:: 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;