:: YELLOW11 semantic presentation begin theorem :: YELLOW11:1 3 = {0,1,2} by CARD_1:51; theorem Th2: :: YELLOW11:2 2 \ 1 = {1} proof thus 2 \ 1 c= {1} :: according to XBOOLE_0:def_10 ::_thesis: {1} c= 2 \ 1 proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in 2 \ 1 or x in {1} ) assume A1: x in 2 \ 1 ; ::_thesis: x in {1} then A2: ( x = 0 or x = 1 ) by CARD_1:50, TARSKI:def_2; not x in {0} by A1, CARD_1:49, XBOOLE_0:def_5; hence x in {1} by A2, TARSKI:def_1; ::_thesis: verum end; let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {1} or x in 2 \ 1 ) assume x in {1} ; ::_thesis: x in 2 \ 1 then A3: x = 1 by TARSKI:def_1; then A4: not x in {0} by TARSKI:def_1; x in {0,1} by A3, TARSKI:def_2; hence x in 2 \ 1 by A4, CARD_1:49, CARD_1:50, XBOOLE_0:def_5; ::_thesis: verum end; theorem Th3: :: YELLOW11:3 3 \ 1 = {1,2} proof thus 3 \ 1 c= {1,2} :: according to XBOOLE_0:def_10 ::_thesis: {1,2} c= 3 \ 1 proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in 3 \ 1 or x in {1,2} ) assume A1: x in 3 \ 1 ; ::_thesis: x in {1,2} then A2: ( x = 0 or x = 1 or x = 2 ) by CARD_1:51, ENUMSET1:def_1; not x in {0} by A1, CARD_1:49, XBOOLE_0:def_5; hence x in {1,2} by A2, TARSKI:def_1, TARSKI:def_2; ::_thesis: verum end; thus {1,2} c= 3 \ 1 ::_thesis: verum proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {1,2} or x in 3 \ 1 ) assume x in {1,2} ; ::_thesis: x in 3 \ 1 then A3: ( x = 1 or x = 2 ) by TARSKI:def_2; then A4: not x in {0} by TARSKI:def_1; x in {0,1,2} by A3, ENUMSET1:def_1; hence x in 3 \ 1 by A4, CARD_1:49, CARD_1:51, XBOOLE_0:def_5; ::_thesis: verum end; end; theorem Th4: :: YELLOW11:4 3 \ 2 = {2} proof thus 3 \ 2 c= {2} :: according to XBOOLE_0:def_10 ::_thesis: {2} c= 3 \ 2 proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in 3 \ 2 or x in {2} ) assume A1: x in 3 \ 2 ; ::_thesis: x in {2} then A2: ( x = 0 or x = 1 or x = 2 ) by CARD_1:51, ENUMSET1:def_1; not x in {0,1} by A1, CARD_1:50, XBOOLE_0:def_5; hence x in {2} by A2, TARSKI:def_1, TARSKI:def_2; ::_thesis: verum end; thus {2} c= 3 \ 2 ::_thesis: verum proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {2} or x in 3 \ 2 ) assume x in {2} ; ::_thesis: x in 3 \ 2 then A3: x = 2 by TARSKI:def_1; then A4: not x in {0,1} by TARSKI:def_2; x in {0,1,2} by A3, ENUMSET1:def_1; hence x in 3 \ 2 by A4, CARD_1:50, CARD_1:51, XBOOLE_0:def_5; ::_thesis: verum end; end; Lm1: 3 \ 2 c= 3 \ 1 proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in 3 \ 2 or x in 3 \ 1 ) assume x in 3 \ 2 ; ::_thesis: x in 3 \ 1 then x = 2 by Th4, TARSKI:def_1; hence x in 3 \ 1 by Th3, TARSKI:def_2; ::_thesis: verum end; begin theorem Th5: :: YELLOW11:5 for L being reflexive antisymmetric with_suprema with_infima RelStr for a, b being Element of L holds ( a "/\" b = b iff a "\/" b = a ) proof let L be reflexive antisymmetric with_suprema with_infima RelStr ; ::_thesis: for a, b being Element of L holds ( a "/\" b = b iff a "\/" b = a ) let a, b be Element of L; ::_thesis: ( a "/\" b = b iff a "\/" b = a ) thus ( a "/\" b = b implies a "\/" b = a ) ::_thesis: ( a "\/" b = a implies a "/\" b = b ) proof assume a "/\" b = b ; ::_thesis: a "\/" b = a then b <= a by YELLOW_0:23; hence a "\/" b = a by YELLOW_0:24; ::_thesis: verum end; assume a "\/" b = a ; ::_thesis: a "/\" b = b then b <= a by YELLOW_0:22; hence a "/\" b = b by YELLOW_0:25; ::_thesis: verum end; theorem Th6: :: YELLOW11:6 for L being LATTICE for a, b, c being Element of L holds (a "/\" b) "\/" (a "/\" c) <= a "/\" (b "\/" c) proof let L be LATTICE; ::_thesis: for a, b, c being Element of L holds (a "/\" b) "\/" (a "/\" c) <= a "/\" (b "\/" c) let a, b, c be Element of L; ::_thesis: (a "/\" b) "\/" (a "/\" c) <= a "/\" (b "\/" c) A1: a <= a ; c <= b "\/" c by YELLOW_0:22; then A2: a "/\" c <= a "/\" (b "\/" c) by A1, YELLOW_3:2; b <= b "\/" c by YELLOW_0:22; then a "/\" b <= a "/\" (b "\/" c) by A1, YELLOW_3:2; hence (a "/\" b) "\/" (a "/\" c) <= a "/\" (b "\/" c) by A2, YELLOW_5:9; ::_thesis: verum end; theorem Th7: :: YELLOW11:7 for L being LATTICE for a, b, c being Element of L holds a "\/" (b "/\" c) <= (a "\/" b) "/\" (a "\/" c) proof let L be LATTICE; ::_thesis: for a, b, c being Element of L holds a "\/" (b "/\" c) <= (a "\/" b) "/\" (a "\/" c) let a, b, c be Element of L; ::_thesis: a "\/" (b "/\" c) <= (a "\/" b) "/\" (a "\/" c) A1: a <= a ; b "/\" c <= c by YELLOW_0:23; then A2: a "\/" (b "/\" c) <= a "\/" c by A1, YELLOW_3:3; b "/\" c <= b by YELLOW_0:23; then a "\/" (b "/\" c) <= a "\/" b by A1, YELLOW_3:3; hence a "\/" (b "/\" c) <= (a "\/" b) "/\" (a "\/" c) by A2, YELLOW_0:23; ::_thesis: verum end; theorem Th8: :: YELLOW11:8 for L being LATTICE for a, b, c being Element of L st a <= c holds a "\/" (b "/\" c) <= (a "\/" b) "/\" c proof let L be LATTICE; ::_thesis: for a, b, c being Element of L st a <= c holds a "\/" (b "/\" c) <= (a "\/" b) "/\" c let a, b, c be Element of L; ::_thesis: ( a <= c implies a "\/" (b "/\" c) <= (a "\/" b) "/\" c ) A1: b "/\" c <= c by YELLOW_0:23; A2: a <= a ; b "/\" c <= b by YELLOW_0:23; then A3: a "\/" (b "/\" c) <= a "\/" b by A2, YELLOW_3:3; assume a <= c ; ::_thesis: a "\/" (b "/\" c) <= (a "\/" b) "/\" c then a "\/" (b "/\" c) <= c by A1, YELLOW_0:22; hence a "\/" (b "/\" c) <= (a "\/" b) "/\" c by A3, YELLOW_0:23; ::_thesis: verum end; definition func N_5 -> RelStr equals :: YELLOW11:def 1 InclPoset {0,(3 \ 1),2,(3 \ 2),3}; correctness coherence InclPoset {0,(3 \ 1),2,(3 \ 2),3} is RelStr ; ; end; :: deftheorem defines N_5 YELLOW11:def_1_:_ N_5 = InclPoset {0,(3 \ 1),2,(3 \ 2),3}; registration cluster N_5 -> strict reflexive transitive antisymmetric ; correctness coherence ( N_5 is strict & N_5 is reflexive & N_5 is transitive & N_5 is antisymmetric ); ; cluster N_5 -> with_suprema with_infima ; correctness coherence ( N_5 is with_infima & N_5 is with_suprema ); proof set Z = {0,(3 \ 1),2,(3 \ 2),3}; set N = InclPoset {0,(3 \ 1),2,(3 \ 2),3}; A1: InclPoset {0,(3 \ 1),2,(3 \ 2),3} is with_infima proof let x, y be Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}); :: according to LATTICE3:def_11 ::_thesis: ex b1 being Element of the carrier of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}) st ( b1 <= x & b1 <= y & ( for b2 being Element of the carrier of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}) holds ( not b2 <= x or not b2 <= y or b2 <= b1 ) ) ) A2: {0,(3 \ 1),2,(3 \ 2),3} = the carrier of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}) by YELLOW_1:1; thus ex z being Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}) st ( z <= x & z <= y & ( for z9 being Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}) st z9 <= x & z9 <= y holds z9 <= z ) ) ::_thesis: verum proof percases ( ( x = 0 & y = 0 ) or ( x = 0 & y = 3 \ 1 ) or ( x = 0 & y = 2 ) or ( x = 0 & y = 3 \ 2 ) or ( x = 0 & y = 3 ) or ( x = 3 \ 1 & y = 0 ) or ( x = 3 \ 1 & y = 3 \ 1 ) or ( x = 3 \ 1 & y = 2 ) or ( x = 3 \ 1 & y = 3 \ 2 ) or ( x = 3 \ 1 & y = 3 ) or ( x = 2 & y = 0 ) or ( x = 2 & y = 3 \ 1 ) or ( x = 2 & y = 2 ) or ( x = 2 & y = 3 \ 2 ) or ( x = 2 & y = 3 ) or ( x = 3 \ 2 & y = 0 ) or ( x = 3 \ 2 & y = 3 \ 1 ) or ( x = 3 \ 2 & y = 2 ) or ( x = 3 \ 2 & y = 3 \ 2 ) or ( x = 3 \ 2 & y = 3 ) or ( x = 3 & y = 0 ) or ( x = 3 & y = 3 \ 1 ) or ( x = 3 & y = 2 ) or ( x = 3 & y = 3 \ 2 ) or ( x = 3 & y = 3 ) ) by A2, ENUMSET1:def_3; suppose ( x = 0 & y = 0 ) ; ::_thesis: ex z being Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}) st ( z <= x & z <= y & ( for z9 being Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}) st z9 <= x & z9 <= y holds z9 <= z ) ) then reconsider z = x /\ y as Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}) ; take z ; ::_thesis: ( z <= x & z <= y & ( for z9 being Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}) st z9 <= x & z9 <= y holds z9 <= z ) ) A3: z c= y by XBOOLE_1:17; z c= x by XBOOLE_1:17; hence ( z <= x & z <= y ) by A3, YELLOW_1:3; ::_thesis: for z9 being Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}) st z9 <= x & z9 <= y holds z9 <= z let w be Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}); ::_thesis: ( w <= x & w <= y implies w <= z ) assume that A4: w <= x and A5: w <= y ; ::_thesis: w <= z A6: w c= y by A5, YELLOW_1:3; w c= x by A4, YELLOW_1:3; then w c= x /\ y by A6, XBOOLE_1:19; hence w <= z by YELLOW_1:3; ::_thesis: verum end; suppose ( x = 0 & y = 3 \ 1 ) ; ::_thesis: ex z being Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}) st ( z <= x & z <= y & ( for z9 being Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}) st z9 <= x & z9 <= y holds z9 <= z ) ) then reconsider z = x /\ y as Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}) ; take z ; ::_thesis: ( z <= x & z <= y & ( for z9 being Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}) st z9 <= x & z9 <= y holds z9 <= z ) ) A7: z c= y by XBOOLE_1:17; z c= x by XBOOLE_1:17; hence ( z <= x & z <= y ) by A7, YELLOW_1:3; ::_thesis: for z9 being Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}) st z9 <= x & z9 <= y holds z9 <= z let w be Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}); ::_thesis: ( w <= x & w <= y implies w <= z ) assume that A8: w <= x and A9: w <= y ; ::_thesis: w <= z A10: w c= y by A9, YELLOW_1:3; w c= x by A8, YELLOW_1:3; then w c= x /\ y by A10, XBOOLE_1:19; hence w <= z by YELLOW_1:3; ::_thesis: verum end; suppose ( x = 0 & y = 2 ) ; ::_thesis: ex z being Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}) st ( z <= x & z <= y & ( for z9 being Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}) st z9 <= x & z9 <= y holds z9 <= z ) ) then reconsider z = x /\ y as Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}) ; take z ; ::_thesis: ( z <= x & z <= y & ( for z9 being Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}) st z9 <= x & z9 <= y holds z9 <= z ) ) A11: z c= y by XBOOLE_1:17; z c= x by XBOOLE_1:17; hence ( z <= x & z <= y ) by A11, YELLOW_1:3; ::_thesis: for z9 being Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}) st z9 <= x & z9 <= y holds z9 <= z let w be Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}); ::_thesis: ( w <= x & w <= y implies w <= z ) assume that A12: w <= x and A13: w <= y ; ::_thesis: w <= z A14: w c= y by A13, YELLOW_1:3; w c= x by A12, YELLOW_1:3; then w c= x /\ y by A14, XBOOLE_1:19; hence w <= z by YELLOW_1:3; ::_thesis: verum end; suppose ( x = 0 & y = 3 \ 2 ) ; ::_thesis: ex z being Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}) st ( z <= x & z <= y & ( for z9 being Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}) st z9 <= x & z9 <= y holds z9 <= z ) ) then reconsider z = x /\ y as Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}) ; take z ; ::_thesis: ( z <= x & z <= y & ( for z9 being Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}) st z9 <= x & z9 <= y holds z9 <= z ) ) A15: z c= y by XBOOLE_1:17; z c= x by XBOOLE_1:17; hence ( z <= x & z <= y ) by A15, YELLOW_1:3; ::_thesis: for z9 being Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}) st z9 <= x & z9 <= y holds z9 <= z let w be Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}); ::_thesis: ( w <= x & w <= y implies w <= z ) assume that A16: w <= x and A17: w <= y ; ::_thesis: w <= z A18: w c= y by A17, YELLOW_1:3; w c= x by A16, YELLOW_1:3; then w c= x /\ y by A18, XBOOLE_1:19; hence w <= z by YELLOW_1:3; ::_thesis: verum end; suppose ( x = 0 & y = 3 ) ; ::_thesis: ex z being Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}) st ( z <= x & z <= y & ( for z9 being Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}) st z9 <= x & z9 <= y holds z9 <= z ) ) then reconsider z = x /\ y as Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}) ; take z ; ::_thesis: ( z <= x & z <= y & ( for z9 being Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}) st z9 <= x & z9 <= y holds z9 <= z ) ) A19: z c= y by XBOOLE_1:17; z c= x by XBOOLE_1:17; hence ( z <= x & z <= y ) by A19, YELLOW_1:3; ::_thesis: for z9 being Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}) st z9 <= x & z9 <= y holds z9 <= z let w be Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}); ::_thesis: ( w <= x & w <= y implies w <= z ) assume that A20: w <= x and A21: w <= y ; ::_thesis: w <= z A22: w c= y by A21, YELLOW_1:3; w c= x by A20, YELLOW_1:3; then w c= x /\ y by A22, XBOOLE_1:19; hence w <= z by YELLOW_1:3; ::_thesis: verum end; suppose ( x = 3 \ 1 & y = 0 ) ; ::_thesis: ex z being Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}) st ( z <= x & z <= y & ( for z9 being Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}) st z9 <= x & z9 <= y holds z9 <= z ) ) then reconsider z = x /\ y as Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}) ; take z ; ::_thesis: ( z <= x & z <= y & ( for z9 being Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}) st z9 <= x & z9 <= y holds z9 <= z ) ) A23: z c= y by XBOOLE_1:17; z c= x by XBOOLE_1:17; hence ( z <= x & z <= y ) by A23, YELLOW_1:3; ::_thesis: for z9 being Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}) st z9 <= x & z9 <= y holds z9 <= z let w be Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}); ::_thesis: ( w <= x & w <= y implies w <= z ) assume that A24: w <= x and A25: w <= y ; ::_thesis: w <= z A26: w c= y by A25, YELLOW_1:3; w c= x by A24, YELLOW_1:3; then w c= x /\ y by A26, XBOOLE_1:19; hence w <= z by YELLOW_1:3; ::_thesis: verum end; suppose ( x = 3 \ 1 & y = 3 \ 1 ) ; ::_thesis: ex z being Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}) st ( z <= x & z <= y & ( for z9 being Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}) st z9 <= x & z9 <= y holds z9 <= z ) ) then reconsider z = x /\ y as Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}) ; take z ; ::_thesis: ( z <= x & z <= y & ( for z9 being Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}) st z9 <= x & z9 <= y holds z9 <= z ) ) A27: z c= y by XBOOLE_1:17; z c= x by XBOOLE_1:17; hence ( z <= x & z <= y ) by A27, YELLOW_1:3; ::_thesis: for z9 being Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}) st z9 <= x & z9 <= y holds z9 <= z let w be Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}); ::_thesis: ( w <= x & w <= y implies w <= z ) assume that A28: w <= x and A29: w <= y ; ::_thesis: w <= z A30: w c= y by A29, YELLOW_1:3; w c= x by A28, YELLOW_1:3; then w c= x /\ y by A30, XBOOLE_1:19; hence w <= z by YELLOW_1:3; ::_thesis: verum end; supposeA31: ( x = 3 \ 1 & y = 2 ) ; ::_thesis: ex z being Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}) st ( z <= x & z <= y & ( for z9 being Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}) st z9 <= x & z9 <= y holds z9 <= z ) ) 0 in {0,(3 \ 1),2,(3 \ 2),3} by ENUMSET1:def_3; then reconsider z = 0 as Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}) by YELLOW_1:1; take z ; ::_thesis: ( z <= x & z <= y & ( for z9 being Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}) st z9 <= x & z9 <= y holds z9 <= z ) ) A32: z c= y by XBOOLE_1:2; z c= x by XBOOLE_1:2; hence ( z <= x & z <= y ) by A32, YELLOW_1:3; ::_thesis: for z9 being Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}) st z9 <= x & z9 <= y holds z9 <= z let z9 be Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}); ::_thesis: ( z9 <= x & z9 <= y implies z9 <= z ) assume that A33: z9 <= x and A34: z9 <= y ; ::_thesis: z9 <= z A35: z9 c= 3 \ 1 by A31, A33, YELLOW_1:3; A36: now__::_thesis:_not_z9_=_2 assume z9 = 2 ; ::_thesis: contradiction then 0 in z9 by CARD_1:50, TARSKI:def_2; hence contradiction by A35, Th3, TARSKI:def_2; ::_thesis: verum end; A37: z9 c= 2 by A31, A34, YELLOW_1:3; A38: now__::_thesis:_not_z9_=_3 assume z9 = 3 ; ::_thesis: contradiction then A39: 2 in z9 by CARD_1:51, ENUMSET1:def_1; not 2 in 2 ; hence contradiction by A37, A39; ::_thesis: verum end; A40: now__::_thesis:_not_z9_=_3_\_2 assume z9 = 3 \ 2 ; ::_thesis: contradiction then A41: 2 in z9 by Th4, TARSKI:def_1; not 2 in 2 ; hence contradiction by A37, A41; ::_thesis: verum end; A42: now__::_thesis:_not_z9_=_3_\_1 assume z9 = 3 \ 1 ; ::_thesis: contradiction then A43: 2 in z9 by Th3, TARSKI:def_2; not 2 in 2 ; hence contradiction by A37, A43; ::_thesis: verum end; z9 is Element of {0,(3 \ 1),2,(3 \ 2),3} by YELLOW_1:1; hence z9 <= z by A42, A36, A40, A38, ENUMSET1:def_3; ::_thesis: verum end; suppose ( x = 3 \ 1 & y = 3 \ 2 ) ; ::_thesis: ex z being Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}) st ( z <= x & z <= y & ( for z9 being Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}) st z9 <= x & z9 <= y holds z9 <= z ) ) then reconsider z = x /\ y as Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}) by Th3, Th4, ZFMISC_1:13; take z ; ::_thesis: ( z <= x & z <= y & ( for z9 being Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}) st z9 <= x & z9 <= y holds z9 <= z ) ) A44: z c= y by XBOOLE_1:17; z c= x by XBOOLE_1:17; hence ( z <= x & z <= y ) by A44, YELLOW_1:3; ::_thesis: for z9 being Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}) st z9 <= x & z9 <= y holds z9 <= z let w be Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}); ::_thesis: ( w <= x & w <= y implies w <= z ) assume that A45: w <= x and A46: w <= y ; ::_thesis: w <= z A47: w c= y by A46, YELLOW_1:3; w c= x by A45, YELLOW_1:3; then w c= x /\ y by A47, XBOOLE_1:19; hence w <= z by YELLOW_1:3; ::_thesis: verum end; supposeA48: ( x = 3 \ 1 & y = 3 ) ; ::_thesis: ex z being Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}) st ( z <= x & z <= y & ( for z9 being Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}) st z9 <= x & z9 <= y holds z9 <= z ) ) (3 \ 1) /\ 3 = (3 /\ 3) \ 1 by XBOOLE_1:49 .= 3 \ 1 ; then reconsider z = x /\ y as Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}) by A48; take z ; ::_thesis: ( z <= x & z <= y & ( for z9 being Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}) st z9 <= x & z9 <= y holds z9 <= z ) ) A49: z c= y by XBOOLE_1:17; z c= x by XBOOLE_1:17; hence ( z <= x & z <= y ) by A49, YELLOW_1:3; ::_thesis: for z9 being Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}) st z9 <= x & z9 <= y holds z9 <= z let w be Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}); ::_thesis: ( w <= x & w <= y implies w <= z ) assume that A50: w <= x and A51: w <= y ; ::_thesis: w <= z A52: w c= y by A51, YELLOW_1:3; w c= x by A50, YELLOW_1:3; then w c= x /\ y by A52, XBOOLE_1:19; hence w <= z by YELLOW_1:3; ::_thesis: verum end; suppose ( x = 2 & y = 0 ) ; ::_thesis: ex z being Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}) st ( z <= x & z <= y & ( for z9 being Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}) st z9 <= x & z9 <= y holds z9 <= z ) ) then reconsider z = x /\ y as Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}) ; take z ; ::_thesis: ( z <= x & z <= y & ( for z9 being Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}) st z9 <= x & z9 <= y holds z9 <= z ) ) A53: z c= y by XBOOLE_1:17; z c= x by XBOOLE_1:17; hence ( z <= x & z <= y ) by A53, YELLOW_1:3; ::_thesis: for z9 being Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}) st z9 <= x & z9 <= y holds z9 <= z let w be Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}); ::_thesis: ( w <= x & w <= y implies w <= z ) assume that A54: w <= x and A55: w <= y ; ::_thesis: w <= z A56: w c= y by A55, YELLOW_1:3; w c= x by A54, YELLOW_1:3; then w c= x /\ y by A56, XBOOLE_1:19; hence w <= z by YELLOW_1:3; ::_thesis: verum end; supposeA57: ( x = 2 & y = 3 \ 1 ) ; ::_thesis: ex z being Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}) st ( z <= x & z <= y & ( for z9 being Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}) st z9 <= x & z9 <= y holds z9 <= z ) ) 0 in {0,(3 \ 1),2,(3 \ 2),3} by ENUMSET1:def_3; then reconsider z = 0 as Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}) by YELLOW_1:1; take z ; ::_thesis: ( z <= x & z <= y & ( for z9 being Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}) st z9 <= x & z9 <= y holds z9 <= z ) ) A58: z c= y by XBOOLE_1:2; z c= x by XBOOLE_1:2; hence ( z <= x & z <= y ) by A58, YELLOW_1:3; ::_thesis: for z9 being Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}) st z9 <= x & z9 <= y holds z9 <= z let z9 be Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}); ::_thesis: ( z9 <= x & z9 <= y implies z9 <= z ) assume that A59: z9 <= x and A60: z9 <= y ; ::_thesis: z9 <= z A61: z9 c= 3 \ 1 by A57, A60, YELLOW_1:3; A62: now__::_thesis:_not_z9_=_2 assume z9 = 2 ; ::_thesis: contradiction then 0 in z9 by CARD_1:50, TARSKI:def_2; hence contradiction by A61, Th3, TARSKI:def_2; ::_thesis: verum end; A63: z9 c= 2 by A57, A59, YELLOW_1:3; A64: now__::_thesis:_not_z9_=_3 assume z9 = 3 ; ::_thesis: contradiction then A65: 2 in z9 by CARD_1:51, ENUMSET1:def_1; not 2 in 2 ; hence contradiction by A63, A65; ::_thesis: verum end; A66: now__::_thesis:_not_z9_=_3_\_2 assume z9 = 3 \ 2 ; ::_thesis: contradiction then A67: 2 in z9 by Th4, TARSKI:def_1; not 2 in 2 ; hence contradiction by A63, A67; ::_thesis: verum end; A68: now__::_thesis:_not_z9_=_3_\_1 assume z9 = 3 \ 1 ; ::_thesis: contradiction then A69: 2 in z9 by Th3, TARSKI:def_2; not 2 in 2 ; hence contradiction by A63, A69; ::_thesis: verum end; z9 is Element of {0,(3 \ 1),2,(3 \ 2),3} by YELLOW_1:1; hence z9 <= z by A68, A62, A66, A64, ENUMSET1:def_3; ::_thesis: verum end; suppose ( x = 2 & y = 2 ) ; ::_thesis: ex z being Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}) st ( z <= x & z <= y & ( for z9 being Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}) st z9 <= x & z9 <= y holds z9 <= z ) ) then reconsider z = x /\ y as Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}) ; take z ; ::_thesis: ( z <= x & z <= y & ( for z9 being Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}) st z9 <= x & z9 <= y holds z9 <= z ) ) A70: z c= y by XBOOLE_1:17; z c= x by XBOOLE_1:17; hence ( z <= x & z <= y ) by A70, YELLOW_1:3; ::_thesis: for z9 being Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}) st z9 <= x & z9 <= y holds z9 <= z let w be Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}); ::_thesis: ( w <= x & w <= y implies w <= z ) assume that A71: w <= x and A72: w <= y ; ::_thesis: w <= z A73: w c= y by A72, YELLOW_1:3; w c= x by A71, YELLOW_1:3; then w c= x /\ y by A73, XBOOLE_1:19; hence w <= z by YELLOW_1:3; ::_thesis: verum end; supposeA74: ( x = 2 & y = 3 \ 2 ) ; ::_thesis: ex z being Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}) st ( z <= x & z <= y & ( for z9 being Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}) st z9 <= x & z9 <= y holds z9 <= z ) ) 2 misses 3 \ 2 by XBOOLE_1:79; then 2 /\ (3 \ 2) = 0 by XBOOLE_0:def_7; then x /\ y in {0,(3 \ 1),2,(3 \ 2),3} by A74, ENUMSET1:def_3; then reconsider z = x /\ y as Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}) by YELLOW_1:1; take z ; ::_thesis: ( z <= x & z <= y & ( for z9 being Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}) st z9 <= x & z9 <= y holds z9 <= z ) ) A75: z c= y by XBOOLE_1:17; z c= x by XBOOLE_1:17; hence ( z <= x & z <= y ) by A75, YELLOW_1:3; ::_thesis: for z9 being Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}) st z9 <= x & z9 <= y holds z9 <= z let w be Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}); ::_thesis: ( w <= x & w <= y implies w <= z ) assume that A76: w <= x and A77: w <= y ; ::_thesis: w <= z A78: w c= y by A77, YELLOW_1:3; w c= x by A76, YELLOW_1:3; then w c= x /\ y by A78, XBOOLE_1:19; hence w <= z by YELLOW_1:3; ::_thesis: verum end; supposeA79: ( x = 2 & y = 3 ) ; ::_thesis: ex z being Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}) st ( z <= x & z <= y & ( for z9 being Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}) st z9 <= x & z9 <= y holds z9 <= z ) ) 2 c= 3 by NAT_1:39; then reconsider z = x /\ y as Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}) by A79, XBOOLE_1:28; take z ; ::_thesis: ( z <= x & z <= y & ( for z9 being Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}) st z9 <= x & z9 <= y holds z9 <= z ) ) A80: z c= y by XBOOLE_1:17; z c= x by XBOOLE_1:17; hence ( z <= x & z <= y ) by A80, YELLOW_1:3; ::_thesis: for z9 being Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}) st z9 <= x & z9 <= y holds z9 <= z let w be Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}); ::_thesis: ( w <= x & w <= y implies w <= z ) assume that A81: w <= x and A82: w <= y ; ::_thesis: w <= z A83: w c= y by A82, YELLOW_1:3; w c= x by A81, YELLOW_1:3; then w c= x /\ y by A83, XBOOLE_1:19; hence w <= z by YELLOW_1:3; ::_thesis: verum end; suppose ( x = 3 \ 2 & y = 0 ) ; ::_thesis: ex z being Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}) st ( z <= x & z <= y & ( for z9 being Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}) st z9 <= x & z9 <= y holds z9 <= z ) ) then reconsider z = x /\ y as Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}) ; take z ; ::_thesis: ( z <= x & z <= y & ( for z9 being Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}) st z9 <= x & z9 <= y holds z9 <= z ) ) A84: z c= y by XBOOLE_1:17; z c= x by XBOOLE_1:17; hence ( z <= x & z <= y ) by A84, YELLOW_1:3; ::_thesis: for z9 being Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}) st z9 <= x & z9 <= y holds z9 <= z let w be Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}); ::_thesis: ( w <= x & w <= y implies w <= z ) assume that A85: w <= x and A86: w <= y ; ::_thesis: w <= z A87: w c= y by A86, YELLOW_1:3; w c= x by A85, YELLOW_1:3; then w c= x /\ y by A87, XBOOLE_1:19; hence w <= z by YELLOW_1:3; ::_thesis: verum end; suppose ( x = 3 \ 2 & y = 3 \ 1 ) ; ::_thesis: ex z being Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}) st ( z <= x & z <= y & ( for z9 being Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}) st z9 <= x & z9 <= y holds z9 <= z ) ) then reconsider z = x /\ y as Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}) by Th3, Th4, ZFMISC_1:13; take z ; ::_thesis: ( z <= x & z <= y & ( for z9 being Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}) st z9 <= x & z9 <= y holds z9 <= z ) ) A88: z c= y by XBOOLE_1:17; z c= x by XBOOLE_1:17; hence ( z <= x & z <= y ) by A88, YELLOW_1:3; ::_thesis: for z9 being Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}) st z9 <= x & z9 <= y holds z9 <= z let w be Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}); ::_thesis: ( w <= x & w <= y implies w <= z ) assume that A89: w <= x and A90: w <= y ; ::_thesis: w <= z A91: w c= y by A90, YELLOW_1:3; w c= x by A89, YELLOW_1:3; then w c= x /\ y by A91, XBOOLE_1:19; hence w <= z by YELLOW_1:3; ::_thesis: verum end; supposeA92: ( x = 3 \ 2 & y = 2 ) ; ::_thesis: ex z being Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}) st ( z <= x & z <= y & ( for z9 being Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}) st z9 <= x & z9 <= y holds z9 <= z ) ) 2 misses 3 \ 2 by XBOOLE_1:79; then 2 /\ (3 \ 2) = 0 by XBOOLE_0:def_7; then x /\ y in {0,(3 \ 1),2,(3 \ 2),3} by A92, ENUMSET1:def_3; then reconsider z = x /\ y as Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}) by YELLOW_1:1; take z ; ::_thesis: ( z <= x & z <= y & ( for z9 being Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}) st z9 <= x & z9 <= y holds z9 <= z ) ) A93: z c= y by XBOOLE_1:17; z c= x by XBOOLE_1:17; hence ( z <= x & z <= y ) by A93, YELLOW_1:3; ::_thesis: for z9 being Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}) st z9 <= x & z9 <= y holds z9 <= z let w be Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}); ::_thesis: ( w <= x & w <= y implies w <= z ) assume that A94: w <= x and A95: w <= y ; ::_thesis: w <= z A96: w c= y by A95, YELLOW_1:3; w c= x by A94, YELLOW_1:3; then w c= x /\ y by A96, XBOOLE_1:19; hence w <= z by YELLOW_1:3; ::_thesis: verum end; suppose ( x = 3 \ 2 & y = 3 \ 2 ) ; ::_thesis: ex z being Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}) st ( z <= x & z <= y & ( for z9 being Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}) st z9 <= x & z9 <= y holds z9 <= z ) ) then reconsider z = x /\ y as Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}) ; take z ; ::_thesis: ( z <= x & z <= y & ( for z9 being Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}) st z9 <= x & z9 <= y holds z9 <= z ) ) A97: z c= y by XBOOLE_1:17; z c= x by XBOOLE_1:17; hence ( z <= x & z <= y ) by A97, YELLOW_1:3; ::_thesis: for z9 being Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}) st z9 <= x & z9 <= y holds z9 <= z let w be Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}); ::_thesis: ( w <= x & w <= y implies w <= z ) assume that A98: w <= x and A99: w <= y ; ::_thesis: w <= z A100: w c= y by A99, YELLOW_1:3; w c= x by A98, YELLOW_1:3; then w c= x /\ y by A100, XBOOLE_1:19; hence w <= z by YELLOW_1:3; ::_thesis: verum end; supposeA101: ( x = 3 \ 2 & y = 3 ) ; ::_thesis: ex z being Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}) st ( z <= x & z <= y & ( for z9 being Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}) st z9 <= x & z9 <= y holds z9 <= z ) ) (3 \ 2) /\ 3 = (3 /\ 3) \ 2 by XBOOLE_1:49 .= 3 \ 2 ; then reconsider z = x /\ y as Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}) by A101; take z ; ::_thesis: ( z <= x & z <= y & ( for z9 being Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}) st z9 <= x & z9 <= y holds z9 <= z ) ) A102: z c= y by XBOOLE_1:17; z c= x by XBOOLE_1:17; hence ( z <= x & z <= y ) by A102, YELLOW_1:3; ::_thesis: for z9 being Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}) st z9 <= x & z9 <= y holds z9 <= z let w be Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}); ::_thesis: ( w <= x & w <= y implies w <= z ) assume that A103: w <= x and A104: w <= y ; ::_thesis: w <= z A105: w c= y by A104, YELLOW_1:3; w c= x by A103, YELLOW_1:3; then w c= x /\ y by A105, XBOOLE_1:19; hence w <= z by YELLOW_1:3; ::_thesis: verum end; suppose ( x = 3 & y = 0 ) ; ::_thesis: ex z being Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}) st ( z <= x & z <= y & ( for z9 being Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}) st z9 <= x & z9 <= y holds z9 <= z ) ) then reconsider z = x /\ y as Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}) ; take z ; ::_thesis: ( z <= x & z <= y & ( for z9 being Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}) st z9 <= x & z9 <= y holds z9 <= z ) ) A106: z c= y by XBOOLE_1:17; z c= x by XBOOLE_1:17; hence ( z <= x & z <= y ) by A106, YELLOW_1:3; ::_thesis: for z9 being Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}) st z9 <= x & z9 <= y holds z9 <= z let w be Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}); ::_thesis: ( w <= x & w <= y implies w <= z ) assume that A107: w <= x and A108: w <= y ; ::_thesis: w <= z A109: w c= y by A108, YELLOW_1:3; w c= x by A107, YELLOW_1:3; then w c= x /\ y by A109, XBOOLE_1:19; hence w <= z by YELLOW_1:3; ::_thesis: verum end; supposeA110: ( x = 3 & y = 3 \ 1 ) ; ::_thesis: ex z being Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}) st ( z <= x & z <= y & ( for z9 being Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}) st z9 <= x & z9 <= y holds z9 <= z ) ) (3 \ 1) /\ 3 = (3 /\ 3) \ 1 by XBOOLE_1:49 .= 3 \ 1 ; then reconsider z = x /\ y as Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}) by A110; take z ; ::_thesis: ( z <= x & z <= y & ( for z9 being Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}) st z9 <= x & z9 <= y holds z9 <= z ) ) A111: z c= y by XBOOLE_1:17; z c= x by XBOOLE_1:17; hence ( z <= x & z <= y ) by A111, YELLOW_1:3; ::_thesis: for z9 being Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}) st z9 <= x & z9 <= y holds z9 <= z let w be Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}); ::_thesis: ( w <= x & w <= y implies w <= z ) assume that A112: w <= x and A113: w <= y ; ::_thesis: w <= z A114: w c= y by A113, YELLOW_1:3; w c= x by A112, YELLOW_1:3; then w c= x /\ y by A114, XBOOLE_1:19; hence w <= z by YELLOW_1:3; ::_thesis: verum end; supposeA115: ( x = 3 & y = 2 ) ; ::_thesis: ex z being Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}) st ( z <= x & z <= y & ( for z9 being Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}) st z9 <= x & z9 <= y holds z9 <= z ) ) 2 c= 3 by NAT_1:39; then reconsider z = x /\ y as Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}) by A115, XBOOLE_1:28; take z ; ::_thesis: ( z <= x & z <= y & ( for z9 being Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}) st z9 <= x & z9 <= y holds z9 <= z ) ) A116: z c= y by XBOOLE_1:17; z c= x by XBOOLE_1:17; hence ( z <= x & z <= y ) by A116, YELLOW_1:3; ::_thesis: for z9 being Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}) st z9 <= x & z9 <= y holds z9 <= z let w be Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}); ::_thesis: ( w <= x & w <= y implies w <= z ) assume that A117: w <= x and A118: w <= y ; ::_thesis: w <= z A119: w c= y by A118, YELLOW_1:3; w c= x by A117, YELLOW_1:3; then w c= x /\ y by A119, XBOOLE_1:19; hence w <= z by YELLOW_1:3; ::_thesis: verum end; supposeA120: ( x = 3 & y = 3 \ 2 ) ; ::_thesis: ex z being Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}) st ( z <= x & z <= y & ( for z9 being Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}) st z9 <= x & z9 <= y holds z9 <= z ) ) (3 \ 2) /\ 3 = (3 /\ 3) \ 2 by XBOOLE_1:49 .= 3 \ 2 ; then reconsider z = x /\ y as Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}) by A120; take z ; ::_thesis: ( z <= x & z <= y & ( for z9 being Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}) st z9 <= x & z9 <= y holds z9 <= z ) ) A121: z c= y by XBOOLE_1:17; z c= x by XBOOLE_1:17; hence ( z <= x & z <= y ) by A121, YELLOW_1:3; ::_thesis: for z9 being Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}) st z9 <= x & z9 <= y holds z9 <= z let w be Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}); ::_thesis: ( w <= x & w <= y implies w <= z ) assume that A122: w <= x and A123: w <= y ; ::_thesis: w <= z A124: w c= y by A123, YELLOW_1:3; w c= x by A122, YELLOW_1:3; then w c= x /\ y by A124, XBOOLE_1:19; hence w <= z by YELLOW_1:3; ::_thesis: verum end; suppose ( x = 3 & y = 3 ) ; ::_thesis: ex z being Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}) st ( z <= x & z <= y & ( for z9 being Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}) st z9 <= x & z9 <= y holds z9 <= z ) ) then reconsider z = x /\ y as Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}) ; take z ; ::_thesis: ( z <= x & z <= y & ( for z9 being Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}) st z9 <= x & z9 <= y holds z9 <= z ) ) A125: z c= y by XBOOLE_1:17; z c= x by XBOOLE_1:17; hence ( z <= x & z <= y ) by A125, YELLOW_1:3; ::_thesis: for z9 being Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}) st z9 <= x & z9 <= y holds z9 <= z let w be Element of (InclPoset {0,(3 \ 1),2,(3 \ 2),3}); ::_thesis: ( w <= x & w <= y implies w <= z ) assume that A126: w <= x and A127: w <= y ; ::_thesis: w <= z A128: w c= y by A127, YELLOW_1:3; w c= x by A126, YELLOW_1:3; then w c= x /\ y by A128, XBOOLE_1:19; hence w <= z by YELLOW_1:3; ::_thesis: verum end; end; end; end; now__::_thesis:_for_x,_y_being_set_st_x_in_{0,(3_\_1),2,(3_\_2),3}_&_y_in_{0,(3_\_1),2,(3_\_2),3}_holds_ x_\/_y_in_{0,(3_\_1),2,(3_\_2),3} let x, y be set ; ::_thesis: ( x in {0,(3 \ 1),2,(3 \ 2),3} & y in {0,(3 \ 1),2,(3 \ 2),3} implies x \/ y in {0,(3 \ 1),2,(3 \ 2),3} ) assume that A129: x in {0,(3 \ 1),2,(3 \ 2),3} and A130: y in {0,(3 \ 1),2,(3 \ 2),3} ; ::_thesis: x \/ y in {0,(3 \ 1),2,(3 \ 2),3} A131: ( x = 0 or x = 3 \ 1 or x = 2 or x = 3 \ 2 or x = 3 ) by A129, ENUMSET1:def_3; 2 c= 3 by NAT_1:39; then A132: 2 \/ 3 = 3 by XBOOLE_1:12; A133: 2 \/ (3 \ 2) = 2 \/ 3 by XBOOLE_1:39; A134: (3 \ 1) \/ 2 = {0,1,1,2} by Th3, CARD_1:50, ENUMSET1:5 .= {1,1,0,2} by ENUMSET1:67 .= {1,0,2} by ENUMSET1:31 .= {0,1,2} by ENUMSET1:58 ; A135: (3 \ 1) \/ 3 = 3 by XBOOLE_1:12; A136: ( y = 0 or y = 3 \ 1 or y = 2 or y = 3 \ 2 or y = 3 ) by A130, ENUMSET1:def_3; A137: (3 \ 2) \/ 3 = 3 by XBOOLE_1:12; (3 \ 1) \/ (3 \ 2) = {1,2} by Th3, Th4, ZFMISC_1:9; hence x \/ y in {0,(3 \ 1),2,(3 \ 2),3} by A131, A136, A134, A135, A133, A132, A137, Th3, CARD_1:51, ENUMSET1:def_3; ::_thesis: verum end; hence ( N_5 is with_infima & N_5 is with_suprema ) by A1, YELLOW_1:11; ::_thesis: verum end; end; definition func M_3 -> RelStr equals :: YELLOW11:def 2 InclPoset {0,1,(2 \ 1),(3 \ 2),3}; correctness coherence InclPoset {0,1,(2 \ 1),(3 \ 2),3} is RelStr ; ; end; :: deftheorem defines M_3 YELLOW11:def_2_:_ M_3 = InclPoset {0,1,(2 \ 1),(3 \ 2),3}; registration cluster M_3 -> strict reflexive transitive antisymmetric ; correctness coherence ( M_3 is strict & M_3 is reflexive & M_3 is transitive & M_3 is antisymmetric ); ; cluster M_3 -> with_suprema with_infima ; correctness coherence ( M_3 is with_infima & M_3 is with_suprema ); proof set Z = {0,1,(2 \ 1),(3 \ 2),3}; set N = InclPoset {0,1,(2 \ 1),(3 \ 2),3}; A1: InclPoset {0,1,(2 \ 1),(3 \ 2),3} is with_suprema proof let x, y be Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}); :: according to LATTICE3:def_10 ::_thesis: ex b1 being Element of the carrier of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}) st ( x <= b1 & y <= b1 & ( for b2 being Element of the carrier of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}) holds ( not x <= b2 or not y <= b2 or b1 <= b2 ) ) ) A2: {0,1,(2 \ 1),(3 \ 2),3} = the carrier of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}) by YELLOW_1:1; thus ex z being Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}) st ( x <= z & y <= z & ( for z9 being Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}) st x <= z9 & y <= z9 holds z <= z9 ) ) ::_thesis: verum proof percases ( ( x = 0 & y = 0 ) or ( x = 0 & y = 1 ) or ( x = 0 & y = 2 \ 1 ) or ( x = 0 & y = 3 \ 2 ) or ( x = 0 & y = 3 ) or ( x = 1 & y = 0 ) or ( x = 1 & y = 1 ) or ( x = 1 & y = 2 \ 1 ) or ( x = 1 & y = 3 \ 2 ) or ( x = 1 & y = 3 ) or ( x = 2 \ 1 & y = 0 ) or ( x = 2 \ 1 & y = 1 ) or ( x = 2 \ 1 & y = 2 \ 1 ) or ( x = 2 \ 1 & y = 3 \ 2 ) or ( x = 2 \ 1 & y = 3 ) or ( x = 3 \ 2 & y = 0 ) or ( x = 3 \ 2 & y = 1 ) or ( x = 3 \ 2 & y = 2 \ 1 ) or ( x = 3 \ 2 & y = 3 \ 2 ) or ( x = 3 \ 2 & y = 3 ) or ( x = 3 & y = 0 ) or ( x = 3 & y = 1 ) or ( x = 3 & y = 2 \ 1 ) or ( x = 3 & y = 3 \ 2 ) or ( x = 3 & y = 3 ) ) by A2, ENUMSET1:def_3; suppose ( x = 0 & y = 0 ) ; ::_thesis: ex z being Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}) st ( x <= z & y <= z & ( for z9 being Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}) st x <= z9 & y <= z9 holds z <= z9 ) ) then reconsider z = x \/ y as Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}) ; take z ; ::_thesis: ( x <= z & y <= z & ( for z9 being Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}) st x <= z9 & y <= z9 holds z <= z9 ) ) A3: y c= z by XBOOLE_1:7; x c= z by XBOOLE_1:7; hence ( x <= z & y <= z ) by A3, YELLOW_1:3; ::_thesis: for z9 being Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}) st x <= z9 & y <= z9 holds z <= z9 let w be Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}); ::_thesis: ( x <= w & y <= w implies z <= w ) assume that A4: x <= w and A5: y <= w ; ::_thesis: z <= w A6: y c= w by A5, YELLOW_1:3; x c= w by A4, YELLOW_1:3; then x \/ y c= w by A6, XBOOLE_1:8; hence z <= w by YELLOW_1:3; ::_thesis: verum end; suppose ( x = 0 & y = 1 ) ; ::_thesis: ex z being Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}) st ( x <= z & y <= z & ( for z9 being Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}) st x <= z9 & y <= z9 holds z <= z9 ) ) then reconsider z = x \/ y as Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}) ; take z ; ::_thesis: ( x <= z & y <= z & ( for z9 being Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}) st x <= z9 & y <= z9 holds z <= z9 ) ) A7: y c= z by XBOOLE_1:7; x c= z by XBOOLE_1:7; hence ( x <= z & y <= z ) by A7, YELLOW_1:3; ::_thesis: for z9 being Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}) st x <= z9 & y <= z9 holds z <= z9 let w be Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}); ::_thesis: ( x <= w & y <= w implies z <= w ) assume that A8: x <= w and A9: y <= w ; ::_thesis: z <= w A10: y c= w by A9, YELLOW_1:3; x c= w by A8, YELLOW_1:3; then x \/ y c= w by A10, XBOOLE_1:8; hence z <= w by YELLOW_1:3; ::_thesis: verum end; suppose ( x = 0 & y = 2 \ 1 ) ; ::_thesis: ex z being Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}) st ( x <= z & y <= z & ( for z9 being Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}) st x <= z9 & y <= z9 holds z <= z9 ) ) then reconsider z = x \/ y as Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}) ; take z ; ::_thesis: ( x <= z & y <= z & ( for z9 being Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}) st x <= z9 & y <= z9 holds z <= z9 ) ) A11: y c= z by XBOOLE_1:7; x c= z by XBOOLE_1:7; hence ( x <= z & y <= z ) by A11, YELLOW_1:3; ::_thesis: for z9 being Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}) st x <= z9 & y <= z9 holds z <= z9 let w be Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}); ::_thesis: ( x <= w & y <= w implies z <= w ) assume that A12: x <= w and A13: y <= w ; ::_thesis: z <= w A14: y c= w by A13, YELLOW_1:3; x c= w by A12, YELLOW_1:3; then x \/ y c= w by A14, XBOOLE_1:8; hence z <= w by YELLOW_1:3; ::_thesis: verum end; suppose ( x = 0 & y = 3 \ 2 ) ; ::_thesis: ex z being Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}) st ( x <= z & y <= z & ( for z9 being Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}) st x <= z9 & y <= z9 holds z <= z9 ) ) then reconsider z = x \/ y as Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}) ; take z ; ::_thesis: ( x <= z & y <= z & ( for z9 being Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}) st x <= z9 & y <= z9 holds z <= z9 ) ) A15: y c= z by XBOOLE_1:7; x c= z by XBOOLE_1:7; hence ( x <= z & y <= z ) by A15, YELLOW_1:3; ::_thesis: for z9 being Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}) st x <= z9 & y <= z9 holds z <= z9 let w be Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}); ::_thesis: ( x <= w & y <= w implies z <= w ) assume that A16: x <= w and A17: y <= w ; ::_thesis: z <= w A18: y c= w by A17, YELLOW_1:3; x c= w by A16, YELLOW_1:3; then x \/ y c= w by A18, XBOOLE_1:8; hence z <= w by YELLOW_1:3; ::_thesis: verum end; suppose ( x = 0 & y = 3 ) ; ::_thesis: ex z being Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}) st ( x <= z & y <= z & ( for z9 being Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}) st x <= z9 & y <= z9 holds z <= z9 ) ) then reconsider z = x \/ y as Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}) ; take z ; ::_thesis: ( x <= z & y <= z & ( for z9 being Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}) st x <= z9 & y <= z9 holds z <= z9 ) ) A19: y c= z by XBOOLE_1:7; x c= z by XBOOLE_1:7; hence ( x <= z & y <= z ) by A19, YELLOW_1:3; ::_thesis: for z9 being Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}) st x <= z9 & y <= z9 holds z <= z9 let w be Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}); ::_thesis: ( x <= w & y <= w implies z <= w ) assume that A20: x <= w and A21: y <= w ; ::_thesis: z <= w A22: y c= w by A21, YELLOW_1:3; x c= w by A20, YELLOW_1:3; then x \/ y c= w by A22, XBOOLE_1:8; hence z <= w by YELLOW_1:3; ::_thesis: verum end; suppose ( x = 1 & y = 0 ) ; ::_thesis: ex z being Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}) st ( x <= z & y <= z & ( for z9 being Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}) st x <= z9 & y <= z9 holds z <= z9 ) ) then reconsider z = x \/ y as Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}) ; take z ; ::_thesis: ( x <= z & y <= z & ( for z9 being Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}) st x <= z9 & y <= z9 holds z <= z9 ) ) A23: y c= z by XBOOLE_1:7; x c= z by XBOOLE_1:7; hence ( x <= z & y <= z ) by A23, YELLOW_1:3; ::_thesis: for z9 being Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}) st x <= z9 & y <= z9 holds z <= z9 let w be Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}); ::_thesis: ( x <= w & y <= w implies z <= w ) assume that A24: x <= w and A25: y <= w ; ::_thesis: z <= w A26: y c= w by A25, YELLOW_1:3; x c= w by A24, YELLOW_1:3; then x \/ y c= w by A26, XBOOLE_1:8; hence z <= w by YELLOW_1:3; ::_thesis: verum end; suppose ( x = 1 & y = 1 ) ; ::_thesis: ex z being Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}) st ( x <= z & y <= z & ( for z9 being Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}) st x <= z9 & y <= z9 holds z <= z9 ) ) then reconsider z = x \/ y as Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}) ; take z ; ::_thesis: ( x <= z & y <= z & ( for z9 being Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}) st x <= z9 & y <= z9 holds z <= z9 ) ) A27: y c= z by XBOOLE_1:7; x c= z by XBOOLE_1:7; hence ( x <= z & y <= z ) by A27, YELLOW_1:3; ::_thesis: for z9 being Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}) st x <= z9 & y <= z9 holds z <= z9 let w be Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}); ::_thesis: ( x <= w & y <= w implies z <= w ) assume that A28: x <= w and A29: y <= w ; ::_thesis: z <= w A30: y c= w by A29, YELLOW_1:3; x c= w by A28, YELLOW_1:3; then x \/ y c= w by A30, XBOOLE_1:8; hence z <= w by YELLOW_1:3; ::_thesis: verum end; supposeA31: ( x = 1 & y = 2 \ 1 ) ; ::_thesis: ex z being Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}) st ( x <= z & y <= z & ( for z9 being Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}) st x <= z9 & y <= z9 holds z <= z9 ) ) 3 in {0,1,(2 \ 1),(3 \ 2),3} by ENUMSET1:def_3; then reconsider z = 3 as Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}) by YELLOW_1:1; take z ; ::_thesis: ( x <= z & y <= z & ( for z9 being Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}) st x <= z9 & y <= z9 holds z <= z9 ) ) ( x c= z & y c= z ) proof thus x c= z ::_thesis: y c= z proof let X be set ; :: according to TARSKI:def_3 ::_thesis: ( not X in x or X in z ) assume X in x ; ::_thesis: X in z then X = 0 by A31, CARD_1:49, TARSKI:def_1; hence X in z by CARD_1:51, ENUMSET1:def_1; ::_thesis: verum end; let X be set ; :: according to TARSKI:def_3 ::_thesis: ( not X in y or X in z ) assume X in y ; ::_thesis: X in z then X = 1 by A31, Th2, TARSKI:def_1; hence X in z by CARD_1:51, ENUMSET1:def_1; ::_thesis: verum end; hence ( x <= z & y <= z ) by YELLOW_1:3; ::_thesis: for z9 being Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}) st x <= z9 & y <= z9 holds z <= z9 let z9 be Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}); ::_thesis: ( x <= z9 & y <= z9 implies z <= z9 ) assume that A32: x <= z9 and A33: y <= z9 ; ::_thesis: z <= z9 A34: z9 is Element of {0,1,(2 \ 1),(3 \ 2),3} by YELLOW_1:1; A35: 2 \ 1 c= z9 by A31, A33, YELLOW_1:3; A36: now__::_thesis:_not_z9_=_1 1 in 2 \ 1 by Th2, TARSKI:def_1; then A37: 1 in z9 by A35; assume z9 = 1 ; ::_thesis: contradiction hence contradiction by A37; ::_thesis: verum end; A38: 1 c= z9 by A31, A32, YELLOW_1:3; A39: now__::_thesis:_not_z9_=_2_\_1 A40: 0 in 1 by CARD_1:49, TARSKI:def_1; assume z9 = 2 \ 1 ; ::_thesis: contradiction hence contradiction by A38, A40, Th2, TARSKI:def_1; ::_thesis: verum end; A41: now__::_thesis:_not_z9_=_3_\_2 A42: 0 in 1 by CARD_1:49, TARSKI:def_1; assume z9 = 3 \ 2 ; ::_thesis: contradiction hence contradiction by A38, A42, Th4, TARSKI:def_1; ::_thesis: verum end; z9 <> 0 by A38; hence z <= z9 by A34, A36, A39, A41, ENUMSET1:def_3; ::_thesis: verum end; supposeA43: ( x = 1 & y = 3 \ 2 ) ; ::_thesis: ex z being Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}) st ( x <= z & y <= z & ( for z9 being Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}) st x <= z9 & y <= z9 holds z <= z9 ) ) 3 in {0,1,(2 \ 1),(3 \ 2),3} by ENUMSET1:def_3; then reconsider z = 3 as Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}) by YELLOW_1:1; take z ; ::_thesis: ( x <= z & y <= z & ( for z9 being Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}) st x <= z9 & y <= z9 holds z <= z9 ) ) x c= z proof let X be set ; :: according to TARSKI:def_3 ::_thesis: ( not X in x or X in z ) assume X in x ; ::_thesis: X in z then X = 0 by A43, CARD_1:49, TARSKI:def_1; hence X in z by CARD_1:51, ENUMSET1:def_1; ::_thesis: verum end; hence ( x <= z & y <= z ) by A43, YELLOW_1:3; ::_thesis: for z9 being Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}) st x <= z9 & y <= z9 holds z <= z9 let z9 be Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}); ::_thesis: ( x <= z9 & y <= z9 implies z <= z9 ) assume that A44: x <= z9 and A45: y <= z9 ; ::_thesis: z <= z9 A46: z9 is Element of {0,1,(2 \ 1),(3 \ 2),3} by YELLOW_1:1; A47: 3 \ 2 c= z9 by A43, A45, YELLOW_1:3; A48: now__::_thesis:_not_z9_=_1 assume A49: z9 = 1 ; ::_thesis: contradiction 2 in 3 \ 2 by Th4, TARSKI:def_1; hence contradiction by A47, A49, CARD_1:49, TARSKI:def_1; ::_thesis: verum end; A50: 1 c= z9 by A43, A44, YELLOW_1:3; A51: now__::_thesis:_not_z9_=_2_\_1 A52: 0 in 1 by CARD_1:49, TARSKI:def_1; assume z9 = 2 \ 1 ; ::_thesis: contradiction hence contradiction by A50, A52, Th2, TARSKI:def_1; ::_thesis: verum end; A53: now__::_thesis:_not_z9_=_3_\_2 A54: 0 in 1 by CARD_1:49, TARSKI:def_1; assume z9 = 3 \ 2 ; ::_thesis: contradiction hence contradiction by A50, A54, Th4, TARSKI:def_1; ::_thesis: verum end; z9 <> 0 by A50; hence z <= z9 by A46, A48, A51, A53, ENUMSET1:def_3; ::_thesis: verum end; supposeA55: ( x = 1 & y = 3 ) ; ::_thesis: ex z being Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}) st ( x <= z & y <= z & ( for z9 being Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}) st x <= z9 & y <= z9 holds z <= z9 ) ) 1 c= 3 proof let X be set ; :: according to TARSKI:def_3 ::_thesis: ( not X in 1 or X in 3 ) assume X in 1 ; ::_thesis: X in 3 then X = 0 by CARD_1:49, TARSKI:def_1; hence X in 3 by CARD_1:51, ENUMSET1:def_1; ::_thesis: verum end; then reconsider z = x \/ y as Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}) by A55, XBOOLE_1:12; take z ; ::_thesis: ( x <= z & y <= z & ( for z9 being Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}) st x <= z9 & y <= z9 holds z <= z9 ) ) A56: y c= z by XBOOLE_1:7; x c= z by XBOOLE_1:7; hence ( x <= z & y <= z ) by A56, YELLOW_1:3; ::_thesis: for z9 being Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}) st x <= z9 & y <= z9 holds z <= z9 let w be Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}); ::_thesis: ( x <= w & y <= w implies z <= w ) assume that A57: x <= w and A58: y <= w ; ::_thesis: z <= w A59: y c= w by A58, YELLOW_1:3; x c= w by A57, YELLOW_1:3; then x \/ y c= w by A59, XBOOLE_1:8; hence z <= w by YELLOW_1:3; ::_thesis: verum end; suppose ( x = 2 \ 1 & y = 0 ) ; ::_thesis: ex z being Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}) st ( x <= z & y <= z & ( for z9 being Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}) st x <= z9 & y <= z9 holds z <= z9 ) ) then reconsider z = x \/ y as Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}) ; take z ; ::_thesis: ( x <= z & y <= z & ( for z9 being Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}) st x <= z9 & y <= z9 holds z <= z9 ) ) A60: y c= z by XBOOLE_1:7; x c= z by XBOOLE_1:7; hence ( x <= z & y <= z ) by A60, YELLOW_1:3; ::_thesis: for z9 being Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}) st x <= z9 & y <= z9 holds z <= z9 let w be Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}); ::_thesis: ( x <= w & y <= w implies z <= w ) assume that A61: x <= w and A62: y <= w ; ::_thesis: z <= w A63: y c= w by A62, YELLOW_1:3; x c= w by A61, YELLOW_1:3; then x \/ y c= w by A63, XBOOLE_1:8; hence z <= w by YELLOW_1:3; ::_thesis: verum end; supposeA64: ( x = 2 \ 1 & y = 1 ) ; ::_thesis: ex z being Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}) st ( x <= z & y <= z & ( for z9 being Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}) st x <= z9 & y <= z9 holds z <= z9 ) ) 3 in {0,1,(2 \ 1),(3 \ 2),3} by ENUMSET1:def_3; then reconsider z = 3 as Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}) by YELLOW_1:1; take z ; ::_thesis: ( x <= z & y <= z & ( for z9 being Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}) st x <= z9 & y <= z9 holds z <= z9 ) ) ( x c= z & y c= z ) proof thus x c= z ::_thesis: y c= z proof let X be set ; :: according to TARSKI:def_3 ::_thesis: ( not X in x or X in z ) assume X in x ; ::_thesis: X in z then X = 1 by A64, Th2, TARSKI:def_1; hence X in z by CARD_1:51, ENUMSET1:def_1; ::_thesis: verum end; let X be set ; :: according to TARSKI:def_3 ::_thesis: ( not X in y or X in z ) assume X in y ; ::_thesis: X in z then X = 0 by A64, CARD_1:49, TARSKI:def_1; hence X in z by CARD_1:51, ENUMSET1:def_1; ::_thesis: verum end; hence ( x <= z & y <= z ) by YELLOW_1:3; ::_thesis: for z9 being Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}) st x <= z9 & y <= z9 holds z <= z9 let z9 be Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}); ::_thesis: ( x <= z9 & y <= z9 implies z <= z9 ) assume that A65: x <= z9 and A66: y <= z9 ; ::_thesis: z <= z9 A67: z9 is Element of {0,1,(2 \ 1),(3 \ 2),3} by YELLOW_1:1; A68: 2 \ 1 c= z9 by A64, A65, YELLOW_1:3; A69: now__::_thesis:_not_z9_=_1 1 in 2 \ 1 by Th2, TARSKI:def_1; then A70: 1 in z9 by A68; assume z9 = 1 ; ::_thesis: contradiction hence contradiction by A70; ::_thesis: verum end; A71: 1 c= z9 by A64, A66, YELLOW_1:3; A72: now__::_thesis:_not_z9_=_2_\_1 A73: 0 in 1 by CARD_1:49, TARSKI:def_1; assume z9 = 2 \ 1 ; ::_thesis: contradiction hence contradiction by A71, A73, Th2, TARSKI:def_1; ::_thesis: verum end; A74: now__::_thesis:_not_z9_=_3_\_2 A75: 0 in 1 by CARD_1:49, TARSKI:def_1; assume z9 = 3 \ 2 ; ::_thesis: contradiction hence contradiction by A71, A75, Th4, TARSKI:def_1; ::_thesis: verum end; z9 <> 0 by A71; hence z <= z9 by A67, A69, A72, A74, ENUMSET1:def_3; ::_thesis: verum end; suppose ( x = 2 \ 1 & y = 2 \ 1 ) ; ::_thesis: ex z being Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}) st ( x <= z & y <= z & ( for z9 being Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}) st x <= z9 & y <= z9 holds z <= z9 ) ) then reconsider z = x \/ y as Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}) ; take z ; ::_thesis: ( x <= z & y <= z & ( for z9 being Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}) st x <= z9 & y <= z9 holds z <= z9 ) ) A76: y c= z by XBOOLE_1:7; x c= z by XBOOLE_1:7; hence ( x <= z & y <= z ) by A76, YELLOW_1:3; ::_thesis: for z9 being Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}) st x <= z9 & y <= z9 holds z <= z9 let w be Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}); ::_thesis: ( x <= w & y <= w implies z <= w ) assume that A77: x <= w and A78: y <= w ; ::_thesis: z <= w A79: y c= w by A78, YELLOW_1:3; x c= w by A77, YELLOW_1:3; then x \/ y c= w by A79, XBOOLE_1:8; hence z <= w by YELLOW_1:3; ::_thesis: verum end; supposeA80: ( x = 2 \ 1 & y = 3 \ 2 ) ; ::_thesis: ex z being Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}) st ( x <= z & y <= z & ( for z9 being Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}) st x <= z9 & y <= z9 holds z <= z9 ) ) 3 in {0,1,(2 \ 1),(3 \ 2),3} by ENUMSET1:def_3; then reconsider z = 3 as Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}) by YELLOW_1:1; take z ; ::_thesis: ( x <= z & y <= z & ( for z9 being Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}) st x <= z9 & y <= z9 holds z <= z9 ) ) ( x c= z & y c= z ) proof thus x c= z ::_thesis: y c= z proof let X be set ; :: according to TARSKI:def_3 ::_thesis: ( not X in x or X in z ) assume X in x ; ::_thesis: X in z then X = 1 by A80, Th2, TARSKI:def_1; hence X in z by CARD_1:51, ENUMSET1:def_1; ::_thesis: verum end; let X be set ; :: according to TARSKI:def_3 ::_thesis: ( not X in y or X in z ) assume X in y ; ::_thesis: X in z hence X in z by A80; ::_thesis: verum end; hence ( x <= z & y <= z ) by YELLOW_1:3; ::_thesis: for z9 being Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}) st x <= z9 & y <= z9 holds z <= z9 let z9 be Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}); ::_thesis: ( x <= z9 & y <= z9 implies z <= z9 ) assume that A81: x <= z9 and A82: y <= z9 ; ::_thesis: z <= z9 A83: z9 is Element of {0,1,(2 \ 1),(3 \ 2),3} by YELLOW_1:1; A84: 3 \ 2 c= z9 by A80, A82, YELLOW_1:3; A85: now__::_thesis:_not_z9_=_2_\_1 assume A86: z9 = 2 \ 1 ; ::_thesis: contradiction 2 in 3 \ 2 by Th4, TARSKI:def_1; hence contradiction by A84, A86, Th2, TARSKI:def_1; ::_thesis: verum end; A87: 2 \ 1 c= z9 by A80, A81, YELLOW_1:3; A88: now__::_thesis:_not_z9_=_3_\_2 assume A89: z9 = 3 \ 2 ; ::_thesis: contradiction 1 in 2 \ 1 by Th2, TARSKI:def_1; hence contradiction by A87, A89, Th4, TARSKI:def_1; ::_thesis: verum end; A90: now__::_thesis:_not_z9_=_1 1 in 2 \ 1 by Th2, TARSKI:def_1; then A91: 1 in z9 by A87; assume z9 = 1 ; ::_thesis: contradiction hence contradiction by A91; ::_thesis: verum end; z9 <> 0 by A87, Th2; hence z <= z9 by A83, A90, A85, A88, ENUMSET1:def_3; ::_thesis: verum end; supposeA92: ( x = 2 \ 1 & y = 3 ) ; ::_thesis: ex z being Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}) st ( x <= z & y <= z & ( for z9 being Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}) st x <= z9 & y <= z9 holds z <= z9 ) ) 2 \ 1 c= 3 proof let X be set ; :: according to TARSKI:def_3 ::_thesis: ( not X in 2 \ 1 or X in 3 ) assume X in 2 \ 1 ; ::_thesis: X in 3 then X = 1 by Th2, TARSKI:def_1; hence X in 3 by CARD_1:51, ENUMSET1:def_1; ::_thesis: verum end; then reconsider z = x \/ y as Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}) by A92, XBOOLE_1:12; take z ; ::_thesis: ( x <= z & y <= z & ( for z9 being Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}) st x <= z9 & y <= z9 holds z <= z9 ) ) A93: y c= z by XBOOLE_1:7; x c= z by XBOOLE_1:7; hence ( x <= z & y <= z ) by A93, YELLOW_1:3; ::_thesis: for z9 being Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}) st x <= z9 & y <= z9 holds z <= z9 let w be Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}); ::_thesis: ( x <= w & y <= w implies z <= w ) assume that A94: x <= w and A95: y <= w ; ::_thesis: z <= w A96: y c= w by A95, YELLOW_1:3; x c= w by A94, YELLOW_1:3; then x \/ y c= w by A96, XBOOLE_1:8; hence z <= w by YELLOW_1:3; ::_thesis: verum end; suppose ( x = 3 \ 2 & y = 0 ) ; ::_thesis: ex z being Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}) st ( x <= z & y <= z & ( for z9 being Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}) st x <= z9 & y <= z9 holds z <= z9 ) ) then reconsider z = x \/ y as Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}) ; take z ; ::_thesis: ( x <= z & y <= z & ( for z9 being Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}) st x <= z9 & y <= z9 holds z <= z9 ) ) A97: y c= z by XBOOLE_1:7; x c= z by XBOOLE_1:7; hence ( x <= z & y <= z ) by A97, YELLOW_1:3; ::_thesis: for z9 being Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}) st x <= z9 & y <= z9 holds z <= z9 let w be Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}); ::_thesis: ( x <= w & y <= w implies z <= w ) assume that A98: x <= w and A99: y <= w ; ::_thesis: z <= w A100: y c= w by A99, YELLOW_1:3; x c= w by A98, YELLOW_1:3; then x \/ y c= w by A100, XBOOLE_1:8; hence z <= w by YELLOW_1:3; ::_thesis: verum end; supposeA101: ( x = 3 \ 2 & y = 1 ) ; ::_thesis: ex z being Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}) st ( x <= z & y <= z & ( for z9 being Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}) st x <= z9 & y <= z9 holds z <= z9 ) ) 3 in {0,1,(2 \ 1),(3 \ 2),3} by ENUMSET1:def_3; then reconsider z = 3 as Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}) by YELLOW_1:1; take z ; ::_thesis: ( x <= z & y <= z & ( for z9 being Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}) st x <= z9 & y <= z9 holds z <= z9 ) ) y c= z proof let X be set ; :: according to TARSKI:def_3 ::_thesis: ( not X in y or X in z ) assume X in y ; ::_thesis: X in z then X = 0 by A101, CARD_1:49, TARSKI:def_1; hence X in z by CARD_1:51, ENUMSET1:def_1; ::_thesis: verum end; hence ( x <= z & y <= z ) by A101, YELLOW_1:3; ::_thesis: for z9 being Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}) st x <= z9 & y <= z9 holds z <= z9 let z9 be Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}); ::_thesis: ( x <= z9 & y <= z9 implies z <= z9 ) assume that A102: x <= z9 and A103: y <= z9 ; ::_thesis: z <= z9 A104: z9 is Element of {0,1,(2 \ 1),(3 \ 2),3} by YELLOW_1:1; A105: 3 \ 2 c= z9 by A101, A102, YELLOW_1:3; A106: now__::_thesis:_not_z9_=_1 assume A107: z9 = 1 ; ::_thesis: contradiction 2 in 3 \ 2 by Th4, TARSKI:def_1; hence contradiction by A105, A107, CARD_1:49, TARSKI:def_1; ::_thesis: verum end; A108: 1 c= z9 by A101, A103, YELLOW_1:3; A109: now__::_thesis:_not_z9_=_2_\_1 A110: 0 in 1 by CARD_1:49, TARSKI:def_1; assume z9 = 2 \ 1 ; ::_thesis: contradiction hence contradiction by A108, A110, Th2, TARSKI:def_1; ::_thesis: verum end; A111: now__::_thesis:_not_z9_=_3_\_2 A112: 0 in 1 by CARD_1:49, TARSKI:def_1; assume z9 = 3 \ 2 ; ::_thesis: contradiction hence contradiction by A108, A112, Th4, TARSKI:def_1; ::_thesis: verum end; z9 <> 0 by A108; hence z <= z9 by A104, A106, A109, A111, ENUMSET1:def_3; ::_thesis: verum end; supposeA113: ( x = 3 \ 2 & y = 2 \ 1 ) ; ::_thesis: ex z being Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}) st ( x <= z & y <= z & ( for z9 being Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}) st x <= z9 & y <= z9 holds z <= z9 ) ) 3 in {0,1,(2 \ 1),(3 \ 2),3} by ENUMSET1:def_3; then reconsider z = 3 as Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}) by YELLOW_1:1; take z ; ::_thesis: ( x <= z & y <= z & ( for z9 being Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}) st x <= z9 & y <= z9 holds z <= z9 ) ) ( x c= z & y c= z ) proof thus x c= z by A113; ::_thesis: y c= z let X be set ; :: according to TARSKI:def_3 ::_thesis: ( not X in y or X in z ) assume X in y ; ::_thesis: X in z then X = 1 by A113, Th2, TARSKI:def_1; hence X in z by CARD_1:51, ENUMSET1:def_1; ::_thesis: verum end; hence ( x <= z & y <= z ) by YELLOW_1:3; ::_thesis: for z9 being Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}) st x <= z9 & y <= z9 holds z <= z9 let z9 be Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}); ::_thesis: ( x <= z9 & y <= z9 implies z <= z9 ) assume that A114: x <= z9 and A115: y <= z9 ; ::_thesis: z <= z9 A116: z9 is Element of {0,1,(2 \ 1),(3 \ 2),3} by YELLOW_1:1; A117: 3 \ 2 c= z9 by A113, A114, YELLOW_1:3; A118: now__::_thesis:_not_z9_=_2_\_1 assume A119: z9 = 2 \ 1 ; ::_thesis: contradiction 2 in 3 \ 2 by Th4, TARSKI:def_1; hence contradiction by A117, A119, Th2, TARSKI:def_1; ::_thesis: verum end; A120: 2 \ 1 c= z9 by A113, A115, YELLOW_1:3; A121: now__::_thesis:_not_z9_=_3_\_2 assume A122: z9 = 3 \ 2 ; ::_thesis: contradiction 1 in 2 \ 1 by Th2, TARSKI:def_1; hence contradiction by A120, A122, Th4, TARSKI:def_1; ::_thesis: verum end; A123: now__::_thesis:_not_z9_=_1 1 in 2 \ 1 by Th2, TARSKI:def_1; then A124: 1 in z9 by A120; assume z9 = 1 ; ::_thesis: contradiction hence contradiction by A124; ::_thesis: verum end; z9 <> 0 by A120, Th2; hence z <= z9 by A116, A123, A118, A121, ENUMSET1:def_3; ::_thesis: verum end; suppose ( x = 3 \ 2 & y = 3 \ 2 ) ; ::_thesis: ex z being Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}) st ( x <= z & y <= z & ( for z9 being Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}) st x <= z9 & y <= z9 holds z <= z9 ) ) then reconsider z = x \/ y as Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}) ; take z ; ::_thesis: ( x <= z & y <= z & ( for z9 being Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}) st x <= z9 & y <= z9 holds z <= z9 ) ) A125: y c= z by XBOOLE_1:7; x c= z by XBOOLE_1:7; hence ( x <= z & y <= z ) by A125, YELLOW_1:3; ::_thesis: for z9 being Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}) st x <= z9 & y <= z9 holds z <= z9 let w be Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}); ::_thesis: ( x <= w & y <= w implies z <= w ) assume that A126: x <= w and A127: y <= w ; ::_thesis: z <= w A128: y c= w by A127, YELLOW_1:3; x c= w by A126, YELLOW_1:3; then x \/ y c= w by A128, XBOOLE_1:8; hence z <= w by YELLOW_1:3; ::_thesis: verum end; suppose ( x = 3 \ 2 & y = 3 ) ; ::_thesis: ex z being Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}) st ( x <= z & y <= z & ( for z9 being Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}) st x <= z9 & y <= z9 holds z <= z9 ) ) then reconsider z = x \/ y as Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}) by XBOOLE_1:12; take z ; ::_thesis: ( x <= z & y <= z & ( for z9 being Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}) st x <= z9 & y <= z9 holds z <= z9 ) ) A129: y c= z by XBOOLE_1:7; x c= z by XBOOLE_1:7; hence ( x <= z & y <= z ) by A129, YELLOW_1:3; ::_thesis: for z9 being Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}) st x <= z9 & y <= z9 holds z <= z9 let w be Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}); ::_thesis: ( x <= w & y <= w implies z <= w ) assume that A130: x <= w and A131: y <= w ; ::_thesis: z <= w A132: y c= w by A131, YELLOW_1:3; x c= w by A130, YELLOW_1:3; then x \/ y c= w by A132, XBOOLE_1:8; hence z <= w by YELLOW_1:3; ::_thesis: verum end; suppose ( x = 3 & y = 0 ) ; ::_thesis: ex z being Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}) st ( x <= z & y <= z & ( for z9 being Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}) st x <= z9 & y <= z9 holds z <= z9 ) ) then reconsider z = x \/ y as Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}) ; take z ; ::_thesis: ( x <= z & y <= z & ( for z9 being Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}) st x <= z9 & y <= z9 holds z <= z9 ) ) A133: y c= z by XBOOLE_1:7; x c= z by XBOOLE_1:7; hence ( x <= z & y <= z ) by A133, YELLOW_1:3; ::_thesis: for z9 being Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}) st x <= z9 & y <= z9 holds z <= z9 let w be Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}); ::_thesis: ( x <= w & y <= w implies z <= w ) assume that A134: x <= w and A135: y <= w ; ::_thesis: z <= w A136: y c= w by A135, YELLOW_1:3; x c= w by A134, YELLOW_1:3; then x \/ y c= w by A136, XBOOLE_1:8; hence z <= w by YELLOW_1:3; ::_thesis: verum end; supposeA137: ( x = 3 & y = 1 ) ; ::_thesis: ex z being Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}) st ( x <= z & y <= z & ( for z9 being Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}) st x <= z9 & y <= z9 holds z <= z9 ) ) 1 c= 3 proof let X be set ; :: according to TARSKI:def_3 ::_thesis: ( not X in 1 or X in 3 ) assume X in 1 ; ::_thesis: X in 3 then X = 0 by CARD_1:49, TARSKI:def_1; hence X in 3 by CARD_1:51, ENUMSET1:def_1; ::_thesis: verum end; then reconsider z = x \/ y as Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}) by A137, XBOOLE_1:12; take z ; ::_thesis: ( x <= z & y <= z & ( for z9 being Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}) st x <= z9 & y <= z9 holds z <= z9 ) ) A138: y c= z by XBOOLE_1:7; x c= z by XBOOLE_1:7; hence ( x <= z & y <= z ) by A138, YELLOW_1:3; ::_thesis: for z9 being Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}) st x <= z9 & y <= z9 holds z <= z9 let w be Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}); ::_thesis: ( x <= w & y <= w implies z <= w ) assume that A139: x <= w and A140: y <= w ; ::_thesis: z <= w A141: y c= w by A140, YELLOW_1:3; x c= w by A139, YELLOW_1:3; then x \/ y c= w by A141, XBOOLE_1:8; hence z <= w by YELLOW_1:3; ::_thesis: verum end; supposeA142: ( x = 3 & y = 2 \ 1 ) ; ::_thesis: ex z being Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}) st ( x <= z & y <= z & ( for z9 being Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}) st x <= z9 & y <= z9 holds z <= z9 ) ) 2 \ 1 c= 3 proof let X be set ; :: according to TARSKI:def_3 ::_thesis: ( not X in 2 \ 1 or X in 3 ) assume X in 2 \ 1 ; ::_thesis: X in 3 then X = 1 by Th2, TARSKI:def_1; hence X in 3 by CARD_1:51, ENUMSET1:def_1; ::_thesis: verum end; then reconsider z = x \/ y as Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}) by A142, XBOOLE_1:12; take z ; ::_thesis: ( x <= z & y <= z & ( for z9 being Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}) st x <= z9 & y <= z9 holds z <= z9 ) ) A143: y c= z by XBOOLE_1:7; x c= z by XBOOLE_1:7; hence ( x <= z & y <= z ) by A143, YELLOW_1:3; ::_thesis: for z9 being Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}) st x <= z9 & y <= z9 holds z <= z9 let w be Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}); ::_thesis: ( x <= w & y <= w implies z <= w ) assume that A144: x <= w and A145: y <= w ; ::_thesis: z <= w A146: y c= w by A145, YELLOW_1:3; x c= w by A144, YELLOW_1:3; then x \/ y c= w by A146, XBOOLE_1:8; hence z <= w by YELLOW_1:3; ::_thesis: verum end; suppose ( x = 3 & y = 3 \ 2 ) ; ::_thesis: ex z being Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}) st ( x <= z & y <= z & ( for z9 being Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}) st x <= z9 & y <= z9 holds z <= z9 ) ) then reconsider z = x \/ y as Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}) by XBOOLE_1:12; take z ; ::_thesis: ( x <= z & y <= z & ( for z9 being Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}) st x <= z9 & y <= z9 holds z <= z9 ) ) A147: y c= z by XBOOLE_1:7; x c= z by XBOOLE_1:7; hence ( x <= z & y <= z ) by A147, YELLOW_1:3; ::_thesis: for z9 being Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}) st x <= z9 & y <= z9 holds z <= z9 let w be Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}); ::_thesis: ( x <= w & y <= w implies z <= w ) assume that A148: x <= w and A149: y <= w ; ::_thesis: z <= w A150: y c= w by A149, YELLOW_1:3; x c= w by A148, YELLOW_1:3; then x \/ y c= w by A150, XBOOLE_1:8; hence z <= w by YELLOW_1:3; ::_thesis: verum end; suppose ( x = 3 & y = 3 ) ; ::_thesis: ex z being Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}) st ( x <= z & y <= z & ( for z9 being Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}) st x <= z9 & y <= z9 holds z <= z9 ) ) then reconsider z = x \/ y as Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}) ; take z ; ::_thesis: ( x <= z & y <= z & ( for z9 being Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}) st x <= z9 & y <= z9 holds z <= z9 ) ) A151: y c= z by XBOOLE_1:7; x c= z by XBOOLE_1:7; hence ( x <= z & y <= z ) by A151, YELLOW_1:3; ::_thesis: for z9 being Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}) st x <= z9 & y <= z9 holds z <= z9 let w be Element of (InclPoset {0,1,(2 \ 1),(3 \ 2),3}); ::_thesis: ( x <= w & y <= w implies z <= w ) assume that A152: x <= w and A153: y <= w ; ::_thesis: z <= w A154: y c= w by A153, YELLOW_1:3; x c= w by A152, YELLOW_1:3; then x \/ y c= w by A154, XBOOLE_1:8; hence z <= w by YELLOW_1:3; ::_thesis: verum end; end; end; end; now__::_thesis:_for_x,_y_being_set_st_x_in_{0,1,(2_\_1),(3_\_2),3}_&_y_in_{0,1,(2_\_1),(3_\_2),3}_holds_ x_/\_y_in_{0,1,(2_\_1),(3_\_2),3} now__::_thesis:_for_x_being_set_holds_not_x_in_(2_\_1)_/\_(3_\_2) let x be set ; ::_thesis: not x in (2 \ 1) /\ (3 \ 2) assume A155: x in (2 \ 1) /\ (3 \ 2) ; ::_thesis: contradiction then x in 2 \ 1 by XBOOLE_0:def_4; then A156: x = 1 by Th2, TARSKI:def_1; x in 3 \ 2 by A155, XBOOLE_0:def_4; hence contradiction by A156, Th4, TARSKI:def_1; ::_thesis: verum end; then A157: (2 \ 1) /\ (3 \ 2) = 0 by XBOOLE_0:def_1; A158: (3 \ 2) /\ 3 = 3 \ 2 by XBOOLE_1:28; 2 \ 1 c= 3 proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in 2 \ 1 or x in 3 ) assume x in 2 \ 1 ; ::_thesis: x in 3 then x = 1 by Th2, TARSKI:def_1; hence x in 3 by CARD_1:51, ENUMSET1:def_1; ::_thesis: verum end; then A159: (2 \ 1) /\ 3 = 2 \ 1 by XBOOLE_1:28; 1 c= 3 by NAT_1:39; then A160: 1 /\ 3 = 1 by XBOOLE_1:28; now__::_thesis:_for_x_being_set_holds_not_x_in_1_/\_(3_\_2) let x be set ; ::_thesis: not x in 1 /\ (3 \ 2) assume A161: x in 1 /\ (3 \ 2) ; ::_thesis: contradiction then x in 1 by XBOOLE_0:def_4; then A162: x = 0 by CARD_1:49, TARSKI:def_1; x in 3 \ 2 by A161, XBOOLE_0:def_4; hence contradiction by A162, Th4, TARSKI:def_1; ::_thesis: verum end; then A163: 1 /\ (3 \ 2) = 0 by XBOOLE_0:def_1; 1 misses 2 \ 1 by XBOOLE_1:79; then A164: 1 /\ (2 \ 1) = 0 by XBOOLE_0:def_7; let x, y be set ; ::_thesis: ( x in {0,1,(2 \ 1),(3 \ 2),3} & y in {0,1,(2 \ 1),(3 \ 2),3} implies x /\ y in {0,1,(2 \ 1),(3 \ 2),3} ) assume that A165: x in {0,1,(2 \ 1),(3 \ 2),3} and A166: y in {0,1,(2 \ 1),(3 \ 2),3} ; ::_thesis: x /\ y in {0,1,(2 \ 1),(3 \ 2),3} A167: ( y = 0 or y = 1 or y = 2 \ 1 or y = 3 \ 2 or y = 3 ) by A166, ENUMSET1:def_3; ( x = 0 or x = 1 or x = 2 \ 1 or x = 3 \ 2 or x = 3 ) by A165, ENUMSET1:def_3; hence x /\ y in {0,1,(2 \ 1),(3 \ 2),3} by A167, A164, A163, A160, A157, A159, A158, ENUMSET1:def_3; ::_thesis: verum end; hence ( M_3 is with_infima & M_3 is with_suprema ) by A1, YELLOW_1:12; ::_thesis: verum end; end; theorem Th9: :: YELLOW11:9 for L being LATTICE holds ( ex K being full Sublattice of L st N_5 ,K are_isomorphic iff ex a, b, c, d, e being Element of L st ( a <> b & a <> c & a <> d & a <> e & b <> c & b <> d & b <> e & c <> d & c <> e & d <> e & a "/\" b = a & a "/\" c = a & c "/\" e = c & d "/\" e = d & b "/\" c = a & b "/\" d = b & c "/\" d = a & b "\/" c = e & c "\/" d = e ) ) proof set cn = the carrier of N_5; let L be LATTICE; ::_thesis: ( ex K being full Sublattice of L st N_5 ,K are_isomorphic iff ex a, b, c, d, e being Element of L st ( a <> b & a <> c & a <> d & a <> e & b <> c & b <> d & b <> e & c <> d & c <> e & d <> e & a "/\" b = a & a "/\" c = a & c "/\" e = c & d "/\" e = d & b "/\" c = a & b "/\" d = b & c "/\" d = a & b "\/" c = e & c "\/" d = e ) ) A1: the carrier of N_5 = {0,(3 \ 1),2,(3 \ 2),3} by YELLOW_1:1; thus ( ex K being full Sublattice of L st N_5 ,K are_isomorphic implies ex a, b, c, d, e being Element of L st ( a <> b & a <> c & a <> d & a <> e & b <> c & b <> d & b <> e & c <> d & c <> e & d <> e & a "/\" b = a & a "/\" c = a & c "/\" e = c & d "/\" e = d & b "/\" c = a & b "/\" d = b & c "/\" d = a & b "\/" c = e & c "\/" d = e ) ) ::_thesis: ( ex a, b, c, d, e being Element of L st ( a <> b & a <> c & a <> d & a <> e & b <> c & b <> d & b <> e & c <> d & c <> e & d <> e & a "/\" b = a & a "/\" c = a & c "/\" e = c & d "/\" e = d & b "/\" c = a & b "/\" d = b & c "/\" d = a & b "\/" c = e & c "\/" d = e ) implies ex K being full Sublattice of L st N_5 ,K are_isomorphic ) proof reconsider td = 3 \ 2 as Element of N_5 by A1, ENUMSET1:def_3; reconsider dw = 2 as Element of N_5 by A1, ENUMSET1:def_3; reconsider t = 3 as Element of N_5 by A1, ENUMSET1:def_3; reconsider tj = 3 \ 1 as Element of N_5 by A1, ENUMSET1:def_3; reconsider cl = the carrier of L as non empty set ; reconsider z = 0 as Element of N_5 by A1, ENUMSET1:def_3; given K being full Sublattice of L such that A2: N_5 ,K are_isomorphic ; ::_thesis: ex a, b, c, d, e being Element of L st ( a <> b & a <> c & a <> d & a <> e & b <> c & b <> d & b <> e & c <> d & c <> e & d <> e & a "/\" b = a & a "/\" c = a & c "/\" e = c & d "/\" e = d & b "/\" c = a & b "/\" d = b & c "/\" d = a & b "\/" c = e & c "\/" d = e ) consider f being Function of N_5,K such that A3: f is isomorphic by A2, WAYBEL_1:def_8; A4: not K is empty by A3, WAYBEL_0:def_38; then A5: ( f is one-to-one & f is monotone ) by A3, WAYBEL_0:def_38; reconsider K = K as non empty SubRelStr of L by A3, WAYBEL_0:def_38; reconsider ck = the carrier of K as non empty set ; A6: ck = rng f by A3, WAYBEL_0:66; reconsider g = f as Function of the carrier of N_5,ck ; reconsider a = g . z, b = g . td, c = g . dw, d = g . tj, e = g . t as Element of K ; reconsider ck = ck as non empty Subset of cl by YELLOW_0:def_13; A7: b in ck ; A8: c in ck ; A9: e in ck ; A10: d in ck ; a in ck ; then reconsider A = a, B = b, C = c, D = d, E = e as Element of L by A7, A8, A10, A9; take A ; ::_thesis: ex b, c, d, e being Element of L st ( A <> b & A <> c & A <> d & A <> e & b <> c & b <> d & b <> e & c <> d & c <> e & d <> e & A "/\" b = A & A "/\" c = A & c "/\" e = c & d "/\" e = d & b "/\" c = A & b "/\" d = b & c "/\" d = A & b "\/" c = e & c "\/" d = e ) take B ; ::_thesis: ex c, d, e being Element of L st ( A <> B & A <> c & A <> d & A <> e & B <> c & B <> d & B <> e & c <> d & c <> e & d <> e & A "/\" B = A & A "/\" c = A & c "/\" e = c & d "/\" e = d & B "/\" c = A & B "/\" d = B & c "/\" d = A & B "\/" c = e & c "\/" d = e ) take C ; ::_thesis: ex d, e being Element of L st ( A <> B & A <> C & A <> d & A <> e & B <> C & B <> d & B <> e & C <> d & C <> e & d <> e & A "/\" B = A & A "/\" C = A & C "/\" e = C & d "/\" e = d & B "/\" C = A & B "/\" d = B & C "/\" d = A & B "\/" C = e & C "\/" d = e ) take D ; ::_thesis: ex e being Element of L st ( A <> B & A <> C & A <> D & A <> e & B <> C & B <> D & B <> e & C <> D & C <> e & D <> e & A "/\" B = A & A "/\" C = A & C "/\" e = C & D "/\" e = D & B "/\" C = A & B "/\" D = B & C "/\" D = A & B "\/" C = e & C "\/" D = e ) take E ; ::_thesis: ( A <> B & A <> C & A <> D & A <> E & B <> C & B <> D & B <> E & C <> D & C <> E & D <> E & A "/\" B = A & A "/\" C = A & C "/\" E = C & D "/\" E = D & B "/\" C = A & B "/\" D = B & C "/\" D = A & B "\/" C = E & C "\/" D = E ) thus A <> B by A5, Th4, WAYBEL_1:def_1; ::_thesis: ( A <> C & A <> D & A <> E & B <> C & B <> D & B <> E & C <> D & C <> E & D <> E & A "/\" B = A & A "/\" C = A & C "/\" E = C & D "/\" E = D & B "/\" C = A & B "/\" D = B & C "/\" D = A & B "\/" C = E & C "\/" D = E ) thus A <> C by A5, WAYBEL_1:def_1; ::_thesis: ( A <> D & A <> E & B <> C & B <> D & B <> E & C <> D & C <> E & D <> E & A "/\" B = A & A "/\" C = A & C "/\" E = C & D "/\" E = D & B "/\" C = A & B "/\" D = B & C "/\" D = A & B "\/" C = E & C "\/" D = E ) thus A <> D by A5, Th3, WAYBEL_1:def_1; ::_thesis: ( A <> E & B <> C & B <> D & B <> E & C <> D & C <> E & D <> E & A "/\" B = A & A "/\" C = A & C "/\" E = C & D "/\" E = D & B "/\" C = A & B "/\" D = B & C "/\" D = A & B "\/" C = E & C "\/" D = E ) thus A <> E by A5, WAYBEL_1:def_1; ::_thesis: ( B <> C & B <> D & B <> E & C <> D & C <> E & D <> E & A "/\" B = A & A "/\" C = A & C "/\" E = C & D "/\" E = D & B "/\" C = A & B "/\" D = B & C "/\" D = A & B "\/" C = E & C "\/" D = E ) now__::_thesis:_not_f_._td_=_f_._dw assume f . td = f . dw ; ::_thesis: contradiction then A11: td = dw by A4, A5, WAYBEL_1:def_1; 2 in td by Th4, TARSKI:def_1; hence contradiction by A11; ::_thesis: verum end; hence B <> C ; ::_thesis: ( B <> D & B <> E & C <> D & C <> E & D <> E & A "/\" B = A & A "/\" C = A & C "/\" E = C & D "/\" E = D & B "/\" C = A & B "/\" D = B & C "/\" D = A & B "\/" C = E & C "\/" D = E ) now__::_thesis:_not_f_._td_=_f_._tj A12: 1 in tj by Th3, TARSKI:def_2; assume A13: f . td = f . tj ; ::_thesis: contradiction not 1 in td by Th4, TARSKI:def_1; hence contradiction by A4, A5, A13, A12, WAYBEL_1:def_1; ::_thesis: verum end; hence B <> D ; ::_thesis: ( B <> E & C <> D & C <> E & D <> E & A "/\" B = A & A "/\" C = A & C "/\" E = C & D "/\" E = D & B "/\" C = A & B "/\" D = B & C "/\" D = A & B "\/" C = E & C "\/" D = E ) now__::_thesis:_not_f_._td_=_f_._t A14: 1 in t by CARD_1:51, ENUMSET1:def_1; assume A15: f . td = f . t ; ::_thesis: contradiction not 1 in td by Th4, TARSKI:def_1; hence contradiction by A4, A5, A15, A14, WAYBEL_1:def_1; ::_thesis: verum end; hence B <> E ; ::_thesis: ( C <> D & C <> E & D <> E & A "/\" B = A & A "/\" C = A & C "/\" E = C & D "/\" E = D & B "/\" C = A & B "/\" D = B & C "/\" D = A & B "\/" C = E & C "\/" D = E ) now__::_thesis:_not_f_._dw_=_f_._tj assume f . dw = f . tj ; ::_thesis: contradiction then A16: dw = tj by A4, A5, WAYBEL_1:def_1; 2 in tj by Th3, TARSKI:def_2; hence contradiction by A16; ::_thesis: verum end; hence C <> D ; ::_thesis: ( C <> E & D <> E & A "/\" B = A & A "/\" C = A & C "/\" E = C & D "/\" E = D & B "/\" C = A & B "/\" D = B & C "/\" D = A & B "\/" C = E & C "\/" D = E ) thus C <> E by A5, WAYBEL_1:def_1; ::_thesis: ( D <> E & A "/\" B = A & A "/\" C = A & C "/\" E = C & D "/\" E = D & B "/\" C = A & B "/\" D = B & C "/\" D = A & B "\/" C = E & C "\/" D = E ) now__::_thesis:_not_f_._tj_=_f_._t A17: 0 in t by CARD_1:51, ENUMSET1:def_1; assume A18: f . tj = f . t ; ::_thesis: contradiction not 0 in tj by Th3, TARSKI:def_2; hence contradiction by A4, A5, A18, A17, WAYBEL_1:def_1; ::_thesis: verum end; hence D <> E ; ::_thesis: ( A "/\" B = A & A "/\" C = A & C "/\" E = C & D "/\" E = D & B "/\" C = A & B "/\" D = B & C "/\" D = A & B "\/" C = E & C "\/" D = E ) z c= td by XBOOLE_1:2; then z <= td by YELLOW_1:3; then a <= b by A3, WAYBEL_0:66; then A <= B by YELLOW_0:59; hence A "/\" B = A by YELLOW_0:25; ::_thesis: ( A "/\" C = A & C "/\" E = C & D "/\" E = D & B "/\" C = A & B "/\" D = B & C "/\" D = A & B "\/" C = E & C "\/" D = E ) z c= dw by XBOOLE_1:2; then z <= dw by YELLOW_1:3; then a <= c by A3, WAYBEL_0:66; then A <= C by YELLOW_0:59; hence A "/\" C = A by YELLOW_0:25; ::_thesis: ( C "/\" E = C & D "/\" E = D & B "/\" C = A & B "/\" D = B & C "/\" D = A & B "\/" C = E & C "\/" D = E ) dw c= t by NAT_1:39; then dw <= t by YELLOW_1:3; then c <= e by A3, WAYBEL_0:66; then C <= E by YELLOW_0:59; hence C "/\" E = C by YELLOW_0:25; ::_thesis: ( D "/\" E = D & B "/\" C = A & B "/\" D = B & C "/\" D = A & B "\/" C = E & C "\/" D = E ) tj <= t by YELLOW_1:3; then d <= e by A3, WAYBEL_0:66; then D <= E by YELLOW_0:59; hence D "/\" E = D by YELLOW_0:25; ::_thesis: ( B "/\" C = A & B "/\" D = B & C "/\" D = A & B "\/" C = E & C "\/" D = E ) thus B "/\" C = A ::_thesis: ( B "/\" D = B & C "/\" D = A & B "\/" C = E & C "\/" D = E ) proof A19: now__::_thesis:_not_B_"/\"_C_=_D assume B "/\" C = D ; ::_thesis: contradiction then D <= C by YELLOW_0:23; then d <= c by YELLOW_0:60; then tj <= dw by A3, WAYBEL_0:66; then A20: tj c= dw by YELLOW_1:3; 2 in tj by Th3, TARSKI:def_2; then 2 in 2 by A20; hence contradiction ; ::_thesis: verum end; A21: now__::_thesis:_not_B_"/\"_C_=_E assume B "/\" C = E ; ::_thesis: contradiction then E <= C by YELLOW_0:23; then e <= c by YELLOW_0:60; then t <= dw by A3, WAYBEL_0:66; then A22: t c= dw by YELLOW_1:3; 2 in t by CARD_1:51, ENUMSET1:def_1; then 2 in 2 by A22; hence contradiction ; ::_thesis: verum end; A23: now__::_thesis:_not_B_"/\"_C_=_B assume B "/\" C = B ; ::_thesis: contradiction then B <= C by YELLOW_0:25; then b <= c by YELLOW_0:60; then td <= dw by A3, WAYBEL_0:66; then A24: td c= dw by YELLOW_1:3; 2 in td by Th4, TARSKI:def_1; then 2 in 2 by A24; hence contradiction ; ::_thesis: verum end; A25: now__::_thesis:_not_B_"/\"_C_=_C assume B "/\" C = C ; ::_thesis: contradiction then C <= B by YELLOW_0:25; then c <= b by YELLOW_0:60; then dw <= td by A3, WAYBEL_0:66; then A26: dw c= td by YELLOW_1:3; 0 in dw by CARD_1:50, TARSKI:def_2; hence contradiction by A26, Th4, TARSKI:def_1; ::_thesis: verum end; ex_inf_of {B,C},L by YELLOW_0:21; then inf {B,C} in the carrier of K by YELLOW_0:def_16; then B "/\" C in rng f by A6, YELLOW_0:40; then ex x being set st ( x in dom f & B "/\" C = f . x ) by FUNCT_1:def_3; hence B "/\" C = A by A1, A23, A25, A19, A21, ENUMSET1:def_3; ::_thesis: verum end; td <= tj by Lm1, YELLOW_1:3; then b <= d by A3, WAYBEL_0:66; then B <= D by YELLOW_0:59; hence B "/\" D = B by YELLOW_0:25; ::_thesis: ( C "/\" D = A & B "\/" C = E & C "\/" D = E ) thus C "/\" D = A ::_thesis: ( B "\/" C = E & C "\/" D = E ) proof A27: now__::_thesis:_not_C_"/\"_D_=_D assume C "/\" D = D ; ::_thesis: contradiction then D <= C by YELLOW_0:23; then d <= c by YELLOW_0:60; then tj <= dw by A3, WAYBEL_0:66; then A28: tj c= dw by YELLOW_1:3; 2 in tj by Th3, TARSKI:def_2; then 2 in 2 by A28; hence contradiction ; ::_thesis: verum end; A29: now__::_thesis:_not_C_"/\"_D_=_E assume C "/\" D = E ; ::_thesis: contradiction then E <= C by YELLOW_0:23; then e <= c by YELLOW_0:60; then t <= dw by A3, WAYBEL_0:66; then A30: t c= dw by YELLOW_1:3; 2 in t by CARD_1:51, ENUMSET1:def_1; then 2 in 2 by A30; hence contradiction ; ::_thesis: verum end; A31: now__::_thesis:_not_C_"/\"_D_=_B assume C "/\" D = B ; ::_thesis: contradiction then B <= C by YELLOW_0:23; then b <= c by YELLOW_0:60; then td <= dw by A3, WAYBEL_0:66; then A32: td c= dw by YELLOW_1:3; 2 in td by Th4, TARSKI:def_1; then 2 in 2 by A32; hence contradiction ; ::_thesis: verum end; A33: now__::_thesis:_not_C_"/\"_D_=_C assume C "/\" D = C ; ::_thesis: contradiction then C <= D by YELLOW_0:25; then c <= d by YELLOW_0:60; then dw <= tj by A3, WAYBEL_0:66; then A34: dw c= tj by YELLOW_1:3; 0 in dw by CARD_1:50, TARSKI:def_2; hence contradiction by A34, Th3, TARSKI:def_2; ::_thesis: verum end; ex_inf_of {C,D},L by YELLOW_0:21; then inf {C,D} in the carrier of K by YELLOW_0:def_16; then C "/\" D in rng f by A6, YELLOW_0:40; then ex x being set st ( x in dom f & C "/\" D = f . x ) by FUNCT_1:def_3; hence C "/\" D = A by A1, A31, A33, A27, A29, ENUMSET1:def_3; ::_thesis: verum end; thus B "\/" C = E ::_thesis: C "\/" D = E proof A35: now__::_thesis:_not_B_"\/"_C_=_C assume B "\/" C = C ; ::_thesis: contradiction then C >= B by YELLOW_0:24; then c >= b by YELLOW_0:60; then dw >= td by A3, WAYBEL_0:66; then A36: td c= dw by YELLOW_1:3; 2 in td by Th4, TARSKI:def_1; then 2 in 2 by A36; hence contradiction ; ::_thesis: verum end; A37: now__::_thesis:_not_B_"\/"_C_=_D assume B "\/" C = D ; ::_thesis: contradiction then D >= C by YELLOW_0:22; then d >= c by YELLOW_0:60; then tj >= dw by A3, WAYBEL_0:66; then A38: dw c= tj by YELLOW_1:3; 0 in dw by CARD_1:50, TARSKI:def_2; hence contradiction by A38, Th3, TARSKI:def_2; ::_thesis: verum end; A39: now__::_thesis:_not_B_"\/"_C_=_B assume B "\/" C = B ; ::_thesis: contradiction then B >= C by YELLOW_0:24; then b >= c by YELLOW_0:60; then td >= dw by A3, WAYBEL_0:66; then A40: dw c= td by YELLOW_1:3; 0 in dw by CARD_1:50, TARSKI:def_2; hence contradiction by A40, Th4, TARSKI:def_1; ::_thesis: verum end; A41: now__::_thesis:_not_B_"\/"_C_=_A assume B "\/" C = A ; ::_thesis: contradiction then A >= C by YELLOW_0:22; then a >= c by YELLOW_0:60; then z >= dw by A3, WAYBEL_0:66; then dw c= z by YELLOW_1:3; hence contradiction ; ::_thesis: verum end; ex_sup_of {B,C},L by YELLOW_0:20; then sup {B,C} in the carrier of K by YELLOW_0:def_17; then B "\/" C in rng f by A6, YELLOW_0:41; then ex x being set st ( x in dom f & B "\/" C = f . x ) by FUNCT_1:def_3; hence B "\/" C = E by A1, A39, A35, A37, A41, ENUMSET1:def_3; ::_thesis: verum end; thus C "\/" D = E ::_thesis: verum proof A42: now__::_thesis:_not_C_"\/"_D_=_D assume C "\/" D = D ; ::_thesis: contradiction then D >= C by YELLOW_0:22; then d >= c by YELLOW_0:60; then tj >= dw by A3, WAYBEL_0:66; then A43: dw c= tj by YELLOW_1:3; 0 in dw by CARD_1:50, TARSKI:def_2; hence contradiction by A43, Th3, TARSKI:def_2; ::_thesis: verum end; A44: now__::_thesis:_not_C_"\/"_D_=_C assume C "\/" D = C ; ::_thesis: contradiction then C >= D by YELLOW_0:24; then c >= d by YELLOW_0:60; then dw >= tj by A3, WAYBEL_0:66; then A45: tj c= dw by YELLOW_1:3; 2 in tj by Th3, TARSKI:def_2; hence contradiction by A45, CARD_1:50, TARSKI:def_2; ::_thesis: verum end; A46: now__::_thesis:_not_C_"\/"_D_=_B assume C "\/" D = B ; ::_thesis: contradiction then B >= C by YELLOW_0:22; then b >= c by YELLOW_0:60; then td >= dw by A3, WAYBEL_0:66; then A47: dw c= td by YELLOW_1:3; 0 in dw by CARD_1:50, TARSKI:def_2; hence contradiction by A47, Th4, TARSKI:def_1; ::_thesis: verum end; A48: now__::_thesis:_not_C_"\/"_D_=_A assume C "\/" D = A ; ::_thesis: contradiction then A >= C by YELLOW_0:22; then a >= c by YELLOW_0:60; then z >= dw by A3, WAYBEL_0:66; then dw c= z by YELLOW_1:3; hence contradiction ; ::_thesis: verum end; ex_sup_of {C,D},L by YELLOW_0:20; then sup {C,D} in the carrier of K by YELLOW_0:def_17; then C "\/" D in rng f by A6, YELLOW_0:41; then ex x being set st ( x in dom f & C "\/" D = f . x ) by FUNCT_1:def_3; hence C "\/" D = E by A1, A46, A44, A42, A48, ENUMSET1:def_3; ::_thesis: verum end; end; thus ( ex a, b, c, d, e being Element of L st ( a <> b & a <> c & a <> d & a <> e & b <> c & b <> d & b <> e & c <> d & c <> e & d <> e & a "/\" b = a & a "/\" c = a & c "/\" e = c & d "/\" e = d & b "/\" c = a & b "/\" d = b & c "/\" d = a & b "\/" c = e & c "\/" d = e ) implies ex K being full Sublattice of L st N_5 ,K are_isomorphic ) ::_thesis: verum proof given a, b, c, d, e being Element of L such that A49: a <> b and A50: a <> c and A51: a <> d and A52: a <> e and A53: b <> c and A54: b <> d and A55: b <> e and A56: c <> d and A57: c <> e and A58: d <> e and A59: a "/\" b = a and A60: a "/\" c = a and A61: c "/\" e = c and A62: d "/\" e = d and A63: b "/\" c = a and A64: b "/\" d = b and A65: c "/\" d = a and A66: b "\/" c = e and A67: c "\/" d = e ; ::_thesis: ex K being full Sublattice of L st N_5 ,K are_isomorphic set ck = {a,b,c,d,e}; reconsider ck = {a,b,c,d,e} as Subset of L ; set K = subrelstr ck; A68: the carrier of (subrelstr ck) = ck by YELLOW_0:def_15; A69: subrelstr ck is meet-inheriting proof let x, y be Element of L; :: according to YELLOW_0:def_16 ::_thesis: ( not x in the carrier of (subrelstr ck) or not y in the carrier of (subrelstr ck) or not ex_inf_of {x,y},L or "/\" ({x,y},L) in the carrier of (subrelstr ck) ) assume that A70: x in the carrier of (subrelstr ck) and A71: y in the carrier of (subrelstr ck) and ex_inf_of {x,y},L ; ::_thesis: "/\" ({x,y},L) in the carrier of (subrelstr ck) percases ( ( x = a & y = a ) or ( x = a & y = b ) or ( x = a & y = c ) or ( x = a & y = d ) or ( x = a & y = e ) or ( x = b & y = a ) or ( x = b & y = b ) or ( x = b & y = c ) or ( x = b & y = d ) or ( x = b & y = e ) or ( x = c & y = a ) or ( x = c & y = b ) or ( x = c & y = c ) or ( x = c & y = d ) or ( x = c & y = e ) or ( x = d & y = a ) or ( x = d & y = b ) or ( x = d & y = c ) or ( x = d & y = d ) or ( x = d & y = e ) or ( x = e & y = a ) or ( x = e & y = b ) or ( x = e & y = c ) or ( x = e & y = d ) or ( x = e & y = e ) ) by A68, A70, A71, ENUMSET1:def_3; suppose ( x = a & y = a ) ; ::_thesis: "/\" ({x,y},L) in the carrier of (subrelstr ck) then inf {x,y} = a "/\" a by YELLOW_0:40; then inf {x,y} = a by YELLOW_5:2; hence "/\" ({x,y},L) in the carrier of (subrelstr ck) by A68, ENUMSET1:def_3; ::_thesis: verum end; suppose ( x = a & y = b ) ; ::_thesis: "/\" ({x,y},L) in the carrier of (subrelstr ck) then inf {x,y} = a "/\" b by YELLOW_0:40; hence "/\" ({x,y},L) in the carrier of (subrelstr ck) by A59, A68, ENUMSET1:def_3; ::_thesis: verum end; suppose ( x = a & y = c ) ; ::_thesis: "/\" ({x,y},L) in the carrier of (subrelstr ck) then inf {x,y} = a "/\" c by YELLOW_0:40; hence "/\" ({x,y},L) in the carrier of (subrelstr ck) by A60, A68, ENUMSET1:def_3; ::_thesis: verum end; supposeA72: ( x = a & y = d ) ; ::_thesis: "/\" ({x,y},L) in the carrier of (subrelstr ck) A73: b <= d by A64, YELLOW_0:25; a <= b by A59, YELLOW_0:25; then a <= d by A73, ORDERS_2:3; then a "/\" d = a by YELLOW_0:25; then inf {x,y} = a by A72, YELLOW_0:40; hence "/\" ({x,y},L) in the carrier of (subrelstr ck) by A68, ENUMSET1:def_3; ::_thesis: verum end; supposeA74: ( x = a & y = e ) ; ::_thesis: "/\" ({x,y},L) in the carrier of (subrelstr ck) A75: c <= e by A61, YELLOW_0:25; a <= c by A60, YELLOW_0:25; then a <= e by A75, ORDERS_2:3; then a "/\" e = a by YELLOW_0:25; then inf {x,y} = a by A74, YELLOW_0:40; hence "/\" ({x,y},L) in the carrier of (subrelstr ck) by A68, ENUMSET1:def_3; ::_thesis: verum end; suppose ( x = b & y = a ) ; ::_thesis: "/\" ({x,y},L) in the carrier of (subrelstr ck) then inf {x,y} = a "/\" b by YELLOW_0:40; hence "/\" ({x,y},L) in the carrier of (subrelstr ck) by A59, A68, ENUMSET1:def_3; ::_thesis: verum end; suppose ( x = b & y = b ) ; ::_thesis: "/\" ({x,y},L) in the carrier of (subrelstr ck) then inf {x,y} = b "/\" b by YELLOW_0:40; then inf {x,y} = b by YELLOW_5:2; hence "/\" ({x,y},L) in the carrier of (subrelstr ck) by A68, ENUMSET1:def_3; ::_thesis: verum end; suppose ( x = b & y = c ) ; ::_thesis: "/\" ({x,y},L) in the carrier of (subrelstr ck) then inf {x,y} = b "/\" c by YELLOW_0:40; hence "/\" ({x,y},L) in the carrier of (subrelstr ck) by A63, A68, ENUMSET1:def_3; ::_thesis: verum end; suppose ( x = b & y = d ) ; ::_thesis: "/\" ({x,y},L) in the carrier of (subrelstr ck) then inf {x,y} = b "/\" d by YELLOW_0:40; hence "/\" ({x,y},L) in the carrier of (subrelstr ck) by A64, A68, ENUMSET1:def_3; ::_thesis: verum end; supposeA76: ( x = b & y = e ) ; ::_thesis: "/\" ({x,y},L) in the carrier of (subrelstr ck) A77: d <= e by A62, YELLOW_0:25; b <= d by A64, YELLOW_0:25; then b <= e by A77, ORDERS_2:3; then b "/\" e = b by YELLOW_0:25; then inf {x,y} = b by A76, YELLOW_0:40; hence "/\" ({x,y},L) in the carrier of (subrelstr ck) by A68, ENUMSET1:def_3; ::_thesis: verum end; suppose ( x = c & y = a ) ; ::_thesis: "/\" ({x,y},L) in the carrier of (subrelstr ck) then inf {x,y} = a "/\" c by YELLOW_0:40; hence "/\" ({x,y},L) in the carrier of (subrelstr ck) by A60, A68, ENUMSET1:def_3; ::_thesis: verum end; suppose ( x = c & y = b ) ; ::_thesis: "/\" ({x,y},L) in the carrier of (subrelstr ck) then inf {x,y} = b "/\" c by YELLOW_0:40; hence "/\" ({x,y},L) in the carrier of (subrelstr ck) by A63, A68, ENUMSET1:def_3; ::_thesis: verum end; suppose ( x = c & y = c ) ; ::_thesis: "/\" ({x,y},L) in the carrier of (subrelstr ck) then inf {x,y} = c "/\" c by YELLOW_0:40; then inf {x,y} = c by YELLOW_5:2; hence "/\" ({x,y},L) in the carrier of (subrelstr ck) by A68, ENUMSET1:def_3; ::_thesis: verum end; suppose ( x = c & y = d ) ; ::_thesis: "/\" ({x,y},L) in the carrier of (subrelstr ck) then inf {x,y} = c "/\" d by YELLOW_0:40; hence "/\" ({x,y},L) in the carrier of (subrelstr ck) by A65, A68, ENUMSET1:def_3; ::_thesis: verum end; suppose ( x = c & y = e ) ; ::_thesis: "/\" ({x,y},L) in the carrier of (subrelstr ck) then inf {x,y} = c "/\" e by YELLOW_0:40; hence "/\" ({x,y},L) in the carrier of (subrelstr ck) by A61, A68, ENUMSET1:def_3; ::_thesis: verum end; supposeA78: ( x = d & y = a ) ; ::_thesis: "/\" ({x,y},L) in the carrier of (subrelstr ck) A79: b <= d by A64, YELLOW_0:25; a <= b by A59, YELLOW_0:25; then a <= d by A79, ORDERS_2:3; then a "/\" d = a by YELLOW_0:25; then inf {x,y} = a by A78, YELLOW_0:40; hence "/\" ({x,y},L) in the carrier of (subrelstr ck) by A68, ENUMSET1:def_3; ::_thesis: verum end; suppose ( x = d & y = b ) ; ::_thesis: "/\" ({x,y},L) in the carrier of (subrelstr ck) then inf {x,y} = b "/\" d by YELLOW_0:40; hence "/\" ({x,y},L) in the carrier of (subrelstr ck) by A64, A68, ENUMSET1:def_3; ::_thesis: verum end; suppose ( x = d & y = c ) ; ::_thesis: "/\" ({x,y},L) in the carrier of (subrelstr ck) then inf {x,y} = c "/\" d by YELLOW_0:40; hence "/\" ({x,y},L) in the carrier of (subrelstr ck) by A65, A68, ENUMSET1:def_3; ::_thesis: verum end; suppose ( x = d & y = d ) ; ::_thesis: "/\" ({x,y},L) in the carrier of (subrelstr ck) then inf {x,y} = d "/\" d by YELLOW_0:40; then inf {x,y} = d by YELLOW_5:2; hence "/\" ({x,y},L) in the carrier of (subrelstr ck) by A68, ENUMSET1:def_3; ::_thesis: verum end; suppose ( x = d & y = e ) ; ::_thesis: "/\" ({x,y},L) in the carrier of (subrelstr ck) then inf {x,y} = d "/\" e by YELLOW_0:40; hence "/\" ({x,y},L) in the carrier of (subrelstr ck) by A62, A68, ENUMSET1:def_3; ::_thesis: verum end; supposeA80: ( x = e & y = a ) ; ::_thesis: "/\" ({x,y},L) in the carrier of (subrelstr ck) A81: c <= e by A61, YELLOW_0:25; a <= c by A60, YELLOW_0:25; then a <= e by A81, ORDERS_2:3; then a "/\" e = a by YELLOW_0:25; then inf {x,y} = a by A80, YELLOW_0:40; hence "/\" ({x,y},L) in the carrier of (subrelstr ck) by A68, ENUMSET1:def_3; ::_thesis: verum end; supposeA82: ( x = e & y = b ) ; ::_thesis: "/\" ({x,y},L) in the carrier of (subrelstr ck) A83: d <= e by A62, YELLOW_0:25; b <= d by A64, YELLOW_0:25; then b <= e by A83, ORDERS_2:3; then b "/\" e = b by YELLOW_0:25; then inf {x,y} = b by A82, YELLOW_0:40; hence "/\" ({x,y},L) in the carrier of (subrelstr ck) by A68, ENUMSET1:def_3; ::_thesis: verum end; suppose ( x = e & y = c ) ; ::_thesis: "/\" ({x,y},L) in the carrier of (subrelstr ck) then inf {x,y} = c "/\" e by YELLOW_0:40; hence "/\" ({x,y},L) in the carrier of (subrelstr ck) by A61, A68, ENUMSET1:def_3; ::_thesis: verum end; suppose ( x = e & y = d ) ; ::_thesis: "/\" ({x,y},L) in the carrier of (subrelstr ck) then inf {x,y} = d "/\" e by YELLOW_0:40; hence "/\" ({x,y},L) in the carrier of (subrelstr ck) by A62, A68, ENUMSET1:def_3; ::_thesis: verum end; suppose ( x = e & y = e ) ; ::_thesis: "/\" ({x,y},L) in the carrier of (subrelstr ck) then inf {x,y} = e "/\" e by YELLOW_0:40; then inf {x,y} = e by YELLOW_5:2; hence "/\" ({x,y},L) in the carrier of (subrelstr ck) by A68, ENUMSET1:def_3; ::_thesis: verum end; end; end; subrelstr ck is join-inheriting proof let x, y be Element of L; :: according to YELLOW_0:def_17 ::_thesis: ( not x in the carrier of (subrelstr ck) or not y in the carrier of (subrelstr ck) or not ex_sup_of {x,y},L or "\/" ({x,y},L) in the carrier of (subrelstr ck) ) assume that A84: x in the carrier of (subrelstr ck) and A85: y in the carrier of (subrelstr ck) and ex_sup_of {x,y},L ; ::_thesis: "\/" ({x,y},L) in the carrier of (subrelstr ck) percases ( ( x = a & y = a ) or ( x = a & y = b ) or ( x = a & y = c ) or ( x = a & y = d ) or ( x = a & y = e ) or ( x = b & y = a ) or ( x = b & y = b ) or ( x = b & y = c ) or ( x = b & y = d ) or ( x = b & y = e ) or ( x = c & y = a ) or ( x = c & y = b ) or ( x = c & y = c ) or ( x = c & y = d ) or ( x = c & y = e ) or ( x = d & y = a ) or ( x = d & y = b ) or ( x = d & y = c ) or ( x = d & y = d ) or ( x = d & y = e ) or ( x = e & y = a ) or ( x = e & y = b ) or ( x = e & y = c ) or ( x = e & y = d ) or ( x = e & y = e ) ) by A68, A84, A85, ENUMSET1:def_3; suppose ( x = a & y = a ) ; ::_thesis: "\/" ({x,y},L) in the carrier of (subrelstr ck) then sup {x,y} = a "\/" a by YELLOW_0:41; then sup {x,y} = a by YELLOW_5:1; hence "\/" ({x,y},L) in the carrier of (subrelstr ck) by A68, ENUMSET1:def_3; ::_thesis: verum end; suppose ( x = a & y = b ) ; ::_thesis: "\/" ({x,y},L) in the carrier of (subrelstr ck) then x "\/" y = b by A59, Th5; then sup {x,y} = b by YELLOW_0:41; hence "\/" ({x,y},L) in the carrier of (subrelstr ck) by A68, ENUMSET1:def_3; ::_thesis: verum end; suppose ( x = a & y = c ) ; ::_thesis: "\/" ({x,y},L) in the carrier of (subrelstr ck) then x "\/" y = c by A60, Th5; then sup {x,y} = c by YELLOW_0:41; hence "\/" ({x,y},L) in the carrier of (subrelstr ck) by A68, ENUMSET1:def_3; ::_thesis: verum end; supposeA86: ( x = a & y = d ) ; ::_thesis: "\/" ({x,y},L) in the carrier of (subrelstr ck) A87: b <= d by A64, YELLOW_0:25; a <= b by A59, YELLOW_0:25; then a <= d by A87, ORDERS_2:3; then a "/\" d = a by YELLOW_0:25; then a "\/" d = d by Th5; then sup {x,y} = d by A86, YELLOW_0:41; hence "\/" ({x,y},L) in the carrier of (subrelstr ck) by A68, ENUMSET1:def_3; ::_thesis: verum end; supposeA88: ( x = a & y = e ) ; ::_thesis: "\/" ({x,y},L) in the carrier of (subrelstr ck) A89: c <= e by A61, YELLOW_0:25; a <= c by A60, YELLOW_0:25; then a <= e by A89, ORDERS_2:3; then a "/\" e = a by YELLOW_0:25; then a "\/" e = e by Th5; then sup {x,y} = e by A88, YELLOW_0:41; hence "\/" ({x,y},L) in the carrier of (subrelstr ck) by A68, ENUMSET1:def_3; ::_thesis: verum end; supposeA90: ( x = b & y = a ) ; ::_thesis: "\/" ({x,y},L) in the carrier of (subrelstr ck) a "\/" b = b by A59, Th5; then sup {x,y} = b by A90, YELLOW_0:41; hence "\/" ({x,y},L) in the carrier of (subrelstr ck) by A68, ENUMSET1:def_3; ::_thesis: verum end; suppose ( x = b & y = b ) ; ::_thesis: "\/" ({x,y},L) in the carrier of (subrelstr ck) then sup {x,y} = b "\/" b by YELLOW_0:41; then sup {x,y} = b by YELLOW_5:1; hence "\/" ({x,y},L) in the carrier of (subrelstr ck) by A68, ENUMSET1:def_3; ::_thesis: verum end; suppose ( x = b & y = c ) ; ::_thesis: "\/" ({x,y},L) in the carrier of (subrelstr ck) then sup {x,y} = b "\/" c by YELLOW_0:41; hence "\/" ({x,y},L) in the carrier of (subrelstr ck) by A66, A68, ENUMSET1:def_3; ::_thesis: verum end; supposeA91: ( x = b & y = d ) ; ::_thesis: "\/" ({x,y},L) in the carrier of (subrelstr ck) b "\/" d = d by A64, Th5; then sup {x,y} = d by A91, YELLOW_0:41; hence "\/" ({x,y},L) in the carrier of (subrelstr ck) by A68, ENUMSET1:def_3; ::_thesis: verum end; supposeA92: ( x = b & y = e ) ; ::_thesis: "\/" ({x,y},L) in the carrier of (subrelstr ck) A93: d <= e by A62, YELLOW_0:25; b <= d by A64, YELLOW_0:25; then b <= e by A93, ORDERS_2:3; then b "/\" e = b by YELLOW_0:25; then b "\/" e = e by Th5; then sup {x,y} = e by A92, YELLOW_0:41; hence "\/" ({x,y},L) in the carrier of (subrelstr ck) by A68, ENUMSET1:def_3; ::_thesis: verum end; supposeA94: ( x = c & y = a ) ; ::_thesis: "\/" ({x,y},L) in the carrier of (subrelstr ck) c "\/" a = c by A60, Th5; then sup {x,y} = c by A94, YELLOW_0:41; hence "\/" ({x,y},L) in the carrier of (subrelstr ck) by A68, ENUMSET1:def_3; ::_thesis: verum end; suppose ( x = c & y = b ) ; ::_thesis: "\/" ({x,y},L) in the carrier of (subrelstr ck) then sup {x,y} = b "\/" c by YELLOW_0:41; hence "\/" ({x,y},L) in the carrier of (subrelstr ck) by A66, A68, ENUMSET1:def_3; ::_thesis: verum end; suppose ( x = c & y = c ) ; ::_thesis: "\/" ({x,y},L) in the carrier of (subrelstr ck) then sup {x,y} = c "\/" c by YELLOW_0:41; then sup {x,y} = c by YELLOW_5:1; hence "\/" ({x,y},L) in the carrier of (subrelstr ck) by A68, ENUMSET1:def_3; ::_thesis: verum end; suppose ( x = c & y = d ) ; ::_thesis: "\/" ({x,y},L) in the carrier of (subrelstr ck) then sup {x,y} = c "\/" d by YELLOW_0:41; hence "\/" ({x,y},L) in the carrier of (subrelstr ck) by A67, A68, ENUMSET1:def_3; ::_thesis: verum end; supposeA95: ( x = c & y = e ) ; ::_thesis: "\/" ({x,y},L) in the carrier of (subrelstr ck) c "\/" e = e by A61, Th5; then sup {x,y} = e by A95, YELLOW_0:41; hence "\/" ({x,y},L) in the carrier of (subrelstr ck) by A68, ENUMSET1:def_3; ::_thesis: verum end; supposeA96: ( x = d & y = a ) ; ::_thesis: "\/" ({x,y},L) in the carrier of (subrelstr ck) A97: b <= d by A64, YELLOW_0:25; a <= b by A59, YELLOW_0:25; then a <= d by A97, ORDERS_2:3; then a "/\" d = a by YELLOW_0:25; then a "\/" d = d by Th5; then sup {x,y} = d by A96, YELLOW_0:41; hence "\/" ({x,y},L) in the carrier of (subrelstr ck) by A68, ENUMSET1:def_3; ::_thesis: verum end; supposeA98: ( x = d & y = b ) ; ::_thesis: "\/" ({x,y},L) in the carrier of (subrelstr ck) b "\/" d = d by A64, Th5; then sup {x,y} = d by A98, YELLOW_0:41; hence "\/" ({x,y},L) in the carrier of (subrelstr ck) by A68, ENUMSET1:def_3; ::_thesis: verum end; suppose ( x = d & y = c ) ; ::_thesis: "\/" ({x,y},L) in the carrier of (subrelstr ck) then sup {x,y} = c "\/" d by YELLOW_0:41; hence "\/" ({x,y},L) in the carrier of (subrelstr ck) by A67, A68, ENUMSET1:def_3; ::_thesis: verum end; suppose ( x = d & y = d ) ; ::_thesis: "\/" ({x,y},L) in the carrier of (subrelstr ck) then sup {x,y} = d "\/" d by YELLOW_0:41; then sup {x,y} = d by YELLOW_5:1; hence "\/" ({x,y},L) in the carrier of (subrelstr ck) by A68, ENUMSET1:def_3; ::_thesis: verum end; supposeA99: ( x = d & y = e ) ; ::_thesis: "\/" ({x,y},L) in the carrier of (subrelstr ck) d "\/" e = e by A62, Th5; then sup {x,y} = e by A99, YELLOW_0:41; hence "\/" ({x,y},L) in the carrier of (subrelstr ck) by A68, ENUMSET1:def_3; ::_thesis: verum end; supposeA100: ( x = e & y = a ) ; ::_thesis: "\/" ({x,y},L) in the carrier of (subrelstr ck) A101: c <= e by A61, YELLOW_0:25; a <= c by A60, YELLOW_0:25; then a <= e by A101, ORDERS_2:3; then a "/\" e = a by YELLOW_0:25; then a "\/" e = e by Th5; then sup {x,y} = e by A100, YELLOW_0:41; hence "\/" ({x,y},L) in the carrier of (subrelstr ck) by A68, ENUMSET1:def_3; ::_thesis: verum end; supposeA102: ( x = e & y = b ) ; ::_thesis: "\/" ({x,y},L) in the carrier of (subrelstr ck) A103: d <= e by A62, YELLOW_0:25; b <= d by A64, YELLOW_0:25; then b <= e by A103, ORDERS_2:3; then b "/\" e = b by YELLOW_0:25; then b "\/" e = e by Th5; then sup {x,y} = e by A102, YELLOW_0:41; hence "\/" ({x,y},L) in the carrier of (subrelstr ck) by A68, ENUMSET1:def_3; ::_thesis: verum end; supposeA104: ( x = e & y = c ) ; ::_thesis: "\/" ({x,y},L) in the carrier of (subrelstr ck) c "\/" e = e by A61, Th5; then sup {x,y} = e by A104, YELLOW_0:41; hence "\/" ({x,y},L) in the carrier of (subrelstr ck) by A68, ENUMSET1:def_3; ::_thesis: verum end; supposeA105: ( x = e & y = d ) ; ::_thesis: "\/" ({x,y},L) in the carrier of (subrelstr ck) d "\/" e = e by A62, Th5; then sup {x,y} = e by A105, YELLOW_0:41; hence "\/" ({x,y},L) in the carrier of (subrelstr ck) by A68, ENUMSET1:def_3; ::_thesis: verum end; suppose ( x = e & y = e ) ; ::_thesis: "\/" ({x,y},L) in the carrier of (subrelstr ck) then sup {x,y} = e "\/" e by YELLOW_0:41; then sup {x,y} = e by YELLOW_5:1; hence "\/" ({x,y},L) in the carrier of (subrelstr ck) by A68, ENUMSET1:def_3; ::_thesis: verum end; end; end; then reconsider K = subrelstr ck as non empty full Sublattice of L by A69, YELLOW_0:def_15; take K ; ::_thesis: N_5 ,K are_isomorphic thus N_5 ,K are_isomorphic ::_thesis: verum proof reconsider t = 3 as Element of N_5 by A1, ENUMSET1:def_3; reconsider td = 3 \ 2 as Element of N_5 by A1, ENUMSET1:def_3; reconsider dw = 2 as Element of N_5 by A1, ENUMSET1:def_3; A106: now__::_thesis:_not_dw_=_td assume A107: dw = td ; ::_thesis: contradiction 2 in td by Th4, TARSKI:def_1; hence contradiction by A107; ::_thesis: verum end; A108: now__::_thesis:_not_td_=_t assume A109: td = t ; ::_thesis: contradiction not 1 in td by Th4, TARSKI:def_1; hence contradiction by A109, CARD_1:51, ENUMSET1:def_1; ::_thesis: verum end; reconsider tj = 3 \ 1 as Element of N_5 by A1, ENUMSET1:def_3; reconsider z = 0 as Element of N_5 by A1, ENUMSET1:def_3; defpred S1[ set , set ] means for X being Element of N_5 st X = $1 holds ( ( X = z implies $2 = a ) & ( X = td implies $2 = b ) & ( X = dw implies $2 = c ) & ( X = tj implies $2 = d ) & ( X = t implies $2 = e ) ); A110: now__::_thesis:_not_tj_=_dw assume A111: tj = dw ; ::_thesis: contradiction 2 in tj by Th3, TARSKI:def_2; hence contradiction by A111; ::_thesis: verum end; A112: now__::_thesis:_not_tj_=_t assume A113: tj = t ; ::_thesis: contradiction not 0 in tj by Th3, TARSKI:def_2; hence contradiction by A113, CARD_1:51, ENUMSET1:def_1; ::_thesis: verum end; A114: now__::_thesis:_not_tj_=_td assume A115: tj = td ; ::_thesis: contradiction not 1 in td by Th4, TARSKI:def_1; hence contradiction by A115, Th3, TARSKI:def_2; ::_thesis: verum end; A116: for x being set st x in the carrier of N_5 holds ex y being set st ( y in ck & S1[x,y] ) proof let x be set ; ::_thesis: ( x in the carrier of N_5 implies ex y being set st ( y in ck & S1[x,y] ) ) assume A117: x in the carrier of N_5 ; ::_thesis: ex y being set st ( y in ck & S1[x,y] ) percases ( x = 0 or x = 3 \ 1 or x = 2 or x = 3 \ 2 or x = 3 ) by A1, A117, ENUMSET1:def_3; supposeA118: x = 0 ; ::_thesis: ex y being set st ( y in ck & S1[x,y] ) take y = a; ::_thesis: ( y in ck & S1[x,y] ) thus y in ck by ENUMSET1:def_3; ::_thesis: S1[x,y] let X be Element of N_5; ::_thesis: ( X = x implies ( ( X = z implies y = a ) & ( X = td implies y = b ) & ( X = dw implies y = c ) & ( X = tj implies y = d ) & ( X = t implies y = e ) ) ) thus ( X = x implies ( ( X = z implies y = a ) & ( X = td implies y = b ) & ( X = dw implies y = c ) & ( X = tj implies y = d ) & ( X = t implies y = e ) ) ) by A118, Th3, Th4; ::_thesis: verum end; supposeA119: x = 3 \ 1 ; ::_thesis: ex y being set st ( y in ck & S1[x,y] ) take y = d; ::_thesis: ( y in ck & S1[x,y] ) thus y in ck by ENUMSET1:def_3; ::_thesis: S1[x,y] let X be Element of N_5; ::_thesis: ( X = x implies ( ( X = z implies y = a ) & ( X = td implies y = b ) & ( X = dw implies y = c ) & ( X = tj implies y = d ) & ( X = t implies y = e ) ) ) thus ( X = x implies ( ( X = z implies y = a ) & ( X = td implies y = b ) & ( X = dw implies y = c ) & ( X = tj implies y = d ) & ( X = t implies y = e ) ) ) by A110, A114, A112, A119, Th3; ::_thesis: verum end; supposeA120: x = 2 ; ::_thesis: ex y being set st ( y in ck & S1[x,y] ) take y = c; ::_thesis: ( y in ck & S1[x,y] ) thus y in ck by ENUMSET1:def_3; ::_thesis: S1[x,y] let X be Element of N_5; ::_thesis: ( X = x implies ( ( X = z implies y = a ) & ( X = td implies y = b ) & ( X = dw implies y = c ) & ( X = tj implies y = d ) & ( X = t implies y = e ) ) ) thus ( X = x implies ( ( X = z implies y = a ) & ( X = td implies y = b ) & ( X = dw implies y = c ) & ( X = tj implies y = d ) & ( X = t implies y = e ) ) ) by A110, A106, A120; ::_thesis: verum end; supposeA121: x = 3 \ 2 ; ::_thesis: ex y being set st ( y in ck & S1[x,y] ) take y = b; ::_thesis: ( y in ck & S1[x,y] ) thus y in ck by ENUMSET1:def_3; ::_thesis: S1[x,y] let X be Element of N_5; ::_thesis: ( X = x implies ( ( X = z implies y = a ) & ( X = td implies y = b ) & ( X = dw implies y = c ) & ( X = tj implies y = d ) & ( X = t implies y = e ) ) ) thus ( X = x implies ( ( X = z implies y = a ) & ( X = td implies y = b ) & ( X = dw implies y = c ) & ( X = tj implies y = d ) & ( X = t implies y = e ) ) ) by A114, A106, A108, A121, Th4; ::_thesis: verum end; supposeA122: x = 3 ; ::_thesis: ex y being set st ( y in ck & S1[x,y] ) take y = e; ::_thesis: ( y in ck & S1[x,y] ) thus y in ck by ENUMSET1:def_3; ::_thesis: S1[x,y] let X be Element of N_5; ::_thesis: ( X = x implies ( ( X = z implies y = a ) & ( X = td implies y = b ) & ( X = dw implies y = c ) & ( X = tj implies y = d ) & ( X = t implies y = e ) ) ) thus ( X = x implies ( ( X = z implies y = a ) & ( X = td implies y = b ) & ( X = dw implies y = c ) & ( X = tj implies y = d ) & ( X = t implies y = e ) ) ) by A112, A108, A122; ::_thesis: verum end; end; end; consider f being Function of the carrier of N_5,ck such that A123: for x being set st x in the carrier of N_5 holds S1[x,f . x] from FUNCT_2:sch_1(A116); reconsider f = f as Function of N_5,K by A68; A124: now__::_thesis:_for_x,_y_being_Element_of_N_5_st_f_._x_=_f_._y_holds_ x_=_y let x, y be Element of N_5; ::_thesis: ( f . x = f . y implies x = y ) assume A125: f . x = f . y ; ::_thesis: x = y thus x = y ::_thesis: verum proof percases ( ( x = z & y = z ) or ( x = tj & y = tj ) or ( x = td & y = td ) or ( x = dw & y = dw ) or ( x = t & y = t ) or ( x = z & y = tj ) or ( x = z & y = dw ) or ( x = z & y = td ) or ( x = z & y = t ) or ( x = tj & y = z ) or ( x = tj & y = dw ) or ( x = tj & y = td ) or ( x = tj & y = t ) or ( x = dw & y = z ) or ( x = dw & y = tj ) or ( x = dw & y = td ) or ( x = dw & y = t ) or ( x = td & y = z ) or ( x = td & y = tj ) or ( x = td & y = dw ) or ( x = td & y = t ) or ( x = t & y = z ) or ( x = t & y = tj ) or ( x = t & y = dw ) or ( x = t & y = td ) ) by A1, ENUMSET1:def_3; suppose ( x = z & y = z ) ; ::_thesis: x = y hence x = y ; ::_thesis: verum end; suppose ( x = tj & y = tj ) ; ::_thesis: x = y hence x = y ; ::_thesis: verum end; suppose ( x = td & y = td ) ; ::_thesis: x = y hence x = y ; ::_thesis: verum end; suppose ( x = dw & y = dw ) ; ::_thesis: x = y hence x = y ; ::_thesis: verum end; suppose ( x = t & y = t ) ; ::_thesis: x = y hence x = y ; ::_thesis: verum end; supposeA126: ( x = z & y = tj ) ; ::_thesis: x = y then f . x = a by A123; hence x = y by A51, A123, A125, A126; ::_thesis: verum end; supposeA127: ( x = z & y = dw ) ; ::_thesis: x = y then f . x = a by A123; hence x = y by A50, A123, A125, A127; ::_thesis: verum end; supposeA128: ( x = z & y = td ) ; ::_thesis: x = y then f . x = a by A123; hence x = y by A49, A123, A125, A128; ::_thesis: verum end; supposeA129: ( x = z & y = t ) ; ::_thesis: x = y then f . x = a by A123; hence x = y by A52, A123, A125, A129; ::_thesis: verum end; supposeA130: ( x = tj & y = z ) ; ::_thesis: x = y then f . x = d by A123; hence x = y by A51, A123, A125, A130; ::_thesis: verum end; supposeA131: ( x = tj & y = dw ) ; ::_thesis: x = y then f . x = d by A123; hence x = y by A56, A123, A125, A131; ::_thesis: verum end; supposeA132: ( x = tj & y = td ) ; ::_thesis: x = y then f . x = d by A123; hence x = y by A54, A123, A125, A132; ::_thesis: verum end; supposeA133: ( x = tj & y = t ) ; ::_thesis: x = y then f . x = d by A123; hence x = y by A58, A123, A125, A133; ::_thesis: verum end; supposeA134: ( x = dw & y = z ) ; ::_thesis: x = y then f . x = c by A123; hence x = y by A50, A123, A125, A134; ::_thesis: verum end; supposeA135: ( x = dw & y = tj ) ; ::_thesis: x = y then f . x = c by A123; hence x = y by A56, A123, A125, A135; ::_thesis: verum end; supposeA136: ( x = dw & y = td ) ; ::_thesis: x = y then f . x = c by A123; hence x = y by A53, A123, A125, A136; ::_thesis: verum end; supposeA137: ( x = dw & y = t ) ; ::_thesis: x = y then f . x = c by A123; hence x = y by A57, A123, A125, A137; ::_thesis: verum end; supposeA138: ( x = td & y = z ) ; ::_thesis: x = y then f . x = b by A123; hence x = y by A49, A123, A125, A138; ::_thesis: verum end; supposeA139: ( x = td & y = tj ) ; ::_thesis: x = y then f . x = b by A123; hence x = y by A54, A123, A125, A139; ::_thesis: verum end; supposeA140: ( x = td & y = dw ) ; ::_thesis: x = y then f . x = b by A123; hence x = y by A53, A123, A125, A140; ::_thesis: verum end; supposeA141: ( x = td & y = t ) ; ::_thesis: x = y then f . x = b by A123; hence x = y by A55, A123, A125, A141; ::_thesis: verum end; supposeA142: ( x = t & y = z ) ; ::_thesis: x = y then f . x = e by A123; hence x = y by A52, A123, A125, A142; ::_thesis: verum end; supposeA143: ( x = t & y = tj ) ; ::_thesis: x = y then f . x = e by A123; hence x = y by A58, A123, A125, A143; ::_thesis: verum end; supposeA144: ( x = t & y = dw ) ; ::_thesis: x = y then f . x = e by A123; hence x = y by A57, A123, A125, A144; ::_thesis: verum end; supposeA145: ( x = t & y = td ) ; ::_thesis: x = y then f . x = e by A123; hence x = y by A55, A123, A125, A145; ::_thesis: verum end; end; end; end; A146: rng f c= ck proof let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng f or y in ck ) assume y in rng f ; ::_thesis: y in ck then consider x being set such that A147: x in dom f and A148: y = f . x by FUNCT_1:def_3; reconsider x = x as Element of N_5 by A147; ( x = z or x = tj or x = dw or x = td or x = t ) by A1, ENUMSET1:def_3; then ( y = a or y = d or y = c or y = b or y = e ) by A123, A148; hence y in ck by ENUMSET1:def_3; ::_thesis: verum end; A149: dom f = the carrier of N_5 by FUNCT_2:def_1; ck c= rng f proof let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in ck or y in rng f ) assume A150: y in ck ; ::_thesis: y in rng f percases ( y = a or y = b or y = c or y = d or y = e ) by A150, ENUMSET1:def_3; supposeA151: y = a ; ::_thesis: y in rng f a = f . z by A123; hence y in rng f by A149, A151, FUNCT_1:def_3; ::_thesis: verum end; supposeA152: y = b ; ::_thesis: y in rng f b = f . td by A123; hence y in rng f by A149, A152, FUNCT_1:def_3; ::_thesis: verum end; supposeA153: y = c ; ::_thesis: y in rng f c = f . dw by A123; hence y in rng f by A149, A153, FUNCT_1:def_3; ::_thesis: verum end; supposeA154: y = d ; ::_thesis: y in rng f d = f . tj by A123; hence y in rng f by A149, A154, FUNCT_1:def_3; ::_thesis: verum end; supposeA155: y = e ; ::_thesis: y in rng f e = f . t by A123; hence y in rng f by A149, A155, FUNCT_1:def_3; ::_thesis: verum end; end; end; then A156: rng f = ck by A146, XBOOLE_0:def_10; A157: for x, y being Element of N_5 holds ( x <= y iff f . x <= f . y ) proof let x, y be Element of N_5; ::_thesis: ( x <= y iff f . x <= f . y ) thus ( x <= y implies f . x <= f . y ) ::_thesis: ( f . x <= f . y implies x <= y ) proof assume A158: x <= y ; ::_thesis: f . x <= f . y percases ( ( x = z & y = z ) or ( x = z & y = td ) or ( x = z & y = dw ) or ( x = z & y = tj ) or ( x = z & y = t ) or ( x = td & y = z ) or ( x = td & y = td ) or ( x = td & y = dw ) or ( x = td & y = z ) or ( x = td & y = tj ) or ( x = td & y = t ) or ( x = dw & y = z ) or ( x = dw & y = td ) or ( x = dw & y = dw ) or ( x = dw & y = tj ) or ( x = dw & y = t ) or ( x = tj & y = z ) or ( x = tj & y = td ) or ( x = tj & y = dw ) or ( x = tj & y = tj ) or ( x = tj & y = t ) or ( x = t & y = z ) or ( x = t & y = td ) or ( x = t & y = dw ) or ( x = t & y = tj ) or ( x = t & y = t ) ) by A1, ENUMSET1:def_3; suppose ( x = z & y = z ) ; ::_thesis: f . x <= f . y hence f . x <= f . y ; ::_thesis: verum end; supposeA159: ( x = z & y = td ) ; ::_thesis: f . x <= f . y then A160: f . y = b by A123; A161: a <= b by A59, YELLOW_0:25; f . x = a by A123, A159; hence f . x <= f . y by A160, A161, YELLOW_0:60; ::_thesis: verum end; supposeA162: ( x = z & y = dw ) ; ::_thesis: f . x <= f . y then A163: f . y = c by A123; A164: a <= c by A60, YELLOW_0:25; f . x = a by A123, A162; hence f . x <= f . y by A163, A164, YELLOW_0:60; ::_thesis: verum end; supposeA165: ( x = z & y = tj ) ; ::_thesis: f . x <= f . y A166: b <= d by A64, YELLOW_0:25; a <= b by A59, YELLOW_0:25; then A167: a <= d by A166, ORDERS_2:3; A168: f . y = d by A123, A165; f . x = a by A123, A165; hence f . x <= f . y by A168, A167, YELLOW_0:60; ::_thesis: verum end; supposeA169: ( x = z & y = t ) ; ::_thesis: f . x <= f . y A170: c <= e by A61, YELLOW_0:25; a <= c by A60, YELLOW_0:25; then A171: a <= e by A170, ORDERS_2:3; A172: f . y = e by A123, A169; f . x = a by A123, A169; hence f . x <= f . y by A172, A171, YELLOW_0:60; ::_thesis: verum end; suppose ( x = td & y = z ) ; ::_thesis: f . x <= f . y then td c= z by A158, YELLOW_1:3; hence f . x <= f . y by Th4; ::_thesis: verum end; suppose ( x = td & y = td ) ; ::_thesis: f . x <= f . y hence f . x <= f . y ; ::_thesis: verum end; supposeA173: ( x = td & y = dw ) ; ::_thesis: f . x <= f . y A174: not 2 in dw ; 2 in td by Th4, TARSKI:def_1; then not td c= dw by A174; hence f . x <= f . y by A158, A173, YELLOW_1:3; ::_thesis: verum end; suppose ( x = td & y = z ) ; ::_thesis: f . x <= f . y then td c= z by A158, YELLOW_1:3; hence f . x <= f . y by Th4; ::_thesis: verum end; supposeA175: ( x = td & y = tj ) ; ::_thesis: f . x <= f . y then A176: f . y = d by A123; A177: b <= d by A64, YELLOW_0:25; f . x = b by A123, A175; hence f . x <= f . y by A176, A177, YELLOW_0:60; ::_thesis: verum end; supposeA178: ( x = td & y = t ) ; ::_thesis: f . x <= f . y A179: d <= e by A62, YELLOW_0:25; b <= d by A64, YELLOW_0:25; then A180: b <= e by A179, ORDERS_2:3; A181: f . y = e by A123, A178; f . x = b by A123, A178; hence f . x <= f . y by A181, A180, YELLOW_0:60; ::_thesis: verum end; suppose ( x = dw & y = z ) ; ::_thesis: f . x <= f . y then dw c= z by A158, YELLOW_1:3; hence f . x <= f . y ; ::_thesis: verum end; supposeA182: ( x = dw & y = td ) ; ::_thesis: f . x <= f . y 0 in dw by CARD_1:50, TARSKI:def_2; then not dw c= td by Th4, TARSKI:def_1; hence f . x <= f . y by A158, A182, YELLOW_1:3; ::_thesis: verum end; suppose ( x = dw & y = dw ) ; ::_thesis: f . x <= f . y hence f . x <= f . y ; ::_thesis: verum end; supposeA183: ( x = dw & y = tj ) ; ::_thesis: f . x <= f . y 0 in dw by CARD_1:50, TARSKI:def_2; then not dw c= tj by Th3, TARSKI:def_2; hence f . x <= f . y by A158, A183, YELLOW_1:3; ::_thesis: verum end; supposeA184: ( x = dw & y = t ) ; ::_thesis: f . x <= f . y then A185: f . y = e by A123; A186: c <= e by A61, YELLOW_0:25; f . x = c by A123, A184; hence f . x <= f . y by A185, A186, YELLOW_0:60; ::_thesis: verum end; suppose ( x = tj & y = z ) ; ::_thesis: f . x <= f . y then tj c= z by A158, YELLOW_1:3; hence f . x <= f . y by Th3; ::_thesis: verum end; supposeA187: ( x = tj & y = td ) ; ::_thesis: f . x <= f . y 1 in tj by Th3, TARSKI:def_2; then not tj c= td by Th4, TARSKI:def_1; hence f . x <= f . y by A158, A187, YELLOW_1:3; ::_thesis: verum end; supposeA188: ( x = tj & y = dw ) ; ::_thesis: f . x <= f . y A189: not 2 in dw ; 2 in tj by Th3, TARSKI:def_2; then not tj c= dw by A189; hence f . x <= f . y by A158, A188, YELLOW_1:3; ::_thesis: verum end; suppose ( x = tj & y = tj ) ; ::_thesis: f . x <= f . y hence f . x <= f . y ; ::_thesis: verum end; supposeA190: ( x = tj & y = t ) ; ::_thesis: f . x <= f . y then A191: f . y = e by A123; A192: d <= e by A62, YELLOW_0:25; f . x = d by A123, A190; hence f . x <= f . y by A191, A192, YELLOW_0:60; ::_thesis: verum end; suppose ( x = t & y = z ) ; ::_thesis: f . x <= f . y then t c= z by A158, YELLOW_1:3; hence f . x <= f . y ; ::_thesis: verum end; supposeA193: ( x = t & y = td ) ; ::_thesis: f . x <= f . y 0 in t by CARD_1:51, ENUMSET1:def_1; then not t c= td by Th4, TARSKI:def_1; hence f . x <= f . y by A158, A193, YELLOW_1:3; ::_thesis: verum end; supposeA194: ( x = t & y = dw ) ; ::_thesis: f . x <= f . y A195: not 2 in dw ; 2 in t by CARD_1:51, ENUMSET1:def_1; then not t c= dw by A195; hence f . x <= f . y by A158, A194, YELLOW_1:3; ::_thesis: verum end; supposeA196: ( x = t & y = tj ) ; ::_thesis: f . x <= f . y 0 in t by CARD_1:51, ENUMSET1:def_1; then not t c= tj by Th3, TARSKI:def_2; hence f . x <= f . y by A158, A196, YELLOW_1:3; ::_thesis: verum end; suppose ( x = t & y = t ) ; ::_thesis: f . x <= f . y hence f . x <= f . y ; ::_thesis: verum end; end; end; thus ( f . x <= f . y implies x <= y ) ::_thesis: verum proof A197: f . y in ck by A149, A156, FUNCT_1:def_3; A198: f . x in ck by A149, A156, FUNCT_1:def_3; assume A199: f . x <= f . y ; ::_thesis: x <= y percases ( ( f . x = a & f . y = a ) or ( f . x = a & f . y = b ) or ( f . x = a & f . y = c ) or ( f . x = a & f . y = d ) or ( f . x = a & f . y = e ) or ( f . x = b & f . y = a ) or ( f . x = b & f . y = b ) or ( f . x = b & f . y = c ) or ( f . x = b & f . y = d ) or ( f . x = b & f . y = e ) or ( f . x = c & f . y = a ) or ( f . x = c & f . y = b ) or ( f . x = c & f . y = c ) or ( f . x = c & f . y = d ) or ( f . x = c & f . y = e ) or ( f . x = d & f . y = a ) or ( f . x = d & f . y = b ) or ( f . x = d & f . y = c ) or ( f . x = d & f . y = d ) or ( f . x = d & f . y = e ) or ( f . x = e & f . y = a ) or ( f . x = e & f . y = b ) or ( f . x = e & f . y = c ) or ( f . x = e & f . y = d ) or ( f . x = e & f . y = e ) ) by A198, A197, ENUMSET1:def_3; suppose ( f . x = a & f . y = a ) ; ::_thesis: x <= y hence x <= y by A124; ::_thesis: verum end; supposeA200: ( f . x = a & f . y = b ) ; ::_thesis: x <= y f . z = a by A123; then z = x by A124, A200; then x c= y by XBOOLE_1:2; hence x <= y by YELLOW_1:3; ::_thesis: verum end; supposeA201: ( f . x = a & f . y = c ) ; ::_thesis: x <= y f . z = a by A123; then z = x by A124, A201; then x c= y by XBOOLE_1:2; hence x <= y by YELLOW_1:3; ::_thesis: verum end; supposeA202: ( f . x = a & f . y = d ) ; ::_thesis: x <= y f . z = a by A123; then z = x by A124, A202; then x c= y by XBOOLE_1:2; hence x <= y by YELLOW_1:3; ::_thesis: verum end; supposeA203: ( f . x = a & f . y = e ) ; ::_thesis: x <= y f . z = a by A123; then z = x by A124, A203; then x c= y by XBOOLE_1:2; hence x <= y by YELLOW_1:3; ::_thesis: verum end; suppose ( f . x = b & f . y = a ) ; ::_thesis: x <= y then b <= a by A199, YELLOW_0:59; hence x <= y by A49, A59, YELLOW_0:25; ::_thesis: verum end; suppose ( f . x = b & f . y = b ) ; ::_thesis: x <= y hence x <= y by A124; ::_thesis: verum end; suppose ( f . x = b & f . y = c ) ; ::_thesis: x <= y then b <= c by A199, YELLOW_0:59; hence x <= y by A49, A63, YELLOW_0:25; ::_thesis: verum end; supposeA204: ( f . x = b & f . y = d ) ; ::_thesis: x <= y f . tj = d by A123; then A205: y = tj by A124, A204; f . td = b by A123; then A206: x = td by A124, A204; 1 c= 2 by NAT_1:39; then x c= y by A206, A205, XBOOLE_1:34; hence x <= y by YELLOW_1:3; ::_thesis: verum end; supposeA207: ( f . x = b & f . y = e ) ; ::_thesis: x <= y f . t = e by A123; then A208: t = y by A124, A207; f . td = b by A123; then td = x by A124, A207; hence x <= y by A208, YELLOW_1:3; ::_thesis: verum end; suppose ( f . x = c & f . y = a ) ; ::_thesis: x <= y then c <= a by A199, YELLOW_0:59; hence x <= y by A50, A60, YELLOW_0:25; ::_thesis: verum end; suppose ( f . x = c & f . y = b ) ; ::_thesis: x <= y then c <= b by A199, YELLOW_0:59; hence x <= y by A50, A63, YELLOW_0:25; ::_thesis: verum end; suppose ( f . x = c & f . y = c ) ; ::_thesis: x <= y hence x <= y by A124; ::_thesis: verum end; suppose ( f . x = c & f . y = d ) ; ::_thesis: x <= y then c <= d by A199, YELLOW_0:59; hence x <= y by A50, A65, YELLOW_0:25; ::_thesis: verum end; supposeA209: ( f . x = c & f . y = e ) ; ::_thesis: x <= y A210: dw c= t proof let X be set ; :: according to TARSKI:def_3 ::_thesis: ( not X in dw or X in t ) assume X in dw ; ::_thesis: X in t then ( X = 0 or X = 1 ) by CARD_1:50, TARSKI:def_2; hence X in t by CARD_1:51, ENUMSET1:def_1; ::_thesis: verum end; f . t = e by A123; then A211: t = y by A124, A209; f . dw = c by A123; then dw = x by A124, A209; hence x <= y by A210, A211, YELLOW_1:3; ::_thesis: verum end; supposeA212: ( f . x = d & f . y = a ) ; ::_thesis: x <= y A213: a <= b by A59, YELLOW_0:25; d <= a by A199, A212, YELLOW_0:59; then d <= b by A213, ORDERS_2:3; hence x <= y by A54, A64, YELLOW_0:25; ::_thesis: verum end; suppose ( f . x = d & f . y = b ) ; ::_thesis: x <= y then d <= b by A199, YELLOW_0:59; hence x <= y by A54, A64, YELLOW_0:25; ::_thesis: verum end; suppose ( f . x = d & f . y = c ) ; ::_thesis: x <= y then d <= c by A199, YELLOW_0:59; hence x <= y by A51, A65, YELLOW_0:25; ::_thesis: verum end; suppose ( f . x = d & f . y = d ) ; ::_thesis: x <= y hence x <= y by A124; ::_thesis: verum end; supposeA214: ( f . x = d & f . y = e ) ; ::_thesis: x <= y f . t = e by A123; then A215: t = y by A124, A214; f . tj = d by A123; then tj = x by A124, A214; hence x <= y by A215, YELLOW_1:3; ::_thesis: verum end; supposeA216: ( f . x = e & f . y = a ) ; ::_thesis: x <= y A217: b <= d by A64, YELLOW_0:25; A218: d <= e by A62, YELLOW_0:25; a <= b by A59, YELLOW_0:25; then a <= d by A217, ORDERS_2:3; then A219: a <= e by A218, ORDERS_2:3; e <= a by A199, A216, YELLOW_0:59; hence x <= y by A52, A219, ORDERS_2:2; ::_thesis: verum end; supposeA220: ( f . x = e & f . y = b ) ; ::_thesis: x <= y A221: d <= e by A62, YELLOW_0:25; b <= d by A64, YELLOW_0:25; then A222: b <= e by A221, ORDERS_2:3; e <= b by A199, A220, YELLOW_0:59; hence x <= y by A55, A222, ORDERS_2:2; ::_thesis: verum end; suppose ( f . x = e & f . y = c ) ; ::_thesis: x <= y then e <= c by A199, YELLOW_0:59; hence x <= y by A57, A61, YELLOW_0:25; ::_thesis: verum end; suppose ( f . x = e & f . y = d ) ; ::_thesis: x <= y then e <= d by A199, YELLOW_0:59; hence x <= y by A58, A62, YELLOW_0:25; ::_thesis: verum end; suppose ( f . x = e & f . y = e ) ; ::_thesis: x <= y hence x <= y by A124; ::_thesis: verum end; end; end; end; take f ; :: according to WAYBEL_1:def_8 ::_thesis: f is isomorphic f is V13() by A124, WAYBEL_1:def_1; hence f is isomorphic by A68, A156, A157, WAYBEL_0:66; ::_thesis: verum end; end; end; theorem Th10: :: YELLOW11:10 for L being LATTICE holds ( ex K being full Sublattice of L st M_3 ,K are_isomorphic iff ex a, b, c, d, e being Element of L st ( a <> b & a <> c & a <> d & a <> e & b <> c & b <> d & b <> e & c <> d & c <> e & d <> e & a "/\" b = a & a "/\" c = a & a "/\" d = a & b "/\" e = b & c "/\" e = c & d "/\" e = d & b "/\" c = a & b "/\" d = a & c "/\" d = a & b "\/" c = e & b "\/" d = e & c "\/" d = e ) ) proof set cn = the carrier of M_3; let L be LATTICE; ::_thesis: ( ex K being full Sublattice of L st M_3 ,K are_isomorphic iff ex a, b, c, d, e being Element of L st ( a <> b & a <> c & a <> d & a <> e & b <> c & b <> d & b <> e & c <> d & c <> e & d <> e & a "/\" b = a & a "/\" c = a & a "/\" d = a & b "/\" e = b & c "/\" e = c & d "/\" e = d & b "/\" c = a & b "/\" d = a & c "/\" d = a & b "\/" c = e & b "\/" d = e & c "\/" d = e ) ) A1: the carrier of M_3 = {0,1,(2 \ 1),(3 \ 2),3} by YELLOW_1:1; thus ( ex K being full Sublattice of L st M_3 ,K are_isomorphic implies ex a, b, c, d, e being Element of L st ( a <> b & a <> c & a <> d & a <> e & b <> c & b <> d & b <> e & c <> d & c <> e & d <> e & a "/\" b = a & a "/\" c = a & a "/\" d = a & b "/\" e = b & c "/\" e = c & d "/\" e = d & b "/\" c = a & b "/\" d = a & c "/\" d = a & b "\/" c = e & b "\/" d = e & c "\/" d = e ) ) ::_thesis: ( ex a, b, c, d, e being Element of L st ( a <> b & a <> c & a <> d & a <> e & b <> c & b <> d & b <> e & c <> d & c <> e & d <> e & a "/\" b = a & a "/\" c = a & a "/\" d = a & b "/\" e = b & c "/\" e = c & d "/\" e = d & b "/\" c = a & b "/\" d = a & c "/\" d = a & b "\/" c = e & b "\/" d = e & c "\/" d = e ) implies ex K being full Sublattice of L st M_3 ,K are_isomorphic ) proof reconsider td = 3 \ 2 as Element of M_3 by A1, ENUMSET1:def_3; reconsider dj = 2 \ 1 as Element of M_3 by A1, ENUMSET1:def_3; reconsider t = 3 as Element of M_3 by A1, ENUMSET1:def_3; reconsider j = 1 as Element of M_3 by A1, ENUMSET1:def_3; reconsider cl = the carrier of L as non empty set ; reconsider z = 0 as Element of M_3 by A1, ENUMSET1:def_3; given K being full Sublattice of L such that A2: M_3 ,K are_isomorphic ; ::_thesis: ex a, b, c, d, e being Element of L st ( a <> b & a <> c & a <> d & a <> e & b <> c & b <> d & b <> e & c <> d & c <> e & d <> e & a "/\" b = a & a "/\" c = a & a "/\" d = a & b "/\" e = b & c "/\" e = c & d "/\" e = d & b "/\" c = a & b "/\" d = a & c "/\" d = a & b "\/" c = e & b "\/" d = e & c "\/" d = e ) consider f being Function of M_3,K such that A3: f is isomorphic by A2, WAYBEL_1:def_8; A4: not K is empty by A3, WAYBEL_0:def_38; then A5: ( f is one-to-one & f is monotone ) by A3, WAYBEL_0:def_38; reconsider K = K as non empty SubRelStr of L by A3, WAYBEL_0:def_38; reconsider ck = the carrier of K as non empty set ; A6: ck = rng f by A3, WAYBEL_0:66; reconsider g = f as Function of the carrier of M_3,ck ; reconsider a = g . z, b = g . j, c = g . dj, d = g . td, e = g . t as Element of K ; reconsider ck = ck as non empty Subset of cl by YELLOW_0:def_13; A7: b in ck ; A8: c in ck ; A9: e in ck ; A10: d in ck ; a in ck ; then reconsider A = a, B = b, C = c, D = d, E = e as Element of L by A7, A8, A10, A9; take A ; ::_thesis: ex b, c, d, e being Element of L st ( A <> b & A <> c & A <> d & A <> e & b <> c & b <> d & b <> e & c <> d & c <> e & d <> e & A "/\" b = A & A "/\" c = A & A "/\" d = A & b "/\" e = b & c "/\" e = c & d "/\" e = d & b "/\" c = A & b "/\" d = A & c "/\" d = A & b "\/" c = e & b "\/" d = e & c "\/" d = e ) take B ; ::_thesis: ex c, d, e being Element of L st ( A <> B & A <> c & A <> d & A <> e & B <> c & B <> d & B <> e & c <> d & c <> e & d <> e & A "/\" B = A & A "/\" c = A & A "/\" d = A & B "/\" e = B & c "/\" e = c & d "/\" e = d & B "/\" c = A & B "/\" d = A & c "/\" d = A & B "\/" c = e & B "\/" d = e & c "\/" d = e ) take C ; ::_thesis: ex d, e being Element of L st ( A <> B & A <> C & A <> d & A <> e & B <> C & B <> d & B <> e & C <> d & C <> e & d <> e & A "/\" B = A & A "/\" C = A & A "/\" d = A & B "/\" e = B & C "/\" e = C & d "/\" e = d & B "/\" C = A & B "/\" d = A & C "/\" d = A & B "\/" C = e & B "\/" d = e & C "\/" d = e ) take D ; ::_thesis: ex e being Element of L st ( A <> B & A <> C & A <> D & A <> e & B <> C & B <> D & B <> e & C <> D & C <> e & D <> e & A "/\" B = A & A "/\" C = A & A "/\" D = A & B "/\" e = B & C "/\" e = C & D "/\" e = D & B "/\" C = A & B "/\" D = A & C "/\" D = A & B "\/" C = e & B "\/" D = e & C "\/" D = e ) take E ; ::_thesis: ( A <> B & A <> C & A <> D & A <> E & B <> C & B <> D & B <> E & C <> D & C <> E & D <> E & A "/\" B = A & A "/\" C = A & A "/\" D = A & B "/\" E = B & C "/\" E = C & D "/\" E = D & B "/\" C = A & B "/\" D = A & C "/\" D = A & B "\/" C = E & B "\/" D = E & C "\/" D = E ) thus A <> B by A5, WAYBEL_1:def_1; ::_thesis: ( A <> C & A <> D & A <> E & B <> C & B <> D & B <> E & C <> D & C <> E & D <> E & A "/\" B = A & A "/\" C = A & A "/\" D = A & B "/\" E = B & C "/\" E = C & D "/\" E = D & B "/\" C = A & B "/\" D = A & C "/\" D = A & B "\/" C = E & B "\/" D = E & C "\/" D = E ) thus A <> C by A5, Th2, WAYBEL_1:def_1; ::_thesis: ( A <> D & A <> E & B <> C & B <> D & B <> E & C <> D & C <> E & D <> E & A "/\" B = A & A "/\" C = A & A "/\" D = A & B "/\" E = B & C "/\" E = C & D "/\" E = D & B "/\" C = A & B "/\" D = A & C "/\" D = A & B "\/" C = E & B "\/" D = E & C "\/" D = E ) thus A <> D by A5, Th4, WAYBEL_1:def_1; ::_thesis: ( A <> E & B <> C & B <> D & B <> E & C <> D & C <> E & D <> E & A "/\" B = A & A "/\" C = A & A "/\" D = A & B "/\" E = B & C "/\" E = C & D "/\" E = D & B "/\" C = A & B "/\" D = A & C "/\" D = A & B "\/" C = E & B "\/" D = E & C "\/" D = E ) thus A <> E by A5, WAYBEL_1:def_1; ::_thesis: ( B <> C & B <> D & B <> E & C <> D & C <> E & D <> E & A "/\" B = A & A "/\" C = A & A "/\" D = A & B "/\" E = B & C "/\" E = C & D "/\" E = D & B "/\" C = A & B "/\" D = A & C "/\" D = A & B "\/" C = E & B "\/" D = E & C "\/" D = E ) now__::_thesis:_not_f_._j_=_f_._dj assume f . j = f . dj ; ::_thesis: contradiction then j = dj by A4, A5, WAYBEL_1:def_1; then 1 in 1 by Th2, TARSKI:def_1; hence contradiction ; ::_thesis: verum end; hence B <> C ; ::_thesis: ( B <> D & B <> E & C <> D & C <> E & D <> E & A "/\" B = A & A "/\" C = A & A "/\" D = A & B "/\" E = B & C "/\" E = C & D "/\" E = D & B "/\" C = A & B "/\" D = A & C "/\" D = A & B "\/" C = E & B "\/" D = E & C "\/" D = E ) now__::_thesis:_not_f_._j_=_f_._td assume f . j = f . td ; ::_thesis: contradiction then A11: j = td by A4, A5, WAYBEL_1:def_1; 0 in j by CARD_1:49, TARSKI:def_1; hence contradiction by A11, Th4, TARSKI:def_1; ::_thesis: verum end; hence B <> D ; ::_thesis: ( B <> E & C <> D & C <> E & D <> E & A "/\" B = A & A "/\" C = A & A "/\" D = A & B "/\" E = B & C "/\" E = C & D "/\" E = D & B "/\" C = A & B "/\" D = A & C "/\" D = A & B "\/" C = E & B "\/" D = E & C "\/" D = E ) thus B <> E by A5, WAYBEL_1:def_1; ::_thesis: ( C <> D & C <> E & D <> E & A "/\" B = A & A "/\" C = A & A "/\" D = A & B "/\" E = B & C "/\" E = C & D "/\" E = D & B "/\" C = A & B "/\" D = A & C "/\" D = A & B "\/" C = E & B "\/" D = E & C "\/" D = E ) now__::_thesis:_not_f_._dj_=_f_._td assume f . dj = f . td ; ::_thesis: contradiction then A12: dj = td by A4, A5, WAYBEL_1:def_1; 1 in dj by Th2, TARSKI:def_1; hence contradiction by A12, Th4, TARSKI:def_1; ::_thesis: verum end; hence C <> D ; ::_thesis: ( C <> E & D <> E & A "/\" B = A & A "/\" C = A & A "/\" D = A & B "/\" E = B & C "/\" E = C & D "/\" E = D & B "/\" C = A & B "/\" D = A & C "/\" D = A & B "\/" C = E & B "\/" D = E & C "\/" D = E ) now__::_thesis:_not_f_._dj_=_f_._t assume f . dj = f . t ; ::_thesis: contradiction then A13: dj = t by A4, A5, WAYBEL_1:def_1; 0 in t by CARD_1:51, ENUMSET1:def_1; hence contradiction by A13, Th2, TARSKI:def_1; ::_thesis: verum end; hence C <> E ; ::_thesis: ( D <> E & A "/\" B = A & A "/\" C = A & A "/\" D = A & B "/\" E = B & C "/\" E = C & D "/\" E = D & B "/\" C = A & B "/\" D = A & C "/\" D = A & B "\/" C = E & B "\/" D = E & C "\/" D = E ) now__::_thesis:_not_f_._td_=_f_._t assume f . td = f . t ; ::_thesis: contradiction then A14: td = t by A4, A5, WAYBEL_1:def_1; 0 in t by CARD_1:51, ENUMSET1:def_1; hence contradiction by A14, Th4, TARSKI:def_1; ::_thesis: verum end; hence D <> E ; ::_thesis: ( A "/\" B = A & A "/\" C = A & A "/\" D = A & B "/\" E = B & C "/\" E = C & D "/\" E = D & B "/\" C = A & B "/\" D = A & C "/\" D = A & B "\/" C = E & B "\/" D = E & C "\/" D = E ) z c= j by XBOOLE_1:2; then z <= j by YELLOW_1:3; then a <= b by A3, WAYBEL_0:66; then A <= B by YELLOW_0:59; hence A "/\" B = A by YELLOW_0:25; ::_thesis: ( A "/\" C = A & A "/\" D = A & B "/\" E = B & C "/\" E = C & D "/\" E = D & B "/\" C = A & B "/\" D = A & C "/\" D = A & B "\/" C = E & B "\/" D = E & C "\/" D = E ) z c= dj by XBOOLE_1:2; then z <= dj by YELLOW_1:3; then a <= c by A3, WAYBEL_0:66; then A <= C by YELLOW_0:59; hence A "/\" C = A by YELLOW_0:25; ::_thesis: ( A "/\" D = A & B "/\" E = B & C "/\" E = C & D "/\" E = D & B "/\" C = A & B "/\" D = A & C "/\" D = A & B "\/" C = E & B "\/" D = E & C "\/" D = E ) z c= td by XBOOLE_1:2; then z <= td by YELLOW_1:3; then a <= d by A3, WAYBEL_0:66; then A <= D by YELLOW_0:59; hence A "/\" D = A by YELLOW_0:25; ::_thesis: ( B "/\" E = B & C "/\" E = C & D "/\" E = D & B "/\" C = A & B "/\" D = A & C "/\" D = A & B "\/" C = E & B "\/" D = E & C "\/" D = E ) j c= t by NAT_1:39; then j <= t by YELLOW_1:3; then b <= e by A3, WAYBEL_0:66; then B <= E by YELLOW_0:59; hence B "/\" E = B by YELLOW_0:25; ::_thesis: ( C "/\" E = C & D "/\" E = D & B "/\" C = A & B "/\" D = A & C "/\" D = A & B "\/" C = E & B "\/" D = E & C "\/" D = E ) dj c= t proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in dj or x in t ) assume x in dj ; ::_thesis: x in t then x = 1 by Th2, TARSKI:def_1; hence x in t by CARD_1:51, ENUMSET1:def_1; ::_thesis: verum end; then dj <= t by YELLOW_1:3; then c <= e by A3, WAYBEL_0:66; then C <= E by YELLOW_0:59; hence C "/\" E = C by YELLOW_0:25; ::_thesis: ( D "/\" E = D & B "/\" C = A & B "/\" D = A & C "/\" D = A & B "\/" C = E & B "\/" D = E & C "\/" D = E ) td <= t by YELLOW_1:3; then d <= e by A3, WAYBEL_0:66; then D <= E by YELLOW_0:59; hence D "/\" E = D by YELLOW_0:25; ::_thesis: ( B "/\" C = A & B "/\" D = A & C "/\" D = A & B "\/" C = E & B "\/" D = E & C "\/" D = E ) thus B "/\" C = A ::_thesis: ( B "/\" D = A & C "/\" D = A & B "\/" C = E & B "\/" D = E & C "\/" D = E ) proof A15: now__::_thesis:_not_B_"/\"_C_=_D assume B "/\" C = D ; ::_thesis: contradiction then D <= C by YELLOW_0:23; then d <= c by YELLOW_0:60; then td <= dj by A3, WAYBEL_0:66; then A16: td c= dj by YELLOW_1:3; 2 in td by Th4, TARSKI:def_1; hence contradiction by A16, Th2, TARSKI:def_1; ::_thesis: verum end; A17: now__::_thesis:_not_B_"/\"_C_=_B assume B "/\" C = B ; ::_thesis: contradiction then B <= C by YELLOW_0:25; then b <= c by YELLOW_0:60; then j <= dj by A3, WAYBEL_0:66; then A18: j c= dj by YELLOW_1:3; 0 in j by CARD_1:49, TARSKI:def_1; hence contradiction by A18, Th2, TARSKI:def_1; ::_thesis: verum end; A19: now__::_thesis:_not_B_"/\"_C_=_E assume B "/\" C = E ; ::_thesis: contradiction then E <= C by YELLOW_0:23; then e <= c by YELLOW_0:60; then t <= dj by A3, WAYBEL_0:66; then A20: t c= dj by YELLOW_1:3; 2 in t by CARD_1:51, ENUMSET1:def_1; hence contradiction by A20, Th2, TARSKI:def_1; ::_thesis: verum end; A21: now__::_thesis:_not_B_"/\"_C_=_C assume B "/\" C = C ; ::_thesis: contradiction then C <= B by YELLOW_0:25; then c <= b by YELLOW_0:60; then dj <= j by A3, WAYBEL_0:66; then A22: dj c= j by YELLOW_1:3; 1 in dj by Th2, TARSKI:def_1; hence contradiction by A22, CARD_1:49, TARSKI:def_1; ::_thesis: verum end; ex_inf_of {B,C},L by YELLOW_0:21; then inf {B,C} in the carrier of K by YELLOW_0:def_16; then B "/\" C in rng f by A6, YELLOW_0:40; then ex x being set st ( x in dom f & B "/\" C = f . x ) by FUNCT_1:def_3; hence B "/\" C = A by A1, A17, A21, A15, A19, ENUMSET1:def_3; ::_thesis: verum end; thus B "/\" D = A ::_thesis: ( C "/\" D = A & B "\/" C = E & B "\/" D = E & C "\/" D = E ) proof A23: now__::_thesis:_not_B_"/\"_D_=_D assume B "/\" D = D ; ::_thesis: contradiction then D <= B by YELLOW_0:23; then d <= b by YELLOW_0:60; then td <= j by A3, WAYBEL_0:66; then A24: td c= j by YELLOW_1:3; 2 in td by Th4, TARSKI:def_1; hence contradiction by A24, CARD_1:49, TARSKI:def_1; ::_thesis: verum end; A25: now__::_thesis:_not_B_"/\"_D_=_C assume B "/\" D = C ; ::_thesis: contradiction then C <= B by YELLOW_0:23; then c <= b by YELLOW_0:60; then dj <= j by A3, WAYBEL_0:66; then A26: dj c= j by YELLOW_1:3; 1 in dj by Th2, TARSKI:def_1; hence contradiction by A26, CARD_1:49, TARSKI:def_1; ::_thesis: verum end; A27: now__::_thesis:_not_B_"/\"_D_=_B assume B "/\" D = B ; ::_thesis: contradiction then B <= D by YELLOW_0:25; then b <= d by YELLOW_0:60; then j <= td by A3, WAYBEL_0:66; then A28: j c= td by YELLOW_1:3; 0 in j by CARD_1:49, TARSKI:def_1; hence contradiction by A28, Th4, TARSKI:def_1; ::_thesis: verum end; A29: now__::_thesis:_not_B_"/\"_D_=_E assume B "/\" D = E ; ::_thesis: contradiction then E <= B by YELLOW_0:23; then e <= b by YELLOW_0:60; then t <= j by A3, WAYBEL_0:66; then A30: t c= j by YELLOW_1:3; 2 in t by CARD_1:51, ENUMSET1:def_1; hence contradiction by A30, CARD_1:49, TARSKI:def_1; ::_thesis: verum end; ex_inf_of {B,D},L by YELLOW_0:21; then inf {B,D} in the carrier of K by YELLOW_0:def_16; then B "/\" D in rng f by A6, YELLOW_0:40; then ex x being set st ( x in dom f & B "/\" D = f . x ) by FUNCT_1:def_3; hence B "/\" D = A by A1, A27, A25, A23, A29, ENUMSET1:def_3; ::_thesis: verum end; thus C "/\" D = A ::_thesis: ( B "\/" C = E & B "\/" D = E & C "\/" D = E ) proof A31: now__::_thesis:_not_C_"/\"_D_=_D assume C "/\" D = D ; ::_thesis: contradiction then D <= C by YELLOW_0:23; then d <= c by YELLOW_0:60; then td <= dj by A3, WAYBEL_0:66; then A32: td c= dj by YELLOW_1:3; 2 in td by Th4, TARSKI:def_1; hence contradiction by A32, Th2, TARSKI:def_1; ::_thesis: verum end; A33: now__::_thesis:_not_C_"/\"_D_=_E assume C "/\" D = E ; ::_thesis: contradiction then E <= C by YELLOW_0:23; then e <= c by YELLOW_0:60; then t <= dj by A3, WAYBEL_0:66; then A34: t c= dj by YELLOW_1:3; 2 in t by CARD_1:51, ENUMSET1:def_1; hence contradiction by A34, Th2, TARSKI:def_1; ::_thesis: verum end; A35: now__::_thesis:_not_C_"/\"_D_=_C assume C "/\" D = C ; ::_thesis: contradiction then C <= D by YELLOW_0:25; then c <= d by YELLOW_0:60; then dj <= td by A3, WAYBEL_0:66; then A36: dj c= td by YELLOW_1:3; 1 in dj by Th2, TARSKI:def_1; hence contradiction by A36, Th4, TARSKI:def_1; ::_thesis: verum end; A37: now__::_thesis:_not_C_"/\"_D_=_B assume C "/\" D = B ; ::_thesis: contradiction then B <= C by YELLOW_0:23; then b <= c by YELLOW_0:60; then j <= dj by A3, WAYBEL_0:66; then A38: j c= dj by YELLOW_1:3; 0 in j by CARD_1:49, TARSKI:def_1; hence contradiction by A38, Th2, TARSKI:def_1; ::_thesis: verum end; ex_inf_of {C,D},L by YELLOW_0:21; then inf {C,D} in the carrier of K by YELLOW_0:def_16; then C "/\" D in rng f by A6, YELLOW_0:40; then ex x being set st ( x in dom f & C "/\" D = f . x ) by FUNCT_1:def_3; hence C "/\" D = A by A1, A37, A35, A31, A33, ENUMSET1:def_3; ::_thesis: verum end; thus B "\/" C = E ::_thesis: ( B "\/" D = E & C "\/" D = E ) proof A39: now__::_thesis:_not_B_"\/"_C_=_C assume B "\/" C = C ; ::_thesis: contradiction then C >= B by YELLOW_0:24; then c >= b by YELLOW_0:60; then dj >= j by A3, WAYBEL_0:66; then A40: j c= dj by YELLOW_1:3; 0 in j by CARD_1:49, TARSKI:def_1; hence contradiction by A40, Th2, TARSKI:def_1; ::_thesis: verum end; A41: now__::_thesis:_not_B_"\/"_C_=_B assume B "\/" C = B ; ::_thesis: contradiction then B >= C by YELLOW_0:24; then b >= c by YELLOW_0:60; then j >= dj by A3, WAYBEL_0:66; then A42: dj c= j by YELLOW_1:3; 1 in dj by Th2, TARSKI:def_1; hence contradiction by A42, CARD_1:49, TARSKI:def_1; ::_thesis: verum end; A43: now__::_thesis:_not_B_"\/"_C_=_D assume B "\/" C = D ; ::_thesis: contradiction then D >= C by YELLOW_0:22; then d >= c by YELLOW_0:60; then td >= dj by A3, WAYBEL_0:66; then A44: dj c= td by YELLOW_1:3; 1 in dj by Th2, TARSKI:def_1; hence contradiction by A44, Th4, TARSKI:def_1; ::_thesis: verum end; A45: now__::_thesis:_not_B_"\/"_C_=_A assume B "\/" C = A ; ::_thesis: contradiction then A >= C by YELLOW_0:22; then a >= c by YELLOW_0:60; then z >= dj by A3, WAYBEL_0:66; then dj c= z by YELLOW_1:3; hence contradiction by Th2; ::_thesis: verum end; ex_sup_of {B,C},L by YELLOW_0:20; then sup {B,C} in the carrier of K by YELLOW_0:def_17; then B "\/" C in rng f by A6, YELLOW_0:41; then ex x being set st ( x in dom f & B "\/" C = f . x ) by FUNCT_1:def_3; hence B "\/" C = E by A1, A41, A39, A43, A45, ENUMSET1:def_3; ::_thesis: verum end; thus B "\/" D = E ::_thesis: C "\/" D = E proof A46: now__::_thesis:_not_B_"\/"_D_=_D assume B "\/" D = D ; ::_thesis: contradiction then D >= B by YELLOW_0:22; then d >= b by YELLOW_0:60; then td >= j by A3, WAYBEL_0:66; then A47: j c= td by YELLOW_1:3; 0 in j by CARD_1:49, TARSKI:def_1; hence contradiction by A47, Th4, TARSKI:def_1; ::_thesis: verum end; A48: now__::_thesis:_not_B_"\/"_D_=_B assume B "\/" D = B ; ::_thesis: contradiction then B >= D by YELLOW_0:22; then b >= d by YELLOW_0:60; then j >= td by A3, WAYBEL_0:66; then A49: td c= j by YELLOW_1:3; 2 in td by Th4, TARSKI:def_1; hence contradiction by A49, CARD_1:49, TARSKI:def_1; ::_thesis: verum end; A50: now__::_thesis:_not_B_"\/"_D_=_C assume B "\/" D = C ; ::_thesis: contradiction then C >= D by YELLOW_0:22; then c >= d by YELLOW_0:60; then dj >= td by A3, WAYBEL_0:66; then A51: td c= dj by YELLOW_1:3; 2 in td by Th4, TARSKI:def_1; hence contradiction by A51, Th2, TARSKI:def_1; ::_thesis: verum end; A52: now__::_thesis:_not_B_"\/"_D_=_A assume B "\/" D = A ; ::_thesis: contradiction then A >= B by YELLOW_0:22; then a >= b by YELLOW_0:60; then z >= j by A3, WAYBEL_0:66; then j c= z by YELLOW_1:3; hence contradiction ; ::_thesis: verum end; ex_sup_of {B,D},L by YELLOW_0:20; then sup {B,D} in the carrier of K by YELLOW_0:def_17; then B "\/" D in rng f by A6, YELLOW_0:41; then ex x being set st ( x in dom f & B "\/" D = f . x ) by FUNCT_1:def_3; hence B "\/" D = E by A1, A48, A50, A46, A52, ENUMSET1:def_3; ::_thesis: verum end; thus C "\/" D = E ::_thesis: verum proof A53: now__::_thesis:_not_C_"\/"_D_=_B assume C "\/" D = B ; ::_thesis: contradiction then B >= C by YELLOW_0:22; then b >= c by YELLOW_0:60; then j >= dj by A3, WAYBEL_0:66; then A54: dj c= j by YELLOW_1:3; 1 in dj by Th2, TARSKI:def_1; then 1 in 1 by A54; hence contradiction ; ::_thesis: verum end; A55: now__::_thesis:_not_C_"\/"_D_=_D assume C "\/" D = D ; ::_thesis: contradiction then D >= C by YELLOW_0:22; then d >= c by YELLOW_0:60; then td >= dj by A3, WAYBEL_0:66; then A56: dj c= td by YELLOW_1:3; 1 in dj by Th2, TARSKI:def_1; hence contradiction by A56, Th4, TARSKI:def_1; ::_thesis: verum end; A57: now__::_thesis:_not_C_"\/"_D_=_C assume C "\/" D = C ; ::_thesis: contradiction then C >= D by YELLOW_0:24; then c >= d by YELLOW_0:60; then dj >= td by A3, WAYBEL_0:66; then A58: td c= dj by YELLOW_1:3; 2 in td by Th4, TARSKI:def_1; hence contradiction by A58, Th2, TARSKI:def_1; ::_thesis: verum end; A59: now__::_thesis:_not_C_"\/"_D_=_A assume C "\/" D = A ; ::_thesis: contradiction then A >= C by YELLOW_0:22; then a >= c by YELLOW_0:60; then z >= dj by A3, WAYBEL_0:66; then dj c= z by YELLOW_1:3; hence contradiction by Th2; ::_thesis: verum end; ex_sup_of {C,D},L by YELLOW_0:20; then sup {C,D} in the carrier of K by YELLOW_0:def_17; then C "\/" D in rng f by A6, YELLOW_0:41; then ex x being set st ( x in dom f & C "\/" D = f . x ) by FUNCT_1:def_3; hence C "\/" D = E by A1, A53, A57, A55, A59, ENUMSET1:def_3; ::_thesis: verum end; end; thus ( ex a, b, c, d, e being Element of L st ( a <> b & a <> c & a <> d & a <> e & b <> c & b <> d & b <> e & c <> d & c <> e & d <> e & a "/\" b = a & a "/\" c = a & a "/\" d = a & b "/\" e = b & c "/\" e = c & d "/\" e = d & b "/\" c = a & b "/\" d = a & c "/\" d = a & b "\/" c = e & b "\/" d = e & c "\/" d = e ) implies ex K being full Sublattice of L st M_3 ,K are_isomorphic ) ::_thesis: verum proof given a, b, c, d, e being Element of L such that A60: a <> b and A61: a <> c and A62: a <> d and A63: a <> e and A64: b <> c and A65: b <> d and A66: b <> e and A67: c <> d and A68: c <> e and A69: d <> e and A70: a "/\" b = a and A71: a "/\" c = a and A72: a "/\" d = a and A73: b "/\" e = b and A74: c "/\" e = c and A75: d "/\" e = d and A76: b "/\" c = a and A77: b "/\" d = a and A78: c "/\" d = a and A79: b "\/" c = e and A80: b "\/" d = e and A81: c "\/" d = e ; ::_thesis: ex K being full Sublattice of L st M_3 ,K are_isomorphic set ck = {a,b,c,d,e}; reconsider ck = {a,b,c,d,e} as Subset of L ; set K = subrelstr ck; A82: the carrier of (subrelstr ck) = ck by YELLOW_0:def_15; A83: subrelstr ck is meet-inheriting proof let x, y be Element of L; :: according to YELLOW_0:def_16 ::_thesis: ( not x in the carrier of (subrelstr ck) or not y in the carrier of (subrelstr ck) or not ex_inf_of {x,y},L or "/\" ({x,y},L) in the carrier of (subrelstr ck) ) assume that A84: x in the carrier of (subrelstr ck) and A85: y in the carrier of (subrelstr ck) and ex_inf_of {x,y},L ; ::_thesis: "/\" ({x,y},L) in the carrier of (subrelstr ck) percases ( ( x = a & y = a ) or ( x = a & y = b ) or ( x = a & y = c ) or ( x = a & y = d ) or ( x = a & y = e ) or ( x = b & y = a ) or ( x = b & y = b ) or ( x = b & y = c ) or ( x = b & y = d ) or ( x = b & y = e ) or ( x = c & y = a ) or ( x = c & y = b ) or ( x = c & y = c ) or ( x = c & y = d ) or ( x = c & y = e ) or ( x = d & y = a ) or ( x = d & y = b ) or ( x = d & y = c ) or ( x = d & y = d ) or ( x = d & y = e ) or ( x = e & y = a ) or ( x = e & y = b ) or ( x = e & y = c ) or ( x = e & y = d ) or ( x = e & y = e ) ) by A82, A84, A85, ENUMSET1:def_3; suppose ( x = a & y = a ) ; ::_thesis: "/\" ({x,y},L) in the carrier of (subrelstr ck) then inf {x,y} = a "/\" a by YELLOW_0:40; then inf {x,y} = a by YELLOW_5:2; hence "/\" ({x,y},L) in the carrier of (subrelstr ck) by A82, ENUMSET1:def_3; ::_thesis: verum end; suppose ( x = a & y = b ) ; ::_thesis: "/\" ({x,y},L) in the carrier of (subrelstr ck) then inf {x,y} = a "/\" b by YELLOW_0:40; hence "/\" ({x,y},L) in the carrier of (subrelstr ck) by A70, A82, ENUMSET1:def_3; ::_thesis: verum end; suppose ( x = a & y = c ) ; ::_thesis: "/\" ({x,y},L) in the carrier of (subrelstr ck) then inf {x,y} = a "/\" c by YELLOW_0:40; hence "/\" ({x,y},L) in the carrier of (subrelstr ck) by A71, A82, ENUMSET1:def_3; ::_thesis: verum end; suppose ( x = a & y = d ) ; ::_thesis: "/\" ({x,y},L) in the carrier of (subrelstr ck) then inf {x,y} = a "/\" d by YELLOW_0:40; hence "/\" ({x,y},L) in the carrier of (subrelstr ck) by A72, A82, ENUMSET1:def_3; ::_thesis: verum end; supposeA86: ( x = a & y = e ) ; ::_thesis: "/\" ({x,y},L) in the carrier of (subrelstr ck) A87: c <= e by A74, YELLOW_0:25; a <= c by A71, YELLOW_0:25; then a <= e by A87, ORDERS_2:3; then a "/\" e = a by YELLOW_0:25; then inf {x,y} = a by A86, YELLOW_0:40; hence "/\" ({x,y},L) in the carrier of (subrelstr ck) by A82, ENUMSET1:def_3; ::_thesis: verum end; suppose ( x = b & y = a ) ; ::_thesis: "/\" ({x,y},L) in the carrier of (subrelstr ck) then inf {x,y} = a "/\" b by YELLOW_0:40; hence "/\" ({x,y},L) in the carrier of (subrelstr ck) by A70, A82, ENUMSET1:def_3; ::_thesis: verum end; suppose ( x = b & y = b ) ; ::_thesis: "/\" ({x,y},L) in the carrier of (subrelstr ck) then inf {x,y} = b "/\" b by YELLOW_0:40; then inf {x,y} = b by YELLOW_5:2; hence "/\" ({x,y},L) in the carrier of (subrelstr ck) by A82, ENUMSET1:def_3; ::_thesis: verum end; suppose ( x = b & y = c ) ; ::_thesis: "/\" ({x,y},L) in the carrier of (subrelstr ck) then inf {x,y} = b "/\" c by YELLOW_0:40; hence "/\" ({x,y},L) in the carrier of (subrelstr ck) by A76, A82, ENUMSET1:def_3; ::_thesis: verum end; suppose ( x = b & y = d ) ; ::_thesis: "/\" ({x,y},L) in the carrier of (subrelstr ck) then inf {x,y} = b "/\" d by YELLOW_0:40; hence "/\" ({x,y},L) in the carrier of (subrelstr ck) by A77, A82, ENUMSET1:def_3; ::_thesis: verum end; suppose ( x = b & y = e ) ; ::_thesis: "/\" ({x,y},L) in the carrier of (subrelstr ck) then inf {x,y} = b "/\" e by YELLOW_0:40; hence "/\" ({x,y},L) in the carrier of (subrelstr ck) by A73, A82, ENUMSET1:def_3; ::_thesis: verum end; suppose ( x = c & y = a ) ; ::_thesis: "/\" ({x,y},L) in the carrier of (subrelstr ck) then inf {x,y} = a "/\" c by YELLOW_0:40; hence "/\" ({x,y},L) in the carrier of (subrelstr ck) by A71, A82, ENUMSET1:def_3; ::_thesis: verum end; suppose ( x = c & y = b ) ; ::_thesis: "/\" ({x,y},L) in the carrier of (subrelstr ck) then inf {x,y} = b "/\" c by YELLOW_0:40; hence "/\" ({x,y},L) in the carrier of (subrelstr ck) by A76, A82, ENUMSET1:def_3; ::_thesis: verum end; suppose ( x = c & y = c ) ; ::_thesis: "/\" ({x,y},L) in the carrier of (subrelstr ck) then inf {x,y} = c "/\" c by YELLOW_0:40; then inf {x,y} = c by YELLOW_5:2; hence "/\" ({x,y},L) in the carrier of (subrelstr ck) by A82, ENUMSET1:def_3; ::_thesis: verum end; suppose ( x = c & y = d ) ; ::_thesis: "/\" ({x,y},L) in the carrier of (subrelstr ck) then inf {x,y} = c "/\" d by YELLOW_0:40; hence "/\" ({x,y},L) in the carrier of (subrelstr ck) by A78, A82, ENUMSET1:def_3; ::_thesis: verum end; suppose ( x = c & y = e ) ; ::_thesis: "/\" ({x,y},L) in the carrier of (subrelstr ck) then inf {x,y} = c "/\" e by YELLOW_0:40; hence "/\" ({x,y},L) in the carrier of (subrelstr ck) by A74, A82, ENUMSET1:def_3; ::_thesis: verum end; suppose ( x = d & y = a ) ; ::_thesis: "/\" ({x,y},L) in the carrier of (subrelstr ck) then inf {x,y} = a "/\" d by YELLOW_0:40; hence "/\" ({x,y},L) in the carrier of (subrelstr ck) by A72, A82, ENUMSET1:def_3; ::_thesis: verum end; suppose ( x = d & y = b ) ; ::_thesis: "/\" ({x,y},L) in the carrier of (subrelstr ck) then inf {x,y} = b "/\" d by YELLOW_0:40; hence "/\" ({x,y},L) in the carrier of (subrelstr ck) by A77, A82, ENUMSET1:def_3; ::_thesis: verum end; suppose ( x = d & y = c ) ; ::_thesis: "/\" ({x,y},L) in the carrier of (subrelstr ck) then inf {x,y} = c "/\" d by YELLOW_0:40; hence "/\" ({x,y},L) in the carrier of (subrelstr ck) by A78, A82, ENUMSET1:def_3; ::_thesis: verum end; suppose ( x = d & y = d ) ; ::_thesis: "/\" ({x,y},L) in the carrier of (subrelstr ck) then inf {x,y} = d "/\" d by YELLOW_0:40; then inf {x,y} = d by YELLOW_5:2; hence "/\" ({x,y},L) in the carrier of (subrelstr ck) by A82, ENUMSET1:def_3; ::_thesis: verum end; suppose ( x = d & y = e ) ; ::_thesis: "/\" ({x,y},L) in the carrier of (subrelstr ck) then inf {x,y} = d "/\" e by YELLOW_0:40; hence "/\" ({x,y},L) in the carrier of (subrelstr ck) by A75, A82, ENUMSET1:def_3; ::_thesis: verum end; supposeA88: ( x = e & y = a ) ; ::_thesis: "/\" ({x,y},L) in the carrier of (subrelstr ck) A89: c <= e by A74, YELLOW_0:25; a <= c by A71, YELLOW_0:25; then a <= e by A89, ORDERS_2:3; then a "/\" e = a by YELLOW_0:25; then inf {x,y} = a by A88, YELLOW_0:40; hence "/\" ({x,y},L) in the carrier of (subrelstr ck) by A82, ENUMSET1:def_3; ::_thesis: verum end; suppose ( x = e & y = b ) ; ::_thesis: "/\" ({x,y},L) in the carrier of (subrelstr ck) then inf {x,y} = b "/\" e by YELLOW_0:40; hence "/\" ({x,y},L) in the carrier of (subrelstr ck) by A73, A82, ENUMSET1:def_3; ::_thesis: verum end; suppose ( x = e & y = c ) ; ::_thesis: "/\" ({x,y},L) in the carrier of (subrelstr ck) then inf {x,y} = c "/\" e by YELLOW_0:40; hence "/\" ({x,y},L) in the carrier of (subrelstr ck) by A74, A82, ENUMSET1:def_3; ::_thesis: verum end; suppose ( x = e & y = d ) ; ::_thesis: "/\" ({x,y},L) in the carrier of (subrelstr ck) then inf {x,y} = d "/\" e by YELLOW_0:40; hence "/\" ({x,y},L) in the carrier of (subrelstr ck) by A75, A82, ENUMSET1:def_3; ::_thesis: verum end; suppose ( x = e & y = e ) ; ::_thesis: "/\" ({x,y},L) in the carrier of (subrelstr ck) then inf {x,y} = e "/\" e by YELLOW_0:40; then inf {x,y} = e by YELLOW_5:2; hence "/\" ({x,y},L) in the carrier of (subrelstr ck) by A82, ENUMSET1:def_3; ::_thesis: verum end; end; end; subrelstr ck is join-inheriting proof let x, y be Element of L; :: according to YELLOW_0:def_17 ::_thesis: ( not x in the carrier of (subrelstr ck) or not y in the carrier of (subrelstr ck) or not ex_sup_of {x,y},L or "\/" ({x,y},L) in the carrier of (subrelstr ck) ) assume that A90: x in the carrier of (subrelstr ck) and A91: y in the carrier of (subrelstr ck) and ex_sup_of {x,y},L ; ::_thesis: "\/" ({x,y},L) in the carrier of (subrelstr ck) percases ( ( x = a & y = a ) or ( x = a & y = b ) or ( x = a & y = c ) or ( x = a & y = d ) or ( x = a & y = e ) or ( x = b & y = a ) or ( x = b & y = b ) or ( x = b & y = c ) or ( x = b & y = d ) or ( x = b & y = e ) or ( x = c & y = a ) or ( x = c & y = b ) or ( x = c & y = c ) or ( x = c & y = d ) or ( x = c & y = e ) or ( x = d & y = a ) or ( x = d & y = b ) or ( x = d & y = c ) or ( x = d & y = d ) or ( x = d & y = e ) or ( x = e & y = a ) or ( x = e & y = b ) or ( x = e & y = c ) or ( x = e & y = d ) or ( x = e & y = e ) ) by A82, A90, A91, ENUMSET1:def_3; suppose ( x = a & y = a ) ; ::_thesis: "\/" ({x,y},L) in the carrier of (subrelstr ck) then sup {x,y} = a "\/" a by YELLOW_0:41; then sup {x,y} = a by YELLOW_5:1; hence "\/" ({x,y},L) in the carrier of (subrelstr ck) by A82, ENUMSET1:def_3; ::_thesis: verum end; suppose ( x = a & y = b ) ; ::_thesis: "\/" ({x,y},L) in the carrier of (subrelstr ck) then x "\/" y = b by A70, Th5; then sup {x,y} = b by YELLOW_0:41; hence "\/" ({x,y},L) in the carrier of (subrelstr ck) by A82, ENUMSET1:def_3; ::_thesis: verum end; suppose ( x = a & y = c ) ; ::_thesis: "\/" ({x,y},L) in the carrier of (subrelstr ck) then x "\/" y = c by A71, Th5; then sup {x,y} = c by YELLOW_0:41; hence "\/" ({x,y},L) in the carrier of (subrelstr ck) by A82, ENUMSET1:def_3; ::_thesis: verum end; suppose ( x = a & y = d ) ; ::_thesis: "\/" ({x,y},L) in the carrier of (subrelstr ck) then x "\/" y = d by A72, Th5; then sup {x,y} = d by YELLOW_0:41; hence "\/" ({x,y},L) in the carrier of (subrelstr ck) by A82, ENUMSET1:def_3; ::_thesis: verum end; supposeA92: ( x = a & y = e ) ; ::_thesis: "\/" ({x,y},L) in the carrier of (subrelstr ck) A93: c <= e by A74, YELLOW_0:25; a <= c by A71, YELLOW_0:25; then a <= e by A93, ORDERS_2:3; then a "/\" e = a by YELLOW_0:25; then a "\/" e = e by Th5; then sup {x,y} = e by A92, YELLOW_0:41; hence "\/" ({x,y},L) in the carrier of (subrelstr ck) by A82, ENUMSET1:def_3; ::_thesis: verum end; supposeA94: ( x = b & y = a ) ; ::_thesis: "\/" ({x,y},L) in the carrier of (subrelstr ck) a "\/" b = b by A70, Th5; then sup {x,y} = b by A94, YELLOW_0:41; hence "\/" ({x,y},L) in the carrier of (subrelstr ck) by A82, ENUMSET1:def_3; ::_thesis: verum end; suppose ( x = b & y = b ) ; ::_thesis: "\/" ({x,y},L) in the carrier of (subrelstr ck) then sup {x,y} = b "\/" b by YELLOW_0:41; then sup {x,y} = b by YELLOW_5:1; hence "\/" ({x,y},L) in the carrier of (subrelstr ck) by A82, ENUMSET1:def_3; ::_thesis: verum end; suppose ( x = b & y = c ) ; ::_thesis: "\/" ({x,y},L) in the carrier of (subrelstr ck) then sup {x,y} = b "\/" c by YELLOW_0:41; hence "\/" ({x,y},L) in the carrier of (subrelstr ck) by A79, A82, ENUMSET1:def_3; ::_thesis: verum end; suppose ( x = b & y = d ) ; ::_thesis: "\/" ({x,y},L) in the carrier of (subrelstr ck) then sup {x,y} = b "\/" d by YELLOW_0:41; hence "\/" ({x,y},L) in the carrier of (subrelstr ck) by A80, A82, ENUMSET1:def_3; ::_thesis: verum end; supposeA95: ( x = b & y = e ) ; ::_thesis: "\/" ({x,y},L) in the carrier of (subrelstr ck) b "\/" e = e by A73, Th5; then sup {x,y} = e by A95, YELLOW_0:41; hence "\/" ({x,y},L) in the carrier of (subrelstr ck) by A82, ENUMSET1:def_3; ::_thesis: verum end; supposeA96: ( x = c & y = a ) ; ::_thesis: "\/" ({x,y},L) in the carrier of (subrelstr ck) c "\/" a = c by A71, Th5; then sup {x,y} = c by A96, YELLOW_0:41; hence "\/" ({x,y},L) in the carrier of (subrelstr ck) by A82, ENUMSET1:def_3; ::_thesis: verum end; suppose ( x = c & y = b ) ; ::_thesis: "\/" ({x,y},L) in the carrier of (subrelstr ck) then sup {x,y} = b "\/" c by YELLOW_0:41; hence "\/" ({x,y},L) in the carrier of (subrelstr ck) by A79, A82, ENUMSET1:def_3; ::_thesis: verum end; suppose ( x = c & y = c ) ; ::_thesis: "\/" ({x,y},L) in the carrier of (subrelstr ck) then sup {x,y} = c "\/" c by YELLOW_0:41; then sup {x,y} = c by YELLOW_5:1; hence "\/" ({x,y},L) in the carrier of (subrelstr ck) by A82, ENUMSET1:def_3; ::_thesis: verum end; suppose ( x = c & y = d ) ; ::_thesis: "\/" ({x,y},L) in the carrier of (subrelstr ck) then sup {x,y} = c "\/" d by YELLOW_0:41; hence "\/" ({x,y},L) in the carrier of (subrelstr ck) by A81, A82, ENUMSET1:def_3; ::_thesis: verum end; supposeA97: ( x = c & y = e ) ; ::_thesis: "\/" ({x,y},L) in the carrier of (subrelstr ck) c "\/" e = e by A74, Th5; then sup {x,y} = e by A97, YELLOW_0:41; hence "\/" ({x,y},L) in the carrier of (subrelstr ck) by A82, ENUMSET1:def_3; ::_thesis: verum end; suppose ( x = d & y = a ) ; ::_thesis: "\/" ({x,y},L) in the carrier of (subrelstr ck) then x "\/" y = d by A72, Th5; then sup {x,y} = d by YELLOW_0:41; hence "\/" ({x,y},L) in the carrier of (subrelstr ck) by A82, ENUMSET1:def_3; ::_thesis: verum end; suppose ( x = d & y = b ) ; ::_thesis: "\/" ({x,y},L) in the carrier of (subrelstr ck) then sup {x,y} = b "\/" d by YELLOW_0:41; hence "\/" ({x,y},L) in the carrier of (subrelstr ck) by A80, A82, ENUMSET1:def_3; ::_thesis: verum end; suppose ( x = d & y = c ) ; ::_thesis: "\/" ({x,y},L) in the carrier of (subrelstr ck) then sup {x,y} = c "\/" d by YELLOW_0:41; hence "\/" ({x,y},L) in the carrier of (subrelstr ck) by A81, A82, ENUMSET1:def_3; ::_thesis: verum end; suppose ( x = d & y = d ) ; ::_thesis: "\/" ({x,y},L) in the carrier of (subrelstr ck) then sup {x,y} = d "\/" d by YELLOW_0:41; then sup {x,y} = d by YELLOW_5:1; hence "\/" ({x,y},L) in the carrier of (subrelstr ck) by A82, ENUMSET1:def_3; ::_thesis: verum end; supposeA98: ( x = d & y = e ) ; ::_thesis: "\/" ({x,y},L) in the carrier of (subrelstr ck) d "\/" e = e by A75, Th5; then sup {x,y} = e by A98, YELLOW_0:41; hence "\/" ({x,y},L) in the carrier of (subrelstr ck) by A82, ENUMSET1:def_3; ::_thesis: verum end; supposeA99: ( x = e & y = a ) ; ::_thesis: "\/" ({x,y},L) in the carrier of (subrelstr ck) A100: c <= e by A74, YELLOW_0:25; a <= c by A71, YELLOW_0:25; then a <= e by A100, ORDERS_2:3; then a "/\" e = a by YELLOW_0:25; then a "\/" e = e by Th5; then sup {x,y} = e by A99, YELLOW_0:41; hence "\/" ({x,y},L) in the carrier of (subrelstr ck) by A82, ENUMSET1:def_3; ::_thesis: verum end; supposeA101: ( x = e & y = b ) ; ::_thesis: "\/" ({x,y},L) in the carrier of (subrelstr ck) b "\/" e = e by A73, Th5; then sup {x,y} = e by A101, YELLOW_0:41; hence "\/" ({x,y},L) in the carrier of (subrelstr ck) by A82, ENUMSET1:def_3; ::_thesis: verum end; supposeA102: ( x = e & y = c ) ; ::_thesis: "\/" ({x,y},L) in the carrier of (subrelstr ck) c "\/" e = e by A74, Th5; then sup {x,y} = e by A102, YELLOW_0:41; hence "\/" ({x,y},L) in the carrier of (subrelstr ck) by A82, ENUMSET1:def_3; ::_thesis: verum end; supposeA103: ( x = e & y = d ) ; ::_thesis: "\/" ({x,y},L) in the carrier of (subrelstr ck) d "\/" e = e by A75, Th5; then sup {x,y} = e by A103, YELLOW_0:41; hence "\/" ({x,y},L) in the carrier of (subrelstr ck) by A82, ENUMSET1:def_3; ::_thesis: verum end; suppose ( x = e & y = e ) ; ::_thesis: "\/" ({x,y},L) in the carrier of (subrelstr ck) then sup {x,y} = e "\/" e by YELLOW_0:41; then sup {x,y} = e by YELLOW_5:1; hence "\/" ({x,y},L) in the carrier of (subrelstr ck) by A82, ENUMSET1:def_3; ::_thesis: verum end; end; end; then reconsider K = subrelstr ck as non empty full Sublattice of L by A83, YELLOW_0:def_15; take K ; ::_thesis: M_3 ,K are_isomorphic thus M_3 ,K are_isomorphic ::_thesis: verum proof reconsider t = 3 as Element of M_3 by A1, ENUMSET1:def_3; reconsider td = 3 \ 2 as Element of M_3 by A1, ENUMSET1:def_3; reconsider dj = 2 \ 1 as Element of M_3 by A1, ENUMSET1:def_3; A104: now__::_thesis:_not_dj_=_t A105: 2 in t by CARD_1:51, ENUMSET1:def_1; assume dj = t ; ::_thesis: contradiction hence contradiction by A105, Th2, TARSKI:def_1; ::_thesis: verum end; A106: now__::_thesis:_not_td_=_t A107: 0 in t by CARD_1:51, ENUMSET1:def_1; assume td = t ; ::_thesis: contradiction hence contradiction by A107, Th4, TARSKI:def_1; ::_thesis: verum end; reconsider j = 1 as Element of M_3 by A1, ENUMSET1:def_3; reconsider z = 0 as Element of M_3 by A1, ENUMSET1:def_3; defpred S1[ set , set ] means for X being Element of M_3 st X = $1 holds ( ( X = z implies $2 = a ) & ( X = j implies $2 = b ) & ( X = dj implies $2 = c ) & ( X = td implies $2 = d ) & ( X = t implies $2 = e ) ); A108: now__::_thesis:_not_j_=_dj assume A109: j = dj ; ::_thesis: contradiction 1 in dj by Th2, TARSKI:def_1; hence contradiction by A109; ::_thesis: verum end; A110: now__::_thesis:_not_j_=_td assume A111: j = td ; ::_thesis: contradiction 2 in td by Th4, TARSKI:def_1; hence contradiction by A111, CARD_1:49, TARSKI:def_1; ::_thesis: verum end; A112: now__::_thesis:_not_dj_=_td assume A113: dj = td ; ::_thesis: contradiction 2 in td by Th4, TARSKI:def_1; hence contradiction by A113, Th2, TARSKI:def_1; ::_thesis: verum end; A114: for x being set st x in the carrier of M_3 holds ex y being set st ( y in ck & S1[x,y] ) proof let x be set ; ::_thesis: ( x in the carrier of M_3 implies ex y being set st ( y in ck & S1[x,y] ) ) assume A115: x in the carrier of M_3 ; ::_thesis: ex y being set st ( y in ck & S1[x,y] ) percases ( x = 0 or x = 1 or x = 2 \ 1 or x = 3 \ 2 or x = 3 ) by A1, A115, ENUMSET1:def_3; supposeA116: x = 0 ; ::_thesis: ex y being set st ( y in ck & S1[x,y] ) take y = a; ::_thesis: ( y in ck & S1[x,y] ) thus y in ck by ENUMSET1:def_3; ::_thesis: S1[x,y] let X be Element of M_3; ::_thesis: ( X = x implies ( ( X = z implies y = a ) & ( X = j implies y = b ) & ( X = dj implies y = c ) & ( X = td implies y = d ) & ( X = t implies y = e ) ) ) thus ( X = x implies ( ( X = z implies y = a ) & ( X = j implies y = b ) & ( X = dj implies y = c ) & ( X = td implies y = d ) & ( X = t implies y = e ) ) ) by A116, Th2, Th4; ::_thesis: verum end; supposeA117: x = 1 ; ::_thesis: ex y being set st ( y in ck & S1[x,y] ) take y = b; ::_thesis: ( y in ck & S1[x,y] ) thus y in ck by ENUMSET1:def_3; ::_thesis: S1[x,y] let X be Element of M_3; ::_thesis: ( X = x implies ( ( X = z implies y = a ) & ( X = j implies y = b ) & ( X = dj implies y = c ) & ( X = td implies y = d ) & ( X = t implies y = e ) ) ) thus ( X = x implies ( ( X = z implies y = a ) & ( X = j implies y = b ) & ( X = dj implies y = c ) & ( X = td implies y = d ) & ( X = t implies y = e ) ) ) by A108, A110, A117; ::_thesis: verum end; supposeA118: x = 2 \ 1 ; ::_thesis: ex y being set st ( y in ck & S1[x,y] ) take y = c; ::_thesis: ( y in ck & S1[x,y] ) thus y in ck by ENUMSET1:def_3; ::_thesis: S1[x,y] let X be Element of M_3; ::_thesis: ( X = x implies ( ( X = z implies y = a ) & ( X = j implies y = b ) & ( X = dj implies y = c ) & ( X = td implies y = d ) & ( X = t implies y = e ) ) ) thus ( X = x implies ( ( X = z implies y = a ) & ( X = j implies y = b ) & ( X = dj implies y = c ) & ( X = td implies y = d ) & ( X = t implies y = e ) ) ) by A108, A112, A104, A118, Th2; ::_thesis: verum end; supposeA119: x = 3 \ 2 ; ::_thesis: ex y being set st ( y in ck & S1[x,y] ) take y = d; ::_thesis: ( y in ck & S1[x,y] ) thus y in ck by ENUMSET1:def_3; ::_thesis: S1[x,y] let X be Element of M_3; ::_thesis: ( X = x implies ( ( X = z implies y = a ) & ( X = j implies y = b ) & ( X = dj implies y = c ) & ( X = td implies y = d ) & ( X = t implies y = e ) ) ) thus ( X = x implies ( ( X = z implies y = a ) & ( X = j implies y = b ) & ( X = dj implies y = c ) & ( X = td implies y = d ) & ( X = t implies y = e ) ) ) by A110, A112, A106, A119, Th4; ::_thesis: verum end; supposeA120: x = 3 ; ::_thesis: ex y being set st ( y in ck & S1[x,y] ) take y = e; ::_thesis: ( y in ck & S1[x,y] ) thus y in ck by ENUMSET1:def_3; ::_thesis: S1[x,y] let X be Element of M_3; ::_thesis: ( X = x implies ( ( X = z implies y = a ) & ( X = j implies y = b ) & ( X = dj implies y = c ) & ( X = td implies y = d ) & ( X = t implies y = e ) ) ) thus ( X = x implies ( ( X = z implies y = a ) & ( X = j implies y = b ) & ( X = dj implies y = c ) & ( X = td implies y = d ) & ( X = t implies y = e ) ) ) by A104, A106, A120; ::_thesis: verum end; end; end; consider f being Function of the carrier of M_3,ck such that A121: for x being set st x in the carrier of M_3 holds S1[x,f . x] from FUNCT_2:sch_1(A114); reconsider f = f as Function of M_3,K by A82; A122: now__::_thesis:_for_x,_y_being_Element_of_M_3_st_f_._x_=_f_._y_holds_ x_=_y let x, y be Element of M_3; ::_thesis: ( f . x = f . y implies x = y ) assume A123: f . x = f . y ; ::_thesis: x = y thus x = y ::_thesis: verum proof percases ( ( x = z & y = z ) or ( x = j & y = j ) or ( x = dj & y = dj ) or ( x = td & y = td ) or ( x = t & y = t ) or ( x = z & y = j ) or ( x = z & y = dj ) or ( x = z & y = td ) or ( x = z & y = t ) or ( x = j & y = z ) or ( x = j & y = dj ) or ( x = j & y = td ) or ( x = j & y = t ) or ( x = dj & y = z ) or ( x = dj & y = j ) or ( x = dj & y = td ) or ( x = dj & y = t ) or ( x = td & y = z ) or ( x = td & y = j ) or ( x = td & y = dj ) or ( x = td & y = t ) or ( x = t & y = z ) or ( x = t & y = j ) or ( x = t & y = dj ) or ( x = t & y = td ) ) by A1, ENUMSET1:def_3; suppose ( x = z & y = z ) ; ::_thesis: x = y hence x = y ; ::_thesis: verum end; suppose ( x = j & y = j ) ; ::_thesis: x = y hence x = y ; ::_thesis: verum end; suppose ( x = dj & y = dj ) ; ::_thesis: x = y hence x = y ; ::_thesis: verum end; suppose ( x = td & y = td ) ; ::_thesis: x = y hence x = y ; ::_thesis: verum end; suppose ( x = t & y = t ) ; ::_thesis: x = y hence x = y ; ::_thesis: verum end; supposeA124: ( x = z & y = j ) ; ::_thesis: x = y then f . x = a by A121; hence x = y by A60, A121, A123, A124; ::_thesis: verum end; supposeA125: ( x = z & y = dj ) ; ::_thesis: x = y then f . x = a by A121; hence x = y by A61, A121, A123, A125; ::_thesis: verum end; supposeA126: ( x = z & y = td ) ; ::_thesis: x = y then f . x = a by A121; hence x = y by A62, A121, A123, A126; ::_thesis: verum end; supposeA127: ( x = z & y = t ) ; ::_thesis: x = y then f . x = a by A121; hence x = y by A63, A121, A123, A127; ::_thesis: verum end; supposeA128: ( x = j & y = z ) ; ::_thesis: x = y then f . x = b by A121; hence x = y by A60, A121, A123, A128; ::_thesis: verum end; supposeA129: ( x = j & y = dj ) ; ::_thesis: x = y then f . x = b by A121; hence x = y by A64, A121, A123, A129; ::_thesis: verum end; supposeA130: ( x = j & y = td ) ; ::_thesis: x = y then f . x = b by A121; hence x = y by A65, A121, A123, A130; ::_thesis: verum end; supposeA131: ( x = j & y = t ) ; ::_thesis: x = y then f . x = b by A121; hence x = y by A66, A121, A123, A131; ::_thesis: verum end; supposeA132: ( x = dj & y = z ) ; ::_thesis: x = y then f . x = c by A121; hence x = y by A61, A121, A123, A132; ::_thesis: verum end; supposeA133: ( x = dj & y = j ) ; ::_thesis: x = y then f . x = c by A121; hence x = y by A64, A121, A123, A133; ::_thesis: verum end; supposeA134: ( x = dj & y = td ) ; ::_thesis: x = y then f . x = c by A121; hence x = y by A67, A121, A123, A134; ::_thesis: verum end; supposeA135: ( x = dj & y = t ) ; ::_thesis: x = y then f . x = c by A121; hence x = y by A68, A121, A123, A135; ::_thesis: verum end; supposeA136: ( x = td & y = z ) ; ::_thesis: x = y then f . x = d by A121; hence x = y by A62, A121, A123, A136; ::_thesis: verum end; supposeA137: ( x = td & y = j ) ; ::_thesis: x = y then f . x = d by A121; hence x = y by A65, A121, A123, A137; ::_thesis: verum end; supposeA138: ( x = td & y = dj ) ; ::_thesis: x = y then f . x = d by A121; hence x = y by A67, A121, A123, A138; ::_thesis: verum end; supposeA139: ( x = td & y = t ) ; ::_thesis: x = y then f . x = d by A121; hence x = y by A69, A121, A123, A139; ::_thesis: verum end; supposeA140: ( x = t & y = z ) ; ::_thesis: x = y then f . x = e by A121; hence x = y by A63, A121, A123, A140; ::_thesis: verum end; supposeA141: ( x = t & y = j ) ; ::_thesis: x = y then f . x = e by A121; hence x = y by A66, A121, A123, A141; ::_thesis: verum end; supposeA142: ( x = t & y = dj ) ; ::_thesis: x = y then f . x = e by A121; hence x = y by A68, A121, A123, A142; ::_thesis: verum end; supposeA143: ( x = t & y = td ) ; ::_thesis: x = y then f . x = e by A121; hence x = y by A69, A121, A123, A143; ::_thesis: verum end; end; end; end; A144: rng f c= ck proof let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng f or y in ck ) assume y in rng f ; ::_thesis: y in ck then consider x being set such that A145: x in dom f and A146: y = f . x by FUNCT_1:def_3; reconsider x = x as Element of M_3 by A145; ( x = z or x = j or x = dj or x = td or x = t ) by A1, ENUMSET1:def_3; then ( y = a or y = d or y = c or y = b or y = e ) by A121, A146; hence y in ck by ENUMSET1:def_3; ::_thesis: verum end; A147: dom f = the carrier of M_3 by FUNCT_2:def_1; ck c= rng f proof let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in ck or y in rng f ) assume A148: y in ck ; ::_thesis: y in rng f percases ( y = a or y = b or y = c or y = d or y = e ) by A148, ENUMSET1:def_3; supposeA149: y = a ; ::_thesis: y in rng f a = f . z by A121; hence y in rng f by A147, A149, FUNCT_1:def_3; ::_thesis: verum end; supposeA150: y = b ; ::_thesis: y in rng f b = f . j by A121; hence y in rng f by A147, A150, FUNCT_1:def_3; ::_thesis: verum end; supposeA151: y = c ; ::_thesis: y in rng f c = f . dj by A121; hence y in rng f by A147, A151, FUNCT_1:def_3; ::_thesis: verum end; supposeA152: y = d ; ::_thesis: y in rng f d = f . td by A121; hence y in rng f by A147, A152, FUNCT_1:def_3; ::_thesis: verum end; supposeA153: y = e ; ::_thesis: y in rng f e = f . t by A121; hence y in rng f by A147, A153, FUNCT_1:def_3; ::_thesis: verum end; end; end; then A154: rng f = ck by A144, XBOOLE_0:def_10; A155: for x, y being Element of M_3 holds ( x <= y iff f . x <= f . y ) proof let x, y be Element of M_3; ::_thesis: ( x <= y iff f . x <= f . y ) thus ( x <= y implies f . x <= f . y ) ::_thesis: ( f . x <= f . y implies x <= y ) proof assume A156: x <= y ; ::_thesis: f . x <= f . y percases ( ( x = z & y = z ) or ( x = z & y = j ) or ( x = z & y = dj ) or ( x = z & y = td ) or ( x = z & y = t ) or ( x = j & y = z ) or ( x = j & y = j ) or ( x = j & y = dj ) or ( x = j & y = td ) or ( x = j & y = t ) or ( x = dj & y = z ) or ( x = dj & y = j ) or ( x = dj & y = dj ) or ( x = dj & y = td ) or ( x = dj & y = t ) or ( x = td & y = z ) or ( x = td & y = j ) or ( x = td & y = dj ) or ( x = td & y = td ) or ( x = td & y = t ) or ( x = t & y = z ) or ( x = t & y = j ) or ( x = t & y = dj ) or ( x = t & y = td ) or ( x = t & y = t ) ) by A1, ENUMSET1:def_3; suppose ( x = z & y = z ) ; ::_thesis: f . x <= f . y hence f . x <= f . y ; ::_thesis: verum end; supposeA157: ( x = z & y = j ) ; ::_thesis: f . x <= f . y then A158: f . y = b by A121; A159: a <= b by A70, YELLOW_0:25; f . x = a by A121, A157; hence f . x <= f . y by A158, A159, YELLOW_0:60; ::_thesis: verum end; supposeA160: ( x = z & y = dj ) ; ::_thesis: f . x <= f . y then A161: f . y = c by A121; A162: a <= c by A71, YELLOW_0:25; f . x = a by A121, A160; hence f . x <= f . y by A161, A162, YELLOW_0:60; ::_thesis: verum end; supposeA163: ( x = z & y = td ) ; ::_thesis: f . x <= f . y then A164: f . y = d by A121; A165: a <= d by A72, YELLOW_0:25; f . x = a by A121, A163; hence f . x <= f . y by A164, A165, YELLOW_0:60; ::_thesis: verum end; supposeA166: ( x = z & y = t ) ; ::_thesis: f . x <= f . y A167: c <= e by A74, YELLOW_0:25; a <= c by A71, YELLOW_0:25; then A168: a <= e by A167, ORDERS_2:3; A169: f . y = e by A121, A166; f . x = a by A121, A166; hence f . x <= f . y by A169, A168, YELLOW_0:60; ::_thesis: verum end; suppose ( x = j & y = z ) ; ::_thesis: f . x <= f . y then j c= z by A156, YELLOW_1:3; hence f . x <= f . y ; ::_thesis: verum end; suppose ( x = j & y = j ) ; ::_thesis: f . x <= f . y hence f . x <= f . y ; ::_thesis: verum end; supposeA170: ( x = j & y = dj ) ; ::_thesis: f . x <= f . y 0 in j by CARD_1:49, TARSKI:def_1; then not j c= dj by Th2, TARSKI:def_1; hence f . x <= f . y by A156, A170, YELLOW_1:3; ::_thesis: verum end; supposeA171: ( x = j & y = td ) ; ::_thesis: f . x <= f . y 0 in j by CARD_1:49, TARSKI:def_1; then not j c= td by Th4, TARSKI:def_1; hence f . x <= f . y by A156, A171, YELLOW_1:3; ::_thesis: verum end; supposeA172: ( x = j & y = t ) ; ::_thesis: f . x <= f . y then A173: f . y = e by A121; A174: b <= e by A73, YELLOW_0:25; f . x = b by A121, A172; hence f . x <= f . y by A173, A174, YELLOW_0:60; ::_thesis: verum end; suppose ( x = dj & y = z ) ; ::_thesis: f . x <= f . y then dj c= z by A156, YELLOW_1:3; hence f . x <= f . y by Th2; ::_thesis: verum end; supposeA175: ( x = dj & y = j ) ; ::_thesis: f . x <= f . y A176: not 1 in j ; 1 in dj by Th2, TARSKI:def_1; then not dj c= j by A176; hence f . x <= f . y by A156, A175, YELLOW_1:3; ::_thesis: verum end; suppose ( x = dj & y = dj ) ; ::_thesis: f . x <= f . y hence f . x <= f . y ; ::_thesis: verum end; supposeA177: ( x = dj & y = td ) ; ::_thesis: f . x <= f . y 1 in dj by Th2, TARSKI:def_1; then not dj c= td by Th4, TARSKI:def_1; hence f . x <= f . y by A156, A177, YELLOW_1:3; ::_thesis: verum end; supposeA178: ( x = dj & y = t ) ; ::_thesis: f . x <= f . y then A179: f . y = e by A121; A180: c <= e by A74, YELLOW_0:25; f . x = c by A121, A178; hence f . x <= f . y by A179, A180, YELLOW_0:60; ::_thesis: verum end; suppose ( x = td & y = z ) ; ::_thesis: f . x <= f . y then td c= z by A156, YELLOW_1:3; hence f . x <= f . y by Th4; ::_thesis: verum end; supposeA181: ( x = td & y = j ) ; ::_thesis: f . x <= f . y 2 in td by Th4, TARSKI:def_1; then not td c= j by CARD_1:49, TARSKI:def_1; hence f . x <= f . y by A156, A181, YELLOW_1:3; ::_thesis: verum end; supposeA182: ( x = td & y = dj ) ; ::_thesis: f . x <= f . y 2 in td by Th4, TARSKI:def_1; then not td c= dj by Th2, TARSKI:def_1; hence f . x <= f . y by A156, A182, YELLOW_1:3; ::_thesis: verum end; suppose ( x = td & y = td ) ; ::_thesis: f . x <= f . y hence f . x <= f . y ; ::_thesis: verum end; supposeA183: ( x = td & y = t ) ; ::_thesis: f . x <= f . y then A184: f . y = e by A121; A185: d <= e by A75, YELLOW_0:25; f . x = d by A121, A183; hence f . x <= f . y by A184, A185, YELLOW_0:60; ::_thesis: verum end; suppose ( x = t & y = z ) ; ::_thesis: f . x <= f . y then t c= z by A156, YELLOW_1:3; hence f . x <= f . y ; ::_thesis: verum end; supposeA186: ( x = t & y = j ) ; ::_thesis: f . x <= f . y A187: not 1 in j ; 1 in t by CARD_1:51, ENUMSET1:def_1; then not t c= j by A187; hence f . x <= f . y by A156, A186, YELLOW_1:3; ::_thesis: verum end; supposeA188: ( x = t & y = dj ) ; ::_thesis: f . x <= f . y 2 in t by CARD_1:51, ENUMSET1:def_1; then not t c= dj by Th2, TARSKI:def_1; hence f . x <= f . y by A156, A188, YELLOW_1:3; ::_thesis: verum end; supposeA189: ( x = t & y = td ) ; ::_thesis: f . x <= f . y 0 in t by CARD_1:51, ENUMSET1:def_1; then not t c= td by Th4, TARSKI:def_1; hence f . x <= f . y by A156, A189, YELLOW_1:3; ::_thesis: verum end; suppose ( x = t & y = t ) ; ::_thesis: f . x <= f . y hence f . x <= f . y ; ::_thesis: verum end; end; end; thus ( f . x <= f . y implies x <= y ) ::_thesis: verum proof A190: dj c= t proof let X be set ; :: according to TARSKI:def_3 ::_thesis: ( not X in dj or X in t ) assume X in dj ; ::_thesis: X in t then X = 1 by Th2, TARSKI:def_1; hence X in t by CARD_1:51, ENUMSET1:def_1; ::_thesis: verum end; A191: f . y in ck by A147, A154, FUNCT_1:def_3; A192: f . x in ck by A147, A154, FUNCT_1:def_3; assume A193: f . x <= f . y ; ::_thesis: x <= y percases ( ( f . x = a & f . y = a ) or ( f . x = a & f . y = b ) or ( f . x = a & f . y = c ) or ( f . x = a & f . y = d ) or ( f . x = a & f . y = e ) or ( f . x = b & f . y = a ) or ( f . x = b & f . y = b ) or ( f . x = b & f . y = c ) or ( f . x = b & f . y = d ) or ( f . x = b & f . y = e ) or ( f . x = c & f . y = a ) or ( f . x = c & f . y = b ) or ( f . x = c & f . y = c ) or ( f . x = c & f . y = d ) or ( f . x = c & f . y = e ) or ( f . x = d & f . y = a ) or ( f . x = d & f . y = b ) or ( f . x = d & f . y = c ) or ( f . x = d & f . y = d ) or ( f . x = d & f . y = e ) or ( f . x = e & f . y = a ) or ( f . x = e & f . y = b ) or ( f . x = e & f . y = c ) or ( f . x = e & f . y = d ) or ( f . x = e & f . y = e ) ) by A192, A191, ENUMSET1:def_3; suppose ( f . x = a & f . y = a ) ; ::_thesis: x <= y hence x <= y by A122; ::_thesis: verum end; supposeA194: ( f . x = a & f . y = b ) ; ::_thesis: x <= y f . z = a by A121; then z = x by A122, A194; then x c= y by XBOOLE_1:2; hence x <= y by YELLOW_1:3; ::_thesis: verum end; supposeA195: ( f . x = a & f . y = c ) ; ::_thesis: x <= y f . z = a by A121; then z = x by A122, A195; then x c= y by XBOOLE_1:2; hence x <= y by YELLOW_1:3; ::_thesis: verum end; supposeA196: ( f . x = a & f . y = d ) ; ::_thesis: x <= y f . z = a by A121; then z = x by A122, A196; then x c= y by XBOOLE_1:2; hence x <= y by YELLOW_1:3; ::_thesis: verum end; supposeA197: ( f . x = a & f . y = e ) ; ::_thesis: x <= y f . z = a by A121; then z = x by A122, A197; then x c= y by XBOOLE_1:2; hence x <= y by YELLOW_1:3; ::_thesis: verum end; suppose ( f . x = b & f . y = a ) ; ::_thesis: x <= y then b <= a by A193, YELLOW_0:59; hence x <= y by A60, A70, YELLOW_0:25; ::_thesis: verum end; suppose ( f . x = b & f . y = b ) ; ::_thesis: x <= y hence x <= y by A122; ::_thesis: verum end; suppose ( f . x = b & f . y = c ) ; ::_thesis: x <= y then b <= c by A193, YELLOW_0:59; hence x <= y by A60, A76, YELLOW_0:25; ::_thesis: verum end; suppose ( f . x = b & f . y = d ) ; ::_thesis: x <= y then b <= d by A193, YELLOW_0:59; hence x <= y by A60, A77, YELLOW_0:25; ::_thesis: verum end; supposeA198: ( f . x = b & f . y = e ) ; ::_thesis: x <= y f . t = e by A121; then A199: t = y by A122, A198; f . j = b by A121; then j = x by A122, A198; then x c= y by A199, NAT_1:39; hence x <= y by YELLOW_1:3; ::_thesis: verum end; suppose ( f . x = c & f . y = a ) ; ::_thesis: x <= y then c <= a by A193, YELLOW_0:59; hence x <= y by A61, A71, YELLOW_0:25; ::_thesis: verum end; suppose ( f . x = c & f . y = b ) ; ::_thesis: x <= y then c <= b by A193, YELLOW_0:59; hence x <= y by A61, A76, YELLOW_0:25; ::_thesis: verum end; suppose ( f . x = c & f . y = c ) ; ::_thesis: x <= y hence x <= y by A122; ::_thesis: verum end; suppose ( f . x = c & f . y = d ) ; ::_thesis: x <= y then c <= d by A193, YELLOW_0:59; hence x <= y by A61, A78, YELLOW_0:25; ::_thesis: verum end; supposeA200: ( f . x = c & f . y = e ) ; ::_thesis: x <= y f . t = e by A121; then A201: t = y by A122, A200; f . dj = c by A121; then dj = x by A122, A200; hence x <= y by A190, A201, YELLOW_1:3; ::_thesis: verum end; suppose ( f . x = d & f . y = a ) ; ::_thesis: x <= y then d <= a by A193, YELLOW_0:59; hence x <= y by A62, A72, YELLOW_0:25; ::_thesis: verum end; suppose ( f . x = d & f . y = b ) ; ::_thesis: x <= y then d <= b by A193, YELLOW_0:59; hence x <= y by A62, A77, YELLOW_0:25; ::_thesis: verum end; suppose ( f . x = d & f . y = c ) ; ::_thesis: x <= y then d <= c by A193, YELLOW_0:59; hence x <= y by A62, A78, YELLOW_0:25; ::_thesis: verum end; suppose ( f . x = d & f . y = d ) ; ::_thesis: x <= y hence x <= y by A122; ::_thesis: verum end; supposeA202: ( f . x = d & f . y = e ) ; ::_thesis: x <= y f . t = e by A121; then A203: t = y by A122, A202; f . td = d by A121; then td = x by A122, A202; hence x <= y by A203, YELLOW_1:3; ::_thesis: verum end; supposeA204: ( f . x = e & f . y = a ) ; ::_thesis: x <= y A205: a <= b by A70, YELLOW_0:25; e <= a by A193, A204, YELLOW_0:59; then e <= b by A205, ORDERS_2:3; hence x <= y by A66, A73, YELLOW_0:25; ::_thesis: verum end; suppose ( f . x = e & f . y = b ) ; ::_thesis: x <= y then e <= b by A193, YELLOW_0:59; hence x <= y by A66, A73, YELLOW_0:25; ::_thesis: verum end; suppose ( f . x = e & f . y = c ) ; ::_thesis: x <= y then e <= c by A193, YELLOW_0:59; hence x <= y by A68, A74, YELLOW_0:25; ::_thesis: verum end; suppose ( f . x = e & f . y = d ) ; ::_thesis: x <= y then e <= d by A193, YELLOW_0:59; hence x <= y by A69, A75, YELLOW_0:25; ::_thesis: verum end; suppose ( f . x = e & f . y = e ) ; ::_thesis: x <= y hence x <= y by A122; ::_thesis: verum end; end; end; end; take f ; :: according to WAYBEL_1:def_8 ::_thesis: f is isomorphic f is V13() by A122, WAYBEL_1:def_1; hence f is isomorphic by A82, A154, A155, WAYBEL_0:66; ::_thesis: verum end; end; end; begin definition let L be non empty RelStr ; attrL is modular means :Def3: :: YELLOW11:def 3 for a, b, c being Element of L st a <= c holds a "\/" (b "/\" c) = (a "\/" b) "/\" c; end; :: deftheorem Def3 defines modular YELLOW11:def_3_:_ for L being non empty RelStr holds ( L is modular iff for a, b, c being Element of L st a <= c holds a "\/" (b "/\" c) = (a "\/" b) "/\" c ); registration cluster non empty reflexive antisymmetric with_infima distributive -> non empty reflexive antisymmetric with_infima modular for RelStr ; coherence for b1 being non empty reflexive antisymmetric with_infima RelStr st b1 is distributive holds b1 is modular proof let L be non empty reflexive antisymmetric with_infima RelStr ; ::_thesis: ( L is distributive implies L is modular ) assume A1: L is distributive ; ::_thesis: L is modular now__::_thesis:_for_a,_b,_c_being_Element_of_L_st_a_<=_c_holds_ a_"\/"_(b_"/\"_c)_=_(a_"\/"_b)_"/\"_c let a, b, c be Element of L; ::_thesis: ( a <= c implies a "\/" (b "/\" c) = (a "\/" b) "/\" c ) assume a <= c ; ::_thesis: a "\/" (b "/\" c) = (a "\/" b) "/\" c hence a "\/" (b "/\" c) = (a "/\" c) "\/" (b "/\" c) by YELLOW_0:25 .= (a "\/" b) "/\" c by A1, WAYBEL_1:def_3 ; ::_thesis: verum end; hence L is modular by Def3; ::_thesis: verum end; end; Lm2: for L being LATTICE holds ( L is modular iff for a, b, c, d, e being Element of L holds ( not a <> b or not a <> c or not a <> d or not a <> e or not b <> c or not b <> d or not b <> e or not c <> d or not c <> e or not d <> e or not a "/\" b = a or not a "/\" c = a or not c "/\" e = c or not d "/\" e = d or not b "/\" c = a or not b "/\" d = b or not c "/\" d = a or not b "\/" c = e or not c "\/" d = e ) ) proof let L be LATTICE; ::_thesis: ( L is modular iff for a, b, c, d, e being Element of L holds ( not a <> b or not a <> c or not a <> d or not a <> e or not b <> c or not b <> d or not b <> e or not c <> d or not c <> e or not d <> e or not a "/\" b = a or not a "/\" c = a or not c "/\" e = c or not d "/\" e = d or not b "/\" c = a or not b "/\" d = b or not c "/\" d = a or not b "\/" c = e or not c "\/" d = e ) ) now__::_thesis:_(_ex_a,_b,_c,_d,_e_being_Element_of_L_st_ (_a_<>_b_&_a_<>_c_&_a_<>_d_&_a_<>_e_&_b_<>_c_&_b_<>_d_&_b_<>_e_&_c_<>_d_&_c_<>_e_&_d_<>_e_&_a_"/\"_b_=_a_&_a_"/\"_c_=_a_&_c_"/\"_e_=_c_&_d_"/\"_e_=_d_&_b_"/\"_c_=_a_&_b_"/\"_d_=_b_&_c_"/\"_d_=_a_&_b_"\/"_c_=_e_&_c_"\/"_d_=_e_)_implies_not_L_is_modular_) given a, b, c, d, e being Element of L such that a <> b and a <> c and a <> d and a <> e and b <> c and A1: b <> d and b <> e and c <> d and c <> e and d <> e and A2: a "/\" b = a and a "/\" c = a and c "/\" e = c and A3: d "/\" e = d and b "/\" c = a and A4: b "/\" d = b and A5: c "/\" d = a and A6: b "\/" c = e and c "\/" d = e ; ::_thesis: not L is modular A7: b <= d by A4, YELLOW_0:23; b "\/" (c "/\" d) = b by A2, A5, Th5; hence not L is modular by A1, A3, A6, A7, Def3; ::_thesis: verum end; hence ( L is modular implies for a, b, c, d, e being Element of L holds ( not a <> b or not a <> c or not a <> d or not a <> e or not b <> c or not b <> d or not b <> e or not c <> d or not c <> e or not d <> e or not a "/\" b = a or not a "/\" c = a or not c "/\" e = c or not d "/\" e = d or not b "/\" c = a or not b "/\" d = b or not c "/\" d = a or not b "\/" c = e or not c "\/" d = e ) ) ; ::_thesis: ( ( for a, b, c, d, e being Element of L holds ( not a <> b or not a <> c or not a <> d or not a <> e or not b <> c or not b <> d or not b <> e or not c <> d or not c <> e or not d <> e or not a "/\" b = a or not a "/\" c = a or not c "/\" e = c or not d "/\" e = d or not b "/\" c = a or not b "/\" d = b or not c "/\" d = a or not b "\/" c = e or not c "\/" d = e ) ) implies L is modular ) now__::_thesis:_(_not_L_is_modular_implies_ex_a,_b,_c,_d,_e_being_Element_of_L_st_ (_a_<>_b_&_a_<>_c_&_a_<>_d_&_a_<>_e_&_b_<>_c_&_b_<>_d_&_b_<>_e_&_c_<>_d_&_c_<>_e_&_d_<>_e_&_a_"/\"_b_=_a_&_a_"/\"_c_=_a_&_c_"/\"_e_=_c_&_d_"/\"_e_=_d_&_b_"/\"_c_=_a_&_b_"/\"_d_=_b_&_c_"/\"_d_=_a_&_b_"\/"_c_=_e_&_c_"\/"_d_=_e_)_) assume not L is modular ; ::_thesis: ex a, b, c, d, e being Element of L st ( a <> b & a <> c & a <> d & a <> e & b <> c & b <> d & b <> e & c <> d & c <> e & d <> e & a "/\" b = a & a "/\" c = a & c "/\" e = c & d "/\" e = d & b "/\" c = a & b "/\" d = b & c "/\" d = a & b "\/" c = e & c "\/" d = e ) then consider x, y, z being Element of L such that A8: x <= z and A9: x "\/" (y "/\" z) <> (x "\/" y) "/\" z by Def3; x "\/" (y "/\" z) <= z "\/" (y "/\" z) by A8, YELLOW_5:7; then A10: x "\/" (y "/\" z) <= z by LATTICE3:17; set z1 = (x "\/" y) "/\" z; set y1 = y; x "\/" y <= x "\/" y ; then (x "\/" y) "/\" x <= (x "\/" y) "/\" z by A8, YELLOW_3:2; then x <= (x "\/" y) "/\" z by LATTICE3:18; then A11: x "\/" y <= ((x "\/" y) "/\" z) "\/" y by YELLOW_5:7; set x1 = x "\/" (y "/\" z); A12: y "/\" z <= y by YELLOW_0:23; y <= y ; then A13: (x "\/" (y "/\" z)) "/\" y <= y "/\" z by A10, YELLOW_3:2; set t = x "\/" y; set b = y "/\" z; A14: now__::_thesis:_not_y_"/\"_z_=_x_"\/"_y assume A15: y "/\" z = x "\/" y ; ::_thesis: contradiction then (x "\/" y) "/\" z = y "/\" (z "/\" z) by LATTICE3:16 .= x "\/" y by A15, YELLOW_5:2 .= (x "\/" x) "\/" y by YELLOW_5:1 .= x "\/" (y "/\" z) by A15, LATTICE3:14 ; hence contradiction by A9; ::_thesis: verum end; y "/\" z <= x "\/" (y "/\" z) by YELLOW_0:22; then (y "/\" z) "/\" (y "/\" z) <= (x "\/" (y "/\" z)) "/\" y by A12, YELLOW_3:2; then A16: y "/\" z <= (x "\/" (y "/\" z)) "/\" y by YELLOW_5:2; A17: (x "\/" y) "/\" z <= x "\/" y by YELLOW_0:23; thus ex a, b, c, d, e being Element of L st ( a <> b & a <> c & a <> d & a <> e & b <> c & b <> d & b <> e & c <> d & c <> e & d <> e & a "/\" b = a & a "/\" c = a & c "/\" e = c & d "/\" e = d & b "/\" c = a & b "/\" d = b & c "/\" d = a & b "\/" c = e & c "\/" d = e ) ::_thesis: verum proof reconsider b = y "/\" z, x1 = x "\/" (y "/\" z), y1 = y, z1 = (x "\/" y) "/\" z, t = x "\/" y as Element of L ; take b ; ::_thesis: ex b, c, d, e being Element of L st ( b <> b & b <> c & b <> d & b <> e & b <> c & b <> d & b <> e & c <> d & c <> e & d <> e & b "/\" b = b & b "/\" c = b & c "/\" e = c & d "/\" e = d & b "/\" c = b & b "/\" d = b & c "/\" d = b & b "\/" c = e & c "\/" d = e ) take x1 ; ::_thesis: ex c, d, e being Element of L st ( b <> x1 & b <> c & b <> d & b <> e & x1 <> c & x1 <> d & x1 <> e & c <> d & c <> e & d <> e & b "/\" x1 = b & b "/\" c = b & c "/\" e = c & d "/\" e = d & x1 "/\" c = b & x1 "/\" d = x1 & c "/\" d = b & x1 "\/" c = e & c "\/" d = e ) take y1 ; ::_thesis: ex d, e being Element of L st ( b <> x1 & b <> y1 & b <> d & b <> e & x1 <> y1 & x1 <> d & x1 <> e & y1 <> d & y1 <> e & d <> e & b "/\" x1 = b & b "/\" y1 = b & y1 "/\" e = y1 & d "/\" e = d & x1 "/\" y1 = b & x1 "/\" d = x1 & y1 "/\" d = b & x1 "\/" y1 = e & y1 "\/" d = e ) take z1 ; ::_thesis: ex e being Element of L st ( b <> x1 & b <> y1 & b <> z1 & b <> e & x1 <> y1 & x1 <> z1 & x1 <> e & y1 <> z1 & y1 <> e & z1 <> e & b "/\" x1 = b & b "/\" y1 = b & y1 "/\" e = y1 & z1 "/\" e = z1 & x1 "/\" y1 = b & x1 "/\" z1 = x1 & y1 "/\" z1 = b & x1 "\/" y1 = e & y1 "\/" z1 = e ) take t ; ::_thesis: ( b <> x1 & b <> y1 & b <> z1 & b <> t & x1 <> y1 & x1 <> z1 & x1 <> t & y1 <> z1 & y1 <> t & z1 <> t & b "/\" x1 = b & b "/\" y1 = b & y1 "/\" t = y1 & z1 "/\" t = z1 & x1 "/\" y1 = b & x1 "/\" z1 = x1 & y1 "/\" z1 = b & x1 "\/" y1 = t & y1 "\/" z1 = t ) A18: y1 <= x "\/" y by YELLOW_0:22; now__::_thesis:_not_b_=_x1 A19: y "/\" z <= y by YELLOW_0:23; assume A20: b = x1 ; ::_thesis: contradiction then x <= y "/\" z by YELLOW_0:22; then x <= y by A19, YELLOW_0:def_2; hence contradiction by A9, A20, YELLOW_5:8; ::_thesis: verum end; hence b <> x1 ; ::_thesis: ( b <> y1 & b <> z1 & b <> t & x1 <> y1 & x1 <> z1 & x1 <> t & y1 <> z1 & y1 <> t & z1 <> t & b "/\" x1 = b & b "/\" y1 = b & y1 "/\" t = y1 & z1 "/\" t = z1 & x1 "/\" y1 = b & x1 "/\" z1 = x1 & y1 "/\" z1 = b & x1 "\/" y1 = t & y1 "\/" z1 = t ) now__::_thesis:_not_b_=_y1 assume A21: b = y1 ; ::_thesis: contradiction then y <= z by YELLOW_0:23; hence contradiction by A8, A9, A21, YELLOW_5:9, YELLOW_5:10; ::_thesis: verum end; hence b <> y1 ; ::_thesis: ( b <> z1 & b <> t & x1 <> y1 & x1 <> z1 & x1 <> t & y1 <> z1 & y1 <> t & z1 <> t & b "/\" x1 = b & b "/\" y1 = b & y1 "/\" t = y1 & z1 "/\" t = z1 & x1 "/\" y1 = b & x1 "/\" z1 = x1 & y1 "/\" z1 = b & x1 "\/" y1 = t & y1 "\/" z1 = t ) now__::_thesis:_not_b_=_z1 assume b = z1 ; ::_thesis: contradiction then A22: (x "\/" y) "/\" z <= x "\/" (y "/\" z) by YELLOW_0:22; x "\/" (y "/\" z) <= (x "\/" y) "/\" z by A8, Th8; hence contradiction by A9, A22, YELLOW_0:def_3; ::_thesis: verum end; hence b <> z1 ; ::_thesis: ( b <> t & x1 <> y1 & x1 <> z1 & x1 <> t & y1 <> z1 & y1 <> t & z1 <> t & b "/\" x1 = b & b "/\" y1 = b & y1 "/\" t = y1 & z1 "/\" t = z1 & x1 "/\" y1 = b & x1 "/\" z1 = x1 & y1 "/\" z1 = b & x1 "\/" y1 = t & y1 "\/" z1 = t ) thus b <> t by A14; ::_thesis: ( x1 <> y1 & x1 <> z1 & x1 <> t & y1 <> z1 & y1 <> t & z1 <> t & b "/\" x1 = b & b "/\" y1 = b & y1 "/\" t = y1 & z1 "/\" t = z1 & x1 "/\" y1 = b & x1 "/\" z1 = x1 & y1 "/\" z1 = b & x1 "\/" y1 = t & y1 "\/" z1 = t ) now__::_thesis:_not_x1_=_y1 A23: x1 "\/" y1 = x "\/" ((y "/\" z) "\/" y) by LATTICE3:14 .= t by LATTICE3:17 ; assume A24: x1 = y1 ; ::_thesis: contradiction then A25: x1 "\/" y1 = x1 by YELLOW_5:1; x1 "/\" y1 = x1 by A24, YELLOW_5:2; hence contradiction by A16, A13, A14, A25, A23, YELLOW_0:def_3; ::_thesis: verum end; hence x1 <> y1 ; ::_thesis: ( x1 <> z1 & x1 <> t & y1 <> z1 & y1 <> t & z1 <> t & b "/\" x1 = b & b "/\" y1 = b & y1 "/\" t = y1 & z1 "/\" t = z1 & x1 "/\" y1 = b & x1 "/\" z1 = x1 & y1 "/\" z1 = b & x1 "\/" y1 = t & y1 "\/" z1 = t ) thus x1 <> z1 by A9; ::_thesis: ( x1 <> t & y1 <> z1 & y1 <> t & z1 <> t & b "/\" x1 = b & b "/\" y1 = b & y1 "/\" t = y1 & z1 "/\" t = z1 & x1 "/\" y1 = b & x1 "/\" z1 = x1 & y1 "/\" z1 = b & x1 "\/" y1 = t & y1 "\/" z1 = t ) now__::_thesis:_not_t_=_x1 assume t = x1 ; ::_thesis: contradiction then A26: (x "\/" y) "/\" z <= x "\/" (y "/\" z) by YELLOW_0:23; x "\/" (y "/\" z) <= (x "\/" y) "/\" z by A8, Th8; hence contradiction by A9, A26, YELLOW_0:def_3; ::_thesis: verum end; hence x1 <> t ; ::_thesis: ( y1 <> z1 & y1 <> t & z1 <> t & b "/\" x1 = b & b "/\" y1 = b & y1 "/\" t = y1 & z1 "/\" t = z1 & x1 "/\" y1 = b & x1 "/\" z1 = x1 & y1 "/\" z1 = b & x1 "\/" y1 = t & y1 "\/" z1 = t ) now__::_thesis:_not_y1_=_z1 A27: y1 "/\" z1 = ((x "\/" y) "/\" y) "/\" z by LATTICE3:16 .= b by LATTICE3:18 ; assume A28: y1 = z1 ; ::_thesis: contradiction then A29: z1 "\/" y1 = z1 by YELLOW_5:1; z1 "/\" y1 = z1 by A28, YELLOW_5:2; hence contradiction by A14, A17, A11, A29, A27, YELLOW_0:def_3; ::_thesis: verum end; hence y1 <> z1 ; ::_thesis: ( y1 <> t & z1 <> t & b "/\" x1 = b & b "/\" y1 = b & y1 "/\" t = y1 & z1 "/\" t = z1 & x1 "/\" y1 = b & x1 "/\" z1 = x1 & y1 "/\" z1 = b & x1 "\/" y1 = t & y1 "\/" z1 = t ) now__::_thesis:_not_y1_=_t assume A30: y1 = t ; ::_thesis: contradiction then x <= y by YELLOW_0:22; then x "/\" x <= y "/\" z by A8, YELLOW_3:2; then x <= y "/\" z by YELLOW_5:2; hence contradiction by A9, A30, YELLOW_5:8; ::_thesis: verum end; hence y1 <> t ; ::_thesis: ( z1 <> t & b "/\" x1 = b & b "/\" y1 = b & y1 "/\" t = y1 & z1 "/\" t = z1 & x1 "/\" y1 = b & x1 "/\" z1 = x1 & y1 "/\" z1 = b & x1 "\/" y1 = t & y1 "\/" z1 = t ) now__::_thesis:_not_z1_=_t A31: y <= x "\/" y by YELLOW_0:22; assume A32: z1 = t ; ::_thesis: contradiction then x "\/" y <= z by YELLOW_0:23; then y <= z by A31, YELLOW_0:def_2; hence contradiction by A9, A32, YELLOW_5:10; ::_thesis: verum end; hence z1 <> t ; ::_thesis: ( b "/\" x1 = b & b "/\" y1 = b & y1 "/\" t = y1 & z1 "/\" t = z1 & x1 "/\" y1 = b & x1 "/\" z1 = x1 & y1 "/\" z1 = b & x1 "\/" y1 = t & y1 "\/" z1 = t ) b <= x1 by YELLOW_0:22; hence b "/\" x1 = b by YELLOW_5:10; ::_thesis: ( b "/\" y1 = b & y1 "/\" t = y1 & z1 "/\" t = z1 & x1 "/\" y1 = b & x1 "/\" z1 = x1 & y1 "/\" z1 = b & x1 "\/" y1 = t & y1 "\/" z1 = t ) b <= y1 by YELLOW_0:23; hence b "/\" y1 = b by YELLOW_5:10; ::_thesis: ( y1 "/\" t = y1 & z1 "/\" t = z1 & x1 "/\" y1 = b & x1 "/\" z1 = x1 & y1 "/\" z1 = b & x1 "\/" y1 = t & y1 "\/" z1 = t ) y1 <= t by YELLOW_0:22; hence y1 "/\" t = y1 by YELLOW_5:10; ::_thesis: ( z1 "/\" t = z1 & x1 "/\" y1 = b & x1 "/\" z1 = x1 & y1 "/\" z1 = b & x1 "\/" y1 = t & y1 "\/" z1 = t ) z1 <= t by YELLOW_0:23; hence z1 "/\" t = z1 by YELLOW_5:10; ::_thesis: ( x1 "/\" y1 = b & x1 "/\" z1 = x1 & y1 "/\" z1 = b & x1 "\/" y1 = t & y1 "\/" z1 = t ) thus x1 "/\" y1 = b by A16, A13, YELLOW_0:def_3; ::_thesis: ( x1 "/\" z1 = x1 & y1 "/\" z1 = b & x1 "\/" y1 = t & y1 "\/" z1 = t ) thus x1 "/\" z1 = x1 by A8, Th8, YELLOW_5:10; ::_thesis: ( y1 "/\" z1 = b & x1 "\/" y1 = t & y1 "\/" z1 = t ) thus y1 "/\" z1 = (y "/\" (x "\/" y)) "/\" z by LATTICE3:16 .= b by LATTICE3:18 ; ::_thesis: ( x1 "\/" y1 = t & y1 "\/" z1 = t ) thus x1 "\/" y1 = x "\/" ((y "/\" z) "\/" y) by LATTICE3:14 .= t by LATTICE3:17 ; ::_thesis: y1 "\/" z1 = t x "\/" y <= x "\/" y ; then (x "\/" y) "/\" x <= (x "\/" y) "/\" z by A8, YELLOW_3:2; then x <= (x "\/" y) "/\" z by LATTICE3:18; then A33: x "\/" y <= z1 "\/" y1 by YELLOW_5:7; z1 <= x "\/" y by YELLOW_0:23; then y1 "\/" z1 <= x "\/" y by A18, YELLOW_5:9; hence y1 "\/" z1 = t by A33, YELLOW_0:def_3; ::_thesis: verum end; end; hence ( ( for a, b, c, d, e being Element of L holds ( not a <> b or not a <> c or not a <> d or not a <> e or not b <> c or not b <> d or not b <> e or not c <> d or not c <> e or not d <> e or not a "/\" b = a or not a "/\" c = a or not c "/\" e = c or not d "/\" e = d or not b "/\" c = a or not b "/\" d = b or not c "/\" d = a or not b "\/" c = e or not c "\/" d = e ) ) implies L is modular ) ; ::_thesis: verum end; theorem :: YELLOW11:11 for L being LATTICE holds ( L is modular iff for K being full Sublattice of L holds not N_5 ,K are_isomorphic ) proof let L be LATTICE; ::_thesis: ( L is modular iff for K being full Sublattice of L holds not N_5 ,K are_isomorphic ) thus ( L is modular implies for K being full Sublattice of L holds not N_5 ,K are_isomorphic ) ::_thesis: ( ( for K being full Sublattice of L holds not N_5 ,K are_isomorphic ) implies L is modular ) proof assume L is modular ; ::_thesis: for K being full Sublattice of L holds not N_5 ,K are_isomorphic then for a, b, c, d, e being Element of L holds ( not a <> b or not a <> c or not a <> d or not a <> e or not b <> c or not b <> d or not b <> e or not c <> d or not c <> e or not d <> e or not a "/\" b = a or not a "/\" c = a or not c "/\" e = c or not d "/\" e = d or not b "/\" c = a or not b "/\" d = b or not c "/\" d = a or not b "\/" c = e or not c "\/" d = e ) by Lm2; hence for K being full Sublattice of L holds not N_5 ,K are_isomorphic by Th9; ::_thesis: verum end; assume for K being full Sublattice of L holds not N_5 ,K are_isomorphic ; ::_thesis: L is modular then for a, b, c, d, e being Element of L holds ( not a <> b or not a <> c or not a <> d or not a <> e or not b <> c or not b <> d or not b <> e or not c <> d or not c <> e or not d <> e or not a "/\" b = a or not a "/\" c = a or not c "/\" e = c or not d "/\" e = d or not b "/\" c = a or not b "/\" d = b or not c "/\" d = a or not b "\/" c = e or not c "\/" d = e ) by Th9; hence L is modular by Lm2; ::_thesis: verum end; Lm3: for L being LATTICE st L is modular holds ( L is distributive iff for a, b, c, d, e being Element of L holds ( not a <> b or not a <> c or not a <> d or not a <> e or not b <> c or not b <> d or not b <> e or not c <> d or not c <> e or not d <> e or not a "/\" b = a or not a "/\" c = a or not a "/\" d = a or not b "/\" e = b or not c "/\" e = c or not d "/\" e = d or not b "/\" c = a or not b "/\" d = a or not c "/\" d = a or not b "\/" c = e or not b "\/" d = e or not c "\/" d = e ) ) proof let L be LATTICE; ::_thesis: ( L is modular implies ( L is distributive iff for a, b, c, d, e being Element of L holds ( not a <> b or not a <> c or not a <> d or not a <> e or not b <> c or not b <> d or not b <> e or not c <> d or not c <> e or not d <> e or not a "/\" b = a or not a "/\" c = a or not a "/\" d = a or not b "/\" e = b or not c "/\" e = c or not d "/\" e = d or not b "/\" c = a or not b "/\" d = a or not c "/\" d = a or not b "\/" c = e or not b "\/" d = e or not c "\/" d = e ) ) ) assume A1: L is modular ; ::_thesis: ( L is distributive iff for a, b, c, d, e being Element of L holds ( not a <> b or not a <> c or not a <> d or not a <> e or not b <> c or not b <> d or not b <> e or not c <> d or not c <> e or not d <> e or not a "/\" b = a or not a "/\" c = a or not a "/\" d = a or not b "/\" e = b or not c "/\" e = c or not d "/\" e = d or not b "/\" c = a or not b "/\" d = a or not c "/\" d = a or not b "\/" c = e or not b "\/" d = e or not c "\/" d = e ) ) now__::_thesis:_(_ex_a,_b,_c,_d,_e_being_Element_of_L_st_ (_a_<>_b_&_a_<>_c_&_a_<>_d_&_a_<>_e_&_b_<>_c_&_b_<>_d_&_b_<>_e_&_c_<>_d_&_c_<>_e_&_d_<>_e_&_a_"/\"_b_=_a_&_a_"/\"_c_=_a_&_a_"/\"_d_=_a_&_b_"/\"_e_=_b_&_c_"/\"_e_=_c_&_d_"/\"_e_=_d_&_b_"/\"_c_=_a_&_b_"/\"_d_=_a_&_c_"/\"_d_=_a_&_b_"\/"_c_=_e_&_b_"\/"_d_=_e_&_c_"\/"_d_=_e_)_implies_not_L_is_distributive_) given a, b, c, d, e being Element of L such that a <> b and a <> c and A2: a <> d and a <> e and b <> c and b <> d and b <> e and c <> d and c <> e and d <> e and a "/\" b = a and a "/\" c = a and a "/\" d = a and b "/\" e = b and c "/\" e = c and A3: d "/\" e = d and A4: b "/\" c = a and A5: b "/\" d = a and A6: c "/\" d = a and A7: b "\/" c = e and b "\/" d = e and c "\/" d = e ; ::_thesis: not L is distributive (b "/\" c) "\/" (b "/\" d) = a by A4, A5, YELLOW_5:1; hence not L is distributive by A2, A3, A4, A6, A7, WAYBEL_1:def_3; ::_thesis: verum end; hence ( L is distributive implies for a, b, c, d, e being Element of L holds ( not a <> b or not a <> c or not a <> d or not a <> e or not b <> c or not b <> d or not b <> e or not c <> d or not c <> e or not d <> e or not a "/\" b = a or not a "/\" c = a or not a "/\" d = a or not b "/\" e = b or not c "/\" e = c or not d "/\" e = d or not b "/\" c = a or not b "/\" d = a or not c "/\" d = a or not b "\/" c = e or not b "\/" d = e or not c "\/" d = e ) ) ; ::_thesis: ( ( for a, b, c, d, e being Element of L holds ( not a <> b or not a <> c or not a <> d or not a <> e or not b <> c or not b <> d or not b <> e or not c <> d or not c <> e or not d <> e or not a "/\" b = a or not a "/\" c = a or not a "/\" d = a or not b "/\" e = b or not c "/\" e = c or not d "/\" e = d or not b "/\" c = a or not b "/\" d = a or not c "/\" d = a or not b "\/" c = e or not b "\/" d = e or not c "\/" d = e ) ) implies L is distributive ) now__::_thesis:_(_not_L_is_distributive_implies_ex_a,_b,_c,_d,_e_being_Element_of_L_st_ (_a_<>_b_&_a_<>_c_&_a_<>_d_&_a_<>_e_&_b_<>_c_&_b_<>_d_&_b_<>_e_&_c_<>_d_&_c_<>_e_&_d_<>_e_&_a_"/\"_b_=_a_&_a_"/\"_c_=_a_&_a_"/\"_d_=_a_&_b_"/\"_e_=_b_&_c_"/\"_e_=_c_&_d_"/\"_e_=_d_&_b_"/\"_c_=_a_&_b_"/\"_d_=_a_&_c_"/\"_d_=_a_&_b_"\/"_c_=_e_&_b_"\/"_d_=_e_&_c_"\/"_d_=_e_)_) assume not L is distributive ; ::_thesis: ex a, b, c, d, e being Element of L st ( a <> b & a <> c & a <> d & a <> e & b <> c & b <> d & b <> e & c <> d & c <> e & d <> e & a "/\" b = a & a "/\" c = a & a "/\" d = a & b "/\" e = b & c "/\" e = c & d "/\" e = d & b "/\" c = a & b "/\" d = a & c "/\" d = a & b "\/" c = e & b "\/" d = e & c "\/" d = e ) then consider x, y, z being Element of L such that A8: x "/\" (y "\/" z) <> (x "/\" y) "\/" (x "/\" z) by WAYBEL_1:def_3; set t = ((x "\/" y) "/\" (y "\/" z)) "/\" (z "\/" x); set b = ((x "/\" y) "\/" (y "/\" z)) "\/" (z "/\" x); A9: x "/\" y <= x by YELLOW_0:23; A10: x "/\" (((x "\/" y) "/\" (y "\/" z)) "/\" (z "\/" x)) = (x "/\" ((x "\/" y) "/\" (y "\/" z))) "/\" (z "\/" x) by LATTICE3:16 .= ((x "/\" (x "\/" y)) "/\" (y "\/" z)) "/\" (z "\/" x) by LATTICE3:16 .= (x "/\" (y "\/" z)) "/\" (z "\/" x) by LATTICE3:18 .= (x "/\" (z "\/" x)) "/\" (y "\/" z) by LATTICE3:16 .= x "/\" (y "\/" z) by LATTICE3:18 ; A11: x <= x ; y "/\" z <= z by YELLOW_0:23; then A12: ((y "/\" z) "/\" x) "\/" (z "/\" x) = z "/\" x by A11, YELLOW_3:2, YELLOW_5:8; A13: z "/\" x <= x by YELLOW_0:23; A14: now__::_thesis:_not_((x_"/\"_y)_"\/"_(y_"/\"_z))_"\/"_(z_"/\"_x)_=_((x_"\/"_y)_"/\"_(y_"\/"_z))_"/\"_(z_"\/"_x) assume ((x "/\" y) "\/" (y "/\" z)) "\/" (z "/\" x) = ((x "\/" y) "/\" (y "\/" z)) "/\" (z "\/" x) ; ::_thesis: contradiction then x "/\" (y "\/" z) = ((x "/\" y) "\/" ((y "/\" z) "\/" (z "/\" x))) "/\" x by A10, LATTICE3:14 .= (x "/\" y) "\/" (((y "/\" z) "\/" (z "/\" x)) "/\" x) by A1, A9, Def3 .= (x "/\" y) "\/" (z "/\" x) by A1, A13, A12, Def3 ; hence contradiction by A8; ::_thesis: verum end; set y1 = (y "\/" (((x "/\" y) "\/" (y "/\" z)) "\/" (z "/\" x))) "/\" (((x "\/" y) "/\" (y "\/" z)) "/\" (z "\/" x)); A15: y "/\" z <= y "\/" z by YELLOW_5:5; y "/\" z <= x "\/" y by YELLOW_5:5; then (y "/\" z) "/\" (y "/\" z) <= (x "\/" y) "/\" (y "\/" z) by A15, YELLOW_3:2; then A16: y "/\" z <= (x "\/" y) "/\" (y "\/" z) by YELLOW_5:2; y "/\" z <= z "\/" x by YELLOW_5:5; then (y "/\" z) "/\" (y "/\" z) <= ((x "\/" y) "/\" (y "\/" z)) "/\" (z "\/" x) by A16, YELLOW_3:2; then A17: y "/\" z <= ((x "\/" y) "/\" (y "\/" z)) "/\" (z "\/" x) by YELLOW_5:2; A18: x "/\" (((x "\/" y) "/\" (y "\/" z)) "/\" (z "\/" x)) = (x "/\" ((x "\/" y) "/\" (y "\/" z))) "/\" (z "\/" x) by LATTICE3:16 .= ((x "/\" (x "\/" y)) "/\" (y "\/" z)) "/\" (z "\/" x) by LATTICE3:16 .= (x "/\" (y "\/" z)) "/\" (z "\/" x) by LATTICE3:18 .= ((z "\/" x) "/\" x) "/\" (y "\/" z) by LATTICE3:16 .= x "/\" (y "\/" z) by LATTICE3:18 ; set z1 = (z "\/" (((x "/\" y) "\/" (y "/\" z)) "\/" (z "/\" x))) "/\" (((x "\/" y) "/\" (y "\/" z)) "/\" (z "\/" x)); A19: z "/\" x <= y "\/" z by YELLOW_5:5; z "/\" x <= x "\/" y by YELLOW_5:5; then (z "/\" x) "/\" (z "/\" x) <= (x "\/" y) "/\" (y "\/" z) by A19, YELLOW_3:2; then A20: z "/\" x <= (x "\/" y) "/\" (y "\/" z) by YELLOW_5:2; z "/\" x <= z "\/" x by YELLOW_5:5; then (z "/\" x) "/\" (z "/\" x) <= ((x "\/" y) "/\" (y "\/" z)) "/\" (z "\/" x) by A20, YELLOW_3:2; then A21: z "/\" x <= ((x "\/" y) "/\" (y "\/" z)) "/\" (z "\/" x) by YELLOW_5:2; A22: y "/\" (((x "\/" y) "/\" (y "\/" z)) "/\" (z "\/" x)) = (y "/\" ((x "\/" y) "/\" (y "\/" z))) "/\" (z "\/" x) by LATTICE3:16 .= ((y "/\" (x "\/" y)) "/\" (y "\/" z)) "/\" (z "\/" x) by LATTICE3:16 .= (y "/\" (y "\/" z)) "/\" (z "\/" x) by LATTICE3:18 .= y "/\" (x "\/" z) by LATTICE3:18 ; A23: x <= x "\/" (y "/\" z) by YELLOW_0:22; x "/\" z <= x by YELLOW_0:23; then A24: x "/\" z <= x "\/" (y "/\" z) by A23, YELLOW_0:def_2; A25: y <= y "\/" z by YELLOW_0:22; A26: z "\/" (x "/\" y) <= (z "\/" x) "/\" (z "\/" y) by Th7; A27: y "\/" (x "/\" z) <= (y "\/" x) "/\" (y "\/" z) by Th7; A28: x <= x "\/" y by YELLOW_0:22; x "/\" (z "\/" y) <= x by YELLOW_0:23; then A29: x "/\" (z "\/" y) <= x "\/" y by A28, YELLOW_0:def_2; A30: y "\/" (((x "/\" y) "\/" (y "/\" z)) "\/" (z "/\" x)) = (y "\/" ((x "/\" y) "\/" (y "/\" z))) "\/" (z "/\" x) by LATTICE3:14 .= ((y "\/" (x "/\" y)) "\/" (y "/\" z)) "\/" (z "/\" x) by LATTICE3:14 .= (y "\/" (y "/\" z)) "\/" (z "/\" x) by LATTICE3:17 .= y "\/" (x "/\" z) by LATTICE3:17 ; A31: x <= x "\/" (z "/\" y) by YELLOW_0:22; x "/\" y <= x by YELLOW_0:23; then A32: x "/\" y <= x "\/" (z "/\" y) by A31, YELLOW_0:def_2; A33: z <= z "\/" y by YELLOW_0:22; A34: y "\/" (z "/\" x) <= (y "\/" z) "/\" (y "\/" x) by Th7; A35: (x "/\" y) "\/" (y "/\" z) <= y "/\" (x "\/" z) by Th6; A36: y "/\" z <= y by YELLOW_0:23; A37: z <= z "\/" x by YELLOW_0:22; z "/\" (y "\/" x) <= z by YELLOW_0:23; then A38: z "/\" (y "\/" x) <= z "\/" x by A37, YELLOW_0:def_2; A39: z "\/" (((x "/\" y) "\/" (y "/\" z)) "\/" (z "/\" x)) = (z "\/" (z "/\" x)) "\/" ((x "/\" y) "\/" (y "/\" z)) by LATTICE3:14 .= z "\/" ((y "/\" z) "\/" (x "/\" y)) by LATTICE3:17 .= (z "\/" (y "/\" z)) "\/" (x "/\" y) by LATTICE3:14 .= z "\/" (x "/\" y) by LATTICE3:17 ; (x "/\" y) "\/" (x "/\" z) <= x "/\" (y "\/" z) by Th6; then ((x "/\" y) "\/" (x "/\" z)) "\/" ((x "/\" y) "\/" (y "/\" z)) <= (x "/\" (y "\/" z)) "\/" (y "/\" (x "\/" z)) by A35, YELLOW_3:3; then (x "/\" z) "\/" ((x "/\" y) "\/" ((x "/\" y) "\/" (y "/\" z))) <= (x "/\" (y "\/" z)) "\/" (y "/\" (x "\/" z)) by LATTICE3:14; then (x "/\" z) "\/" (((x "/\" y) "\/" (x "/\" y)) "\/" (y "/\" z)) <= (x "/\" (y "\/" z)) "\/" (y "/\" (x "\/" z)) by LATTICE3:14; then (((x "/\" y) "\/" (x "/\" y)) "\/" (x "/\" z)) "\/" (y "/\" z) <= (x "/\" (y "\/" z)) "\/" (y "/\" (x "\/" z)) by LATTICE3:14; then ((x "/\" y) "\/" (x "/\" z)) "\/" (y "/\" z) <= (x "/\" (y "\/" z)) "\/" (y "/\" (x "\/" z)) by YELLOW_5:1; then A40: ((x "/\" y) "\/" (y "/\" z)) "\/" (z "/\" x) <= (x "/\" (((x "\/" y) "/\" (y "\/" z)) "/\" (z "\/" x))) "\/" (y "/\" (((x "\/" y) "/\" (y "\/" z)) "/\" (z "\/" x))) by A18, A22, LATTICE3:14; (z "\/" (((x "/\" y) "\/" (y "/\" z)) "\/" (z "/\" x))) "/\" (((x "\/" y) "/\" (y "\/" z)) "/\" (z "\/" x)) <= ((x "\/" y) "/\" (y "\/" z)) "/\" (z "\/" x) by YELLOW_0:23; then A41: (((x "\/" y) "/\" (y "\/" z)) "/\" (z "\/" x)) "/\" ((z "\/" (((x "/\" y) "\/" (y "/\" z)) "\/" (z "/\" x))) "/\" (((x "\/" y) "/\" (y "\/" z)) "/\" (z "\/" x))) = (z "\/" (((x "/\" y) "\/" (y "/\" z)) "\/" (z "/\" x))) "/\" (((x "\/" y) "/\" (y "\/" z)) "/\" (z "\/" x)) by YELLOW_5:10; A42: z <= z "\/" (y "/\" x) by YELLOW_0:22; z "/\" x <= z by YELLOW_0:23; then A43: z "/\" x <= z "\/" (y "/\" x) by A42, YELLOW_0:def_2; A44: y <= y "\/" x by YELLOW_0:22; A45: x <= x "\/" z by YELLOW_0:22; x "/\" (y "\/" z) <= x by YELLOW_0:23; then A46: x "/\" (y "\/" z) <= x "\/" z by A45, YELLOW_0:def_2; A47: x "\/" (((x "/\" y) "\/" (y "/\" z)) "\/" (z "/\" x)) = (x "\/" ((x "/\" y) "\/" (y "/\" z))) "\/" (z "/\" x) by LATTICE3:14 .= ((x "\/" (x "/\" y)) "\/" (y "/\" z)) "\/" (z "/\" x) by LATTICE3:14 .= (x "\/" (y "/\" z)) "\/" (z "/\" x) by LATTICE3:17 .= ((z "/\" x) "\/" x) "\/" (y "/\" z) by LATTICE3:14 .= x "\/" (y "/\" z) by LATTICE3:17 ; z "\/" (y "/\" x) <= (z "\/" y) "/\" (z "\/" x) by Th7; then (z "\/" (y "/\" x)) "/\" (y "\/" (z "/\" x)) <= ((z "\/" y) "/\" (z "\/" x)) "/\" ((y "\/" z) "/\" (y "\/" x)) by A34, YELLOW_3:2; then (z "\/" (((x "/\" y) "\/" (y "/\" z)) "\/" (z "/\" x))) "/\" (y "\/" (((x "/\" y) "\/" (y "/\" z)) "\/" (z "/\" x))) <= (((z "\/" x) "/\" (z "\/" y)) "/\" (z "\/" y)) "/\" (y "\/" x) by A30, A39, LATTICE3:16; then (z "\/" (((x "/\" y) "\/" (y "/\" z)) "\/" (z "/\" x))) "/\" (y "\/" (((x "/\" y) "\/" (y "/\" z)) "\/" (z "/\" x))) <= ((z "\/" x) "/\" ((z "\/" y) "/\" (z "\/" y))) "/\" (y "\/" x) by LATTICE3:16; then (z "\/" (((x "/\" y) "\/" (y "/\" z)) "\/" (z "/\" x))) "/\" (y "\/" (((x "/\" y) "\/" (y "/\" z)) "\/" (z "/\" x))) <= ((z "\/" x) "/\" (z "\/" y)) "/\" (y "\/" x) by YELLOW_5:2; then A48: (z "\/" (((x "/\" y) "\/" (y "/\" z)) "\/" (z "/\" x))) "/\" (y "\/" (((x "/\" y) "\/" (y "/\" z)) "\/" (z "/\" x))) <= ((x "\/" y) "/\" (y "\/" z)) "/\" (z "\/" x) by LATTICE3:16; (y "\/" (((x "/\" y) "\/" (y "/\" z)) "\/" (z "/\" x))) "/\" (((x "\/" y) "/\" (y "\/" z)) "/\" (z "\/" x)) <= ((x "\/" y) "/\" (y "\/" z)) "/\" (z "\/" x) by YELLOW_0:23; then A49: (((x "\/" y) "/\" (y "\/" z)) "/\" (z "\/" x)) "/\" ((y "\/" (((x "/\" y) "\/" (y "/\" z)) "\/" (z "/\" x))) "/\" (((x "\/" y) "/\" (y "\/" z)) "/\" (z "\/" x))) = (y "\/" (((x "/\" y) "\/" (y "/\" z)) "\/" (z "/\" x))) "/\" (((x "\/" y) "/\" (y "\/" z)) "/\" (z "\/" x)) by YELLOW_5:10; set x1 = (x "\/" (((x "/\" y) "\/" (y "/\" z)) "\/" (z "/\" x))) "/\" (((x "\/" y) "/\" (y "\/" z)) "/\" (z "\/" x)); A50: x "/\" y <= y "\/" z by YELLOW_5:5; x "/\" y <= x "\/" y by YELLOW_5:5; then (x "/\" y) "/\" (x "/\" y) <= (x "\/" y) "/\" (y "\/" z) by A50, YELLOW_3:2; then A51: x "/\" y <= (x "\/" y) "/\" (y "\/" z) by YELLOW_5:2; A52: z "/\" (((x "\/" y) "/\" (y "\/" z)) "/\" (z "\/" x)) = (z "/\" (z "\/" x)) "/\" ((x "\/" y) "/\" (y "\/" z)) by LATTICE3:16 .= z "/\" ((y "\/" z) "/\" (x "\/" y)) by LATTICE3:18 .= (z "/\" (y "\/" z)) "/\" (x "\/" y) by LATTICE3:16 .= z "/\" (x "\/" y) by LATTICE3:18 ; x "\/" (y "/\" z) <= (x "\/" y) "/\" (x "\/" z) by Th7; then (x "\/" (y "/\" z)) "/\" (y "\/" (x "/\" z)) <= ((x "\/" y) "/\" (x "\/" z)) "/\" ((y "\/" x) "/\" (y "\/" z)) by A27, YELLOW_3:2; then (x "\/" (((x "/\" y) "\/" (y "/\" z)) "\/" (z "/\" x))) "/\" (y "\/" (((x "/\" y) "\/" (y "/\" z)) "\/" (z "/\" x))) <= (((x "\/" z) "/\" (x "\/" y)) "/\" (x "\/" y)) "/\" (y "\/" z) by A47, A30, LATTICE3:16; then (x "\/" (((x "/\" y) "\/" (y "/\" z)) "\/" (z "/\" x))) "/\" (y "\/" (((x "/\" y) "\/" (y "/\" z)) "\/" (z "/\" x))) <= ((x "\/" z) "/\" ((x "\/" y) "/\" (x "\/" y))) "/\" (y "\/" z) by LATTICE3:16; then (x "\/" (((x "/\" y) "\/" (y "/\" z)) "\/" (z "/\" x))) "/\" (y "\/" (((x "/\" y) "\/" (y "/\" z)) "\/" (z "/\" x))) <= ((x "\/" z) "/\" (x "\/" y)) "/\" (y "\/" z) by YELLOW_5:2; then A53: (x "\/" (((x "/\" y) "\/" (y "/\" z)) "\/" (z "/\" x))) "/\" (y "\/" (((x "/\" y) "\/" (y "/\" z)) "\/" (z "/\" x))) <= ((x "\/" y) "/\" (y "\/" z)) "/\" (z "\/" x) by LATTICE3:16; A54: (z "/\" y) "\/" (y "/\" x) <= y "/\" (z "\/" x) by Th6; A55: y "/\" x <= y by YELLOW_0:23; A56: ((x "\/" (((x "/\" y) "\/" (y "/\" z)) "\/" (z "/\" x))) "/\" (((x "\/" y) "/\" (y "\/" z)) "/\" (z "\/" x))) "/\" ((y "\/" (((x "/\" y) "\/" (y "/\" z)) "\/" (z "/\" x))) "/\" (((x "\/" y) "/\" (y "\/" z)) "/\" (z "\/" x))) = (x "\/" (((x "/\" y) "\/" (y "/\" z)) "\/" (z "/\" x))) "/\" ((((x "\/" y) "/\" (y "\/" z)) "/\" (z "\/" x)) "/\" ((y "\/" (((x "/\" y) "\/" (y "/\" z)) "\/" (z "/\" x))) "/\" (((x "\/" y) "/\" (y "\/" z)) "/\" (z "\/" x)))) by LATTICE3:16 .= (x "\/" (((x "/\" y) "\/" (y "/\" z)) "\/" (z "/\" x))) "/\" (((((x "\/" y) "/\" (y "\/" z)) "/\" (z "\/" x)) "/\" (((x "\/" y) "/\" (y "\/" z)) "/\" (z "\/" x))) "/\" (y "\/" (((x "/\" y) "\/" (y "/\" z)) "\/" (z "/\" x)))) by LATTICE3:16 .= (x "\/" (((x "/\" y) "\/" (y "/\" z)) "\/" (z "/\" x))) "/\" ((y "\/" (((x "/\" y) "\/" (y "/\" z)) "\/" (z "/\" x))) "/\" (((x "\/" y) "/\" (y "\/" z)) "/\" (z "\/" x))) by YELLOW_5:2 .= ((x "\/" (((x "/\" y) "\/" (y "/\" z)) "\/" (z "/\" x))) "/\" (y "\/" (((x "/\" y) "\/" (y "/\" z)) "\/" (z "/\" x)))) "/\" (((x "\/" y) "/\" (y "\/" z)) "/\" (z "\/" x)) by LATTICE3:16 .= (x "\/" (y "/\" z)) "/\" (y "\/" (x "/\" z)) by A47, A30, A53, YELLOW_5:10 .= (y "/\" (x "\/" (y "/\" z))) "\/" (x "/\" z) by A1, A24, Def3 .= ((x "/\" y) "\/" (y "/\" z)) "\/" (z "/\" x) by A1, A36, Def3 ; then ((x "/\" y) "\/" (y "/\" z)) "\/" (z "/\" x) <= (y "\/" (((x "/\" y) "\/" (y "/\" z)) "\/" (z "/\" x))) "/\" (((x "\/" y) "/\" (y "\/" z)) "/\" (z "\/" x)) by YELLOW_0:23; then A57: (((x "/\" y) "\/" (y "/\" z)) "\/" (z "/\" x)) "\/" ((y "\/" (((x "/\" y) "\/" (y "/\" z)) "\/" (z "/\" x))) "/\" (((x "\/" y) "/\" (y "\/" z)) "/\" (z "\/" x))) = (y "\/" (((x "/\" y) "\/" (y "/\" z)) "\/" (z "/\" x))) "/\" (((x "\/" y) "/\" (y "\/" z)) "/\" (z "\/" x)) by YELLOW_5:8; x "\/" (z "/\" y) <= (x "\/" z) "/\" (x "\/" y) by Th7; then (x "\/" (z "/\" y)) "/\" (z "\/" (x "/\" y)) <= ((x "\/" z) "/\" (x "\/" y)) "/\" ((z "\/" x) "/\" (z "\/" y)) by A26, YELLOW_3:2; then (x "\/" (((x "/\" y) "\/" (y "/\" z)) "\/" (z "/\" x))) "/\" (z "\/" (((x "/\" y) "\/" (y "/\" z)) "\/" (z "/\" x))) <= (((x "\/" y) "/\" (x "\/" z)) "/\" (x "\/" z)) "/\" (z "\/" y) by A47, A39, LATTICE3:16; then (x "\/" (((x "/\" y) "\/" (y "/\" z)) "\/" (z "/\" x))) "/\" (z "\/" (((x "/\" y) "\/" (y "/\" z)) "\/" (z "/\" x))) <= ((x "\/" y) "/\" ((x "\/" z) "/\" (x "\/" z))) "/\" (z "\/" y) by LATTICE3:16; then (x "\/" (((x "/\" y) "\/" (y "/\" z)) "\/" (z "/\" x))) "/\" (z "\/" (((x "/\" y) "\/" (y "/\" z)) "\/" (z "/\" x))) <= ((x "\/" y) "/\" (x "\/" z)) "/\" (z "\/" y) by YELLOW_5:2; then A58: (x "\/" (((x "/\" y) "\/" (y "/\" z)) "\/" (z "/\" x))) "/\" (z "\/" (((x "/\" y) "\/" (y "/\" z)) "\/" (z "/\" x))) <= ((x "\/" y) "/\" (y "\/" z)) "/\" (z "\/" x) by LATTICE3:16; (x "\/" (((x "/\" y) "\/" (y "/\" z)) "\/" (z "/\" x))) "/\" (((x "\/" y) "/\" (y "\/" z)) "/\" (z "\/" x)) <= ((x "\/" y) "/\" (y "\/" z)) "/\" (z "\/" x) by YELLOW_0:23; then A59: (((x "\/" y) "/\" (y "\/" z)) "/\" (z "\/" x)) "/\" ((x "\/" (((x "/\" y) "\/" (y "/\" z)) "\/" (z "/\" x))) "/\" (((x "\/" y) "/\" (y "\/" z)) "/\" (z "\/" x))) = (x "\/" (((x "/\" y) "\/" (y "/\" z)) "\/" (z "/\" x))) "/\" (((x "\/" y) "/\" (y "\/" z)) "/\" (z "\/" x)) by YELLOW_5:10; A60: ((z "\/" (((x "/\" y) "\/" (y "/\" z)) "\/" (z "/\" x))) "/\" (((x "\/" y) "/\" (y "\/" z)) "/\" (z "\/" x))) "/\" ((y "\/" (((x "/\" y) "\/" (y "/\" z)) "\/" (z "/\" x))) "/\" (((x "\/" y) "/\" (y "\/" z)) "/\" (z "\/" x))) = (z "\/" (((x "/\" y) "\/" (y "/\" z)) "\/" (z "/\" x))) "/\" ((((x "\/" y) "/\" (y "\/" z)) "/\" (z "\/" x)) "/\" ((y "\/" (((x "/\" y) "\/" (y "/\" z)) "\/" (z "/\" x))) "/\" (((x "\/" y) "/\" (y "\/" z)) "/\" (z "\/" x)))) by LATTICE3:16 .= (z "\/" (((x "/\" y) "\/" (y "/\" z)) "\/" (z "/\" x))) "/\" (((((x "\/" y) "/\" (y "\/" z)) "/\" (z "\/" x)) "/\" (((x "\/" y) "/\" (y "\/" z)) "/\" (z "\/" x))) "/\" (y "\/" (((x "/\" y) "\/" (y "/\" z)) "\/" (z "/\" x)))) by LATTICE3:16 .= (z "\/" (((x "/\" y) "\/" (y "/\" z)) "\/" (z "/\" x))) "/\" ((y "\/" (((x "/\" y) "\/" (y "/\" z)) "\/" (z "/\" x))) "/\" (((x "\/" y) "/\" (y "\/" z)) "/\" (z "\/" x))) by YELLOW_5:2 .= ((z "\/" (((x "/\" y) "\/" (y "/\" z)) "\/" (z "/\" x))) "/\" (y "\/" (((x "/\" y) "\/" (y "/\" z)) "\/" (z "/\" x)))) "/\" (((x "\/" y) "/\" (y "\/" z)) "/\" (z "\/" x)) by LATTICE3:16 .= (z "\/" (y "/\" x)) "/\" (y "\/" (z "/\" x)) by A30, A39, A48, YELLOW_5:10 .= (y "/\" (z "\/" (y "/\" x))) "\/" (z "/\" x) by A1, A43, Def3 .= ((x "/\" y) "\/" (y "/\" z)) "\/" (z "/\" x) by A1, A55, Def3 ; (z "/\" y) "\/" (z "/\" x) <= z "/\" (y "\/" x) by Th6; then ((z "/\" y) "\/" (z "/\" x)) "\/" ((z "/\" y) "\/" (y "/\" x)) <= (z "/\" (y "\/" x)) "\/" (y "/\" (z "\/" x)) by A54, YELLOW_3:3; then (z "/\" x) "\/" ((z "/\" y) "\/" ((z "/\" y) "\/" (y "/\" x))) <= (z "/\" (y "\/" x)) "\/" (y "/\" (z "\/" x)) by LATTICE3:14; then (z "/\" x) "\/" (((z "/\" y) "\/" (z "/\" y)) "\/" (y "/\" x)) <= (z "/\" (y "\/" x)) "\/" (y "/\" (z "\/" x)) by LATTICE3:14; then (((z "/\" y) "\/" (z "/\" y)) "\/" (z "/\" x)) "\/" (y "/\" x) <= (z "/\" (y "\/" x)) "\/" (y "/\" (z "\/" x)) by LATTICE3:14; then ((z "/\" y) "\/" (z "/\" x)) "\/" (y "/\" x) <= (z "/\" (y "\/" x)) "\/" (y "/\" (z "\/" x)) by YELLOW_5:1; then A61: ((x "/\" y) "\/" (y "/\" z)) "\/" (z "/\" x) <= (z "/\" (((x "\/" y) "/\" (y "\/" z)) "/\" (z "\/" x))) "\/" (y "/\" (((x "\/" y) "/\" (y "\/" z)) "/\" (z "\/" x))) by A22, A52, LATTICE3:14; A62: (x "/\" z) "\/" (z "/\" y) <= z "/\" (x "\/" y) by Th6; A63: z "/\" y <= z by YELLOW_0:23; (x "/\" z) "\/" (x "/\" y) <= x "/\" (z "\/" y) by Th6; then ((x "/\" z) "\/" (x "/\" y)) "\/" ((x "/\" z) "\/" (z "/\" y)) <= (x "/\" (z "\/" y)) "\/" (z "/\" (x "\/" y)) by A62, YELLOW_3:3; then (x "/\" y) "\/" ((x "/\" z) "\/" ((x "/\" z) "\/" (z "/\" y))) <= (x "/\" (z "\/" y)) "\/" (z "/\" (x "\/" y)) by LATTICE3:14; then (x "/\" y) "\/" (((x "/\" z) "\/" (x "/\" z)) "\/" (z "/\" y)) <= (x "/\" (z "\/" y)) "\/" (z "/\" (x "\/" y)) by LATTICE3:14; then (((x "/\" z) "\/" (x "/\" z)) "\/" (x "/\" y)) "\/" (z "/\" y) <= (x "/\" (z "\/" y)) "\/" (z "/\" (x "\/" y)) by LATTICE3:14; then ((x "/\" z) "\/" (x "/\" y)) "\/" (z "/\" y) <= (x "/\" (z "\/" y)) "\/" (z "/\" (x "\/" y)) by YELLOW_5:1; then A64: ((x "/\" y) "\/" (y "/\" z)) "\/" (z "/\" x) <= (x "/\" (((x "\/" y) "/\" (y "\/" z)) "/\" (z "\/" x))) "\/" (z "/\" (((x "\/" y) "/\" (y "\/" z)) "/\" (z "\/" x))) by A18, A52, LATTICE3:14; x "/\" y <= z "\/" x by YELLOW_5:5; then (x "/\" y) "/\" (x "/\" y) <= ((x "\/" y) "/\" (y "\/" z)) "/\" (z "\/" x) by A51, YELLOW_3:2; then x "/\" y <= ((x "\/" y) "/\" (y "\/" z)) "/\" (z "\/" x) by YELLOW_5:2; then (x "/\" y) "\/" (y "/\" z) <= (((x "\/" y) "/\" (y "\/" z)) "/\" (z "\/" x)) "\/" (((x "\/" y) "/\" (y "\/" z)) "/\" (z "\/" x)) by A17, YELLOW_3:3; then (x "/\" y) "\/" (y "/\" z) <= ((x "\/" y) "/\" (y "\/" z)) "/\" (z "\/" x) by YELLOW_5:1; then ((x "/\" y) "\/" (y "/\" z)) "\/" (z "/\" x) <= (((x "\/" y) "/\" (y "\/" z)) "/\" (z "\/" x)) "\/" (((x "\/" y) "/\" (y "\/" z)) "/\" (z "\/" x)) by A21, YELLOW_3:3; then A65: ((x "/\" y) "\/" (y "/\" z)) "\/" (z "/\" x) <= ((x "\/" y) "/\" (y "\/" z)) "/\" (z "\/" x) by YELLOW_5:1; then (z "\/" (((x "/\" y) "\/" (y "/\" z)) "\/" (z "/\" x))) "/\" (((x "\/" y) "/\" (y "\/" z)) "/\" (z "\/" x)) = (z "/\" (((x "\/" y) "/\" (y "\/" z)) "/\" (z "\/" x))) "\/" (((x "/\" y) "\/" (y "/\" z)) "\/" (z "/\" x)) by A1, Def3; then A66: ((x "\/" (((x "/\" y) "\/" (y "/\" z)) "\/" (z "/\" x))) "/\" (((x "\/" y) "/\" (y "\/" z)) "/\" (z "\/" x))) "\/" ((z "\/" (((x "/\" y) "\/" (y "/\" z)) "\/" (z "/\" x))) "/\" (((x "\/" y) "/\" (y "\/" z)) "/\" (z "\/" x))) = ((x "/\" (((x "\/" y) "/\" (y "\/" z)) "/\" (z "\/" x))) "\/" (((x "/\" y) "\/" (y "/\" z)) "\/" (z "/\" x))) "\/" ((z "/\" (((x "\/" y) "/\" (y "\/" z)) "/\" (z "\/" x))) "\/" (((x "/\" y) "\/" (y "/\" z)) "\/" (z "/\" x))) by A1, A65, Def3 .= (x "/\" (((x "\/" y) "/\" (y "\/" z)) "/\" (z "\/" x))) "\/" ((((x "/\" y) "\/" (y "/\" z)) "\/" (z "/\" x)) "\/" ((z "/\" (((x "\/" y) "/\" (y "\/" z)) "/\" (z "\/" x))) "\/" (((x "/\" y) "\/" (y "/\" z)) "\/" (z "/\" x)))) by LATTICE3:14 .= (x "/\" (((x "\/" y) "/\" (y "\/" z)) "/\" (z "\/" x))) "\/" (((((x "/\" y) "\/" (y "/\" z)) "\/" (z "/\" x)) "\/" (((x "/\" y) "\/" (y "/\" z)) "\/" (z "/\" x))) "\/" (z "/\" (((x "\/" y) "/\" (y "\/" z)) "/\" (z "\/" x)))) by LATTICE3:14 .= (x "/\" (((x "\/" y) "/\" (y "\/" z)) "/\" (z "\/" x))) "\/" ((z "/\" (((x "\/" y) "/\" (y "\/" z)) "/\" (z "\/" x))) "\/" (((x "/\" y) "\/" (y "/\" z)) "\/" (z "/\" x))) by YELLOW_5:1 .= ((x "/\" (((x "\/" y) "/\" (y "\/" z)) "/\" (z "\/" x))) "\/" (z "/\" (((x "\/" y) "/\" (y "\/" z)) "/\" (z "\/" x)))) "\/" (((x "/\" y) "\/" (y "/\" z)) "\/" (z "/\" x)) by LATTICE3:14 .= (x "/\" (z "\/" y)) "\/" (z "/\" (x "\/" y)) by A18, A52, A64, YELLOW_5:8 .= (z "\/" (x "/\" (z "\/" y))) "/\" (x "\/" y) by A1, A29, Def3 .= ((z "\/" x) "/\" (z "\/" y)) "/\" (x "\/" y) by A1, A33, Def3 .= ((x "\/" y) "/\" (y "\/" z)) "/\" (z "\/" x) by LATTICE3:16 ; A67: (y "\/" (((x "/\" y) "\/" (y "/\" z)) "\/" (z "/\" x))) "/\" (((x "\/" y) "/\" (y "\/" z)) "/\" (z "\/" x)) = (y "/\" (((x "\/" y) "/\" (y "\/" z)) "/\" (z "\/" x))) "\/" (((x "/\" y) "\/" (y "/\" z)) "\/" (z "/\" x)) by A1, A65, Def3; then A68: ((x "\/" (((x "/\" y) "\/" (y "/\" z)) "\/" (z "/\" x))) "/\" (((x "\/" y) "/\" (y "\/" z)) "/\" (z "\/" x))) "\/" ((y "\/" (((x "/\" y) "\/" (y "/\" z)) "\/" (z "/\" x))) "/\" (((x "\/" y) "/\" (y "\/" z)) "/\" (z "\/" x))) = ((x "/\" (((x "\/" y) "/\" (y "\/" z)) "/\" (z "\/" x))) "\/" (((x "/\" y) "\/" (y "/\" z)) "\/" (z "/\" x))) "\/" ((y "/\" (((x "\/" y) "/\" (y "\/" z)) "/\" (z "\/" x))) "\/" (((x "/\" y) "\/" (y "/\" z)) "\/" (z "/\" x))) by A1, A65, Def3 .= (x "/\" (((x "\/" y) "/\" (y "\/" z)) "/\" (z "\/" x))) "\/" ((((x "/\" y) "\/" (y "/\" z)) "\/" (z "/\" x)) "\/" ((y "/\" (((x "\/" y) "/\" (y "\/" z)) "/\" (z "\/" x))) "\/" (((x "/\" y) "\/" (y "/\" z)) "\/" (z "/\" x)))) by LATTICE3:14 .= (x "/\" (((x "\/" y) "/\" (y "\/" z)) "/\" (z "\/" x))) "\/" (((((x "/\" y) "\/" (y "/\" z)) "\/" (z "/\" x)) "\/" (((x "/\" y) "\/" (y "/\" z)) "\/" (z "/\" x))) "\/" (y "/\" (((x "\/" y) "/\" (y "\/" z)) "/\" (z "\/" x)))) by LATTICE3:14 .= (x "/\" (((x "\/" y) "/\" (y "\/" z)) "/\" (z "\/" x))) "\/" ((y "/\" (((x "\/" y) "/\" (y "\/" z)) "/\" (z "\/" x))) "\/" (((x "/\" y) "\/" (y "/\" z)) "\/" (z "/\" x))) by YELLOW_5:1 .= ((x "/\" (((x "\/" y) "/\" (y "\/" z)) "/\" (z "\/" x))) "\/" (y "/\" (((x "\/" y) "/\" (y "\/" z)) "/\" (z "\/" x)))) "\/" (((x "/\" y) "\/" (y "/\" z)) "\/" (z "/\" x)) by LATTICE3:14 .= (x "/\" (y "\/" z)) "\/" (y "/\" (x "\/" z)) by A18, A22, A40, YELLOW_5:8 .= (y "\/" (x "/\" (y "\/" z))) "/\" (x "\/" z) by A1, A46, Def3 .= ((x "\/" y) "/\" (y "\/" z)) "/\" (z "\/" x) by A1, A25, Def3 ; A69: ((z "\/" (((x "/\" y) "\/" (y "/\" z)) "\/" (z "/\" x))) "/\" (((x "\/" y) "/\" (y "\/" z)) "/\" (z "\/" x))) "\/" ((y "\/" (((x "/\" y) "\/" (y "/\" z)) "\/" (z "/\" x))) "/\" (((x "\/" y) "/\" (y "\/" z)) "/\" (z "\/" x))) = ((z "/\" (((x "\/" y) "/\" (y "\/" z)) "/\" (z "\/" x))) "\/" (((x "/\" y) "\/" (y "/\" z)) "\/" (z "/\" x))) "\/" ((y "/\" (((x "\/" y) "/\" (y "\/" z)) "/\" (z "\/" x))) "\/" (((x "/\" y) "\/" (y "/\" z)) "\/" (z "/\" x))) by A1, A65, A67, Def3 .= (z "/\" (((x "\/" y) "/\" (y "\/" z)) "/\" (z "\/" x))) "\/" ((((x "/\" y) "\/" (y "/\" z)) "\/" (z "/\" x)) "\/" ((y "/\" (((x "\/" y) "/\" (y "\/" z)) "/\" (z "\/" x))) "\/" (((x "/\" y) "\/" (y "/\" z)) "\/" (z "/\" x)))) by LATTICE3:14 .= (z "/\" (((x "\/" y) "/\" (y "\/" z)) "/\" (z "\/" x))) "\/" (((((x "/\" y) "\/" (y "/\" z)) "\/" (z "/\" x)) "\/" (((x "/\" y) "\/" (y "/\" z)) "\/" (z "/\" x))) "\/" (y "/\" (((x "\/" y) "/\" (y "\/" z)) "/\" (z "\/" x)))) by LATTICE3:14 .= (z "/\" (((x "\/" y) "/\" (y "\/" z)) "/\" (z "\/" x))) "\/" ((y "/\" (((x "\/" y) "/\" (y "\/" z)) "/\" (z "\/" x))) "\/" (((x "/\" y) "\/" (y "/\" z)) "\/" (z "/\" x))) by YELLOW_5:1 .= ((z "/\" (((x "\/" y) "/\" (y "\/" z)) "/\" (z "\/" x))) "\/" (y "/\" (((x "\/" y) "/\" (y "\/" z)) "/\" (z "\/" x)))) "\/" (((x "/\" y) "\/" (y "/\" z)) "\/" (z "/\" x)) by LATTICE3:14 .= (z "/\" (y "\/" x)) "\/" (y "/\" (z "\/" x)) by A22, A52, A61, YELLOW_5:8 .= (y "\/" (z "/\" (y "\/" x))) "/\" (z "\/" x) by A1, A38, Def3 .= ((x "\/" y) "/\" (y "\/" z)) "/\" (z "\/" x) by A1, A44, Def3 ; A70: ((x "\/" (((x "/\" y) "\/" (y "/\" z)) "\/" (z "/\" x))) "/\" (((x "\/" y) "/\" (y "\/" z)) "/\" (z "\/" x))) "/\" ((z "\/" (((x "/\" y) "\/" (y "/\" z)) "\/" (z "/\" x))) "/\" (((x "\/" y) "/\" (y "\/" z)) "/\" (z "\/" x))) = (x "\/" (((x "/\" y) "\/" (y "/\" z)) "\/" (z "/\" x))) "/\" ((((x "\/" y) "/\" (y "\/" z)) "/\" (z "\/" x)) "/\" ((z "\/" (((x "/\" y) "\/" (y "/\" z)) "\/" (z "/\" x))) "/\" (((x "\/" y) "/\" (y "\/" z)) "/\" (z "\/" x)))) by LATTICE3:16 .= (x "\/" (((x "/\" y) "\/" (y "/\" z)) "\/" (z "/\" x))) "/\" (((((x "\/" y) "/\" (y "\/" z)) "/\" (z "\/" x)) "/\" (((x "\/" y) "/\" (y "\/" z)) "/\" (z "\/" x))) "/\" (z "\/" (((x "/\" y) "\/" (y "/\" z)) "\/" (z "/\" x)))) by LATTICE3:16 .= (x "\/" (((x "/\" y) "\/" (y "/\" z)) "\/" (z "/\" x))) "/\" ((z "\/" (((x "/\" y) "\/" (y "/\" z)) "\/" (z "/\" x))) "/\" (((x "\/" y) "/\" (y "\/" z)) "/\" (z "\/" x))) by YELLOW_5:2 .= ((x "\/" (((x "/\" y) "\/" (y "/\" z)) "\/" (z "/\" x))) "/\" (z "\/" (((x "/\" y) "\/" (y "/\" z)) "\/" (z "/\" x)))) "/\" (((x "\/" y) "/\" (y "\/" z)) "/\" (z "\/" x)) by LATTICE3:16 .= (x "\/" (z "/\" y)) "/\" (z "\/" (x "/\" y)) by A47, A39, A58, YELLOW_5:10 .= (z "/\" (x "\/" (z "/\" y))) "\/" (x "/\" y) by A1, A32, Def3 .= ((z "/\" x) "\/" (z "/\" y)) "\/" (x "/\" y) by A1, A63, Def3 .= ((x "/\" y) "\/" (y "/\" z)) "\/" (z "/\" x) by LATTICE3:14 ; then ((x "/\" y) "\/" (y "/\" z)) "\/" (z "/\" x) <= (z "\/" (((x "/\" y) "\/" (y "/\" z)) "\/" (z "/\" x))) "/\" (((x "\/" y) "/\" (y "\/" z)) "/\" (z "\/" x)) by YELLOW_0:23; then A71: (((x "/\" y) "\/" (y "/\" z)) "\/" (z "/\" x)) "\/" ((z "\/" (((x "/\" y) "\/" (y "/\" z)) "\/" (z "/\" x))) "/\" (((x "\/" y) "/\" (y "\/" z)) "/\" (z "\/" x))) = (z "\/" (((x "/\" y) "\/" (y "/\" z)) "\/" (z "/\" x))) "/\" (((x "\/" y) "/\" (y "\/" z)) "/\" (z "\/" x)) by YELLOW_5:8; ((x "/\" y) "\/" (y "/\" z)) "\/" (z "/\" x) <= (x "\/" (((x "/\" y) "\/" (y "/\" z)) "\/" (z "/\" x))) "/\" (((x "\/" y) "/\" (y "\/" z)) "/\" (z "\/" x)) by A56, YELLOW_0:23; then A72: (((x "/\" y) "\/" (y "/\" z)) "\/" (z "/\" x)) "\/" ((x "\/" (((x "/\" y) "\/" (y "/\" z)) "\/" (z "/\" x))) "/\" (((x "\/" y) "/\" (y "\/" z)) "/\" (z "\/" x))) = (x "\/" (((x "/\" y) "\/" (y "/\" z)) "\/" (z "/\" x))) "/\" (((x "\/" y) "/\" (y "\/" z)) "/\" (z "\/" x)) by YELLOW_5:8; thus ex a, b, c, d, e being Element of L st ( a <> b & a <> c & a <> d & a <> e & b <> c & b <> d & b <> e & c <> d & c <> e & d <> e & a "/\" b = a & a "/\" c = a & a "/\" d = a & b "/\" e = b & c "/\" e = c & d "/\" e = d & b "/\" c = a & b "/\" d = a & c "/\" d = a & b "\/" c = e & b "\/" d = e & c "\/" d = e ) ::_thesis: verum proof reconsider b = ((x "/\" y) "\/" (y "/\" z)) "\/" (z "/\" x), x1 = (x "\/" (((x "/\" y) "\/" (y "/\" z)) "\/" (z "/\" x))) "/\" (((x "\/" y) "/\" (y "\/" z)) "/\" (z "\/" x)), y1 = (y "\/" (((x "/\" y) "\/" (y "/\" z)) "\/" (z "/\" x))) "/\" (((x "\/" y) "/\" (y "\/" z)) "/\" (z "\/" x)), z1 = (z "\/" (((x "/\" y) "\/" (y "/\" z)) "\/" (z "/\" x))) "/\" (((x "\/" y) "/\" (y "\/" z)) "/\" (z "\/" x)), t = ((x "\/" y) "/\" (y "\/" z)) "/\" (z "\/" x) as Element of L ; take b ; ::_thesis: ex b, c, d, e being Element of L st ( b <> b & b <> c & b <> d & b <> e & b <> c & b <> d & b <> e & c <> d & c <> e & d <> e & b "/\" b = b & b "/\" c = b & b "/\" d = b & b "/\" e = b & c "/\" e = c & d "/\" e = d & b "/\" c = b & b "/\" d = b & c "/\" d = b & b "\/" c = e & b "\/" d = e & c "\/" d = e ) take x1 ; ::_thesis: ex c, d, e being Element of L st ( b <> x1 & b <> c & b <> d & b <> e & x1 <> c & x1 <> d & x1 <> e & c <> d & c <> e & d <> e & b "/\" x1 = b & b "/\" c = b & b "/\" d = b & x1 "/\" e = x1 & c "/\" e = c & d "/\" e = d & x1 "/\" c = b & x1 "/\" d = b & c "/\" d = b & x1 "\/" c = e & x1 "\/" d = e & c "\/" d = e ) take y1 ; ::_thesis: ex d, e being Element of L st ( b <> x1 & b <> y1 & b <> d & b <> e & x1 <> y1 & x1 <> d & x1 <> e & y1 <> d & y1 <> e & d <> e & b "/\" x1 = b & b "/\" y1 = b & b "/\" d = b & x1 "/\" e = x1 & y1 "/\" e = y1 & d "/\" e = d & x1 "/\" y1 = b & x1 "/\" d = b & y1 "/\" d = b & x1 "\/" y1 = e & x1 "\/" d = e & y1 "\/" d = e ) take z1 ; ::_thesis: ex e being Element of L st ( b <> x1 & b <> y1 & b <> z1 & b <> e & x1 <> y1 & x1 <> z1 & x1 <> e & y1 <> z1 & y1 <> e & z1 <> e & b "/\" x1 = b & b "/\" y1 = b & b "/\" z1 = b & x1 "/\" e = x1 & y1 "/\" e = y1 & z1 "/\" e = z1 & x1 "/\" y1 = b & x1 "/\" z1 = b & y1 "/\" z1 = b & x1 "\/" y1 = e & x1 "\/" z1 = e & y1 "\/" z1 = e ) take t ; ::_thesis: ( b <> x1 & b <> y1 & b <> z1 & b <> t & x1 <> y1 & x1 <> z1 & x1 <> t & y1 <> z1 & y1 <> t & z1 <> t & b "/\" x1 = b & b "/\" y1 = b & b "/\" z1 = b & x1 "/\" t = x1 & y1 "/\" t = y1 & z1 "/\" t = z1 & x1 "/\" y1 = b & x1 "/\" z1 = b & y1 "/\" z1 = b & x1 "\/" y1 = t & x1 "\/" z1 = t & y1 "\/" z1 = t ) thus b <> x1 by A14, A68, A66, A60, A57, A71, YELLOW_5:2; ::_thesis: ( b <> y1 & b <> z1 & b <> t & x1 <> y1 & x1 <> z1 & x1 <> t & y1 <> z1 & y1 <> t & z1 <> t & b "/\" x1 = b & b "/\" y1 = b & b "/\" z1 = b & x1 "/\" t = x1 & y1 "/\" t = y1 & z1 "/\" t = z1 & x1 "/\" y1 = b & x1 "/\" z1 = b & y1 "/\" z1 = b & x1 "\/" y1 = t & x1 "\/" z1 = t & y1 "\/" z1 = t ) thus b <> y1 by A14, A68, A70, A69, A72, A71, YELLOW_5:2; ::_thesis: ( b <> z1 & b <> t & x1 <> y1 & x1 <> z1 & x1 <> t & y1 <> z1 & y1 <> t & z1 <> t & b "/\" x1 = b & b "/\" y1 = b & b "/\" z1 = b & x1 "/\" t = x1 & y1 "/\" t = y1 & z1 "/\" t = z1 & x1 "/\" y1 = b & x1 "/\" z1 = b & y1 "/\" z1 = b & x1 "\/" y1 = t & x1 "\/" z1 = t & y1 "\/" z1 = t ) thus b <> z1 by A14, A56, A66, A69, A72, A57, YELLOW_5:2; ::_thesis: ( b <> t & x1 <> y1 & x1 <> z1 & x1 <> t & y1 <> z1 & y1 <> t & z1 <> t & b "/\" x1 = b & b "/\" y1 = b & b "/\" z1 = b & x1 "/\" t = x1 & y1 "/\" t = y1 & z1 "/\" t = z1 & x1 "/\" y1 = b & x1 "/\" z1 = b & y1 "/\" z1 = b & x1 "\/" y1 = t & x1 "\/" z1 = t & y1 "\/" z1 = t ) thus b <> t by A14; ::_thesis: ( x1 <> y1 & x1 <> z1 & x1 <> t & y1 <> z1 & y1 <> t & z1 <> t & b "/\" x1 = b & b "/\" y1 = b & b "/\" z1 = b & x1 "/\" t = x1 & y1 "/\" t = y1 & z1 "/\" t = z1 & x1 "/\" y1 = b & x1 "/\" z1 = b & y1 "/\" z1 = b & x1 "\/" y1 = t & x1 "\/" z1 = t & y1 "\/" z1 = t ) now__::_thesis:_not_x1_=_y1 assume A73: x1 = y1 ; ::_thesis: contradiction then x1 "/\" y1 = x1 by YELLOW_5:2; hence contradiction by A14, A68, A56, A73, YELLOW_5:1; ::_thesis: verum end; hence x1 <> y1 ; ::_thesis: ( x1 <> z1 & x1 <> t & y1 <> z1 & y1 <> t & z1 <> t & b "/\" x1 = b & b "/\" y1 = b & b "/\" z1 = b & x1 "/\" t = x1 & y1 "/\" t = y1 & z1 "/\" t = z1 & x1 "/\" y1 = b & x1 "/\" z1 = b & y1 "/\" z1 = b & x1 "\/" y1 = t & x1 "\/" z1 = t & y1 "\/" z1 = t ) now__::_thesis:_not_x1_=_z1 assume A74: x1 = z1 ; ::_thesis: contradiction then x1 "/\" z1 = x1 by YELLOW_5:2; hence contradiction by A14, A66, A70, A74, YELLOW_5:1; ::_thesis: verum end; hence x1 <> z1 ; ::_thesis: ( x1 <> t & y1 <> z1 & y1 <> t & z1 <> t & b "/\" x1 = b & b "/\" y1 = b & b "/\" z1 = b & x1 "/\" t = x1 & y1 "/\" t = y1 & z1 "/\" t = z1 & x1 "/\" y1 = b & x1 "/\" z1 = b & y1 "/\" z1 = b & x1 "\/" y1 = t & x1 "\/" z1 = t & y1 "\/" z1 = t ) thus x1 <> t by A14, A56, A70, A69, A49, A41, YELLOW_5:1; ::_thesis: ( y1 <> z1 & y1 <> t & z1 <> t & b "/\" x1 = b & b "/\" y1 = b & b "/\" z1 = b & x1 "/\" t = x1 & y1 "/\" t = y1 & z1 "/\" t = z1 & x1 "/\" y1 = b & x1 "/\" z1 = b & y1 "/\" z1 = b & x1 "\/" y1 = t & x1 "\/" z1 = t & y1 "\/" z1 = t ) now__::_thesis:_not_y1_=_z1 assume A75: y1 = z1 ; ::_thesis: contradiction then y1 "/\" z1 = y1 by YELLOW_5:2; hence contradiction by A14, A69, A60, A75, YELLOW_5:1; ::_thesis: verum end; hence y1 <> z1 ; ::_thesis: ( y1 <> t & z1 <> t & b "/\" x1 = b & b "/\" y1 = b & b "/\" z1 = b & x1 "/\" t = x1 & y1 "/\" t = y1 & z1 "/\" t = z1 & x1 "/\" y1 = b & x1 "/\" z1 = b & y1 "/\" z1 = b & x1 "\/" y1 = t & x1 "\/" z1 = t & y1 "\/" z1 = t ) thus y1 <> t by A14, A56, A66, A60, A59, A41, YELLOW_5:1; ::_thesis: ( z1 <> t & b "/\" x1 = b & b "/\" y1 = b & b "/\" z1 = b & x1 "/\" t = x1 & y1 "/\" t = y1 & z1 "/\" t = z1 & x1 "/\" y1 = b & x1 "/\" z1 = b & y1 "/\" z1 = b & x1 "\/" y1 = t & x1 "\/" z1 = t & y1 "\/" z1 = t ) thus z1 <> t by A14, A68, A70, A60, A59, A49, YELLOW_5:1; ::_thesis: ( b "/\" x1 = b & b "/\" y1 = b & b "/\" z1 = b & x1 "/\" t = x1 & y1 "/\" t = y1 & z1 "/\" t = z1 & x1 "/\" y1 = b & x1 "/\" z1 = b & y1 "/\" z1 = b & x1 "\/" y1 = t & x1 "\/" z1 = t & y1 "\/" z1 = t ) b <= x1 by A56, YELLOW_0:23; hence b "/\" x1 = b by YELLOW_5:10; ::_thesis: ( b "/\" y1 = b & b "/\" z1 = b & x1 "/\" t = x1 & y1 "/\" t = y1 & z1 "/\" t = z1 & x1 "/\" y1 = b & x1 "/\" z1 = b & y1 "/\" z1 = b & x1 "\/" y1 = t & x1 "\/" z1 = t & y1 "\/" z1 = t ) b <= y1 by A56, YELLOW_0:23; hence b "/\" y1 = b by YELLOW_5:10; ::_thesis: ( b "/\" z1 = b & x1 "/\" t = x1 & y1 "/\" t = y1 & z1 "/\" t = z1 & x1 "/\" y1 = b & x1 "/\" z1 = b & y1 "/\" z1 = b & x1 "\/" y1 = t & x1 "\/" z1 = t & y1 "\/" z1 = t ) b <= z1 by A70, YELLOW_0:23; hence b "/\" z1 = b by YELLOW_5:10; ::_thesis: ( x1 "/\" t = x1 & y1 "/\" t = y1 & z1 "/\" t = z1 & x1 "/\" y1 = b & x1 "/\" z1 = b & y1 "/\" z1 = b & x1 "\/" y1 = t & x1 "\/" z1 = t & y1 "\/" z1 = t ) x1 <= t by A68, YELLOW_0:22; hence x1 "/\" t = x1 by YELLOW_5:10; ::_thesis: ( y1 "/\" t = y1 & z1 "/\" t = z1 & x1 "/\" y1 = b & x1 "/\" z1 = b & y1 "/\" z1 = b & x1 "\/" y1 = t & x1 "\/" z1 = t & y1 "\/" z1 = t ) y1 <= t by A68, YELLOW_0:22; hence y1 "/\" t = y1 by YELLOW_5:10; ::_thesis: ( z1 "/\" t = z1 & x1 "/\" y1 = b & x1 "/\" z1 = b & y1 "/\" z1 = b & x1 "\/" y1 = t & x1 "\/" z1 = t & y1 "\/" z1 = t ) z1 <= t by A66, YELLOW_0:22; hence z1 "/\" t = z1 by YELLOW_5:10; ::_thesis: ( x1 "/\" y1 = b & x1 "/\" z1 = b & y1 "/\" z1 = b & x1 "\/" y1 = t & x1 "\/" z1 = t & y1 "\/" z1 = t ) thus x1 "/\" y1 = b by A56; ::_thesis: ( x1 "/\" z1 = b & y1 "/\" z1 = b & x1 "\/" y1 = t & x1 "\/" z1 = t & y1 "\/" z1 = t ) thus x1 "/\" z1 = b by A70; ::_thesis: ( y1 "/\" z1 = b & x1 "\/" y1 = t & x1 "\/" z1 = t & y1 "\/" z1 = t ) thus y1 "/\" z1 = b by A60; ::_thesis: ( x1 "\/" y1 = t & x1 "\/" z1 = t & y1 "\/" z1 = t ) thus x1 "\/" y1 = t by A68; ::_thesis: ( x1 "\/" z1 = t & y1 "\/" z1 = t ) thus x1 "\/" z1 = t by A66; ::_thesis: y1 "\/" z1 = t thus y1 "\/" z1 = t by A69; ::_thesis: verum end; end; hence ( ( for a, b, c, d, e being Element of L holds ( not a <> b or not a <> c or not a <> d or not a <> e or not b <> c or not b <> d or not b <> e or not c <> d or not c <> e or not d <> e or not a "/\" b = a or not a "/\" c = a or not a "/\" d = a or not b "/\" e = b or not c "/\" e = c or not d "/\" e = d or not b "/\" c = a or not b "/\" d = a or not c "/\" d = a or not b "\/" c = e or not b "\/" d = e or not c "\/" d = e ) ) implies L is distributive ) ; ::_thesis: verum end; theorem :: YELLOW11:12 for L being LATTICE st L is modular holds ( L is distributive iff for K being full Sublattice of L holds not M_3 ,K are_isomorphic ) proof let L be LATTICE; ::_thesis: ( L is modular implies ( L is distributive iff for K being full Sublattice of L holds not M_3 ,K are_isomorphic ) ) assume A1: L is modular ; ::_thesis: ( L is distributive iff for K being full Sublattice of L holds not M_3 ,K are_isomorphic ) thus ( L is distributive implies for K being full Sublattice of L holds not M_3 ,K are_isomorphic ) ::_thesis: ( ( for K being full Sublattice of L holds not M_3 ,K are_isomorphic ) implies L is distributive ) proof assume L is distributive ; ::_thesis: for K being full Sublattice of L holds not M_3 ,K are_isomorphic then for a, b, c, d, e being Element of L holds ( not a <> b or not a <> c or not a <> d or not a <> e or not b <> c or not b <> d or not b <> e or not c <> d or not c <> e or not d <> e or not a "/\" b = a or not a "/\" c = a or not a "/\" d = a or not b "/\" e = b or not c "/\" e = c or not d "/\" e = d or not b "/\" c = a or not b "/\" d = a or not c "/\" d = a or not b "\/" c = e or not b "\/" d = e or not c "\/" d = e ) by Lm3; hence for K being full Sublattice of L holds not M_3 ,K are_isomorphic by Th10; ::_thesis: verum end; assume for K being full Sublattice of L holds not M_3 ,K are_isomorphic ; ::_thesis: L is distributive then for a, b, c, d, e being Element of L holds ( not a <> b or not a <> c or not a <> d or not a <> e or not b <> c or not b <> d or not b <> e or not c <> d or not c <> e or not d <> e or not a "/\" b = a or not a "/\" c = a or not a "/\" d = a or not b "/\" e = b or not c "/\" e = c or not d "/\" e = d or not b "/\" c = a or not b "/\" d = a or not c "/\" d = a or not b "\/" c = e or not b "\/" d = e or not c "\/" d = e ) by Th10; hence L is distributive by A1, Lm3; ::_thesis: verum end; begin definition let L be non empty RelStr ; let a, b be Element of L; func[#a,b#] -> Subset of L means :Def4: :: YELLOW11:def 4 for c being Element of L holds ( c in it iff ( a <= c & c <= b ) ); existence ex b1 being Subset of L st for c being Element of L holds ( c in b1 iff ( a <= c & c <= b ) ) proof defpred S1[ set ] means ex c1 being Element of L st ( c1 = $1 & a <= c1 & c1 <= b ); consider S being set such that A1: for c being set holds ( c in S iff ( c in the carrier of L & S1[c] ) ) from XBOOLE_0:sch_1(); for c being set st c in S holds c in the carrier of L by A1; then reconsider S = S as Subset of L by TARSKI:def_3; reconsider S = S as Subset of L ; take S ; ::_thesis: for c being Element of L holds ( c in S iff ( a <= c & c <= b ) ) thus for c being Element of L holds ( c in S iff ( a <= c & c <= b ) ) ::_thesis: verum proof let c be Element of L; ::_thesis: ( c in S iff ( a <= c & c <= b ) ) thus ( c in S implies ( a <= c & c <= b ) ) ::_thesis: ( a <= c & c <= b implies c in S ) proof assume c in S ; ::_thesis: ( a <= c & c <= b ) then ex c1 being Element of L st ( c1 = c & a <= c1 & c1 <= b ) by A1; hence ( a <= c & c <= b ) ; ::_thesis: verum end; thus ( a <= c & c <= b implies c in S ) by A1; ::_thesis: verum end; end; uniqueness for b1, b2 being Subset of L st ( for c being Element of L holds ( c in b1 iff ( a <= c & c <= b ) ) ) & ( for c being Element of L holds ( c in b2 iff ( a <= c & c <= b ) ) ) holds b1 = b2 proof let x, y be Subset of L; ::_thesis: ( ( for c being Element of L holds ( c in x iff ( a <= c & c <= b ) ) ) & ( for c being Element of L holds ( c in y iff ( a <= c & c <= b ) ) ) implies x = y ) assume that A2: for c being Element of L holds ( c in x iff ( a <= c & c <= b ) ) and A3: for c being Element of L holds ( c in y iff ( a <= c & c <= b ) ) ; ::_thesis: x = y now__::_thesis:_for_c1_being_set_st_c1_in_y_holds_ c1_in_x let c1 be set ; ::_thesis: ( c1 in y implies c1 in x ) assume A4: c1 in y ; ::_thesis: c1 in x then reconsider c = c1 as Element of L ; ( c in y iff ( a <= c & c <= b ) ) by A3; hence c1 in x by A2, A4; ::_thesis: verum end; then A5: y c= x by TARSKI:def_3; now__::_thesis:_for_c1_being_set_st_c1_in_x_holds_ c1_in_y let c1 be set ; ::_thesis: ( c1 in x implies c1 in y ) assume A6: c1 in x ; ::_thesis: c1 in y then reconsider c = c1 as Element of L ; ( c in x iff ( a <= c & c <= b ) ) by A2; hence c1 in y by A3, A6; ::_thesis: verum end; then x c= y by TARSKI:def_3; hence x = y by A5, XBOOLE_0:def_10; ::_thesis: verum end; end; :: deftheorem Def4 defines [# YELLOW11:def_4_:_ for L being non empty RelStr for a, b being Element of L for b4 being Subset of L holds ( b4 = [#a,b#] iff for c being Element of L holds ( c in b4 iff ( a <= c & c <= b ) ) ); definition let L be non empty RelStr ; let IT be Subset of L; attrIT is interval means :Def5: :: YELLOW11:def 5 ex a, b being Element of L st IT = [#a,b#]; end; :: deftheorem Def5 defines interval YELLOW11:def_5_:_ for L being non empty RelStr for IT being Subset of L holds ( IT is interval iff ex a, b being Element of L st IT = [#a,b#] ); registration let L be non empty reflexive transitive RelStr ; cluster non empty interval -> directed for Element of K32( the carrier of L); coherence for b1 being Subset of L st not b1 is empty & b1 is interval holds b1 is directed proof let M be Subset of L; ::_thesis: ( not M is empty & M is interval implies M is directed ) assume A1: ( not M is empty & M is interval ) ; ::_thesis: M is directed then consider z being set such that A2: z in M by XBOOLE_0:def_1; reconsider z = z as Element of L by A2; consider a, b being Element of L such that A3: M = [#a,b#] by A1, Def5; A4: z <= b by A3, A2, Def4; a <= z by A3, A2, Def4; then A5: a <= b by A4, ORDERS_2:3; let x, y be Element of L; :: according to WAYBEL_0:def_1 ::_thesis: ( not x in M or not y in M or ex b1 being Element of the carrier of L st ( b1 in M & x <= b1 & y <= b1 ) ) assume that A6: x in M and A7: y in M ; ::_thesis: ex b1 being Element of the carrier of L st ( b1 in M & x <= b1 & y <= b1 ) take b ; ::_thesis: ( b in M & x <= b & y <= b ) b <= b ; hence b in M by A3, A5, Def4; ::_thesis: ( x <= b & y <= b ) thus ( x <= b & y <= b ) by A3, A6, A7, Def4; ::_thesis: verum end; cluster non empty interval -> filtered for Element of K32( the carrier of L); coherence for b1 being Subset of L st not b1 is empty & b1 is interval holds b1 is filtered proof let M be Subset of L; ::_thesis: ( not M is empty & M is interval implies M is filtered ) assume A8: ( not M is empty & M is interval ) ; ::_thesis: M is filtered then consider z being set such that A9: z in M by XBOOLE_0:def_1; reconsider z = z as Element of L by A9; consider a, b being Element of L such that A10: M = [#a,b#] by A8, Def5; A11: z <= b by A10, A9, Def4; a <= z by A10, A9, Def4; then A12: a <= b by A11, ORDERS_2:3; let x, y be Element of L; :: according to WAYBEL_0:def_2 ::_thesis: ( not x in M or not y in M or ex b1 being Element of the carrier of L st ( b1 in M & b1 <= x & b1 <= y ) ) assume that A13: x in M and A14: y in M ; ::_thesis: ex b1 being Element of the carrier of L st ( b1 in M & b1 <= x & b1 <= y ) take a ; ::_thesis: ( a in M & a <= x & a <= y ) a <= a ; hence a in M by A10, A12, Def4; ::_thesis: ( a <= x & a <= y ) thus ( a <= x & a <= y ) by A10, A13, A14, Def4; ::_thesis: verum end; end; registration let L be non empty RelStr ; let a, b be Element of L; cluster[#a,b#] -> interval ; coherence [#a,b#] is interval by Def5; end; theorem :: YELLOW11:13 for L being non empty reflexive transitive RelStr for a, b being Element of L holds [#a,b#] = (uparrow a) /\ (downarrow b) proof let L be non empty reflexive transitive RelStr ; ::_thesis: for a, b being Element of L holds [#a,b#] = (uparrow a) /\ (downarrow b) let a, b be Element of L; ::_thesis: [#a,b#] = (uparrow a) /\ (downarrow b) thus [#a,b#] c= (uparrow a) /\ (downarrow b) :: according to XBOOLE_0:def_10 ::_thesis: (uparrow a) /\ (downarrow b) c= [#a,b#] proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in [#a,b#] or x in (uparrow a) /\ (downarrow b) ) A1: a in {a} by TARSKI:def_1; A2: b in {b} by TARSKI:def_1; assume A3: x in [#a,b#] ; ::_thesis: x in (uparrow a) /\ (downarrow b) then reconsider y = x as Element of L ; y <= b by A3, Def4; then y in { z where z is Element of L : ex w being Element of L st ( z <= w & w in {b} ) } by A2; then A4: y in downarrow {b} by WAYBEL_0:14; a <= y by A3, Def4; then y in { z where z is Element of L : ex w being Element of L st ( z >= w & w in {a} ) } by A1; then y in uparrow {a} by WAYBEL_0:15; hence x in (uparrow a) /\ (downarrow b) by A4, XBOOLE_0:def_4; ::_thesis: verum end; thus (uparrow a) /\ (downarrow b) c= [#a,b#] ::_thesis: verum proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (uparrow a) /\ (downarrow b) or x in [#a,b#] ) assume A5: x in (uparrow a) /\ (downarrow b) ; ::_thesis: x in [#a,b#] then x in uparrow {a} by XBOOLE_0:def_4; then x in { z where z is Element of L : ex w being Element of L st ( z >= w & w in {a} ) } by WAYBEL_0:15; then consider y1 being Element of L such that A6: x = y1 and A7: ex w being Element of L st ( y1 >= w & w in {a} ) ; A8: a <= y1 by A7, TARSKI:def_1; x in downarrow {b} by A5, XBOOLE_0:def_4; then x in { z where z is Element of L : ex w being Element of L st ( z <= w & w in {b} ) } by WAYBEL_0:14; then ex y2 being Element of L st ( x = y2 & ex w being Element of L st ( y2 <= w & w in {b} ) ) ; then y1 <= b by A6, TARSKI:def_1; hence x in [#a,b#] by A6, A8, Def4; ::_thesis: verum end; end; registration let L be with_infima Poset; let a, b be Element of L; cluster subrelstr [#a,b#] -> meet-inheriting ; coherence subrelstr [#a,b#] is meet-inheriting proof let x, y be Element of L; :: according to YELLOW_0:def_16 ::_thesis: ( not x in the carrier of (subrelstr [#a,b#]) or not y in the carrier of (subrelstr [#a,b#]) or not ex_inf_of {x,y},L or "/\" ({x,y},L) in the carrier of (subrelstr [#a,b#]) ) set ab = subrelstr [#a,b#]; assume that A1: x in the carrier of (subrelstr [#a,b#]) and A2: y in the carrier of (subrelstr [#a,b#]) and ex_inf_of {x,y},L ; ::_thesis: "/\" ({x,y},L) in the carrier of (subrelstr [#a,b#]) A3: x in [#a,b#] by A1, YELLOW_0:def_15; then A4: x <= b by Def4; A5: inf {x,y} = x "/\" y by YELLOW_0:40; then inf {x,y} <= x by YELLOW_0:23; then A6: inf {x,y} <= b by A4, YELLOW_0:def_2; y in [#a,b#] by A2, YELLOW_0:def_15; then A7: a <= y by Def4; a <= x by A3, Def4; then a <= inf {x,y} by A7, A5, YELLOW_0:23; then inf {x,y} in [#a,b#] by A6, Def4; hence "/\" ({x,y},L) in the carrier of (subrelstr [#a,b#]) by YELLOW_0:def_15; ::_thesis: verum end; end; registration let L be with_suprema Poset; let a, b be Element of L; cluster subrelstr [#a,b#] -> join-inheriting ; coherence subrelstr [#a,b#] is join-inheriting proof let x, y be Element of L; :: according to YELLOW_0:def_17 ::_thesis: ( not x in the carrier of (subrelstr [#a,b#]) or not y in the carrier of (subrelstr [#a,b#]) or not ex_sup_of {x,y},L or "\/" ({x,y},L) in the carrier of (subrelstr [#a,b#]) ) set ab = subrelstr [#a,b#]; assume that A1: x in the carrier of (subrelstr [#a,b#]) and A2: y in the carrier of (subrelstr [#a,b#]) and ex_sup_of {x,y},L ; ::_thesis: "\/" ({x,y},L) in the carrier of (subrelstr [#a,b#]) A3: x in [#a,b#] by A1, YELLOW_0:def_15; then A4: a <= x by Def4; A5: sup {x,y} = x "\/" y by YELLOW_0:41; then x <= sup {x,y} by YELLOW_0:22; then A6: a <= sup {x,y} by A4, YELLOW_0:def_2; y in [#a,b#] by A2, YELLOW_0:def_15; then A7: y <= b by Def4; x <= b by A3, Def4; then sup {x,y} <= b by A7, A5, YELLOW_0:22; then sup {x,y} in [#a,b#] by A6, Def4; hence "\/" ({x,y},L) in the carrier of (subrelstr [#a,b#]) by YELLOW_0:def_15; ::_thesis: verum end; end; theorem :: YELLOW11:14 for L being LATTICE for a, b being Element of L st L is modular holds subrelstr [#b,(a "\/" b)#], subrelstr [#(a "/\" b),a#] are_isomorphic proof let L be LATTICE; ::_thesis: for a, b being Element of L st L is modular holds subrelstr [#b,(a "\/" b)#], subrelstr [#(a "/\" b),a#] are_isomorphic let a, b be Element of L; ::_thesis: ( L is modular implies subrelstr [#b,(a "\/" b)#], subrelstr [#(a "/\" b),a#] are_isomorphic ) assume A1: L is modular ; ::_thesis: subrelstr [#b,(a "\/" b)#], subrelstr [#(a "/\" b),a#] are_isomorphic defpred S1[ set , set ] means ( $2 is Element of L & ( for X, Y being Element of L st $1 = X & $2 = Y holds Y = X "/\" a ) ); A2: for x being set st x in the carrier of (subrelstr [#b,(a "\/" b)#]) holds ex y being set st ( y in the carrier of (subrelstr [#(a "/\" b),a#]) & S1[x,y] ) proof let x be set ; ::_thesis: ( x in the carrier of (subrelstr [#b,(a "\/" b)#]) implies ex y being set st ( y in the carrier of (subrelstr [#(a "/\" b),a#]) & S1[x,y] ) ) assume x in the carrier of (subrelstr [#b,(a "\/" b)#]) ; ::_thesis: ex y being set st ( y in the carrier of (subrelstr [#(a "/\" b),a#]) & S1[x,y] ) then A3: x in [#b,(a "\/" b)#] by YELLOW_0:def_15; then reconsider x1 = x as Element of L ; take y = a "/\" x1; ::_thesis: ( y in the carrier of (subrelstr [#(a "/\" b),a#]) & S1[x,y] ) x1 <= a "\/" b by A3, Def4; then y <= a "/\" (a "\/" b) by YELLOW_5:6; then A4: y <= a by LATTICE3:18; b <= x1 by A3, Def4; then a "/\" b <= y by YELLOW_5:6; then y in [#(a "/\" b),a#] by A4, Def4; hence y in the carrier of (subrelstr [#(a "/\" b),a#]) by YELLOW_0:def_15; ::_thesis: S1[x,y] thus S1[x,y] ; ::_thesis: verum end; consider f being Function of the carrier of (subrelstr [#b,(a "\/" b)#]), the carrier of (subrelstr [#(a "/\" b),a#]) such that A5: for x being set st x in the carrier of (subrelstr [#b,(a "\/" b)#]) holds S1[x,f . x] from FUNCT_2:sch_1(A2); reconsider f = f as Function of (subrelstr [#b,(a "\/" b)#]),(subrelstr [#(a "/\" b),a#]) ; take f ; :: according to WAYBEL_1:def_8 ::_thesis: f is isomorphic thus f is isomorphic ::_thesis: verum proof A6: b <= a "\/" b by YELLOW_0:22; b <= b ; then b in [#b,(a "\/" b)#] by A6, Def4; then reconsider s1 = subrelstr [#b,(b "\/" a)#] as non empty full Sublattice of L by YELLOW_0:def_15; A7: a "/\" b <= a by YELLOW_0:23; a "/\" b <= a "/\" b ; then a "/\" b in [#(a "/\" b),a#] by A7, Def4; then reconsider s2 = subrelstr [#(a "/\" b),a#] as non empty full Sublattice of L by YELLOW_0:def_15; reconsider f1 = f as Function of s1,s2 ; dom f1 = the carrier of (subrelstr [#b,(a "\/" b)#]) by FUNCT_2:def_1; then A8: dom f1 = [#b,(a "\/" b)#] by YELLOW_0:def_15; the carrier of (subrelstr [#(a "/\" b),a#]) c= rng f1 proof let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in the carrier of (subrelstr [#(a "/\" b),a#]) or y in rng f1 ) assume y in the carrier of (subrelstr [#(a "/\" b),a#]) ; ::_thesis: y in rng f1 then A9: y in [#(a "/\" b),a#] by YELLOW_0:def_15; then reconsider Y = y as Element of L ; A10: a "/\" b <= Y by A9, Def4; then (a "/\" b) "\/" b <= Y "\/" b by WAYBEL_1:2; then A11: b <= Y "\/" b by LATTICE3:17; A12: Y <= a by A9, Def4; then Y "\/" b <= a "\/" b by WAYBEL_1:2; then A13: Y "\/" b in [#b,(a "\/" b)#] by A11, Def4; then A14: Y "\/" b in the carrier of (subrelstr [#b,(a "\/" b)#]) by YELLOW_0:def_15; then reconsider f1yb = f1 . (Y "\/" b) as Element of L by A5; f1yb = (Y "\/" b) "/\" a by A5, A14 .= Y "\/" (b "/\" a) by A1, A12, Def3 .= Y by A10, YELLOW_5:8 ; hence y in rng f1 by A8, A13, FUNCT_1:def_3; ::_thesis: verum end; then A15: rng f1 = the carrier of (subrelstr [#(a "/\" b),a#]) by XBOOLE_0:def_10; A16: for x, y being Element of s1 holds ( x <= y iff f1 . x <= f1 . y ) proof let x, y be Element of s1; ::_thesis: ( x <= y iff f1 . x <= f1 . y ) A17: the carrier of s1 = [#b,(a "\/" b)#] by YELLOW_0:def_15; then x in [#b,(a "\/" b)#] ; then reconsider X = x as Element of L ; y in [#b,(a "\/" b)#] by A17; then reconsider Y = y as Element of L ; reconsider f1Y = f1 . Y as Element of L by A5; reconsider f1X = f1 . X as Element of L by A5; thus ( x <= y implies f1 . x <= f1 . y ) ::_thesis: ( f1 . x <= f1 . y implies x <= y ) proof assume x <= y ; ::_thesis: f1 . x <= f1 . y then A18: [x,y] in the InternalRel of s1 by ORDERS_2:def_5; the InternalRel of s1 c= the InternalRel of L by YELLOW_0:def_13; then A19: X <= Y by A18, ORDERS_2:def_5; A20: f1Y = Y "/\" a by A5; f1X = X "/\" a by A5; then f1X <= f1Y by A19, A20, WAYBEL_1:1; hence f1 . x <= f1 . y by YELLOW_0:60; ::_thesis: verum end; thus ( f1 . x <= f1 . y implies x <= y ) ::_thesis: verum proof assume f1 . x <= f1 . y ; ::_thesis: x <= y then A21: [(f1 . x),(f1 . y)] in the InternalRel of s2 by ORDERS_2:def_5; the InternalRel of s2 c= the InternalRel of L by YELLOW_0:def_13; then A22: f1X <= f1Y by A21, ORDERS_2:def_5; A23: f1Y = Y "/\" a by A5; A24: b <= X by A17, Def4; f1X = X "/\" a by A5; then b "\/" (a "/\" X) <= b "\/" (a "/\" Y) by A22, A23, WAYBEL_1:2; then A25: (b "\/" a) "/\" X <= b "\/" (a "/\" Y) by A1, A24, Def3; A26: X <= b "\/" a by A17, Def4; b <= Y by A17, Def4; then (b "\/" a) "/\" X <= (b "\/" a) "/\" Y by A1, A25, Def3; then A27: X <= (b "\/" a) "/\" Y by A26, YELLOW_5:10; Y <= b "\/" a by A17, Def4; then X <= Y by A27, YELLOW_5:10; hence x <= y by YELLOW_0:60; ::_thesis: verum end; end; f1 is V13() proof let x1, x2 be set ; :: according to FUNCT_1:def_4 ::_thesis: ( not x1 in K21(f1) or not x2 in K21(f1) or not f1 . x1 = f1 . x2 or x1 = x2 ) assume that A28: x1 in dom f1 and A29: x2 in dom f1 and A30: f1 . x1 = f1 . x2 ; ::_thesis: x1 = x2 reconsider X2 = x2 as Element of L by A8, A29; A31: b <= X2 by A8, A29, Def4; reconsider X1 = x1 as Element of L by A8, A28; A32: b <= X1 by A8, A28, Def4; reconsider f1X1 = f1 . X1 as Element of L by A5, A28; A33: f1X1 = X1 "/\" a by A5, A28; reconsider f1X2 = f1 . X2 as Element of L by A5, A29; A34: f1X2 = X2 "/\" a by A5, A29; A35: X2 <= a "\/" b by A8, A29, Def4; X1 <= a "\/" b by A8, A28, Def4; then X1 = (b "\/" a) "/\" X1 by YELLOW_5:10 .= b "\/" (a "/\" X2) by A1, A30, A32, A33, A34, Def3 .= (b "\/" a) "/\" X2 by A1, A31, Def3 .= X2 by A35, YELLOW_5:10 ; hence x1 = x2 ; ::_thesis: verum end; hence f is isomorphic by A15, A16, WAYBEL_0:66; ::_thesis: verum end; end; registration cluster non empty finite V58() reflexive transitive antisymmetric with_suprema with_infima for RelStr ; existence ex b1 being LATTICE st ( b1 is finite & not b1 is empty ) proof set D = {{}}; set R = the Order of {{}}; reconsider A = RelStr(# {{}}, the Order of {{}} #) as with_suprema with_infima Poset ; take A ; ::_thesis: ( A is finite & not A is empty ) thus ( A is finite & not A is empty ) ; ::_thesis: verum end; end; registration cluster finite reflexive transitive antisymmetric with_infima -> lower-bounded for RelStr ; coherence for b1 being Semilattice st b1 is finite holds b1 is lower-bounded proof let L be Semilattice; ::_thesis: ( L is finite implies L is lower-bounded ) defpred S1[ set ] means ex x being Element of L st x is_<=_than c1; A1: S1[ {} ] proof set a = the Element of L; take the Element of L ; ::_thesis: the Element of L is_<=_than {} let b be Element of L; :: according to LATTICE3:def_8 ::_thesis: ( not b in {} or the Element of L <= b ) assume b in {} ; ::_thesis: the Element of L <= b hence the Element of L <= b ; ::_thesis: verum end; A2: for x, B being set st x in the carrier of L & B c= the carrier of L & S1[B] holds S1[B \/ {x}] proof let x, B be set ; ::_thesis: ( x in the carrier of L & B c= the carrier of L & S1[B] implies S1[B \/ {x}] ) assume that A3: x in the carrier of L and B c= the carrier of L ; ::_thesis: ( not S1[B] or S1[B \/ {x}] ) reconsider y = x as Element of L by A3; given a being Element of L such that A4: a is_<=_than B ; ::_thesis: S1[B \/ {x}] take b = a "/\" y; ::_thesis: b is_<=_than B \/ {x} let c be Element of L; :: according to LATTICE3:def_8 ::_thesis: ( not c in B \/ {x} or b <= c ) A5: now__::_thesis:_(_c_in_B_implies_a_"/\"_y_<=_c_) assume c in B ; ::_thesis: a "/\" y <= c then A6: a <= c by A4, LATTICE3:def_8; a "/\" y <= a by YELLOW_0:23; hence a "/\" y <= c by A6, ORDERS_2:3; ::_thesis: verum end; A7: now__::_thesis:_(_c_in_{x}_implies_b_<=_c_) assume c in {x} ; ::_thesis: b <= c then c = y by TARSKI:def_1; hence b <= c by YELLOW_0:23; ::_thesis: verum end; assume c in B \/ {x} ; ::_thesis: b <= c hence b <= c by A5, A7, XBOOLE_0:def_3; ::_thesis: verum end; assume L is finite ; ::_thesis: L is lower-bounded then A8: the carrier of L is finite ; thus S1[ the carrier of L] from FINSET_1:sch_2(A8, A1, A2); :: according to YELLOW_0:def_4 ::_thesis: verum end; end; registration cluster finite reflexive transitive antisymmetric with_suprema with_infima -> complete for RelStr ; coherence for b1 being LATTICE st b1 is finite holds b1 is complete proof let L be LATTICE; ::_thesis: ( L is finite implies L is complete ) assume A1: L is finite ; ::_thesis: L is complete for x being Subset of L holds ex_sup_of x,L proof let x be Subset of L; ::_thesis: ex_sup_of x,L percases ( x = {} or x <> {} ) ; suppose x = {} ; ::_thesis: ex_sup_of x,L hence ex_sup_of x,L by A1, YELLOW_0:42; ::_thesis: verum end; supposeA2: x <> {} ; ::_thesis: ex_sup_of x,L x is finite by A1, FINSET_1:1; hence ex_sup_of x,L by A2, YELLOW_0:54; ::_thesis: verum end; end; end; hence L is complete by YELLOW_0:53; ::_thesis: verum end; end;