be Element of L such that A2: y1 = a & [y1,x] in AR by A1; y1 <= x by A2,Def4; hence thesis by A2,WAYBEL_0:17; end; definition let L be lower-bounded sup-Semilattice, x be Element of L; let AR be auxiliary(iv) Relation of L; cluster AR-below x -> non empty; coherence proof set I = AR-below x; A1:I = {y where y is Element of L: [y,x] in AR} by Def11; [Bottom L,x] in AR by Def7; then Bottom L in I by A1; hence thesis; end; end; definition let L be lower-bounded sup-Semilattice, x be Element of L; let AR be auxiliary(ii) Relation of L; cluster AR-below x -> lower; coherence proof set I = AR-below x; A1:I = {y where y is Element of L: [y,x] in AR} by Def11; let x1, y1 be Element of L; assume A2: x1 in I & y1 <= x1; then consider v1 be Element of L such that A3: v1 = x1 & [v1, x] in AR by A1; x <= x; then [y1, x] in AR by A2,A3,Def5; hence thesis by A1; end; end; definition let L be lower-bounded sup-Semilattice, x be Element of L; let AR be auxiliary(iii) Relation of L; cluster AR-below x -> directed; coherence proof set I = AR-below x; A1:I = {y where y is Element of L: [y,x] in AR} by Def11; let u1,u2 be Element of L; assume A2: u1 in I & u2 in I; then consider v1 be Element of L such that A3: v1 = u1 & [v1,x] in AR by A1; consider v2 be Element of L such that A4: v2 = u2 & [v2,x] in AR by A1,A2; take u12 = u1 "\/" u2; [u12,x] in AR by A3,A4,Def6; hence thesis by A1,YELLOW_0:22; end; end; definition let L be lower-bounded sup-Semilattice; let AR be auxiliary(ii) auxiliary(iii) auxiliary(iv) Relation of L; func AR-below -> map of L, InclPoset Ids L means :Def13: for x be Element of L holds it.x = AR-below x; existence proof deffunc F(Element of L) = AR-below $1; A1: for x being Element of L holds F(x) is Element of InclPoset Ids L proof let x be Element of L; reconsider I = F(x) as Ideal of L; I in {X where X is Ideal of L: not contradiction}; then I in Ids L by WAYBEL_0:def 23; then I in the carrier of InclPoset Ids L by YELLOW_1:1; hence thesis; end; consider f being map of L, InclPoset Ids L such that A2: for x being Element of L holds f.x = F(x) from KappaMD(A1); take f; thus thesis by A2; end; uniqueness proof let M1, M2 be map of L, InclPoset Ids L; assume A3: for x be Element of L holds M1.x = AR-below x; assume A4: for x be Element of L holds M2.x = AR-below x; for x be set st x in the carrier of L holds M1.x = M2.x proof let x be set; assume x in the carrier of L; then reconsider x' = x as Element of L; thus M1.x = AR-below x' by A3 .= M2.x by A4; end; hence thesis by FUNCT_2:18; end; end; theorem Th13: for L being non empty RelStr, AR being Relation of L for a being set, y being Element of L holds a in AR-below y iff [a,y] in AR proof let L be non empty RelStr, AR be Relation of L; let a be set, y be Element of L; A1: AR-below y = {z where z is Element of L: [z,y] in AR} by Def11; a in AR-below y iff [a,y] in AR proof hereby assume a in AR-below y; then consider z being Element of L such that A2: a = z & [z,y] in AR by A1 ; thus [a,y] in AR by A2; end; assume A3: [a,y] in AR; then a in the carrier of L by ZFMISC_1:106; then reconsider x' = a as Element of L; ex z being Element of L st a = z & [z,y] in AR proof take x'; thus thesis by A3; end; hence a in AR-below y by A1; end; hence thesis; end; theorem for L being sup-Semilattice, AR being Relation of L for y being Element of L holds a in AR-above y iff [y,a] in AR proof let L be with_suprema Poset, AR be Relation of L; let y be Element of L; A1: AR-above y = {z where z is Element of L: [y,z] in AR} by Def12; a in AR-above y iff [y,a] in AR proof hereby assume a in AR-above y; then consider z being Element of L such that A2: a = z & [y,z] in AR by A1 ; thus [y,a] in AR by A2; end; assume A3: [y,a] in AR; then a in the carrier of L by ZFMISC_1:106; then reconsider x' = a as Element of L; ex z being Element of L st a = z & [y,z] in AR proof take x'; thus thesis by A3; end; hence a in AR-above y by A1; end; hence thesis; end; Lm4: for L be with_suprema lower-bounded Poset, AR be Relation of L for a be set, y be Element of L holds a in downarrow y iff [a,y] in the InternalRel of L proof let L be with_suprema lower-bounded Poset, AR be Relation of L; let a be set, y be Element of L; hereby assume A1: a in downarrow y; then reconsider a' = a as Element of L; a' <= y by A1,WAYBEL_0:17; hence [a,y] in the InternalRel of L by ORDERS_1:def 9; end; assume A2: [a,y] in the InternalRel of L; then a in the carrier of L by ZFMISC_1:106; then reconsider a' = a as Element of L; a' <= y by A2,ORDERS_1:def 9; hence thesis by WAYBEL_0:17; end; theorem for L being lower-bounded sup-Semilattice, AR being auxiliary(i) Relation of L for x being Element of L holds AR = the InternalRel of L implies AR-below x = downarrow x proof let L be lower-bounded sup-Semilattice, AR be auxiliary(i) Relation of L; let x be Element of L; assume A1: AR = the InternalRel of L; thus AR-below x c= downarrow x by Th12; thus downarrow x c= AR-below x proof let a; assume a in downarrow x; then [a,x] in AR by A1,Lm4; hence thesis by Th13; end; end; definition let L be non empty Poset; func MonSet L -> strict RelStr means :Def14: ( a in the carrier of it iff ex s be map of L, InclPoset Ids L st a = s & s is monotone & for x be Element of L holds s.x c= downarrow x ) & for c, d be set holds [c,d] in the InternalRel of it iff ex f,g be map of L, InclPoset Ids L st c = f & d = g & c in the carrier of it & d in the carrier of it & f <= g; existence proof defpred P[set] means ex s be map of L, InclPoset Ids L st $1 = s & s is monotone & for x be Element of L holds s.x c= downarrow x; consider S be set such that A1: a in S iff a in PFuncs (the carrier of L, the carrier of InclPoset Ids L) & P[a] from Separation; A2: a in S iff ex s be map of L, InclPoset Ids L st a = s & s is monotone & for x be Element of L holds s.x c= downarrow x proof thus a in S implies ex s be map of L, InclPoset Ids L st a = s & s is monotone & for x be Element of L holds s.x c= downarrow x by A1; given s be map of L, InclPoset Ids L such that A3: a = s & s is monotone & for x be Element of L holds s.x c= downarrow x; s in PFuncs (the carrier of L, the carrier of InclPoset Ids L) by PARTFUN1:119; hence a in S by A1,A3; end; defpred P[set,set] means ex f,g be map of L, InclPoset Ids L st $1 = f & $2 = g & f <= g; consider R be Relation of S, S such that A4: for c, d be set holds [c,d] in R iff c in S & d in S & P[c,d] from Rel_On_Set_Ex; A5: for c, d be set holds [c,d] in R iff ex f,g be map of L, InclPoset Ids L st c = f & d = g & c in S & d in S & f <= g proof let c,d be set; hereby assume A6: [c,d] in R; then A7: c in S & d in S by A4; consider f,g be map of L, InclPoset Ids L such that A8: c = f & d = g & f <= g by A4,A6; thus ex f,g be map of L, InclPoset Ids L st c = f & d = g & c in S & d in S & f <= g by A7,A8; end; given f,g be map of L, InclPoset Ids L such that A9: c = f & d = g & c in S & d in S & f <= g; thus [c,d] in R by A4,A9; end; take RelStr (#S,R#); thus thesis by A2,A5; end; uniqueness proof let R1, R2 be strict RelStr; assume A10: ( a in the carrier of R1 iff ex s be map of L, InclPoset Ids L st a = s & s is monotone & for x be Element of L holds s.x c= downarrow x ) & for c, d be set holds [c,d] in the InternalRel of R1 iff ex f,g be map of L, InclPoset Ids L st c = f & d = g & c in the carrier of R1 & d in the carrier of R1 & f <= g; assume A11: ( a in the carrier of R2 iff ex s be map of L, InclPoset Ids L st a = s & s is monotone & for x be Element of L holds s.x c= downarrow x ) & for c, d be set holds [c,d] in the InternalRel of R2 iff ex f,g be map of L, InclPoset Ids L st c = f & d = g & c in the carrier of R2 & d in the carrier of R2 & f <= g; A12: the carrier of R1 = the carrier of R2 proof thus the carrier of R1 c= the carrier of R2 proof let b be set; assume b in the carrier of R1; then consider s be map of L, InclPoset Ids L such that A13: b = s & s is monotone & for x be Element of L holds s.x c= downarrow x by A10; thus b in the carrier of R2 by A11,A13; end; thus the carrier of R2 c= the carrier of R1 proof let b be set; assume b in the carrier of R2; then consider s be map of L, InclPoset Ids L such that A14: b = s & s is monotone & for x be Element of L holds s.x c= downarrow x by A11; thus b in the carrier of R1 by A10,A14; end; end; the InternalRel of R1 = the InternalRel of R2 proof let c,d be set; A15: now assume [c,d] in the InternalRel of R1; then consider f,g be map of L, InclPoset Ids L such that A16: c = f & d = g & c in the carrier of R1 & d in the carrier of R1 & f <= g by A10; thus [c,d] in the InternalRel of R2 by A11,A12,A16; end; now assume [c,d] in the InternalRel of R2; then consider f,g be map of L, InclPoset Ids L such that A17: c = f & d = g & c in the carrier of R2 & d in the carrier of R2 & f <= g by A11; thus [c,d] in the InternalRel of R1 by A10,A12,A17; end; hence thesis by A15; end; hence thesis by A12; end; end; theorem for L being lower-bounded sup-Semilattice holds MonSet L is full SubRelStr of (InclPoset Ids L)|^(the carrier of L) proof let L be lower-bounded sup-Semilattice; set J = ((the carrier of L) --> InclPoset Ids L); A1: the carrier of MonSet L c= the carrier of (InclPoset Ids L)|^(the carrier of L) proof let a be set; assume a in the carrier of MonSet L; then consider s be map of L, InclPoset Ids L such that A2: a = s & s is monotone & for x be Element of L holds s.x c= downarrow x by Def14; s in Funcs (the carrier of L, the carrier of InclPoset Ids L) by FUNCT_2:11; hence thesis by A2,YELLOW_1:28; end; A3:the InternalRel of MonSet L c= the InternalRel of (InclPoset Ids L)|^(the carrier of L) proof let a,b be set; assume [a,b] in the InternalRel of MonSet L; then consider f,g be map of L, InclPoset Ids L such that A4: a = f & b = g & a in the carrier of MonSet L & b in the carrier of MonSet L & f <= g by Def14; set AG = product ((the carrier of L) --> InclPoset Ids L); A5: AG = ((InclPoset Ids L) |^ the carrier of L) by YELLOW_1:def 5; A6: f in Funcs (the carrier of L, the carrier of InclPoset Ids L) & g in Funcs (the carrier of L, the carrier of InclPoset Ids L) by FUNCT_2:11; then A7: f in the carrier of AG & g in the carrier of AG by A5,YELLOW_1:28; reconsider f' = f, g' = g as Element of AG by A5,A6,YELLOW_1:28; A8: f' in product Carrier ((the carrier of L) --> InclPoset Ids L) by A7,YELLOW_1:def 4; ex ff,gg being Function st ff = f' & gg = g' & for i be set st i in the carrier of L ex R being RelStr, xi,yi being Element of R st R = J.i & xi = ff.i & yi = gg.i & xi <= yi proof take f,g; thus f = f' & g = g'; let i be set; assume A9: i in the carrier of L; then reconsider i' = i as Element of L; take R = InclPoset Ids L; reconsider xi = f.i', yi = g.i' as Element of R; take xi, yi; thus R = J.i & xi = f.i & yi = g.i by A9,FUNCOP_1:13; reconsider i' = i as Element of L by A9; consider a, b be Element of InclPoset Ids L such that A10: a = f.i' & b = g.i' & a <= b by A4,YELLOW_2:def 1; thus thesis by A10; end; then f' <= g' by A8,YELLOW_1:def 4; then [a,b] in the InternalRel of product ((the carrier of L) --> InclPoset Ids L) by A4,ORDERS_1:def 9; hence thesis by YELLOW_1:def 5; end; set J = ((the carrier of L) --> InclPoset Ids L); the InternalRel of MonSet L = (the InternalRel of (InclPoset Ids L)|^(the carrier of L) )|_2 the carrier of MonSet L proof let a,b be set; thus [a,b] in the InternalRel of MonSet L implies [a,b] in (the InternalRel of (InclPoset Ids L)|^(the carrier of L))|_2 the carrier of MonSet L by A3,WELLORD1:16; assume [a,b] in (the InternalRel of (InclPoset Ids L)|^ (the carrier of L) )|_2 the carrier of MonSet L; then A11: [a,b] in (the InternalRel of (InclPoset Ids L)|^ (the carrier of L) ) & [a,b] in [:the carrier of MonSet L, the carrier of MonSet L:] by WELLORD1:16; then A12: a in the carrier of (InclPoset Ids L)|^(the carrier of L) & b in the carrier of (InclPoset Ids L)|^(the carrier of L) by ZFMISC_1:106; then A13: a in the carrier of product J & b in the carrier of product J by YELLOW_1:def 5; reconsider a' = a, b' = b as Element of product J by A12,YELLOW_1:def 5; [a',b'] in (the InternalRel of product J) by A11,YELLOW_1:def 5; then A14: a' <= b' by ORDERS_1:def 9; a' in product Carrier J by A13,YELLOW_1:def 4; then consider f,g being Function such that A15: f = a' & g = b' & for i be set st i in the carrier of L ex R being RelStr, xi,yi being Element of R st R = ((the carrier of L) --> InclPoset Ids L).i & xi = f.i & yi = g.i & xi <= yi by A14,YELLOW_1:def 4; f in Funcs (the carrier of L, the carrier of InclPoset Ids L) & g in Funcs (the carrier of L, the carrier of InclPoset Ids L) by A12,A15,YELLOW_1:28; then reconsider f, g as Function of the carrier of L, the carrier of InclPoset Ids L by FUNCT_2:121; reconsider f, g as map of L, InclPoset Ids L ; now take f, g; f <= g proof let j be set; assume j in the carrier of L; then reconsider j' = j as Element of L; take f.j', g.j'; consider R being RelStr, xi,yi being Element of R such that A16: R = ((the carrier of L) --> InclPoset Ids L).j' & xi = f.j' & yi = g.j' & xi <= yi by A15; thus thesis by A16,FUNCOP_1:13; end; hence a' = f & b' = g & a' in the carrier of MonSet L & b' in the carrier of MonSet L & f <= g by A11,A15,ZFMISC_1:106; end; hence [a,b] in the InternalRel of MonSet L by Def14; end; hence MonSet L is full SubRelStr of (InclPoset Ids L)|^(the carrier of L) by A1,A3,YELLOW_0:def 13,def 14; end; theorem Th17: for L being lower-bounded sup-Semilattice, AR being auxiliary(ii) Relation of L for x, y being Element of L holds x <= y implies AR-below x c= AR-below y proof let L be lower-bounded sup-Semilattice, AR be auxiliary(ii) Relation of L; let x, y be Element of L; assume A1: x <= y; A2:AR-below x = {y1 where y1 is Element of L: [y1,x] in AR} by Def11; let a; assume a in AR-below x; then consider y2 be Element of L such that A3: y2 = a & [y2,x] in AR by A2; y2 <= y2; then [y2,y] in AR by A1,A3,Def5; hence thesis by A3,Th13; end; definition let L be lower-bounded sup-Semilattice; let AR be auxiliary(ii) auxiliary(iii) auxiliary(iv) Relation of L; cluster AR-below -> monotone; coherence proof set s = AR-below; let x, y be Element of L; A1: s.x = AR-below x & s.y = AR-below y by Def13; assume x <= y; then AR-below x c= AR-below y by Th17; hence thesis by A1,YELLOW_1:3; end; end; theorem Th18: for L being lower-bounded sup-Semilattice, AR being auxiliary Relation of L holds AR-below in the carrier of MonSet L proof let L be lower-bounded sup-Semilattice, AR be auxiliary Relation of L; set s = AR-below; ex s be map of L, InclPoset Ids L st AR-below = s & s is monotone & for x be Element of L holds s.x c= downarrow x proof take s; for x be Element of L holds s.x c= downarrow x proof let x be Element of L; s.x = AR-below x by Def13; hence s.x c= downarrow x by Th12; end; hence thesis; end; hence thesis by Def14; end; definition let L be lower-bounded sup-Semilattice; cluster MonSet L -> non empty; coherence proof consider AR be auxiliary Relation of L; AR-below in the carrier of MonSet L by Th18; hence thesis by STRUCT_0:def 1; end; end; theorem Th19: for L being lower-bounded sup-Semilattice holds IdsMap L in the carrier of MonSet L proof let L be lower-bounded sup-Semilattice; set s = IdsMap L; ex s' be map of L, InclPoset Ids L st s = s' & s' is monotone & for x be Element of L holds s'.x c= downarrow x proof take s; thus thesis by YELLOW_2:def 4; end; hence thesis by Def14; end; theorem for L being lower-bounded sup-Semilattice, AR being auxiliary Relation of L holds AR-below <= IdsMap L proof let L be lower-bounded sup-Semilattice, AR be auxiliary Relation of L; let x be set; assume x in the carrier of L; then reconsider x' = x as Element of L; A1: (AR-below).x' = AR-below x' by Def13; (IdsMap L).x' = downarrow x' by YELLOW_2:def 4; then A2: (AR-below).x c= (IdsMap L).x by A1,Th12; reconsider a = (AR-below).x', b = (IdsMap L).x' as Element of InclPoset Ids L; take a, b; thus thesis by A2,YELLOW_1:3; end; theorem Th21: for L being lower-bounded non empty Poset for I being Ideal of L holds Bottom L in I proof let L be lower-bounded non empty Poset; let I be Ideal of L; consider x be Element of I; Bottom L <= x by YELLOW_0:44; hence thesis by WAYBEL_0:def 19; end; theorem for L being upper-bounded non empty Poset for F being Filter of L holds Top L in F proof let L be upper-bounded non empty Poset; let F be Filter of L; consider x be Element of F; Top L >= x by YELLOW_0:45; hence thesis by WAYBEL_0:def 20; end; theorem Th23: for L being lower-bounded non empty Poset holds downarrow Bottom L = {Bottom L} proof let L be lower-bounded non empty Poset; thus downarrow Bottom L c= {Bottom L} proof let a; assume a in downarrow Bottom L; then a in downarrow {Bottom L} by WAYBEL_0:def 17; then a in {x where x is Element of L : ex y being Element of L st x <= y & y in {Bottom L}} by WAYBEL_0:14; then consider a' be Element of L such that A1: a' = a & ex y being Element of L st a' <= y & y in {Bottom L}; consider y being Element of L such that A2: a' <= y & y in {Bottom L} by A1; A3: Bottom L <= a' by YELLOW_0:44; y = Bottom L by A2,TARSKI:def 1; hence thesis by A1,A2,A3,ORDERS_1:25; end; let a; assume a in {Bottom L}; then A4: a = Bottom L by TARSKI:def 1; Bottom L <= Bottom L; hence thesis by A4,WAYBEL_0:17; end; theorem for L being upper-bounded non empty Poset holds uparrow Top L = {Top L} proof let L be upper-bounded non empty Poset; thus uparrow Top L c= {Top L} proof let x be set; assume A1: x in uparrow Top L; then reconsider x as Element of L; x >= Top L & x <= Top L by A1,WAYBEL_0:18,YELLOW_0:45; then x = Top L by ORDERS_1:25; hence thesis by TARSKI:def 1; end; let x be set; assume x in {Top L}; then x = Top L & Top L <= Top L by TARSKI:def 1; hence thesis by WAYBEL_0:18; end; Lm5: for L being lower-bounded sup-Semilattice, I being Ideal of L holds {Bottom L} c= I proof let L be lower-bounded sup-Semilattice; let I be Ideal of L; let a; assume a in {Bottom L}; then a = Bottom L by TARSKI:def 1; hence thesis by Th21; end; reserve L for lower-bounded sup-Semilattice; reserve x for Element of L; theorem Th25: (the carrier of L) --> {Bottom L} is map of L, InclPoset Ids L proof {Bottom L} = downarrow Bottom L by Th23; then {Bottom L} in {X where X is Ideal of L: not contradiction}; then {Bottom L} in Ids L by WAYBEL_0:def 23; then {Bottom L} in the carrier of InclPoset Ids L by YELLOW_1:1; then (the carrier of L) --> {Bottom L} is Function of the carrier of L, the carrier of InclPoset Ids L by FUNCOP_1:57; hence thesis ; end; Lm6: for f be map of L, InclPoset Ids L st f = (the carrier of L) --> {Bottom L} holds f is monotone proof let f be map of L, InclPoset Ids L; assume A1: f = (the carrier of L) --> {Bottom L}; let x,y be Element of L; f.x = {Bottom L} & f.y = {Bottom L} by A1,FUNCOP_1:13; hence thesis by YELLOW_1:3; end; theorem Th26: (the carrier of L) --> {Bottom L} in the carrier of MonSet L proof set s = (the carrier of L) --> {Bottom L}; ex s' be map of L, InclPoset Ids L st s = s' & s' is monotone & for x be Element of L holds s'.x c= downarrow x proof reconsider s as map of L, InclPoset Ids L by Th25; take s; for x holds s.x c= downarrow x proof let x be Element of L; s.x = {Bottom L} by FUNCOP_1:13; hence s.x c= downarrow x by Lm5; end; hence thesis by Lm6; end; hence thesis by Def14; end; theorem for AR being auxiliary Relation of L holds [(the carrier of L) --> {Bottom L} , AR-below] in the InternalRel of MonSet L proof let AR be auxiliary Relation of L; set c = (the carrier of L) --> {Bottom L}, d = AR-below; ex f,g be map of L, InclPoset Ids L st c = f & d = g & c in the carrier of MonSet L & d in the carrier of MonSet L & f <= g proof reconsider c as map of L, InclPoset Ids L by Th25; take c,d; c <= d proof let x be set; assume x in the carrier of L; then reconsider x' = x as Element of L; d.x = AR-below x' by Def13; then reconsider dx = d.x' as Ideal of L; reconsider dx' = dx as Element of InclPoset Ids L; c.x' = {Bottom L} by FUNCOP_1:13; then A1: c.x c= dx by Lm5; take c.x', dx'; thus thesis by A1,YELLOW_1:3; end; hence thesis by Th18,Th26; end; hence thesis by Def14; end; Lm7: for L ex x be Element of MonSet L st x is_>=_than the carrier of MonSet L proof let L; set x = IdsMap L; x in the carrier of MonSet L by Th19; then reconsider x as Element of MonSet L; x is_>=_than the carrier of MonSet L proof let b be Element of MonSet L; assume b in the carrier of MonSet L; consider s be map of L, InclPoset Ids L such that A1: b = s & s is monotone & for x be Element of L holds s.x c= downarrow x by Def14; IdsMap L >= s proof let a be set; assume a in the carrier of L; then reconsider a' = a as Element of L; A2: (IdsMap L).a = downarrow a' by YELLOW_2:def 4; A3: s.a c= downarrow a' by A1; reconsider A = s.a', B = (IdsMap L).a' as Element of InclPoset Ids L; take A, B; thus thesis by A2,A3,YELLOW_1:3; end; then [b,x] in the InternalRel of MonSet L by A1,Def14; hence thesis by ORDERS_1:def 9; end; hence thesis; end; definition let L; cluster MonSet L -> upper-bounded; coherence proof consider x be Element of MonSet L such that A1: x is_>=_than the carrier of MonSet L by Lm7; take x; thus thesis by A1; end; end; definition let L; func Rel2Map L -> map of InclPoset Aux L, MonSet L means :Def15: for AR being auxiliary Relation of L holds it.AR = AR-below; existence proof defpred P[set,set] means ex AR be auxiliary Relation of L st AR = $1 & $2 = AR-below; A1: for a st a in the carrier of InclPoset Aux L ex y be set st y in the carrier of MonSet L & P[a,y] proof let a; assume a in the carrier of InclPoset Aux L; then a in Aux L by YELLOW_1:1; then reconsider AR = a as auxiliary Relation of L by Def9; take AR-below; thus thesis by Th18; end; consider f being Function of the carrier of InclPoset Aux L, the carrier of MonSet L such that A2: for a st a in the carrier of InclPoset Aux L holds P[a,f.a] from FuncEx1(A1); take f; now let AR be auxiliary Relation of L; AR in Aux L by Def9; then AR in the carrier of InclPoset Aux L by YELLOW_1:1; then P[AR,f.AR] by A2; hence f.AR = AR-below; end; hence thesis ; end; uniqueness proof let I1, I2 be map of InclPoset Aux L, MonSet L; assume A3: for AR be auxiliary Relation of L holds I1.AR = AR-below; assume A4: for AR be auxiliary Relation of L holds I2.AR = AR-below; dom I1 = the carrier of InclPoset Aux L by FUNCT_2:def 1; then A5: dom I1 = Aux L by YELLOW_1:1; dom I2 = the carrier of InclPoset Aux L by FUNCT_2:def 1; then A6: dom I2 = Aux L by YELLOW_1:1; now let a; assume a in Aux L; then reconsider AR = a as auxiliary Relation of L by Def9; I1.AR = AR-below by A3 .= I2.AR by A4; hence I1.a = I2.a; end; hence thesis by A5,A6,FUNCT_1:9; end; end; theorem for R1, R2 being auxiliary Relation of L st R1 c= R2 holds R1-below <= R2-below proof let R1, R2 be auxiliary Relation of L; assume A1: R1 c= R2; let x be set; assume x in the carrier of L; then reconsider x' = x as Element of L; A2: R1-below x' c= R2-below x' proof let a; assume a in R1-below x'; then a in {y where y is Element of L: [y,x'] in R1} by Def11; then consider b be Element of L such that A3: b = a & [b,x'] in R1; b in {y where y is Element of L: [y,x'] in R2} by A1,A3; hence thesis by A3,Def11; end; reconsider A1 = (R1-below).x', A2 = (R2-below).x' as Element of InclPoset Ids L; take A1, A2; (R1-below).x = R1-below x' & (R2-below).x = R2-below x' by Def13; hence thesis by A2,YELLOW_1:3; end; theorem Th29: for R1, R2 being Relation of L st R1 c= R2 holds R1-below x c= R2-below x proof let R1, R2 be Relation of L; assume A1: R1 c= R2; let a; assume a in R1-below x; then a in {z where z is Element of L: [z,x] in R1} by Def11; then consider b be Element of L such that A2: b = a & [b,x] in R1; b in {z where z is Element of L: [z,x] in R2} by A1,A2; hence thesis by A2,Def11; end; Lm8: Rel2Map L is monotone proof let x,y be Element of InclPoset Aux L; x in the carrier of InclPoset Aux L & y in the carrier of InclPoset Aux L; then x in Aux L & y in Aux L by YELLOW_1:1; then reconsider R1 = x, R2 = y as auxiliary Relation of L by Def9; assume x <= y; then A1: x c= y by YELLOW_1:3; let a, b be Element of MonSet L; assume A2: a = (Rel2Map L).x & b = (Rel2Map L).y; thus a <= b proof A3: (Rel2Map L).x = R1-below & (Rel2Map L).y = R2-below by Def15; consider s be map of L, InclPoset Ids L such that A4: (Rel2Map L).x = s & s is monotone & for x be Element of L holds s.x c= downarrow x by Def14; consider t be map of L, InclPoset Ids L such that A5: (Rel2Map L).y = t & t is monotone & for x be Element of L holds t.x c= downarrow x by Def14; s <= t proof let q be set; assume q in the carrier of L; then reconsider q' = q as Element of L; A6: s.q = R1-below q' by A3,A4,Def13; A7: t.q = R2-below q' by A3,A5,Def13; A8: R1-below q' c= R2-below q' by A1,Th29; reconsider sq = s.q', tq = t.q' as Element of InclPoset Ids L; sq <= tq by A6,A7,A8,YELLOW_1:3; hence thesis; end; then [R1-below, R2-below] in the InternalRel of MonSet L by A3,A4,A5,Def14; hence thesis by A2,A3,ORDERS_1:def 9; end; end; definition let L; cluster Rel2Map L -> monotone; coherence by Lm8; end; definition let L; func Map2Rel L -> map of MonSet L, InclPoset Aux L means :Def16: for s be set st s in the carrier of MonSet L ex AR be auxiliary Relation of L st AR = it.s & for x,y be set holds [x,y] in AR iff ex x',y' be Element of L, s' be map of L, InclPoset Ids L st x' = x & y' = y & s' = s & x' in s'.y'; existence proof defpred P[set, set] means ex AR be auxiliary Relation of L st AR = $2 & for x,y be set holds [x,y] in AR iff ex x',y' be Element of L, s' be map of L, InclPoset Ids L st x' = x & y' = y & s' = $1 & x' in s'.y'; A1: for a st a in the carrier of MonSet L ex b be set st b in the carrier of InclPoset Aux L & P[a,b] proof let a; assume A2: a in the carrier of MonSet L; defpred Q[set,set] means ex x', y' be Element of L, s' be map of L, InclPoset Ids L st x' = $1 & y' = $2 & s' = a & x' in s'.y'; consider R being Relation of the carrier of L, the carrier of L such that A3: for x,y be set holds [x,y] in R iff x in the carrier of L & y in the carrier of L & Q[x,y] from Rel_On_Set_Ex; reconsider R as Relation of L ; now thus R is auxiliary(i) proof let x, y be Element of L; assume [x,y] in R; then consider x', y' be Element of L, s' be map of L, InclPoset Ids L such that A4: x' = x & y' = y & s' = a & x' in s'.y' by A3; consider s be map of L, InclPoset Ids L such that A5: a = s & s is monotone & for x be Element of L holds s.x c= downarrow x by A2,Def14; s'.y c= downarrow y by A4,A5; hence x <= y by A4,WAYBEL_0:17; end; thus R is auxiliary(ii) proof let x, y, z, u be Element of L; assume A6: u <= x & [x,y] in R & y <= z; then consider x', y' be Element of L, s' be map of L, InclPoset Ids L such that A7: x' = x & y' = y & s' = a & x' in s'.y' by A3; consider s be map of L, InclPoset Ids L such that A8: a = s & s is monotone & for x be Element of L holds s.x c= downarrow x by A2,Def14; reconsider sy = s.y, sz = s.z as Element of InclPoset Ids L; sy <= sz by A6,A8,ORDERS_3:def 5; then A9: s.y c= s.z by YELLOW_1:3; s.z in the carrier of InclPoset Ids L; then s.z in Ids L by YELLOW_1:1; then s.z in {X where X is Ideal of L: not contradiction} by WAYBEL_0:def 23; then consider sz be Ideal of L such that A10: s.z = sz; u in sz by A6,A7,A8,A9,A10,WAYBEL_0:def 19; hence thesis by A3,A8,A10; end; thus R is auxiliary(iii) proof let x, y, z be Element of L; assume A11: [x,z] in R & [y,z] in R; then consider x1, z1 be Element of L, s1 be map of L, InclPoset Ids L such that A12: x1 = x & z1 = z & s1 = a & x1 in s1.z1 by A3; consider y2, z2 be Element of L, s2 be map of L, InclPoset Ids L such that A13: y2 = y & z2 = z & s2 = a & y2 in s2.z2 by A3,A11; s1.z in the carrier of InclPoset Ids L; then s1.z in Ids L by YELLOW_1:1; then s1.z in {X where X is Ideal of L: not contradiction} by WAYBEL_0: def 23; then consider sz be Ideal of L such that A14: s1.z = sz; consider z3 be Element of L such that A15: z3 in sz & x <= z3 & y <= z3 by A12,A13,A14,WAYBEL_0:def 1; ex q being Element of L st x <= q & y <= q & for c being Element of L st x <= c & y <= c holds q <= c by LATTICE3:def 10; then x "\/" y <= z3 by A15,LATTICE3:def 13; then x "\/" y in sz by A15,WAYBEL_0:def 19; hence thesis by A3,A12,A14; end; thus R is auxiliary(iv) proof let x be Element of L; reconsider x' = Bottom L, y' = x as Element of L; consider s' be map of L, InclPoset Ids L such that A16: a = s' & s' is monotone & for x be Element of L holds s'.x c= downarrow x by A2,Def14; s'.y' in the carrier of InclPoset Ids L; then s'.y' in Ids L by YELLOW_1:1; then s'.y' in {X where X is Ideal of L: not contradiction} by WAYBEL_0:def 23; then consider sy be Ideal of L such that A17: sy = s'.y'; x' in sy by Th21; hence thesis by A3,A16,A17; end; end; then reconsider R as auxiliary Relation of L by Def8; A18: for x,y be set holds [x,y] in R iff ex x',y' be Element of L, s' be map of L, InclPoset Ids L st x' = x & y' = y & s' = a & x' in s'.y' by A3; take b = R; b in Aux L by Def9; hence thesis by A18,YELLOW_1:1; end; consider f being Function of the carrier of MonSet L, the carrier of InclPoset Aux L such that A19: for a st a in the carrier of MonSet L holds P[a,f.a] from FuncEx1 (A1); reconsider f' = f as map of MonSet L, InclPoset Aux L ; take f'; let s be set; assume A20: s in the carrier of MonSet L; then reconsider s' = s as Element of MonSet L; f'.s' in the carrier of InclPoset Aux L; then f'.s' in Aux L by YELLOW_1:1; then reconsider fs = f'.s' as auxiliary Relation of L by Def9; take fs; ex AR be auxiliary Relation of L st AR = f'.s & for x,y be set holds [x,y] in AR iff ex x',y' be Element of L, s' be map of L, InclPoset Ids L st x' = x & y' = y & s' = s & x' in s'.y' by A19,A20; hence thesis; end; uniqueness proof let M1, M2 be map of MonSet L, InclPoset Aux L; assume A21: for s be set st s in the carrier of MonSet L ex AR be auxiliary Relation of L st AR = M1.s & for x,y be set holds [x,y] in AR iff ex x',y' be Element of L, s' be map of L, InclPoset Ids L st x' = x & y' = y & s' = s & x' in s'.y'; assume A22: for s be set st s in the carrier of MonSet L ex AR be auxiliary Relation of L st AR = M2.s & for x,y be set holds [x,y] in AR iff ex x',y' be Element of L, s' be map of L, InclPoset Ids L st x' = x & y' = y & s' = s & x' in s'.y'; M1 = M2 proof A23: dom M1 = the carrier of MonSet L by FUNCT_2:def 1; A24: dom M2 = the carrier of MonSet L by FUNCT_2:def 1; for s be set st s in the carrier of MonSet L holds M1.s = M2.s proof let s be set; assume A25: s in the carrier of MonSet L; then consider AR1 be auxiliary Relation of L such that A26: AR1 = M1.s & for x,y be set holds [x,y] in AR1 iff ex x',y' be Element of L, s' be map of L, InclPoset Ids L st x' = x & y' = y & s' = s & x' in s'.y' by A21; consider AR2 be auxiliary Relation of L such that A27: AR2 = M2.s & for x,y be set holds [x,y] in AR2 iff ex x',y' be Element of L, s' be map of L, InclPoset Ids L st x' = x & y' = y & s' = s & x' in s'.y' by A22,A25; AR1 = AR2 proof let x,y be set; hereby assume [x,y] in AR1; then consider x',y' be Element of L, s' be map of L, InclPoset Ids L such that A28: x' = x & y' = y & s' = s & x' in s'.y' by A26; thus [x,y] in AR2 by A27,A28; end; assume [x,y] in AR2; then consider x',y' be Element of L, s' be map of L, InclPoset Ids L such that A29: x' = x & y' = y & s' = s & x' in s'.y' by A27; thus [x,y] in AR1 by A26,A29; end; hence thesis by A26,A27; end; hence thesis by A23,A24,FUNCT_1:9; end; hence thesis; end; end; Lm9: Map2Rel L is monotone proof set f = Map2Rel L; A1:InclPoset Aux L = RelStr (#Aux L, RelIncl Aux L#) by YELLOW_1:def 1; let x,y be Element of MonSet L; assume x <= y; then [x,y] in the InternalRel of MonSet L by ORDERS_1:def 9; then consider s,t be map of L, InclPoset Ids L such that A2:x = s & y = t & x in the carrier of MonSet L & y in the carrier of MonSet L & s <= t by Def14; A3: f.s in the carrier of InclPoset Aux L & f.t in the carrier of InclPoset Aux L by A2,FUNCT_2:7; then A4: f.s in Aux L & f.t in Aux L by YELLOW_1:1; then reconsider AS = f.s, AT = f.t as auxiliary Relation of L by Def9; reconsider AS' = AS, AT' = AT as Element of InclPoset Aux L by A3; AS' c= AT' proof for a,b be set st [a,b] in AS holds [a,b] in AT proof let a,b be set; assume A5: [a,b] in AS; then A6: a in the carrier of L & b in the carrier of L by ZFMISC_1:106; then reconsider a' = a, b' = b as Element of L; reconsider sb = s.b', tb = t.b' as Element of InclPoset Ids L; consider a1, b1 being Element of InclPoset Ids L such that A7: a1 = s.b & b1 = t.b & a1 <= b1 by A2,A6,YELLOW_2:def 1; A8: sb c= tb by A7,YELLOW_1:3; consider AR be auxiliary Relation of L such that A9: AR = f.x & for a',b' be set holds [a',b'] in AR iff ex x',y' be Element of L, s' be map of L, InclPoset Ids L st x' = a' & y' = b' & s' = x & x' in s'.y' by Def16; consider x',y' be Element of L, s' be map of L, InclPoset Ids L such that A10: x' = a' & y' = b' & s' = s & x' in s'.y' by A2,A5,A9; consider AR1 be auxiliary Relation of L such that A11: AR1 = f.y & for a',b' be set holds [a',b'] in AR1 iff ex x',y' be Element of L, s' be map of L, InclPoset Ids L st x' = a' & y' = b' & s' = y & x' in s'.y' by Def16; thus thesis by A2,A8,A10,A11; end; hence thesis by RELAT_1:def 3; end; then [AS',AT'] in RelIncl Aux L by A4,WELLORD2:def 1; hence thesis by A1,A2,ORDERS_1:def 9; end; definition let L; cluster Map2Rel L -> monotone; coherence by Lm9; end; theorem Th30: (Map2Rel L) * (Rel2Map L) = id dom (Rel2Map L) proof set LS = (Map2Rel L) * (Rel2Map L); set R = id dom (Rel2Map L); dom LS = the carrier of InclPoset Aux L by FUNCT_2:def 1; then A1: dom LS = Aux L by YELLOW_1:1; dom R = dom (Rel2Map L) by FUNCT_1:34; then dom R = the carrier of InclPoset Aux L by FUNCT_2:def 1; then A2: dom R = Aux L by YELLOW_1:1; for a st a in Aux L holds LS.a = R.a proof let a; assume A3: a in Aux L; then reconsider AR = a as auxiliary Relation of L by Def9; A4: a in the carrier of InclPoset Aux L by A3,YELLOW_1:1; then A5: a in dom (Rel2Map L) by FUNCT_2:def 1; then LS.a = (Map2Rel L).((Rel2Map L).AR) by FUNCT_1:23; then A6: LS.a = (Map2Rel L).(AR-below) by Def15; LS.a in the carrier of InclPoset Aux L by A4,FUNCT_2:7; then LS.a in Aux L by YELLOW_1:1; then reconsider La = LS.a as auxiliary Relation of L by Def9; A7: AR-below in the carrier of MonSet L by Th18; La = AR proof for c,b be set holds [c,b] in La iff [c,b] in AR proof let c,b be set; hereby assume A8: [c,b] in La; consider AR' be auxiliary Relation of L such that A9: AR' = (Map2Rel L).(AR-below) & for c,b be set holds [c,b] in AR' iff ex c',b' be Element of L, s' be map of L, InclPoset Ids L st c' = c & b' = b & s' = AR-below & c' in s'.b' by A7,Def16; consider c',b' be Element of L, s' be map of L, InclPoset Ids L such that A10: c' = c & b' = b & s' = AR-below & c' in s'.b' by A6,A8,A9; c' in AR-below b' by A10,Def13; hence [c,b] in AR by A10,Th13; end; assume A11: [c,b] in AR; then c in the carrier of L & b in the carrier of L by ZFMISC_1:106; then reconsider c' = c, b' = b as Element of L; c' in AR-below b' by A11,Th13; then A12: c' in (AR-below).b' by Def13; consider AR' be auxiliary Relation of L such that A13: AR' = (Map2Rel L).(AR-below) & for c,b be set holds [c,b] in AR' iff ex c',b' be Element of L, s' be map of L, InclPoset Ids L st c' = c & b' = b & s' = AR-below & c' in s'.b' by A7,Def16; thus [c,b] in La by A6,A12,A13; end; hence thesis by RELAT_1:def 2; end; hence thesis by A5,FUNCT_1:35; end; hence LS = R by A1,A2,FUNCT_1:9; end; theorem Th31: (Rel2Map L) * (Map2Rel L) = id (the carrier of MonSet L) proof set LS = (Rel2Map L) * (Map2Rel L); set R = id (the carrier of MonSet L); A1: dom LS = the carrier of MonSet L by FUNCT_2:def 1; A2: dom R = the carrier of MonSet L by FUNCT_1:34; for a st a in the carrier of MonSet L holds LS.a = R.a proof let a; assume A3: a in the carrier of MonSet L; then consider s be map of L, InclPoset Ids L such that A4: a = s & s is monotone & for x be Element of L holds s.x c= downarrow x by Def14; LS.s in the carrier of MonSet L by A3,A4,FUNCT_2:7; then consider Ls be map of L, InclPoset Ids L such that A5: LS.s = Ls & Ls is monotone & for x be Element of L holds Ls.x c= downarrow x by Def14; set AR = (Map2Rel L).s; AR in the carrier of InclPoset Aux L by A3,A4,FUNCT_2:7; then AR in Aux L by YELLOW_1:1; then reconsider AR as auxiliary Relation of L by Def9; dom (Map2Rel L) = the carrier of MonSet L by FUNCT_2:def 1; then Ls = (Rel2Map L).AR by A3,A4,A5,FUNCT_1:23; then A6: Ls = AR-below by Def15; Ls = s proof A7: dom Ls = the carrier of L by FUNCT_2:def 1; A8: dom s = the carrier of L by FUNCT_2:def 1; now let b be set; assume A9: b in the carrier of L; then reconsider b' = b as Element of L; thus Ls.b = s.b proof thus Ls.b c= s.b proof let d be set; assume d in Ls.b; then d in AR-below b' by A6,Def13; then A10: [d,b'] in AR by Th13; consider AR' be auxiliary Relation of L such that A11: AR' = (Map2Rel L).s & for d,b' be set holds [d,b'] in AR' iff ex d',b'' be Element of L, s' be map of L, InclPoset Ids L st d' = d & b'' = b' & s' = s & d' in s'.b'' by A3,A4,Def16; consider d',b'' be Element of L, s' be map of L, InclPoset Ids L such that A12: d' = d & b'' = b' & s' = s & d' in s'.b'' by A10,A11; thus thesis by A12; end; thus s.b c= Ls.b proof let d be set; assume A13: d in s.b; s.b in the carrier of InclPoset Ids L by A9,FUNCT_2:7; then s.b in Ids L by YELLOW_1:1; then s.b in {X where X is Ideal of L: not contradiction} by WAYBEL_0: def 23; then consider X be Ideal of L such that A14: s.b = X; reconsider d' = d as Element of L by A13,A14; consider AR' be auxiliary Relation of L such that A15: AR' = (Map2Rel L).s & for d,b' be set holds [d,b'] in AR' iff ex d',b'' be Element of L, s' be map of L, InclPoset Ids L st d' = d & b'' = b' & s' = s & d' in s'.b'' by A3,A4,Def16; [d',b'] in AR by A13,A15; then d' in AR-below b' by Th13; hence thesis by A6,Def13; end; end; end; hence thesis by A7,A8,FUNCT_1:9; end; hence LS.a = R.a by A3,A4,A5,FUNCT_1:35; end; hence thesis by A1,A2,FUNCT_1:9; end; definition let L; cluster Rel2Map L -> one-to-one; coherence proof ex g be Function st g*(Rel2Map L) = id dom (Rel2Map L) proof take Map2Rel L; thus thesis by Th30; end; hence Rel2Map L is one-to-one by FUNCT_1:53; end; end; :: Proposition 1.10 (i) p.43 theorem Th32: (Rel2Map L)" = Map2Rel L proof (Rel2Map L)*(Map2Rel L) = id (the carrier of MonSet L) implies rng (Rel2Map L) = the carrier of MonSet L by FUNCT_2:24; then Rel2Map L is one-to-one & rng (Rel2Map L) = dom (Map2Rel L) & (Map2Rel L)*(Rel2Map L) = id dom (Rel2Map L) by Th30,Th31,FUNCT_2:def 1; hence Map2Rel L = (Rel2Map L)" by FUNCT_1:63; end; :: Proposition 1.10 (ii) p.43 theorem Rel2Map L is isomorphic proof ex g being map of MonSet L, InclPoset Aux L st g = (Rel2Map L)" & g is monotone proof reconsider g = Map2Rel L as map of MonSet L, InclPoset Aux L; take g; thus thesis by Th32; end; hence thesis by WAYBEL_0:def 38; end; theorem Th34: for L being complete LATTICE, x being Element of L holds meet { I where I is Ideal of L : x <= sup I } = waybelow x proof let L be complete LATTICE, x be Element of L; set X = { I where I is Ideal of L : x <= sup I }; X c= bool the carrier of L proof let a; assume a in X; then consider I be Ideal of L such that A1: a = I & x <= sup I; thus thesis by A1; end; then reconsider X' = X as Subset-Family of L by SETFAM_1:def 7 ; sup downarrow x = x by WAYBEL_0:34; then A2:downarrow x in X; thus meet X c= waybelow x proof let a; assume A3: a in meet X; then a in meet X'; then reconsider y = a as Element of L; now let I be Ideal of L; assume x <= sup I; then I in X; hence y in I by A3,SETFAM_1:def 1; end; then y << x by WAYBEL_3:21; hence thesis by WAYBEL_3:7; end; thus waybelow x c= meet X proof let a; assume a in waybelow x; then a in {y where y is Element of L: y << x} by WAYBEL_3:def 3; then consider a' be Element of L such that A4: a' = a & a' << x; for Y be set holds Y in X implies a in Y proof let Y be set; assume Y in X; then consider I be Ideal of L such that A5: Y = I & x <= sup I; thus thesis by A4,A5,WAYBEL_3:20; end; hence thesis by A2,SETFAM_1:def 1; end; end; scheme LambdaC'{ A() -> non empty RelStr, C[set], F(set) -> set, G(set) -> set } : ex f being Function st dom f = the carrier of A() & for x be Element of A() holds (C[x] implies f.x = F(x)) & (not C[x] implies f.x = G(x)) proof defpred c[set] means C[$1]; deffunc f(set) = F($1); deffunc g(set) = G($1); A1: ex f being Function st dom f = the carrier of A() & for x be set st x in the carrier of A() holds (c[x] implies f.x = f(x)) & (not c[x] implies f.x = g(x)) from LambdaC; thus ex f being Function st dom f = the carrier of A() & for x be Element of A() holds (C[x] implies f.x = F(x)) & (not C[x] implies f.x = G(x)) proof consider f being Function such that A2: dom f = the carrier of A() & for x be set st x in the carrier of A() holds (C[x] implies f.x = F(x)) & (not C[x] implies f.x = G(x)) by A1; take f; thus dom f = the carrier of A() by A2; let x be Element of A(); thus (C[x] implies f.x = F(x)) & (not C[x] implies f.x = G(x)) by A2; end; end; definition let L be Semilattice, I be Ideal of L; func DownMap I -> map of L, InclPoset Ids L means :Def17: for x be Element of L holds ( x <= sup I implies it.x = ( downarrow x ) /\ I ) & ( not x <= sup I implies it.x = downarrow x ); existence proof defpred P[set,set] means for x be Element of L st x = $1 holds ( x <= sup I implies $2 = ( downarrow x ) /\ I ) & ( not x <= sup I implies $2 = downarrow x ); defpred C[Element of L] means $1 <= sup I; deffunc F(Element of L) = ( downarrow $1 ) /\ I; deffunc G(Element of L) = downarrow $1; consider f being Function such that A1: dom f = the carrier of L & for x be Element of L holds ( C[x] implies f.x = F(x) ) & ( not C[x] implies f.x = G(x) ) from LambdaC'; A2: for a be set st a in the carrier of L ex y be set st y in the carrier of InclPoset Ids L & P[a,y] proof let a; assume a in the carrier of L; then reconsider x = a as Element of L; take y = f.x; thus y in the carrier of InclPoset Ids L proof per cases; suppose x <= sup I; then y = ( downarrow x ) /\ I by A1; then y is Ideal of L by YELLOW_2:42; then y in {X where X is Ideal of L: not contradiction}; then y in Ids L by WAYBEL_0:def 23; hence thesis by YELLOW_1:1; suppose not x <= sup I; then y = downarrow x by A1; then y in {X where X is Ideal of L: not contradiction}; then y in Ids L by WAYBEL_0:def 23; hence thesis by YELLOW_1:1; end; thus P[a,y] by A1; end; consider f being Function of the carrier of L, the carrier of InclPoset Ids L such that A3: for a st a in the carrier of L holds P[a,f.a] from FuncEx1(A2); reconsider f as map of L, InclPoset Ids L ; for x be Element of L holds ( x <= sup I implies f.x = ( downarrow x ) /\ I ) & ( not x <= sup I implies f.x = downarrow x ) by A3; hence thesis; end; uniqueness proof let M1, M2 be map of L, InclPoset Ids L; assume A4: for x be Element of L holds ( x <= sup I implies M1.x = ( downarrow x ) /\ I ) & ( not x <= sup I implies M1.x = downarrow x ); assume A5: for x be Element of L holds ( x <= sup I implies M2.x = ( downarrow x ) /\ I ) & ( not x <= sup I implies M2.x = downarrow x ); A6: dom M1 = the carrier of L by FUNCT_2:def 1; A7: dom M2 = the carrier of L by FUNCT_2:def 1; now let x be set; assume x in the carrier of L; then reconsider x' = x as Element of L; A8: now assume A9: x' <= sup I; then M1.x' = ( downarrow x' ) /\ I by A4; hence M1.x' = M2.x' by A5,A9; end; now assume A10: not x' <= sup I; then M1.x' = downarrow x' by A4; hence M1.x' = M2.x' by A5,A10; end; hence M1.x = M2.x by A8; end; hence thesis by A6,A7,FUNCT_1:9; end; end; Lm10: for L be Semilattice, I be Ideal of L holds DownMap I is monotone proof let L be Semilattice, I be Ideal of L; let x,y be Element of L; assume A1: x <= y; per cases; suppose x <= sup I & y <= sup I; then A2: (DownMap I).x = (downarrow x) /\ I & (DownMap I).y = (downarrow y) /\ I by Def17; downarrow x c= downarrow y by A1,WAYBEL_0:21; then (DownMap I).x c= (DownMap I).y by A2,XBOOLE_1:26; hence thesis by YELLOW_1:3; suppose x <= sup I & not y <= sup I; then A3: (DownMap I).x = (downarrow x) /\ I & (DownMap I).y = downarrow y by Def17; A4:downarrow x c= downarrow y by A1,WAYBEL_0:21; (downarrow x) /\ I c= downarrow x by XBOOLE_1:17; then (DownMap I).x c= (DownMap I).y by A3,A4,XBOOLE_1:1; hence thesis by YELLOW_1:3; suppose not x <= sup I & y <= sup I; hence thesis by A1,ORDERS_1:26; suppose not x <= sup I & not y <= sup I; then (DownMap I).x = downarrow x & (DownMap I).y = downarrow y by Def17; then (DownMap I).x c= (DownMap I).y by A1,WAYBEL_0:21; hence thesis by YELLOW_1:3; end; Lm11: for L be Semilattice, x be Element of L for I be Ideal of L holds (DownMap I).x c= downarrow x proof let L be Semilattice, x be Element of L; let I be Ideal of L; per cases; suppose x <= sup I; then (DownMap I).x = (downarrow x) /\ I by Def17; hence thesis by XBOOLE_1:17; suppose not x <= sup I; hence thesis by Def17; end; theorem Th35: for L being Semilattice, I being Ideal of L holds DownMap I in the carrier of MonSet L proof let L be Semilattice, I be Ideal of L; reconsider mI = DownMap I as map of L, InclPoset Ids L; ex s be map of L, InclPoset Ids L st mI = s & s is monotone & for x be Element of L holds s.x c= downarrow x proof take mI; thus thesis by Lm10,Lm11; end; hence thesis by Def14; end; theorem Th36: for L being with_infima antisymmetric reflexive RelStr, x being Element of L for D being non empty lower Subset of L holds {x} "/\" D = (downarrow x) /\ D proof let L be with_infima antisymmetric reflexive RelStr, x be Element of L; let D be non empty lower Subset of L; A1: {x} "/\" D = { x' "/\" y where x', y is Element of L : x' in {x} & y in D } by YELLOW_4:def 4; thus {x} "/\" D c= (downarrow x) /\ D proof let a; assume a in {x} "/\" D; then a in { x' "/\" y where x', y is Element of L : x' in {x} & y in D } by YELLOW_4:def 4; then consider x', y be Element of L such that A2: a = x' "/\" y & x' in {x} & y in D; reconsider a' = a as Element of L by A2; A3: x' = x by A2,TARSKI:def 1; ex v being Element of L st x' >= v & y >= v & for c being Element of L st x' >= c & y >= c holds v >= c by LATTICE3:def 11; then A4: x' "/\" y <= x' & x' "/\" y <= y by LATTICE3:def 14; then A5: a in downarrow x by A2,A3,WAYBEL_0:17; a' in D by A2,A4,WAYBEL_0:def 19; hence a in (downarrow x) /\ D by A5,XBOOLE_0:def 3; end; thus (downarrow x) /\ D c= {x} "/\" D proof let a; assume A6: a in (downarrow x) /\ D; then A7: a in downarrow x & a in D by XBOOLE_0:def 3; reconsider a' = a as Element of D by A6,XBOOLE_0:def 3; A8: x in {x} by TARSKI:def 1; a' <= x by A7,WAYBEL_0:17; then a' = a' "/\" x by YELLOW_0:25; hence a in {x} "/\" D by A1,A8; end; end; begin :: Approximating Relations :: Definition 1.11 p.44 definition let L be non empty RelStr, AR be Relation of L; attr AR is approximating means :Def18: for x be Element of L holds x = sup (AR-below x); end; definition let L be non empty Poset; let mp be map of L, InclPoset Ids L; attr mp is approximating means :Def19: for x be Element of L ex ii be Subset of L st ii = mp.x & x = sup ii; end; :: Lemma 1.12 (i) p.44 theorem Th37: for L being lower-bounded meet-continuous Semilattice, I being Ideal of L holds DownMap I is approximating proof let L be lower-bounded meet-continuous Semilattice; let I be Ideal of L; for x be Element of L ex ii be Subset of L st ii = (DownMap I).x & x = sup ii proof let x be Element of L; set ii = (DownMap I).x; ii in the carrier of InclPoset Ids L; then ii in Ids L by YELLOW_1:1; then ii in {X where X is Ideal of L: not contradiction} by WAYBEL_0:def 23 ; then consider ii' be Ideal of L such that A1: ii' = ii; reconsider dI = ( downarrow x ) /\ I as Subset of L; per cases; suppose A2: x <= sup I; then sup ii' = sup dI by A1,Def17 .= sup ({x} "/\" I) by Th36 .= x "/\" sup I by WAYBEL_2:def 6 .= x by A2,YELLOW_0:25; hence thesis by A1; suppose not x <= sup I; then sup ii' = sup downarrow x by A1,Def17 .= x by WAYBEL_0:34; hence thesis by A1; end; hence thesis by Def19; end; Lm12: for L be complete LATTICE, x be Element of L for D be directed Subset of L holds sup ({x} "/\" D) <= x proof let L be complete LATTICE, x be Element of L; let D be directed Subset of L; A1: {x} "/\" D = { a "/\" b where a, b is Element of L : a in {x} & b in D } by YELLOW_4:def 4; A2: ex_sup_of ({x} "/\" D),L by YELLOW_0:17; for c being Element of L st c in ({x} "/\" D) holds c <= x proof let c be Element of L; assume c in ({x} "/\" D); then consider a, b be Element of L such that A3: c = a "/\" b & a in {x} & b in D by A1; a = x by A3,TARSKI:def 1; hence c <= x by A3,YELLOW_0:23; end; then x is_>=_than ({x} "/\" D) by LATTICE3:def 9; hence sup ({x} "/\" D) <= x by A2,YELLOW_0:30; end; :: Lemma 1.12 (ii) p.44 theorem Th38: for L being lower-bounded continuous LATTICE holds L is meet-continuous proof let L be lower-bounded continuous LATTICE; now let D be non empty directed Subset of L; let x be Element of L; assume A1: x <= sup D; A2: ex_sup_of waybelow x,L & ex_sup_of downarrow (sup ({x} "/\" D)),L by YELLOW_0:17; waybelow x c= downarrow (sup ({x} "/\" D)) proof let q be set; assume q in waybelow x; then q in {y where y is Element of L: y << x} by WAYBEL_3:def 3; then consider q' be Element of L such that A3: q' = q & q' << x; A4: q' <= x by A3,WAYBEL_3:1; consider d be Element of L such that A5: d in D & q' <= d by A1,A3,WAYBEL_3:def 1; x in {x} by TARSKI:def 1; then x "/\" d in { a "/\" b where a, b is Element of L : a in {x} & b in D } by A5; then A6: x "/\" d in {x} "/\" D by YELLOW_4:def 4; ex_inf_of {x,d},L by YELLOW_0:17; then A7: q' <= x "/\" d by A4,A5,YELLOW_0:19; sup ({x} "/\" D) is_>=_than {x} "/\" D by YELLOW_0:32; then x "/\" d <= sup ({x} "/\" D) by A6,LATTICE3:def 9; then q' <= sup ({x} "/\" D) by A7,ORDERS_1:26; hence thesis by A3,WAYBEL_0:17; end; then sup waybelow x <= sup downarrow (sup ({x} "/\" D)) by A2,YELLOW_0:34; then sup waybelow x <= sup ({x} "/\" D) by WAYBEL_0:34; then A8: x <= sup ({x} "/\" D) by WAYBEL_3:def 5; sup ({x} "/\" D) <= x by Lm12; hence x = sup ({x} "/\" D) by A8,ORDERS_1:25; end; then for x being Element of L, D being non empty directed Subset of L st x <= sup D holds x = sup ({x} "/\" D); hence thesis by WAYBEL_2:52; end; definition cluster continuous -> meet-continuous (lower-bounded LATTICE); coherence by Th38; end; :: Lemma 1.12 (iii) p.44 theorem for L being lower-bounded continuous LATTICE, I being Ideal of L holds DownMap I is approximating by Th37; definition let L be non empty reflexive antisymmetric RelStr; cluster L-waybelow -> auxiliary(i); coherence proof let x, y be Element of L; assume [x,y] in L-waybelow; then x << y by Def2; hence thesis by WAYBEL_3:1; end; end; definition let L be non empty reflexive transitive RelStr; cluster L-waybelow -> auxiliary(ii); coherence proof set AR = L-waybelow; let x, y, z, u be Element of L; assume u <= x & [x,y] in AR & y <= z; then u <= x & x << y & y <= z by Def2; then u << z by WAYBEL_3:2; hence [u,z] in AR by Def2; end; end; definition let L be with_suprema Poset; cluster L-waybelow -> auxiliary(iii); coherence proof set AR = L-waybelow; let x, y, z be Element of L; assume [x,z] in AR & [y,z] in AR; then x << z & y << z by Def2; then (x "\/" y) << z by WAYBEL_3:3; hence [(x "\/" y),z] in AR by Def2; end; end; definition let L be /\-complete (non empty Poset); cluster L-waybelow -> auxiliary(iii); coherence proof set AR = L-waybelow; let x, y, z be Element of L; assume [x,z] in AR & [y,z] in AR; then x << z & y << z by Def2; then (x "\/" y) << z by WAYBEL_3:3; hence [(x "\/" y),z] in AR by Def2; end; end; definition let L be lower-bounded antisymmetric reflexive non empty RelStr; cluster L-waybelow -> auxiliary(iv); coherence proof let x be Element of L; Bottom L << x by WAYBEL_3:4; hence thesis by Def2; end; end; theorem Th40: for L being complete LATTICE, x being Element of L holds (L-waybelow)-below x = waybelow x proof let L be complete LATTICE, x be Element of L; set AR = L-waybelow; A1: AR-below x = {y where y is Element of L: [y,x] in AR} by Def11; set A = {y where y is Element of L: [y,x] in AR}; set B = {y where y is Element of L: y << x}; A = B proof thus A c= B proof let a be set; assume a in A; then consider v being Element of L such that A2: a = v and A3: [v,x] in AR; v << x by A3,Def2; hence a in { u' where u' is Element of L : u' << x } by A2; end; thus B c= A proof let a be set; assume a in B; then consider v being Element of L such that A4: a = v and A5: v << x; [v,x] in AR by A5,Def2; hence a in { u' where u' is Element of L : [u',x] in AR } by A4; end; end; hence thesis by A1,WAYBEL_3:def 3; end; theorem Th41: for L being LATTICE holds IntRel L is approximating proof let L be LATTICE; set AR = IntRel L; let x be Element of L; set A = {y where y is Element of L: [y,x] in AR}; set E = { u where u is Element of L : u <= x }; A1: {y where y is Element of L: [y,x] in AR} = E proof thus A c= E proof let a be set; assume a in A; then consider v being Element of L such that A2: a = v and A3: [v,x] in AR; v <= x by A3,Lm1; hence a in E by A2; end; thus E c= A proof let a be set; assume a in E; then consider v being Element of L such that A4: a = v and A5: v <= x; [v,x] in AR by A5,Lm1; hence a in A by A4; end; end; {y where y is Element of L: y <= x } c= the carrier of L proof let z be set; assume z in {y where y is Element of L: y <= x }; then ex y being Element of L st z = y & y <= x; hence thesis; end; then reconsider E as Subset of L; A6: x is_>=_than E proof let b be Element of L; assume b in E; then consider b' be Element of L such that A7: b' = b & b' <= x; thus b <= x by A7; end; now let b be Element of L; assume A8: b is_>=_than E; x in E; hence x <= b by A8,LATTICE3:def 9; end; then sup E = x by A6,YELLOW_0:30; hence thesis by A1,Def11; end; Lm13: for L be lower-bounded continuous LATTICE holds L-waybelow is approximating proof let L be lower-bounded continuous LATTICE; let x be Element of L; x = sup waybelow x by WAYBEL_3:def 5; hence thesis by Th40; end; definition let L be lower-bounded continuous LATTICE; cluster L-waybelow -> approximating; coherence by Lm13; end; definition let L be complete LATTICE; cluster approximating auxiliary Relation of L; existence proof reconsider A = IntRel L as auxiliary Relation of L; A is approximating by Th41; hence thesis; end; end; definition let L be complete LATTICE; defpred P[set] means $1 is approximating auxiliary Relation of L; func App L means :Def20: a in it iff a is approximating auxiliary Relation of L; existence proof consider X be set such that A1: for a holds a in X iff a in Aux L & P[a] from Separation; take X; a is approximating auxiliary Relation of L implies a in X proof assume A2: a is approximating auxiliary Relation of L; then a in Aux L by Def9; hence thesis by A1,A2; end; hence thesis by A1; end; uniqueness proof thus for X1,X2 being set st (for x being set holds x in X1 iff P[x]) & (for x being set holds x in X2 iff P[x]) holds X1 = X2 from SetEq; end; end; definition let L be complete LATTICE; cluster App L -> non empty; coherence proof consider a be approximating auxiliary Relation of L; a in App L by Def20; hence thesis; end; end; theorem Th42: for L being complete LATTICE for mp being map of L, InclPoset Ids L st mp is approximating & mp in the carrier of MonSet L ex AR being approximating auxiliary Relation of L st AR = (Map2Rel L).mp proof let L be complete LATTICE; let mp be map of L, InclPoset Ids L; assume A1: mp is approximating & mp in the carrier of MonSet L; then consider AR be auxiliary Relation of L such that A2: AR = (Map2Rel L).mp & for x,y be set holds [x,y] in AR iff ex x',y' be Element of L, s' be map of L, InclPoset Ids L st x' = x & y' = y & s' = mp & x' in s'.y' by Def16; now let x be Element of L; consider ii be Subset of L such that A3: ii = mp.x & x = sup ii by A1,Def19; AR-below x = mp.x proof thus AR-below x c= mp.x proof let a; assume a in AR-below x; then [a,x] in AR by Th13; then consider x',y' be Element of L, s' be map of L, InclPoset Ids L such that A4: x' = a & y' = x & s' = mp & x' in s'.y' by A2; thus thesis by A4; end; thus mp.x c= AR-below x proof let a; assume A5: a in mp.x; then reconsider a' = a as Element of L by A3; [a',x] in AR by A2,A5; hence thesis by Th13; end; end; hence x = sup (AR-below x) by A3; end; then reconsider AR as approximating auxiliary Relation of L by Def18; take AR; thus thesis by A2; end; theorem Th43: for L being complete LATTICE, x being Element of L holds meet { (DownMap I).x where I is Ideal of L : not contradiction} = waybelow x proof let L be complete LATTICE, x be Element of L; set A = { (DownMap I).x where I is Ideal of L : not contradiction}; set A1 = { (DownMap I).x where I is Ideal of L : x <= sup I}; A1: A1 = { (downarrow x) /\ I where I is Ideal of L : x <= sup I} proof thus A1 c= { (downarrow x) /\ I where I is Ideal of L : x <= sup I} proof let a; assume a in A1; then consider I1 be Ideal of L such that A2: a = (DownMap I1).x & x <= sup I1; a = (downarrow x) /\ I1 by A2,Def17; hence a in { (downarrow x) /\ I where I is Ideal of L : x <= sup I} by A2 ; end; thus { (downarrow x) /\ I where I is Ideal of L : x <= sup I} c= A1 proof let a; assume a in { (downarrow x) /\ I where I is Ideal of L : x <= sup I}; then consider I1 be Ideal of L such that A3: a = (downarrow x) /\ I1 & x <= sup I1; a = (DownMap I1).x by A3,Def17; hence a in A1 by A3; end; end; set A2 = { (DownMap I).x where I is Ideal of L : not x <= sup I}; A4: A2 = { downarrow x where I is Ideal of L : not x <= sup I} proof thus A2 c= { downarrow x where I is Ideal of L : not x <= sup I} proof let a; assume a in A2; then consider I1 be Ideal of L such that A5: a = (DownMap I1).x & not x <= sup I1; a = (downarrow x) by A5,Def17; hence a in { (downarrow x) where I is Ideal of L : not x <= sup I} by A5; end; thus { (downarrow x) where I is Ideal of L : not x <= sup I} c= A2 proof let a; assume a in { (downarrow x) where I is Ideal of L : not x <= sup I}; then consider I1 be Ideal of L such that A6: a = (downarrow x) & not x <= sup I1; a = (DownMap I1).x by A6,Def17; hence a in A2 by A6; end; end; A7: A = A1 \/ A2 proof thus A c= A1 \/ A2 proof let a; assume a in A; then consider I2 be Ideal of L such that A8: a = (DownMap I2).x; x <= sup I2 or not x <= sup I2; then a in A1 or a in A2 by A8; hence thesis by XBOOLE_0:def 2; end; thus A1 \/ A2 c= A proof let a; assume a in A1 \/ A2; then a in A1 or a in A2 by XBOOLE_0:def 2; then consider I2,I3 be Ideal of L such that A9: ( a = (DownMap I2).x & x <= sup I2) or ( a = (DownMap I3).x & not x <= sup I3); per cases by A9; suppose ( a = (DownMap I2).x & x <= sup I2 ); hence thesis; suppose ( a = (DownMap I3).x & not x <= sup I3); hence thesis; end; end; per cases; suppose A10: x = Bottom L; A11: A2 = {} proof assume A2 <> {}; then reconsider A2' = A2 as non empty set; consider a be Element of A2'; a in A2'; then consider I1 be Ideal of L such that A12: a = downarrow x & not x <= sup I1 by A4; thus contradiction by A10,A12,YELLOW_0:44; end; set Dx = downarrow x; x <= sup Dx by A10,YELLOW_0:44; then A13: Dx /\ Dx in A1 by A1; A1 c= { K where K is Ideal of L : x <= sup K} proof let a; assume a in A1; then consider H be Ideal of L such that A14: a = (downarrow x) /\ H & x <= sup H by A1; reconsider a' = a as Ideal of L by A14,YELLOW_2:42; x <= sup a' by A10,YELLOW_0:44; hence thesis; end; then A15: meet { K where K is Ideal of L : x <= sup K} c= meet A1 by A13, SETFAM_1:7; A16: meet A1 = {x} proof reconsider I' = downarrow x as Ideal of L; x <= sup I' by A10,YELLOW_0:44; then (downarrow x) /\ I' in A1 by A1; then {x} in A1 by A10,Th23; hence meet A1 c= {x} by SETFAM_1:4; for Z1 be set st Z1 in A1 holds {x} c= Z1 proof let Z1 be set; assume Z1 in A1; then consider Z1' be Ideal of L such that A17: Z1 = (downarrow x) /\ Z1' & x <= sup Z1' by A1; A18: {x} c= Z1' by A10,Lm5; {x} c= downarrow x by A10,Th23; hence thesis by A17,A18,XBOOLE_1:19; end; hence {x} c= meet A1 by A13,SETFAM_1:6; end; consider E be Ideal of L; x <= sup E by A10,YELLOW_0:44; then A19: E in { K where K is Ideal of L : x <= sup K}; for Z1 be set st Z1 in { K where K is Ideal of L : x <= sup K} holds meet A1 c= Z1 proof let Z1 be set; assume Z1 in { K where K is Ideal of L : x <= sup K}; then consider K1 be Ideal of L such that A20: K1 = Z1 & x <= sup K1; thus thesis by A10,A16,A20,Lm5; end; then meet A1 c= meet { K where K is Ideal of L : x <= sup K} by A19,SETFAM_1 :6; then meet A1 = meet { K where K is Ideal of L : x <= sup K} by A15,XBOOLE_0: def 10; hence thesis by A7,A11,Th34; suppose A21: Bottom L <> x; set O = downarrow Bottom L; A22: sup O = Bottom L by WAYBEL_0:34; then sup O <= x by YELLOW_0:44; then not x < sup O by ORDERS_1:30; then not x <= sup O by A21,A22,ORDERS_1:def 10; then A23: downarrow x in A2 by A4; reconsider O1 = downarrow x as Ideal of L; A24: x <= sup O1 by WAYBEL_0:34; downarrow x = downarrow x /\ O1; then A25: downarrow x in A1 & downarrow x in { downarrow x where I is Ideal of L : x <= sup I } by A1,A24; A26: meet A2 = downarrow x proof thus meet A2 c= downarrow x by A23,SETFAM_1:4; now let Z1 be set; assume Z1 in A2; then consider L1 be Ideal of L such that A27: Z1 = downarrow x & not x <= sup L1 by A4; thus downarrow x c= Z1 by A27; end; hence downarrow x c= meet A2 by A23,SETFAM_1:6; end; A28: meet A = (meet A1) /\ (meet A2) by A7,A23,A25,SETFAM_1:10; A29: meet A1 c= downarrow x by A25,SETFAM_1:4; then A30: meet A = meet A1 by A26,A28,XBOOLE_1:28; A31: meet A1 c= (downarrow x) /\ meet { I where I is Ideal of L : x <= sup I} proof let a; assume A32: a in meet A1; thus a in (downarrow x) /\ meet { I where I is Ideal of L : x <= sup I} proof reconsider L' = [#]L as Subset of L; x in the carrier of L; then A33: x in L' by PRE_TOPC:12; ex_sup_of L',L by YELLOW_0:17; then L' is_<=_than sup L' by YELLOW_0:def 9; then x <= sup L' by A33,LATTICE3:def 9; then A34: L' in { I where I is Ideal of L : x <= sup I}; a in meet { I where I is Ideal of L : x <= sup I} proof now let Y1 be set; assume Y1 in { I where I is Ideal of L : x <= sup I}; then consider Y1' be Ideal of L such that A35: Y1 = Y1' & x <= sup Y1'; A36: (downarrow x) /\ Y1' c= Y1' by XBOOLE_1:17; (downarrow x) /\ Y1' in A1 by A1,A35; then a in (downarrow x) /\ Y1' by A32,SETFAM_1:def 1; hence a in Y1 by A35,A36; end; hence thesis by A34,SETFAM_1:def 1; end; hence thesis by A29,A32,XBOOLE_0:def 3; end; end; (downarrow x) /\ meet { I where I is Ideal of L : x <= sup I} c= meet A1 proof let a; assume a in (downarrow x) /\ meet { I where I is Ideal of L : x <= sup I}; then A37: a in downarrow x & a in meet { I where I is Ideal of L : x <= sup I} by XBOOLE_0:def 3; thus a in meet A1 proof now let Y1 be set; assume Y1 in { (downarrow x) /\ I where I is Ideal of L : x <= sup I}; then consider I1 be Ideal of L such that A38: Y1 = (downarrow x) /\ I1 & x <= sup I1; I1 in { I where I is Ideal of L : x <= sup I} by A38; then a in I1 by A37,SETFAM_1:def 1; hence a in Y1 by A37,A38,XBOOLE_0:def 3; end; hence thesis by A1,A25,SETFAM_1:def 1; end; end; then (downarrow x) /\ meet { I where I is Ideal of L : x <= sup I} = meet A1 by A31,XBOOLE_0:def 10; then A39: (downarrow x) /\ waybelow x = meet A1 by Th34; waybelow x c= downarrow x by WAYBEL_3:11; hence thesis by A30,A39,XBOOLE_1:28; end; :: Proposition 1.13 p.45 theorem Th44: for L being lower-bounded meet-continuous LATTICE, x being Element of L holds meet {AR-below x where AR is auxiliary Relation of L : AR in App L} = waybelow x proof let L be lower-bounded meet-continuous LATTICE, x be Element of L; set A = {AR-below x where AR is auxiliary Relation of L : AR in App L}; consider AA be approximating auxiliary Relation of L; AA in App L by Def20; then A1:AA-below x in A; A2: meet { I where I is Ideal of L : x <= sup I } = waybelow x by Th34; A3: meet { (DownMap I).x where I is Ideal of L : not contradiction } = waybelow x by Th43; consider I1 be Ideal of L; A4: (DownMap I1).x in { (DownMap I).x where I is Ideal of L : not contradiction }; { (DownMap I).x where I is Ideal of L : not contradiction } c= A proof let a be set; assume a in { (DownMap I).x where I is Ideal of L : not contradiction }; then consider I be Ideal of L such that A5: a = (DownMap I).x; A6: DownMap I is approximating by Th37; A7: DownMap I in the carrier of MonSet L by Th35; then consider AR be auxiliary Relation of L such that A8: AR = (Map2Rel L).(DownMap I) & for x,y be set holds [x,y] in AR iff ex x',y' be Element of L, s' be map of L, InclPoset Ids L st x' = x & y' = y & s' = DownMap I & x' in s'.y' by Def16; consider AR1 be approximating auxiliary Relation of L such that A9: AR1 = (Map2Rel L).(DownMap I) by A6,A7,Th42; consider ii be Subset of L such that A10: ii = (DownMap I).x & x = sup ii by A6,Def19; A11: AR-below x = (DownMap I).x proof thus AR-below x c= (DownMap I).x proof let a; assume a in AR-below x; then [a,x] in AR by Th13; then consider x',y' be Element of L, s' be map of L, InclPoset Ids L such that A12: x' = a & y' = x & s' = DownMap I & x' in s'.y' by A8; thus thesis by A12; end; thus (DownMap I).x c= AR-below x proof let a; assume A13: a in (DownMap I).x; then reconsider a' = a as Element of L by A10; [a',x] in AR by A8,A13; hence thesis by Th13; end; end; AR in App L by A8,A9,Def20; hence a in A by A5,A11; end; hence meet A c= waybelow x by A3,A4,SETFAM_1:7; thus waybelow x c= meet A proof A c= { I where I is Ideal of L : x <= sup I } proof let a; assume a in A; then consider AR be auxiliary Relation of L such that A14: a = AR-below x & AR in App L; reconsider AR as approximating auxiliary Relation of L by A14,Def20; sup (AR-below x) = x by Def18; hence a in { I where I is Ideal of L : x <= sup I } by A14; end; hence thesis by A1,A2,SETFAM_1:7; end; end; reserve L for complete LATTICE; :: Proposition 1.14 p.45 ( 1 <=> 2 ) theorem Th45: L is continuous iff (for R being approximating auxiliary Relation of L holds L-waybelow c= R & L-waybelow is approximating) proof set AR = L-waybelow; hereby assume A1: L is continuous; then reconsider L' = L as lower-bounded meet-continuous LATTICE by Th38; thus for R be approximating auxiliary Relation of L holds AR c= R & AR is approximating proof let R be approximating auxiliary Relation of L; reconsider R' = R as approximating auxiliary Relation of L'; for a,b be set st [a,b] in AR holds [a,b] in R proof let a,b be set; assume A2: [a,b] in AR; then a in the carrier of L & b in the carrier of L by ZFMISC_1:106; then reconsider a' = a, b' = b as Element of L'; a' << b' by A2,Def2; then A3: a' in waybelow b' by WAYBEL_3:7; A4: meet { A1-below b' where A1 is auxiliary Relation of L' : A1 in App L' } = waybelow b' by Th44; R' in App L' by Def20; then R'-below b' in { A1-below b' where A1 is auxiliary Relation of L' : A1 in App L' }; then waybelow b' c= R'-below b' by A4,SETFAM_1:4; hence thesis by A3,Th13; end; hence AR c= R by RELAT_1:def 3; thus AR is approximating by A1,Lm13; end; end; assume A5: for R be approximating auxiliary Relation of L holds AR c= R & AR is approximating; for x being Element of L holds x = sup waybelow x proof let x be Element of L; x = sup (AR-below x) by A5,Def18; hence x = sup waybelow x by Th40; end; then A6:L is satisfying_axiom_of_approximation by WAYBEL_3:def 5; for x being Element of L holds waybelow x is non empty directed; hence L is continuous by A6,WAYBEL_3:def 6; end; :: Proposition 1.14 p.45 ( 1 <=> 3 ) theorem L is continuous iff (L is meet-continuous & ex R being approximating auxiliary Relation of L st for R' being approximating auxiliary Relation of L holds R c= R') proof hereby assume A1: L is continuous; hence L is meet-continuous by Th38; reconsider R = L-waybelow as approximating auxiliary Relation of L by A1,Lm13; take R; thus for R' be approximating auxiliary Relation of L holds R c= R' by A1,Th45; end; assume A2: L is meet-continuous; given R be approximating auxiliary Relation of L such that A3: for R' be approximating auxiliary Relation of L holds R c= R'; for x being Element of L holds x = sup waybelow x proof let x be Element of L; set K = {AR-below x where AR is auxiliary Relation of L : AR in App L}; A4: meet K = waybelow x by A2,Th44; R in App L by Def20; then A5: R-below x in K; then A6: waybelow x c= R-below x by A4,SETFAM_1:4; A7: sup (R-below x) = x by Def18; for a st a in K holds R-below x c= a proof let a; assume a in K; then consider AA be auxiliary Relation of L such that A8: a = AA-below x & AA in App L; reconsider AA as approximating auxiliary Relation of L by A8,Def20; let b be set; assume A9: b in R-below x; R c= AA by A3; then R-below x c= AA-below x by Th29; hence thesis by A8,A9; end; then R-below x c= meet K by A5,SETFAM_1:6; hence thesis by A4,A6,A7,XBOOLE_0:def 10; end; then A10: L is satisfying_axiom_of_approximation by WAYBEL_3:def 5; for x being Element of L holds waybelow x is non empty directed; hence L is continuous by A10,WAYBEL_3:def 6; end; :: Definition 1.15 (SI) p.46 definition let L be RelStr, AR be Relation of L; attr AR is satisfying_SI means :Def21: for x, z be Element of L holds ( [x,z] in AR & x <> z implies ex y be Element of L st ( [x,y] in AR & [y,z] in AR & x <> y ) ); synonym AR satisfies_SI; end; :: Definition 1.15 (INT) p.46 definition let L be RelStr, AR be Relation of L; attr AR is satisfying_INT means :Def22: for x, z be Element of L holds ( [x,z] in AR implies ex y be Element of L st ( [x,y] in AR & [y,z] in AR ) ); synonym AR satisfies_INT; end; canceled; theorem Th48: for L being RelStr, AR being Relation of L holds AR satisfies_SI implies AR satisfies_INT proof let L be RelStr, AR be Relation of L; assume A1: AR satisfies_SI; let x, z be Element of L; [x,z] in AR implies ex y be Element of L st ( [x,y] in AR & [y,z] in AR ) proof assume A2: [x,z] in AR; per cases; suppose x <> z; then consider y be Element of L such that A3: ( [x,y] in AR & [y,z] in AR & x <> y ) by A1,A2,Def21; thus ex y be Element of L st ( [x,y] in AR & [y,z] in AR ) by A3; suppose x = z; hence ex y be Element of L st ( [x,y] in AR & [y,z] in AR ) by A2; end; hence thesis; end; definition let L be non empty RelStr; cluster satisfying_SI -> satisfying_INT Relation of L; coherence by Th48; end; reserve AR for Relation of L; reserve x, y, z for Element of L; theorem Th49: for AR being approximating Relation of L st not x <= y holds ex u being Element of L st [u,x] in AR & not u <= y proof let AR be approximating Relation of L; assume A1: not x <= y; x = sup (AR-below x) & ex_sup_of (AR-below x),L by Def18,YELLOW_0:17; then y is_>=_than (AR-below x) implies y >= x by YELLOW_0:def 9; then consider u being Element of L such that A2: u in AR-below x & not u <= y by A1,LATTICE3:def 9; take u; thus [u,x] in AR & not u <= y by A2,Th13; end; :: Lemma 1.16 p.46 theorem Th50: for R being approximating auxiliary(i) auxiliary(iii) Relation of L holds ( [x,z] in R & x <> z ) implies ex y st x <= y & [y,z] in R & x <> y proof let R be approximating auxiliary(i) auxiliary(iii) Relation of L; assume A1: [x,z] in R & x <> z; then x <= z by Def4; then x < z by A1,ORDERS_1:def 10; then not z < x by ORDERS_1:28; then not z <= x by A1,ORDERS_1:def 10; then consider u be Element of L such that A2: [u,z] in R & not u <= x by Th49; take y = x "\/" u; thus x <= y by YELLOW_0:22; thus [y,z] in R by A1,A2,Def6; thus x <> y by A2,YELLOW_0:24; end; :: Lemma 1.17 p.46 theorem Th51: for R being approximating auxiliary Relation of L holds ( x << z & x <> z ) implies ex y being Element of L st [x,y] in R & [y,z] in R & x <> y proof let R be approximating auxiliary Relation of L; assume A1: x << z & x <> z; set I = {u where u is Element of L : ex y be Element of L st [u,y] in R & [y,z] in R }; A2:[Bottom L,Bottom L] in R by Def7; [Bottom L,z] in R by Def7; then A3:Bottom L in I by A2; I c= the carrier of L proof let v be set; assume v in I; then ex u1 being Element of L st v = u1 & ex y be Element of L st [u1,y] in R & [y,z] in R; hence thesis; end; then reconsider I as non empty Subset of L by A3; A4:I is lower proof let x1,y1 be Element of L; assume A5: x1 in I & y1 <= x1; then consider v1 be Element of L such that A6: v1 = x1 & ex s1 be Element of L st [v1,s1] in R & [s1,z] in R; consider s1 be Element of L such that A7: [v1,s1] in R & [s1,z] in R by A6; s1 <= s1; then [y1, s1] in R by A5,A6,A7,Def5; hence thesis by A7; end; I is directed proof let u1,u2 be Element of L; assume A8: u1 in I & u2 in I; then consider v1 be Element of L such that A9: v1 = u1 & ex y1 be Element of L st [v1,y1] in R & [y1,z] in R; consider v2 be Element of L such that A10: v2 = u2 & ex y2 be Element of L st [v2,y2] in R & [y2,z] in R by A8; consider y1 be Element of L such that A11: [v1,y1] in R & [y1,z] in R by A9; consider y2 be Element of L such that A12: [v2,y2] in R & [y2,z] in R by A10; take u1 "\/" u2; A13: [u1 "\/" u2 , y1 "\/" y2] in R by A9,A10,A11,A12,Th1; [y1 "\/" y2 , z] in R by A11,A12,Def6; hence thesis by A13,YELLOW_0:22; end; then reconsider I as Ideal of L by A4; sup I = z proof set z' = sup I; assume A14: z' <> z; A15:I c= R-below z proof let a be set; assume a in I; then consider u be Element of L such that A16: a = u & ex y2 be Element of L st [u,y2] in R & [y2,z] in R; consider y2 be Element of L such that A17: [u,y2] in R & [y2,z] in R by A16; A18: u <= y2 by A17,Def4; z <= z; then [u,z] in R by A17,A18,Def5; then a in {y where y is Element of L: [y,z] in R} by A16; hence thesis by Def11; end; ex_sup_of I,L & ex_sup_of (R-below z),L by YELLOW_0:17; then A19:sup I <= sup (R-below z) by A15,YELLOW_0:34; z = sup (R-below z) by Def18; then z' < z by A14,A19,ORDERS_1:def 10; then not z <= z' by ORDERS_1:30; then consider y be Element of L such that A20: [y, z] in R & not y <= z' by Th49; consider u be Element of L such that A21: [u, y] in R & not u <= z' by A20,Th49; A22:u in I by A20,A21; z' = "\/"(I,L) & ex_sup_of I,L iff z' is_>=_than I & for b being Element of L st b is_>=_than I holds z' <= b by YELLOW_0:30; hence contradiction by A21,A22,LATTICE3:def 9,YELLOW_0:17; end; then x in I by A1,WAYBEL_3:20; then consider v be Element of L such that A23: v = x & ex y' be Element of L st [v,y'] in R & [y',z] in R; consider y' be Element of L such that A24: [v,y'] in R & [y',z] in R by A23; A25: x <= y' by A23,A24,Def4; z <= z; then [x,z] in R by A24,A25,Def5; then consider y'' be Element of L such that A26: x <= y'' & [y'',z] in R & x <> y'' by A1,Th50; A27: x < y'' by A26,ORDERS_1:def 10; set Y = y' "\/" y''; A28: y' <= Y & y'' <= Y by YELLOW_0:22; then A29: x <> Y by A27,ORDERS_1:32; x <= x; then A30: [x,Y] in R by A23,A24,A28,Def5; [Y,z] in R by A24,A26,Def6; hence thesis by A29,A30; end; :: Theorem 1.18 p.47 theorem Th52: for L being lower-bounded continuous LATTICE holds L-waybelow satisfies_SI proof let L be lower-bounded continuous LATTICE; set R = L-waybelow; thus R satisfies_SI proof let x,z be Element of L; assume A1: [x,z] in R & x <> z; then x << z by Def2; hence thesis by A1,Th51; end; end; definition let L be lower-bounded continuous LATTICE; cluster L-waybelow -> satisfying_SI; coherence by Th52; end; theorem Th53: for L being lower-bounded continuous LATTICE, x,y being Element of L st x << y ex x' being Element of L st x << x' & x' << y proof let L be lower-bounded continuous LATTICE; let x,y be Element of L; set R = L-waybelow; assume x << y; then [x,y] in R by Def2; then consider x' be Element of L such that A1: [x,x'] in R & [x',y] in R by Def22; x << x' & x' << y by A1,Def2; hence thesis; end; :: Corollary 1.19 p.47 theorem for L being lower-bounded continuous LATTICE holds for x,y being Element of L holds ( x << y iff for D being non empty directed Subset of L st y <= sup D ex d being Element of L st d in D & x << d ) proof let L be lower-bounded continuous LATTICE; let x,y be Element of L; hereby assume A1: x << y; let D be non empty directed Subset of L; assume A2: y <= sup D; consider x' be Element of L such that A3: x << x' & x' << y by A1,Th53; consider d be Element of L such that A4: d in D & x' <= d by A2,A3,WAYBEL_3:def 1; x << d by A3,A4,WAYBEL_3:2; hence ex d be Element of L st d in D & x << d by A4; end; assume A5: for D be non empty directed Subset of L st y <= sup D ex d be Element of L st d in D & x << d; for D being non empty directed Subset of L st y <= sup D ex d being Element of L st d in D & x <= d proof let D be non empty directed Subset of L; assume y <= sup D; then consider d be Element of L such that A6: d in D & x << d by A5; x <= d by A6,WAYBEL_3:1; hence thesis by A6; end; hence thesis by WAYBEL_3:def 1; end; begin :: Exercises definition let L be RelStr, X be Subset of L, R be Relation of the carrier of L; pred X is_directed_wrt R means :Def23: for x,y being Element of L st x in X & y in X ex z being Element of L st z in X & [x,z] in R & [y,z] in R; end; theorem for L being RelStr, X being Subset of L st X is_directed_wrt (the InternalRel of L) holds X is directed proof let L be RelStr, X be Subset of L; assume A1: X is_directed_wrt (the InternalRel of L); let x,y be Element of L; assume x in X & y in X; then consider z being Element of L such that A2: z in X & [x,z] in the InternalRel of L & [y,z] in the InternalRel of L by A1,Def23; take z; thus z in X by A2; thus x <= z & y <= z by A2,ORDERS_1:def 9; end; definition let X, x be set; let R be Relation; pred x is_maximal_wrt X,R means :Def24: x in X & not ex y being set st y in X & y <> x & [x,y] in R; end; definition let L be RelStr, X be set, x be Element of L; pred x is_maximal_in X means :Def25: x is_maximal_wrt X, the InternalRel of L; end; theorem for L being RelStr, X being set, x being Element of L holds x is_maximal_in X iff x in X & not ex y being Element of L st y in X & x < y proof let L be RelStr, X be set, x be Element of L; hereby assume x is_maximal_in X; then A1:x is_maximal_wrt X, the InternalRel of L by Def25; hence x in X by Def24; let y be Element of L; per cases by A1,Def24; suppose not y in X; hence not y in X or not x < y; suppose y = x; hence not y in X or not x < y; suppose not [x,y] in the InternalRel of L; then not x <= y by ORDERS_1:def 9; hence not y in X or not x < y by ORDERS_1:def 10; end; assume A2: x in X & not ex y be Element of L st y in X & x < y; assume not x is_maximal_in X; then not x is_maximal_wrt X, the InternalRel of L by Def25; then consider y be set such that A3: y in X & y <> x & [x,y] in the InternalRel of L by A2,Def24; y in the carrier of L by A3,ZFMISC_1:106; then reconsider y as Element of L; x <= y by A3,ORDERS_1:def 9; then x < y by A3,ORDERS_1:def 10; hence thesis by A2,A3; end; definition let X, x be set; let R be Relation; pred x is_minimal_wrt X,R means :Def26: x in X & not ex y being set st y in X & y <> x & [y,x] in R; end; definition let L be RelStr, X be set, x be Element of L; pred x is_minimal_in X means :Def27: x is_minimal_wrt X, the InternalRel of L; end; theorem for L being RelStr, X being set, x being Element of L holds x is_minimal_in X iff x in X & not ex y being Element of L st y in X & x > y proof let L be RelStr, X be set, x be Element of L; hereby assume x is_minimal_in X; then A1: x is_minimal_wrt X, the InternalRel of L by Def27; hence x in X by Def26; let y be Element of L; per cases by A1,Def26; suppose not y in X; hence not y in X or not x > y; suppose y = x; hence not y in X or not x > y; suppose not [y,x] in the InternalRel of L; then not y <= x by ORDERS_1:def 9; hence not y in X or not y < x by ORDERS_1:def 10; end; assume A2: x in X & not ex y be Element of L st y in X & x > y; assume not x is_minimal_in X; then not x is_minimal_wrt X, the InternalRel of L by Def27; then consider y be set such that A3: y in X & y <> x & [y,x] in the InternalRel of L by A2,Def26; y in the carrier of L by A3,ZFMISC_1:106; then reconsider y as Element of L; y <= x by A3,ORDERS_1:def 9; then y < x by A3,ORDERS_1:def 10; hence thesis by A2,A3; end; :: Exercise 1.23 (i) ( 1 => 2 ) theorem AR satisfies_SI implies ( for x holds ( ex y st y is_maximal_wrt (AR-below x), AR ) implies [x,x] in AR ) proof assume A1: AR satisfies_SI; let x; given y such that A2: y is_maximal_wrt (AR-below x), AR; A3: y in AR-below x & not ex y' be Element of L st y' in AR-below x & y <> y' & [y,y'] in AR by A2,Def24; assume A4: not [x,x] in AR; A5: [y,x] in AR by A3,Th13; per cases; suppose x = y; hence contradiction by A3,A4,Th13; suppose x <> y; then consider z such that A6: [y,z] in AR & [z,x] in AR & z <> y by A1,A5, Def21; z in AR-below x by A6,Th13; hence contradiction by A2,A6,Def24; end; :: Exercise 1.23 (i) ( 2 => 1 ) theorem ( for x holds ( ex y st y is_maximal_wrt (AR-below x), AR ) implies [x,x] in AR ) implies AR satisfies_SI proof assume A1: for x holds ( ex y st y is_maximal_wrt (AR-below x), AR ) implies [x,x] in AR; now let z,x; assume A2: [z,x] in AR & z <> x; then A3: z in AR-below x by Th13; per cases; suppose [x,x] in AR; hence ex y st ( [z,y] in AR & [y,x] in AR & z <> y ) by A2; suppose not [x,x] in AR; then not z is_maximal_wrt (AR-below x), AR by A1; then consider y being set such that A4: y in AR-below x & y <> z & [z,y] in AR by A3,Def24; [y,x] in AR by A4,Th13; hence ex y st [z,y] in AR & [y,x] in AR & z <> y by A4; end; hence thesis by Def21; end; :: Exercise 1.23 (ii) ( 3 => 4 ) theorem for AR being auxiliary(ii) auxiliary(iii) Relation of L holds AR satisfies_INT implies ( for x holds AR-below x is_directed_wrt AR) proof let AR be auxiliary(ii) auxiliary(iii) Relation of L; assume A1: AR satisfies_INT; let x,y,z; assume y in AR-below x & z in AR-below x; then A2: [y,x] in AR & [z,x] in AR by Th13; then consider y' be Element of L such that A3: [y,y'] in AR & [y',x] in AR by A1,Def22; consider z' be Element of L such that A4: [z,z'] in AR & [z',x] in AR by A1,A2,Def22; take u = y' "\/" z'; [u,x] in AR by A3,A4,Def6; hence u in AR-below x by Th13; y <= y & y' <= u by YELLOW_0:22; hence [y,u] in AR by A3,Def5; z <= z & z' <= u by YELLOW_0:22; hence [z,u] in AR by A4,Def5; end; :: Exercise 1.23 (ii) ( 4 => 3 ) theorem ( for x holds AR-below x is_directed_wrt AR) implies AR satisfies_INT proof assume A1: for x holds AR-below x is_directed_wrt AR; let X,Z be Element of L; assume [X,Z] in AR; then A2: X in AR-below Z by Th13; AR-below Z is_directed_wrt AR by A1; then consider u be Element of L such that A3: u in AR-below Z & [X,u] in AR & [X,u] in AR by A2,Def23; take u; thus [X,u] in AR by A3; thus [u,Z] in AR by A3,Th13; end; :: Exercise 1.23 (iii) p.51 theorem Th62: for R being approximating auxiliary(i) auxiliary(ii) auxiliary(iii) Relation of L st R satisfies_INT holds R satisfies_SI proof let R be approximating auxiliary(i) auxiliary(ii) auxiliary(iii) Relation of L; assume A1: R satisfies_INT; let x, z; assume A2: [x,z] in R & x <> z; then consider y such that A3: ( [x,y] in R & [y,z] in R ) by A1,Def22; consider y' be Element of L such that A4: x <= y' & [y',z] in R & x <> y' by A2,Th50; A5: x < y' by A4,ORDERS_1:def 10; take Y = y "\/" y'; A6: x <= y by A3,Def4; A7: x <= x & y <= Y by YELLOW_0:22; per cases; suppose y' <= y; then x < y by A5,ORDERS_1:32; hence thesis by A3,A4,A7,Def5,Def6,ORDERS_1:32; suppose not y' <= y; then y <> Y by YELLOW_0:24; then y < Y by A7,ORDERS_1:def 10; hence thesis by A3,A4,A6,A7,Def5,Def6,ORDERS_1:32; end; definition let L; cluster satisfying_INT -> satisfying_SI (approximating auxiliary Relation of L); coherence by Th62; end;

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