:: FILTER_2 semantic presentation

K121() is set
bool K121() is non empty set
K162() is set
bool K162() is non empty set
K166() is Element of bool K162()
[:K166(),K166():] is set
[:[:K166(),K166():],K166():] is set
bool [:[:K166(),K166():],K166():] is non empty set
Nat_Lattice is non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded LattStr
the carrier of Nat_Lattice is non empty set
bool K166() is non empty set
NATPLUS is non empty Element of bool K166()
[:NATPLUS,NATPLUS:] is non empty set
[:[:NATPLUS,NATPLUS:],NATPLUS:] is non empty set
bool [:[:NATPLUS,NATPLUS:],NATPLUS:] is non empty set
{} is empty set
the empty set is empty set
1 is non empty set
L is non empty set
bool L is non empty set
[:L,L:] is non empty set
[:[:L,L:],L:] is non empty set
bool [:[:L,L:],L:] is non empty set
p is non empty Element of bool L
[:p,p:] is non empty set
[:[:p,p:],p:] is non empty set
bool [:[:p,p:],p:] is non empty set
q is Relation-like [:L,L:] -defined L -valued Function-like V18([:L,L:],L) Element of bool [:[:L,L:],L:]
q || p is Relation-like Function-like set
q | [:p,p:] is Relation-like set
K is Relation-like [:p,p:] -defined p -valued Function-like V18([:p,p:],p) Element of bool [:[:p,p:],p:]
dom K is Relation-like set
[:p,p:] is Relation-like L -defined L -valued non empty Element of bool [:L,L:]
bool [:L,L:] is non empty set
b9 is Element of p
b is Element of p
K . (b9,b) is Element of p
[b9,b] is set
{b9,b} is non empty set
{b9} is non empty set
{{b9,b},{b9}} is non empty set
K . [b9,b] is set
K . (b,b9) is Element of p
[b,b9] is set
{b,b9} is non empty set
{b} is non empty set
{{b,b9},{b}} is non empty set
K . [b,b9] is set
a is Element of p
a9 is Element of p
K . (b9,b) is Element of p
[b9,b] is Element of [:p,p:]
q . [b9,b] is set
c₉ is Element of L
c is Element of L
q . (c₉,c) is Element of L
[c₉,c] is set
{c₉,c} is non empty set
{c₉} is non empty set
{{c₉,c},{c₉}} is non empty set
q . [c₉,c] is set
q . (c,c₉) is Element of L
[c,c₉] is set
{c,c₉} is non empty set
{c} is non empty set
{{c,c₉},{c}} is non empty set
q . [c,c₉] is set
[b,b9] is Element of [:p,p:]
K . [b,b9] is set
K . (b,b9) is Element of p
b9 is Element of p
K . (b9,b9) is Element of p
[b9,b9] is set
{b9,b9} is non empty set
{b9} is non empty set
{{b9,b9},{b9}} is non empty set
K . [b9,b9] is set
K . (b9,b9) is Element of p
[b9,b9] is Element of [:p,p:]
q . [b9,b9] is set
q . (b9,b9) is Element of L
q . [b9,b9] is set
b9 is Element of p
b is Element of p
a is Element of p
K . (b,a) is Element of p
[b,a] is set
{b,a} is non empty set
{b} is non empty set
{{b,a},{b}} is non empty set
K . [b,a] is set
K . (b9,(K . (b,a))) is Element of p
[b9,(K . (b,a))] is set
{b9,(K . (b,a))} is non empty set
{b9} is non empty set
{{b9,(K . (b,a))},{b9}} is non empty set
K . [b9,(K . (b,a))] is set
K . (b9,b) is Element of p
[b9,b] is set
{b9,b} is non empty set
{{b9,b},{b9}} is non empty set
K . [b9,b] is set
K . ((K . (b9,b)),a) is Element of p
[(K . (b9,b)),a] is set
{(K . (b9,b)),a} is non empty set
{(K . (b9,b))} is non empty set
{{(K . (b9,b)),a},{(K . (b9,b))}} is non empty set
K . [(K . (b9,b)),a] is set
a9 is Element of p
c₉ is Element of p
c is Element of p
K . (b9,b) is Element of p
K . ((K . (b9,b)),a) is Element of p
[(K . (b9,b)),a] is set
{(K . (b9,b)),a} is non empty set
{(K . (b9,b))} is non empty set
{{(K . (b9,b)),a},{(K . (b9,b))}} is non empty set
K . [(K . (b9,b)),a] is set
[(K . (b9,b)),a] is Element of [:p,p:]
q . [(K . (b9,b)),a] is set
[b9,b] is Element of [:p,p:]
q . [b9,b] is set
[(q . [b9,b]),a] is set
{(q . [b9,b]),a} is non empty set
{(q . [b9,b])} is non empty set
{{(q . [b9,b]),a},{(q . [b9,b])}} is non empty set
q . [(q . [b9,b]),a] is set
q . (b9,b) is set
q . [b9,b] is set
q . ((q . (b9,b)),a) is set
[(q . (b9,b)),a] is set
{(q . (b9,b)),a} is non empty set
{(q . (b9,b))} is non empty set
{{(q . (b9,b)),a},{(q . (b9,b))}} is non empty set
q . [(q . (b9,b)),a] is set
d is Element of L
d9 is Element of L
e9 is Element of L
q . (d9,e9) is Element of L
[d9,e9] is set
{d9,e9} is non empty set
{d9} is non empty set
{{d9,e9},{d9}} is non empty set
q . [d9,e9] is set
q . (d,(q . (d9,e9))) is Element of L
[d,(q . (d9,e9))] is set
{d,(q . (d9,e9))} is non empty set
{d} is non empty set
{{d,(q . (d9,e9))},{d}} is non empty set
q . [d,(q . (d9,e9))] is set
[b,a] is Element of [:p,p:]
K . [b,a] is set
q . (b9,(K . [b,a])) is set
[b9,(K . [b,a])] is set
{b9,(K . [b,a])} is non empty set
{{b9,(K . [b,a])},{b9}} is non empty set
q . [b9,(K . [b,a])] is set
K . (b,a) is Element of p
[b9,(K . (b,a))] is Element of [:p,p:]
{b9,(K . (b,a))} is non empty set
{{b9,(K . (b,a))},{b9}} is non empty set
q . [b9,(K . (b,a))] is set
K . (b9,(K . (b,a))) is Element of p
[b9,(K . (b,a))] is set
K . [b9,(K . (b,a))] is set
L is non empty set
bool L is non empty set
[:L,L:] is non empty set
[:[:L,L:],L:] is non empty set
bool [:[:L,L:],L:] is non empty set
p is non empty Element of bool L
[:p,p:] is non empty set
[:[:p,p:],p:] is non empty set
bool [:[:p,p:],p:] is non empty set
q is Relation-like [:L,L:] -defined L -valued Function-like V18([:L,L:],L) Element of bool [:[:L,L:],L:]
q || p is Relation-like Function-like set
q | [:p,p:] is Relation-like set
K is Relation-like [:p,p:] -defined p -valued Function-like V18([:p,p:],p) Element of bool [:[:p,p:],p:]
b9 is Element of L
b is Element of p
dom K is Relation-like set
[:p,p:] is Relation-like L -defined L -valued non empty Element of bool [:L,L:]
bool [:L,L:] is non empty set
a is Element of p
K . (b,a) is Element of p
[b,a] is set
{b,a} is non empty set
{b} is non empty set
{{b,a},{b}} is non empty set
K . [b,a] is set
K . (b,a) is Element of p
[b,a] is Element of [:p,p:]
q . [b,a] is set
q . (b9,a) is Element of L
[b9,a] is set
{b9,a} is non empty set
{b9} is non empty set
{{b9,a},{b9}} is non empty set
q . [b9,a] is set
a is Element of p
K . (a,b) is Element of p
[a,b] is set
{a,b} is non empty set
{a} is non empty set
{{a,b},{a}} is non empty set
K . [a,b] is set
K . (a,b) is Element of p
[a,b] is Element of [:p,p:]
q . [a,b] is set
q . (a,b9) is Element of L
[a,b9] is set
{a,b9} is non empty set
{{a,b9},{a}} is non empty set
q . [a,b9] is set
a is Element of p
[b,a] is Element of [:p,p:]
{b,a} is non empty set
{b} is non empty set
{{b,a},{b}} is non empty set
K . (b,a) is Element of p
[b,a] is set
K . [b,a] is set
a9 is Element of L
q . (b9,a9) is Element of L
[b9,a9] is set
{b9,a9} is non empty set
{b9} is non empty set
{{b9,a9},{b9}} is non empty set
q . [b9,a9] is set
[a,b] is Element of [:p,p:]
{a,b} is non empty set
{a} is non empty set
{{a,b},{a}} is non empty set
K . (a,b) is Element of p
[a,b] is set
K . [a,b] is set
q . (a9,b9) is Element of L
[a9,b9] is set
{a9,b9} is non empty set
{a9} is non empty set
{{a9,b9},{a9}} is non empty set
q . [a9,b9] is set
L is non empty set
bool L is non empty set
[:L,L:] is non empty set
[:[:L,L:],L:] is non empty set
bool [:[:L,L:],L:] is non empty set
p is non empty Element of bool L
[:p,p:] is non empty set
[:[:p,p:],p:] is non empty set
bool [:[:p,p:],p:] is non empty set
q is Relation-like [:L,L:] -defined L -valued Function-like V18([:L,L:],L) Element of bool [:[:L,L:],L:]
q || p is Relation-like Function-like set
q | [:p,p:] is Relation-like set
K is Relation-like [:L,L:] -defined L -valued Function-like V18([:L,L:],L) Element of bool [:[:L,L:],L:]
K || p is Relation-like Function-like set
K | [:p,p:] is Relation-like set
b9 is Relation-like [:p,p:] -defined p -valued Function-like V18([:p,p:],p) Element of bool [:[:p,p:],p:]
b is Relation-like [:p,p:] -defined p -valued Function-like V18([:p,p:],p) Element of bool [:[:p,p:],p:]
a is Element of p
a9 is Element of p
c₉ is Element of p
b . (a9,c₉) is Element of p
[a9,c₉] is set
{a9,c₉} is non empty set
{a9} is non empty set
{{a9,c₉},{a9}} is non empty set
b . [a9,c₉] is set
b9 . (a,(b . (a9,c₉))) is Element of p
[a,(b . (a9,c₉))] is set
{a,(b . (a9,c₉))} is non empty set
{a} is non empty set
{{a,(b . (a9,c₉))},{a}} is non empty set
b9 . [a,(b . (a9,c₉))] is set
b9 . (a,a9) is Element of p
[a,a9] is set
{a,a9} is non empty set
{{a,a9},{a}} is non empty set
b9 . [a,a9] is set
b9 . (a,c₉) is Element of p
[a,c₉] is set
{a,c₉} is non empty set
{{a,c₉},{a}} is non empty set
b9 . [a,c₉] is set
b . ((b9 . (a,a9)),(b9 . (a,c₉))) is Element of p
[(b9 . (a,a9)),(b9 . (a,c₉))] is set
{(b9 . (a,a9)),(b9 . (a,c₉))} is non empty set
{(b9 . (a,a9))} is non empty set
{{(b9 . (a,a9)),(b9 . (a,c₉))},{(b9 . (a,a9))}} is non empty set
b . [(b9 . (a,a9)),(b9 . (a,c₉))] is set
b9 . (a,a9) is Element of p
b9 . (a,c₉) is Element of p
b . (a9,c₉) is Element of p
e is Element of L
ae is Element of L
[e,ae] is Element of [:L,L:]
{e,ae} is non empty set
{e} is non empty set
{{e,ae},{e}} is non empty set
K . [e,ae] is set
K . (e,ae) is Element of L
[e,ae] is set
K . [e,ae] is set
e9 is Element of L
[e9,e] is Element of [:L,L:]
{e9,e} is non empty set
{e9} is non empty set
{{e9,e},{e9}} is non empty set
q . [e9,e] is set
q . (e9,e) is Element of L
[e9,e] is set
q . [e9,e] is set
[e9,ae] is Element of [:L,L:]
{e9,ae} is non empty set
{{e9,ae},{e9}} is non empty set
q . [e9,ae] is set
q . (e9,ae) is Element of L
[e9,ae] is set
q . [e9,ae] is set
[a,(b . (a9,c₉))] is Element of [:p,p:]
{a,(b . (a9,c₉))} is non empty set
{{a,(b . (a9,c₉))},{a}} is non empty set
q . [a,(b . (a9,c₉))] is set
q . (a,(b . (a9,c₉))) is set
[a,(b . (a9,c₉))] is set
q . [a,(b . (a9,c₉))] is set
dom b is Relation-like set
[:p,p:] is Relation-like L -defined L -valued non empty Element of bool [:L,L:]
bool [:L,L:] is non empty set
[a9,c₉] is Element of [:p,p:]
b . [a9,c₉] is set
K . [a9,c₉] is set
[(b9 . (a,a9)),(b9 . (a,c₉))] is Element of [:p,p:]
{(b9 . (a,a9)),(b9 . (a,c₉))} is non empty set
{(b9 . (a,a9))} is non empty set
{{(b9 . (a,a9)),(b9 . (a,c₉))},{(b9 . (a,a9))}} is non empty set
b . [(b9 . (a,a9)),(b9 . (a,c₉))] is set
K . [(b9 . (a,a9)),(b9 . (a,c₉))] is set
K . ((b9 . (a,a9)),(b9 . (a,c₉))) is set
[(b9 . (a,a9)),(b9 . (a,c₉))] is set
K . [(b9 . (a,a9)),(b9 . (a,c₉))] is set
dom b9 is Relation-like set
b9 . [a,(b . (a9,c₉))] is set
[a,a9] is Element of [:p,p:]
b9 . [a,a9] is set
q . [a,a9] is set
[a,c₉] is Element of [:p,p:]
b9 . [a,c₉] is set
q . [a,c₉] is set
b9 . (a,(b . (a9,c₉))) is Element of p
b9 . [a,(b . (a9,c₉))] is set
b . ((b9 . (a,a9)),(b9 . (a,c₉))) is Element of p
b . [(b9 . (a,a9)),(b9 . (a,c₉))] is set
a is Element of p
a9 is Element of p
b . (a,a9) is Element of p
[a,a9] is set
{a,a9} is non empty set
{a} is non empty set
{{a,a9},{a}} is non empty set
b . [a,a9] is set
c₉ is Element of p
b9 . ((b . (a,a9)),c₉) is Element of p
[(b . (a,a9)),c₉] is set
{(b . (a,a9)),c₉} is non empty set
{(b . (a,a9))} is non empty set
{{(b . (a,a9)),c₉},{(b . (a,a9))}} is non empty set
b9 . [(b . (a,a9)),c₉] is set
b9 . (a,c₉) is Element of p
[a,c₉] is set
{a,c₉} is non empty set
{{a,c₉},{a}} is non empty set
b9 . [a,c₉] is set
b9 . (a9,c₉) is Element of p
[a9,c₉] is set
{a9,c₉} is non empty set
{a9} is non empty set
{{a9,c₉},{a9}} is non empty set
b9 . [a9,c₉] is set
b . ((b9 . (a,c₉)),(b9 . (a9,c₉))) is Element of p
[(b9 . (a,c₉)),(b9 . (a9,c₉))] is set
{(b9 . (a,c₉)),(b9 . (a9,c₉))} is non empty set
{(b9 . (a,c₉))} is non empty set
{{(b9 . (a,c₉)),(b9 . (a9,c₉))},{(b9 . (a,c₉))}} is non empty set
b . [(b9 . (a,c₉)),(b9 . (a9,c₉))] is set
b . (a,a9) is Element of p
b9 . (a,c₉) is Element of p
b9 . (a9,c₉) is Element of p
[(b9 . (a,c₉)),(b9 . (a9,c₉))] is Element of [:p,p:]
{(b9 . (a,c₉)),(b9 . (a9,c₉))} is non empty set
{(b9 . (a,c₉))} is non empty set
{{(b9 . (a,c₉)),(b9 . (a9,c₉))},{(b9 . (a,c₉))}} is non empty set
K . [(b9 . (a,c₉)),(b9 . (a9,c₉))] is set
K . ((b9 . (a,c₉)),(b9 . (a9,c₉))) is set
[(b9 . (a,c₉)),(b9 . (a9,c₉))] is set
K . [(b9 . (a,c₉)),(b9 . (a9,c₉))] is set
dom b9 is Relation-like set
[:p,p:] is Relation-like L -defined L -valued non empty Element of bool [:L,L:]
bool [:L,L:] is non empty set
[(b . (a,a9)),c₉] is Element of [:p,p:]
{(b . (a,a9)),c₉} is non empty set
{(b . (a,a9))} is non empty set
{{(b . (a,a9)),c₉},{(b . (a,a9))}} is non empty set
b9 . [(b . (a,a9)),c₉] is set
q . [(b . (a,a9)),c₉] is set
dom b is Relation-like set
[a,a9] is Element of [:p,p:]
b . [a,a9] is set
K . [a,a9] is set
b . [(b9 . (a,c₉)),(b9 . (a9,c₉))] is set
e is Element of L
ae is Element of L
[e,ae] is Element of [:L,L:]
{e,ae} is non empty set
{e} is non empty set
{{e,ae},{e}} is non empty set
q . [e,ae] is set
q . (e,ae) is Element of L
[e,ae] is set
q . [e,ae] is set
e9 is Element of L
[e9,e] is Element of [:L,L:]
{e9,e} is non empty set
{e9} is non empty set
{{e9,e},{e9}} is non empty set
K . [e9,e] is set
K . (e9,e) is Element of L
[e9,e] is set
K . [e9,e] is set
[e9,ae] is Element of [:L,L:]
{e9,ae} is non empty set
{{e9,ae},{e9}} is non empty set
q . [e9,ae] is set
q . (e9,ae) is Element of L
[e9,ae] is set
q . [e9,ae] is set
q . ((b . (a,a9)),c₉) is set
[(b . (a,a9)),c₉] is set
q . [(b . (a,a9)),c₉] is set
[a9,c₉] is Element of [:p,p:]
b9 . [a9,c₉] is set
q . [a9,c₉] is set
[a,c₉] is Element of [:p,p:]
b9 . [a,c₉] is set
q . [a,c₉] is set
b9 . ((b . (a,a9)),c₉) is Element of p
b9 . [(b . (a,a9)),c₉] is set
b . ((b9 . (a,c₉)),(b9 . (a9,c₉))) is Element of p
b . [(b9 . (a,c₉)),(b9 . (a9,c₉))] is set
L is non empty set
bool L is non empty set
[:L,L:] is non empty set
[:[:L,L:],L:] is non empty set
bool [:[:L,L:],L:] is non empty set
p is non empty Element of bool L
[:p,p:] is non empty set
[:[:p,p:],p:] is non empty set
bool [:[:p,p:],p:] is non empty set
q is Relation-like [:L,L:] -defined L -valued Function-like V18([:L,L:],L) Element of bool [:[:L,L:],L:]
q || p is Relation-like Function-like set
q | [:p,p:] is Relation-like set
K is Relation-like [:L,L:] -defined L -valued Function-like V18([:L,L:],L) Element of bool [:[:L,L:],L:]
K || p is Relation-like Function-like set
K | [:p,p:] is Relation-like set
b9 is Relation-like [:p,p:] -defined p -valued Function-like V18([:p,p:],p) Element of bool [:[:p,p:],p:]
b is Relation-like [:p,p:] -defined p -valued Function-like V18([:p,p:],p) Element of bool [:[:p,p:],p:]
L is non empty set
bool L is non empty set
[:L,L:] is non empty set
[:[:L,L:],L:] is non empty set
bool [:[:L,L:],L:] is non empty set
p is non empty Element of bool L
[:p,p:] is non empty set
[:[:p,p:],p:] is non empty set
bool [:[:p,p:],p:] is non empty set
q is Relation-like [:L,L:] -defined L -valued Function-like V18([:L,L:],L) Element of bool [:[:L,L:],L:]
q || p is Relation-like Function-like set
q | [:p,p:] is Relation-like set
K is Relation-like [:L,L:] -defined L -valued Function-like V18([:L,L:],L) Element of bool [:[:L,L:],L:]
K || p is Relation-like Function-like set
K | [:p,p:] is Relation-like set
b9 is Relation-like [:p,p:] -defined p -valued Function-like V18([:p,p:],p) Element of bool [:[:p,p:],p:]
b is Relation-like [:p,p:] -defined p -valued Function-like V18([:p,p:],p) Element of bool [:[:p,p:],p:]
a is Element of p
a9 is Element of p
b . (a,a9) is Element of p
[a,a9] is set
{a,a9} is non empty set
{a} is non empty set
{{a,a9},{a}} is non empty set
b . [a,a9] is set
b9 . (a,(b . (a,a9))) is Element of p
[a,(b . (a,a9))] is set
{a,(b . (a,a9))} is non empty set
{{a,(b . (a,a9))},{a}} is non empty set
b9 . [a,(b . (a,a9))] is set
dom b is Relation-like set
[:p,p:] is Relation-like L -defined L -valued non empty Element of bool [:L,L:]
bool [:L,L:] is non empty set
dom b9 is Relation-like set
[a,(b . (a,a9))] is Element of [:p,p:]
q . [a,(b . (a,a9))] is set
[a,a9] is Element of [:p,p:]
K . [a,a9] is set
[a,(K . [a,a9])] is set
{a,(K . [a,a9])} is non empty set
{{a,(K . [a,a9])},{a}} is non empty set
q . [a,(K . [a,a9])] is set
K . (a,a9) is Element of L
K . [a,a9] is set
q . (a,(K . (a,a9))) is Element of L
[a,(K . (a,a9))] is set
{a,(K . (a,a9))} is non empty set
{{a,(K . (a,a9))},{a}} is non empty set
q . [a,(K . (a,a9))] is set
L is non empty set
bool L is non empty set
p is Element of bool L
q is Element of bool L
K is Element of L
b9 is Element of L
L is LattStr
the carrier of L is set
the L_join of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
[: the carrier of L, the carrier of L:] is set
[:[: the carrier of L, the carrier of L:], the carrier of L:] is set
bool [:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
the L_meet of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
LattStr(# the carrier of L, the L_join of L, the L_meet of L #) is strict LattStr
p is LattStr
the carrier of p is set
the L_join of p is Relation-like [: the carrier of p, the carrier of p:] -defined the carrier of p -valued Function-like V18([: the carrier of p, the carrier of p:], the carrier of p) Element of bool [:[: the carrier of p, the carrier of p:], the carrier of p:]
[: the carrier of p, the carrier of p:] is set
[:[: the carrier of p, the carrier of p:], the carrier of p:] is set
bool [:[: the carrier of p, the carrier of p:], the carrier of p:] is non empty set
the L_meet of p is Relation-like [: the carrier of p, the carrier of p:] -defined the carrier of p -valued Function-like V18([: the carrier of p, the carrier of p:], the carrier of p) Element of bool [:[: the carrier of p, the carrier of p:], the carrier of p:]
LattStr(# the carrier of p, the L_join of p, the L_meet of p #) is strict LattStr
L .: is strict LattStr
LattStr(# the carrier of L, the L_meet of L, the L_join of L #) is strict LattStr
p .: is strict LattStr
LattStr(# the carrier of p, the L_meet of p, the L_join of p #) is strict LattStr
L is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
L .: is non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the carrier of L is non empty set
the L_meet of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
[: the carrier of L, the carrier of L:] is non empty set
[:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
bool [:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
the L_join of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
LattStr(# the carrier of L, the L_meet of L, the L_join of L #) is non empty strict LattStr
(L .:) .: is non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the carrier of (L .:) is non empty set
the L_meet of (L .:) is Relation-like [: the carrier of (L .:), the carrier of (L .:):] -defined the carrier of (L .:) -valued Function-like V18([: the carrier of (L .:), the carrier of (L .:):], the carrier of (L .:)) commutative associative idempotent Element of bool [:[: the carrier of (L .:), the carrier of (L .:):], the carrier of (L .:):]
[: the carrier of (L .:), the carrier of (L .:):] is non empty set
[:[: the carrier of (L .:), the carrier of (L .:):], the carrier of (L .:):] is non empty set
bool [:[: the carrier of (L .:), the carrier of (L .:):], the carrier of (L .:):] is non empty set
the L_join of (L .:) is Relation-like [: the carrier of (L .:), the carrier of (L .:):] -defined the carrier of (L .:) -valued Function-like V18([: the carrier of (L .:), the carrier of (L .:):], the carrier of (L .:)) commutative associative idempotent Element of bool [:[: the carrier of (L .:), the carrier of (L .:):], the carrier of (L .:):]
LattStr(# the carrier of (L .:), the L_meet of (L .:), the L_join of (L .:) #) is non empty strict LattStr
LattStr(# the carrier of L, the L_join of L, the L_meet of L #) is non empty strict LattStr
L is non empty LattStr
the carrier of L is non empty set
the L_join of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
[: the carrier of L, the carrier of L:] is non empty set
[:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
bool [:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
the L_meet of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
LattStr(# the carrier of L, the L_join of L, the L_meet of L #) is non empty strict LattStr
p is non empty LattStr
the carrier of p is non empty set
the L_join of p is Relation-like [: the carrier of p, the carrier of p:] -defined the carrier of p -valued Function-like V18([: the carrier of p, the carrier of p:], the carrier of p) Element of bool [:[: the carrier of p, the carrier of p:], the carrier of p:]
[: the carrier of p, the carrier of p:] is non empty set
[:[: the carrier of p, the carrier of p:], the carrier of p:] is non empty set
bool [:[: the carrier of p, the carrier of p:], the carrier of p:] is non empty set
the L_meet of p is Relation-like [: the carrier of p, the carrier of p:] -defined the carrier of p -valued Function-like V18([: the carrier of p, the carrier of p:], the carrier of p) Element of bool [:[: the carrier of p, the carrier of p:], the carrier of p:]
LattStr(# the carrier of p, the L_join of p, the L_meet of p #) is non empty strict LattStr
q is Element of the carrier of L
K is Element of the carrier of L
q "\/" K is Element of the carrier of L
the L_join of L . (q,K) is Element of the carrier of L
[q,K] is set
{q,K} is non empty set
{q} is non empty set
{{q,K},{q}} is non empty set
the L_join of L . [q,K] is set
q "/\" K is Element of the carrier of L
the L_meet of L . (q,K) is Element of the carrier of L
the L_meet of L . [q,K] is set
b9 is Element of the carrier of p
b is Element of the carrier of p
b9 "\/" b is Element of the carrier of p
the L_join of p . (b9,b) is Element of the carrier of p
[b9,b] is set
{b9,b} is non empty set
{b9} is non empty set
{{b9,b},{b9}} is non empty set
the L_join of p . [b9,b] is set
b9 "/\" b is Element of the carrier of p
the L_meet of p . (b9,b) is Element of the carrier of p
the L_meet of p . [b9,b] is set
L is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded LattStr
the carrier of L is non empty set
the L_join of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
[: the carrier of L, the carrier of L:] is non empty set
[:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
bool [:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
the L_meet of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
LattStr(# the carrier of L, the L_join of L, the L_meet of L #) is non empty strict LattStr
p is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded LattStr
the carrier of p is non empty set
the L_join of p is Relation-like [: the carrier of p, the carrier of p:] -defined the carrier of p -valued Function-like V18([: the carrier of p, the carrier of p:], the carrier of p) commutative associative idempotent Element of bool [:[: the carrier of p, the carrier of p:], the carrier of p:]
[: the carrier of p, the carrier of p:] is non empty set
[:[: the carrier of p, the carrier of p:], the carrier of p:] is non empty set
bool [:[: the carrier of p, the carrier of p:], the carrier of p:] is non empty set
the L_meet of p is Relation-like [: the carrier of p, the carrier of p:] -defined the carrier of p -valued Function-like V18([: the carrier of p, the carrier of p:], the carrier of p) commutative associative idempotent Element of bool [:[: the carrier of p, the carrier of p:], the carrier of p:]
LattStr(# the carrier of p, the L_join of p, the L_meet of p #) is non empty strict LattStr
Bottom L is Element of the carrier of L
Bottom p is Element of the carrier of p
K is Element of the carrier of p
b9 is Element of the carrier of L
(Bottom L) "/\" b9 is Element of the carrier of L
the L_meet of L . ((Bottom L),b9) is Element of the carrier of L
[(Bottom L),b9] is set
{(Bottom L),b9} is non empty set
{(Bottom L)} is non empty set
{{(Bottom L),b9},{(Bottom L)}} is non empty set
the L_meet of L . [(Bottom L),b9] is set
q is Element of the carrier of p
q "/\" K is Element of the carrier of p
the L_meet of p . (q,K) is Element of the carrier of p
[q,K] is set
{q,K} is non empty set
{q} is non empty set
{{q,K},{q}} is non empty set
the L_meet of p . [q,K] is set
K "/\" q is Element of the carrier of p
the L_meet of p . (K,q) is Element of the carrier of p
[K,q] is set
{K,q} is non empty set
{K} is non empty set
{{K,q},{K}} is non empty set
the L_meet of p . [K,q] is set
L is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like upper-bounded LattStr
the carrier of L is non empty set
the L_join of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
[: the carrier of L, the carrier of L:] is non empty set
[:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
bool [:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
the L_meet of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
LattStr(# the carrier of L, the L_join of L, the L_meet of L #) is non empty strict LattStr
p is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like upper-bounded LattStr
the carrier of p is non empty set
the L_join of p is Relation-like [: the carrier of p, the carrier of p:] -defined the carrier of p -valued Function-like V18([: the carrier of p, the carrier of p:], the carrier of p) commutative associative idempotent Element of bool [:[: the carrier of p, the carrier of p:], the carrier of p:]
[: the carrier of p, the carrier of p:] is non empty set
[:[: the carrier of p, the carrier of p:], the carrier of p:] is non empty set
bool [:[: the carrier of p, the carrier of p:], the carrier of p:] is non empty set
the L_meet of p is Relation-like [: the carrier of p, the carrier of p:] -defined the carrier of p -valued Function-like V18([: the carrier of p, the carrier of p:], the carrier of p) commutative associative idempotent Element of bool [:[: the carrier of p, the carrier of p:], the carrier of p:]
LattStr(# the carrier of p, the L_join of p, the L_meet of p #) is non empty strict LattStr
Top L is Element of the carrier of L
Top p is Element of the carrier of p
K is Element of the carrier of p
b9 is Element of the carrier of L
(Top L) "\/" b9 is Element of the carrier of L
the L_join of L . ((Top L),b9) is Element of the carrier of L
[(Top L),b9] is set
{(Top L),b9} is non empty set
{(Top L)} is non empty set
{{(Top L),b9},{(Top L)}} is non empty set
the L_join of L . [(Top L),b9] is set
q is Element of the carrier of p
q "\/" K is Element of the carrier of p
the L_join of p . (q,K) is Element of the carrier of p
[q,K] is set
{q,K} is non empty set
{q} is non empty set
{{q,K},{q}} is non empty set
the L_join of p . [q,K] is set
K "\/" q is Element of the carrier of p
the L_join of p . (K,q) is Element of the carrier of p
[K,q] is set
{K,q} is non empty set
{K} is non empty set
{{K,q},{K}} is non empty set
the L_join of p . [K,q] is set
L is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded upper-bounded bounded complemented LattStr
the carrier of L is non empty set
the L_join of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
[: the carrier of L, the carrier of L:] is non empty set
[:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
bool [:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
the L_meet of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
LattStr(# the carrier of L, the L_join of L, the L_meet of L #) is non empty strict LattStr
p is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded upper-bounded bounded complemented LattStr
the carrier of p is non empty set
the L_join of p is Relation-like [: the carrier of p, the carrier of p:] -defined the carrier of p -valued Function-like V18([: the carrier of p, the carrier of p:], the carrier of p) commutative associative idempotent Element of bool [:[: the carrier of p, the carrier of p:], the carrier of p:]
[: the carrier of p, the carrier of p:] is non empty set
[:[: the carrier of p, the carrier of p:], the carrier of p:] is non empty set
bool [:[: the carrier of p, the carrier of p:], the carrier of p:] is non empty set
the L_meet of p is Relation-like [: the carrier of p, the carrier of p:] -defined the carrier of p -valued Function-like V18([: the carrier of p, the carrier of p:], the carrier of p) commutative associative idempotent Element of bool [:[: the carrier of p, the carrier of p:], the carrier of p:]
LattStr(# the carrier of p, the L_join of p, the L_meet of p #) is non empty strict LattStr
q is Element of the carrier of L
K is Element of the carrier of L
b9 is Element of the carrier of p
b is Element of the carrier of p
b9 "\/" b is Element of the carrier of p
the L_join of p . (b9,b) is Element of the carrier of p
[b9,b] is set
{b9,b} is non empty set
{b9} is non empty set
{{b9,b},{b9}} is non empty set
the L_join of p . [b9,b] is set
q "\/" K is Element of the carrier of L
the L_join of L . (q,K) is Element of the carrier of L
[q,K] is set
{q,K} is non empty set
{q} is non empty set
{{q,K},{q}} is non empty set
the L_join of L . [q,K] is set
b9 "/\" b is Element of the carrier of p
the L_meet of p . (b9,b) is Element of the carrier of p
the L_meet of p . [b9,b] is set
q "/\" K is Element of the carrier of L
the L_meet of L . (q,K) is Element of the carrier of L
the L_meet of L . [q,K] is set
Top L is Element of the carrier of L
Bottom L is Element of the carrier of L
Top p is Element of the carrier of p
b "\/" b9 is Element of the carrier of p
the L_join of p . (b,b9) is Element of the carrier of p
[b,b9] is set
{b,b9} is non empty set
{b} is non empty set
{{b,b9},{b}} is non empty set
the L_join of p . [b,b9] is set
Bottom p is Element of the carrier of p
b "/\" b9 is Element of the carrier of p
the L_meet of p . (b,b9) is Element of the carrier of p
the L_meet of p . [b,b9] is set
L is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded upper-bounded bounded complemented Boolean implicative Heyting LattStr
the carrier of L is non empty set
the L_join of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
[: the carrier of L, the carrier of L:] is non empty set
[:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
bool [:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
the L_meet of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
LattStr(# the carrier of L, the L_join of L, the L_meet of L #) is non empty strict LattStr
p is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded upper-bounded bounded complemented Boolean implicative Heyting LattStr
the carrier of p is non empty set
the L_join of p is Relation-like [: the carrier of p, the carrier of p:] -defined the carrier of p -valued Function-like V18([: the carrier of p, the carrier of p:], the carrier of p) commutative associative idempotent Element of bool [:[: the carrier of p, the carrier of p:], the carrier of p:]
[: the carrier of p, the carrier of p:] is non empty set
[:[: the carrier of p, the carrier of p:], the carrier of p:] is non empty set
bool [:[: the carrier of p, the carrier of p:], the carrier of p:] is non empty set
the L_meet of p is Relation-like [: the carrier of p, the carrier of p:] -defined the carrier of p -valued Function-like V18([: the carrier of p, the carrier of p:], the carrier of p) commutative associative idempotent Element of bool [:[: the carrier of p, the carrier of p:], the carrier of p:]
LattStr(# the carrier of p, the L_join of p, the L_meet of p #) is non empty strict LattStr
q is Element of the carrier of L
q ` is Element of the carrier of L
K is Element of the carrier of p
K ` is Element of the carrier of p
b9 is Element of the carrier of p
L is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the carrier of L is non empty set
bool the carrier of L is non empty set
p is Element of bool the carrier of L
q is Element of the carrier of L
K is Element of the carrier of L
q "\/" K is Element of the carrier of L
the L_join of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
[: the carrier of L, the carrier of L:] is non empty set
[:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
bool [:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
the L_join of L . (q,K) is Element of the carrier of L
[q,K] is set
{q,K} is non empty set
{q} is non empty set
{{q,K},{q}} is non empty set
the L_join of L . [q,K] is set
q "/\" (q "\/" K) is Element of the carrier of L
the L_meet of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
the L_meet of L . (q,(q "\/" K)) is Element of the carrier of L
[q,(q "\/" K)] is set
{q,(q "\/" K)} is non empty set
{{q,(q "\/" K)},{q}} is non empty set
the L_meet of L . [q,(q "\/" K)] is set
L is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the carrier of L is non empty set
bool the carrier of L is non empty set
p is Element of bool the carrier of L
q is Element of the carrier of L
K is Element of the carrier of L
q "/\" K is Element of the carrier of L
the L_meet of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
[: the carrier of L, the carrier of L:] is non empty set
[:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
bool [:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
the L_meet of L . (q,K) is Element of the carrier of L
[q,K] is set
{q,K} is non empty set
{q} is non empty set
{{q,K},{q}} is non empty set
the L_meet of L . [q,K] is set
(q "/\" K) "\/" K is Element of the carrier of L
the L_join of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
the L_join of L . ((q "/\" K),K) is Element of the carrier of L
[(q "/\" K),K] is set
{(q "/\" K),K} is non empty set
{(q "/\" K)} is non empty set
{{(q "/\" K),K},{(q "/\" K)}} is non empty set
the L_join of L . [(q "/\" K),K] is set
L is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the carrier of L is non empty set
bool the carrier of L is non empty set
p is non empty Element of bool the carrier of L
q is Element of the carrier of L
K is Element of the carrier of L
q "\/" K is Element of the carrier of L
the L_join of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
[: the carrier of L, the carrier of L:] is non empty set
[:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
bool [:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
the L_join of L . (q,K) is Element of the carrier of L
[q,K] is set
{q,K} is non empty set
{q} is non empty set
{{q,K},{q}} is non empty set
the L_join of L . [q,K] is set
q is Element of the carrier of L
K is Element of the carrier of L
q "\/" K is Element of the carrier of L
the L_join of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
[: the carrier of L, the carrier of L:] is non empty set
[:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
bool [:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
the L_join of L . (q,K) is Element of the carrier of L
[q,K] is set
{q,K} is non empty set
{q} is non empty set
{{q,K},{q}} is non empty set
the L_join of L . [q,K] is set
L is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the carrier of L is non empty set
the L_join of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
[: the carrier of L, the carrier of L:] is non empty set
[:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
bool [:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
the L_meet of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
LattStr(# the carrier of L, the L_join of L, the L_meet of L #) is non empty strict LattStr
p is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the carrier of p is non empty set
the L_join of p is Relation-like [: the carrier of p, the carrier of p:] -defined the carrier of p -valued Function-like V18([: the carrier of p, the carrier of p:], the carrier of p) commutative associative idempotent Element of bool [:[: the carrier of p, the carrier of p:], the carrier of p:]
[: the carrier of p, the carrier of p:] is non empty set
[:[: the carrier of p, the carrier of p:], the carrier of p:] is non empty set
bool [:[: the carrier of p, the carrier of p:], the carrier of p:] is non empty set
the L_meet of p is Relation-like [: the carrier of p, the carrier of p:] -defined the carrier of p -valued Function-like V18([: the carrier of p, the carrier of p:], the carrier of p) commutative associative idempotent Element of bool [:[: the carrier of p, the carrier of p:], the carrier of p:]
LattStr(# the carrier of p, the L_join of p, the L_meet of p #) is non empty strict LattStr
bool the carrier of L is non empty set
bool the carrier of p is non empty set
q is set
b9 is Element of the carrier of p
b is Element of the carrier of p
b9 "/\" b is Element of the carrier of p
the L_meet of p . (b9,b) is Element of the carrier of p
[b9,b] is set
{b9,b} is non empty set
{b9} is non empty set
{{b9,b},{b9}} is non empty set
the L_meet of p . [b9,b] is set
a is Element of the carrier of L
a9 is Element of the carrier of L
a "/\" a9 is Element of the carrier of L
the L_meet of L . (a,a9) is Element of the carrier of L
[a,a9] is set
{a,a9} is non empty set
{a} is non empty set
{{a,a9},{a}} is non empty set
the L_meet of L . [a,a9] is set
K is non empty final meet-closed join-closed Element of bool the carrier of L
L is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the carrier of L is non empty set
the L_join of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
[: the carrier of L, the carrier of L:] is non empty set
[:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
bool [:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
the L_meet of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
LattStr(# the carrier of L, the L_join of L, the L_meet of L #) is non empty strict LattStr
p is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the carrier of p is non empty set
the L_join of p is Relation-like [: the carrier of p, the carrier of p:] -defined the carrier of p -valued Function-like V18([: the carrier of p, the carrier of p:], the carrier of p) commutative associative idempotent Element of bool [:[: the carrier of p, the carrier of p:], the carrier of p:]
[: the carrier of p, the carrier of p:] is non empty set
[:[: the carrier of p, the carrier of p:], the carrier of p:] is non empty set
bool [:[: the carrier of p, the carrier of p:], the carrier of p:] is non empty set
the L_meet of p is Relation-like [: the carrier of p, the carrier of p:] -defined the carrier of p -valued Function-like V18([: the carrier of p, the carrier of p:], the carrier of p) commutative associative idempotent Element of bool [:[: the carrier of p, the carrier of p:], the carrier of p:]
LattStr(# the carrier of p, the L_join of p, the L_meet of p #) is non empty strict LattStr
bool the carrier of L is non empty set
bool the carrier of p is non empty set
q is set
b9 is Element of the carrier of p
b is Element of the carrier of p
b9 "\/" b is Element of the carrier of p
the L_join of p . (b9,b) is Element of the carrier of p
[b9,b] is set
{b9,b} is non empty set
{b9} is non empty set
{{b9,b},{b9}} is non empty set
the L_join of p . [b9,b] is set
a is Element of the carrier of L
a9 is Element of the carrier of L
a "\/" a9 is Element of the carrier of L
the L_join of L . (a,a9) is Element of the carrier of L
[a,a9] is set
{a,a9} is non empty set
{a} is non empty set
{{a,a9},{a}} is non empty set
the L_join of L . [a,a9] is set
K is non empty initial meet-closed join-closed Element of bool the carrier of L
L is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the carrier of L is non empty set
p is Element of the carrier of L
L .: is non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the L_meet of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
[: the carrier of L, the carrier of L:] is non empty set
[:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
bool [:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
the L_join of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
LattStr(# the carrier of L, the L_meet of L, the L_join of L #) is non empty strict LattStr
the carrier of (L .:) is non empty set
L is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
L .: is non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the carrier of L is non empty set
the L_meet of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
[: the carrier of L, the carrier of L:] is non empty set
[:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
bool [:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
the L_join of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
LattStr(# the carrier of L, the L_meet of L, the L_join of L #) is non empty strict LattStr
the carrier of (L .:) is non empty set
p is Element of the carrier of (L .:)
L is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the carrier of L is non empty set
q is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
q .: is non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the carrier of q is non empty set
the L_meet of q is Relation-like [: the carrier of q, the carrier of q:] -defined the carrier of q -valued Function-like V18([: the carrier of q, the carrier of q:], the carrier of q) commutative associative idempotent Element of bool [:[: the carrier of q, the carrier of q:], the carrier of q:]
[: the carrier of q, the carrier of q:] is non empty set
[:[: the carrier of q, the carrier of q:], the carrier of q:] is non empty set
bool [:[: the carrier of q, the carrier of q:], the carrier of q:] is non empty set
the L_join of q is Relation-like [: the carrier of q, the carrier of q:] -defined the carrier of q -valued Function-like V18([: the carrier of q, the carrier of q:], the carrier of q) commutative associative idempotent Element of bool [:[: the carrier of q, the carrier of q:], the carrier of q:]
LattStr(# the carrier of q, the L_meet of q, the L_join of q #) is non empty strict LattStr
the carrier of (q .:) is non empty set
p is Element of the carrier of L
(L,p) is Element of the carrier of (L .:)
L .: is non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the L_meet of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
[: the carrier of L, the carrier of L:] is non empty set
[:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
bool [:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
the L_join of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
LattStr(# the carrier of L, the L_meet of L, the L_join of L #) is non empty strict LattStr
the carrier of (L .:) is non empty set
(L,(L,p)) is Element of the carrier of L
K is Element of the carrier of (q .:)
(q,K) is Element of the carrier of q
(q,(q,K)) is Element of the carrier of (q .:)
L is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the carrier of L is non empty set
K is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the carrier of K is non empty set
a is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
a .: is non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the carrier of a is non empty set
the L_meet of a is Relation-like [: the carrier of a, the carrier of a:] -defined the carrier of a -valued Function-like V18([: the carrier of a, the carrier of a:], the carrier of a) commutative associative idempotent Element of bool [:[: the carrier of a, the carrier of a:], the carrier of a:]
[: the carrier of a, the carrier of a:] is non empty set
[:[: the carrier of a, the carrier of a:], the carrier of a:] is non empty set
bool [:[: the carrier of a, the carrier of a:], the carrier of a:] is non empty set
the L_join of a is Relation-like [: the carrier of a, the carrier of a:] -defined the carrier of a -valued Function-like V18([: the carrier of a, the carrier of a:], the carrier of a) commutative associative idempotent Element of bool [:[: the carrier of a, the carrier of a:], the carrier of a:]
LattStr(# the carrier of a, the L_meet of a, the L_join of a #) is non empty strict LattStr
the carrier of (a .:) is non empty set
c is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
c .: is non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the carrier of c is non empty set
the L_meet of c is Relation-like [: the carrier of c, the carrier of c:] -defined the carrier of c -valued Function-like V18([: the carrier of c, the carrier of c:], the carrier of c) commutative associative idempotent Element of bool [:[: the carrier of c, the carrier of c:], the carrier of c:]
[: the carrier of c, the carrier of c:] is non empty set
[:[: the carrier of c, the carrier of c:], the carrier of c:] is non empty set
bool [:[: the carrier of c, the carrier of c:], the carrier of c:] is non empty set
the L_join of c is Relation-like [: the carrier of c, the carrier of c:] -defined the carrier of c -valued Function-like V18([: the carrier of c, the carrier of c:], the carrier of c) commutative associative idempotent Element of bool [:[: the carrier of c, the carrier of c:], the carrier of c:]
LattStr(# the carrier of c, the L_meet of c, the L_join of c #) is non empty strict LattStr
the carrier of (c .:) is non empty set
p is Element of the carrier of L
q is Element of the carrier of L
p "/\" q is Element of the carrier of L
the L_meet of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
[: the carrier of L, the carrier of L:] is non empty set
[:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
bool [:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
the L_meet of L . (p,q) is Element of the carrier of L
[p,q] is set
{p,q} is non empty set
{p} is non empty set
{{p,q},{p}} is non empty set
the L_meet of L . [p,q] is set
L .: is non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the L_join of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
LattStr(# the carrier of L, the L_meet of L, the L_join of L #) is non empty strict LattStr
(L,p) is Element of the carrier of (L .:)
the carrier of (L .:) is non empty set
(L,q) is Element of the carrier of (L .:)
(L,p) "\/" (L,q) is Element of the carrier of (L .:)
the L_join of (L .:) is Relation-like [: the carrier of (L .:), the carrier of (L .:):] -defined the carrier of (L .:) -valued Function-like V18([: the carrier of (L .:), the carrier of (L .:):], the carrier of (L .:)) commutative associative idempotent Element of bool [:[: the carrier of (L .:), the carrier of (L .:):], the carrier of (L .:):]
[: the carrier of (L .:), the carrier of (L .:):] is non empty set
[:[: the carrier of (L .:), the carrier of (L .:):], the carrier of (L .:):] is non empty set
bool [:[: the carrier of (L .:), the carrier of (L .:):], the carrier of (L .:):] is non empty set
the L_join of (L .:) . ((L,p),(L,q)) is Element of the carrier of (L .:)
[(L,p),(L,q)] is set
{(L,p),(L,q)} is non empty set
{(L,p)} is non empty set
{{(L,p),(L,q)},{(L,p)}} is non empty set
the L_join of (L .:) . [(L,p),(L,q)] is set
b9 is Element of the carrier of K
b is Element of the carrier of K
b9 "\/" b is Element of the carrier of K
the L_join of K is Relation-like [: the carrier of K, the carrier of K:] -defined the carrier of K -valued Function-like V18([: the carrier of K, the carrier of K:], the carrier of K) commutative associative idempotent Element of bool [:[: the carrier of K, the carrier of K:], the carrier of K:]
[: the carrier of K, the carrier of K:] is non empty set
[:[: the carrier of K, the carrier of K:], the carrier of K:] is non empty set
bool [:[: the carrier of K, the carrier of K:], the carrier of K:] is non empty set
the L_join of K . (b9,b) is Element of the carrier of K
[b9,b] is set
{b9,b} is non empty set
{b9} is non empty set
{{b9,b},{b9}} is non empty set
the L_join of K . [b9,b] is set
K .: is non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the L_meet of K is Relation-like [: the carrier of K, the carrier of K:] -defined the carrier of K -valued Function-like V18([: the carrier of K, the carrier of K:], the carrier of K) commutative associative idempotent Element of bool [:[: the carrier of K, the carrier of K:], the carrier of K:]
LattStr(# the carrier of K, the L_meet of K, the L_join of K #) is non empty strict LattStr
(K,b9) is Element of the carrier of (K .:)
the carrier of (K .:) is non empty set
(K,b) is Element of the carrier of (K .:)
(K,b9) "/\" (K,b) is Element of the carrier of (K .:)
the L_meet of (K .:) is Relation-like [: the carrier of (K .:), the carrier of (K .:):] -defined the carrier of (K .:) -valued Function-like V18([: the carrier of (K .:), the carrier of (K .:):], the carrier of (K .:)) commutative associative idempotent Element of bool [:[: the carrier of (K .:), the carrier of (K .:):], the carrier of (K .:):]
[: the carrier of (K .:), the carrier of (K .:):] is non empty set
[:[: the carrier of (K .:), the carrier of (K .:):], the carrier of (K .:):] is non empty set
bool [:[: the carrier of (K .:), the carrier of (K .:):], the carrier of (K .:):] is non empty set
the L_meet of (K .:) . ((K,b9),(K,b)) is Element of the carrier of (K .:)
[(K,b9),(K,b)] is set
{(K,b9),(K,b)} is non empty set
{(K,b9)} is non empty set
{{(K,b9),(K,b)},{(K,b9)}} is non empty set
the L_meet of (K .:) . [(K,b9),(K,b)] is set
a9 is Element of the carrier of (a .:)
c₉ is Element of the carrier of (a .:)
a9 "/\" c₉ is Element of the carrier of (a .:)
the L_meet of (a .:) is Relation-like [: the carrier of (a .:), the carrier of (a .:):] -defined the carrier of (a .:) -valued Function-like V18([: the carrier of (a .:), the carrier of (a .:):], the carrier of (a .:)) commutative associative idempotent Element of bool [:[: the carrier of (a .:), the carrier of (a .:):], the carrier of (a .:):]
[: the carrier of (a .:), the carrier of (a .:):] is non empty set
[:[: the carrier of (a .:), the carrier of (a .:):], the carrier of (a .:):] is non empty set
bool [:[: the carrier of (a .:), the carrier of (a .:):], the carrier of (a .:):] is non empty set
the L_meet of (a .:) . (a9,c₉) is Element of the carrier of (a .:)
[a9,c₉] is set
{a9,c₉} is non empty set
{a9} is non empty set
{{a9,c₉},{a9}} is non empty set
the L_meet of (a .:) . [a9,c₉] is set
(a,a9) is Element of the carrier of a
(a,c₉) is Element of the carrier of a
(a,a9) "\/" (a,c₉) is Element of the carrier of a
the L_join of a . ((a,a9),(a,c₉)) is Element of the carrier of a
[(a,a9),(a,c₉)] is set
{(a,a9),(a,c₉)} is non empty set
{(a,a9)} is non empty set
{{(a,a9),(a,c₉)},{(a,a9)}} is non empty set
the L_join of a . [(a,a9),(a,c₉)] is set
d is Element of the carrier of (c .:)
d9 is Element of the carrier of (c .:)
d "\/" d9 is Element of the carrier of (c .:)
the L_join of (c .:) is Relation-like [: the carrier of (c .:), the carrier of (c .:):] -defined the carrier of (c .:) -valued Function-like V18([: the carrier of (c .:), the carrier of (c .:):], the carrier of (c .:)) commutative associative idempotent Element of bool [:[: the carrier of (c .:), the carrier of (c .:):], the carrier of (c .:):]
[: the carrier of (c .:), the carrier of (c .:):] is non empty set
[:[: the carrier of (c .:), the carrier of (c .:):], the carrier of (c .:):] is non empty set
bool [:[: the carrier of (c .:), the carrier of (c .:):], the carrier of (c .:):] is non empty set
the L_join of (c .:) . (d,d9) is Element of the carrier of (c .:)
[d,d9] is set
{d,d9} is non empty set
{d} is non empty set
{{d,d9},{d}} is non empty set
the L_join of (c .:) . [d,d9] is set
(c,d) is Element of the carrier of c
(c,d9) is Element of the carrier of c
(c,d) "/\" (c,d9) is Element of the carrier of c
the L_meet of c . ((c,d),(c,d9)) is Element of the carrier of c
[(c,d),(c,d9)] is set
{(c,d),(c,d9)} is non empty set
{(c,d)} is non empty set
{{(c,d),(c,d9)},{(c,d)}} is non empty set
the L_meet of c . [(c,d),(c,d9)] is set
L is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the carrier of L is non empty set
K is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the carrier of K is non empty set
a is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
a .: is non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the carrier of a is non empty set
the L_meet of a is Relation-like [: the carrier of a, the carrier of a:] -defined the carrier of a -valued Function-like V18([: the carrier of a, the carrier of a:], the carrier of a) commutative associative idempotent Element of bool [:[: the carrier of a, the carrier of a:], the carrier of a:]
[: the carrier of a, the carrier of a:] is non empty set
[:[: the carrier of a, the carrier of a:], the carrier of a:] is non empty set
bool [:[: the carrier of a, the carrier of a:], the carrier of a:] is non empty set
the L_join of a is Relation-like [: the carrier of a, the carrier of a:] -defined the carrier of a -valued Function-like V18([: the carrier of a, the carrier of a:], the carrier of a) commutative associative idempotent Element of bool [:[: the carrier of a, the carrier of a:], the carrier of a:]
LattStr(# the carrier of a, the L_meet of a, the L_join of a #) is non empty strict LattStr
the carrier of (a .:) is non empty set
c is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
c .: is non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the carrier of c is non empty set
the L_meet of c is Relation-like [: the carrier of c, the carrier of c:] -defined the carrier of c -valued Function-like V18([: the carrier of c, the carrier of c:], the carrier of c) commutative associative idempotent Element of bool [:[: the carrier of c, the carrier of c:], the carrier of c:]
[: the carrier of c, the carrier of c:] is non empty set
[:[: the carrier of c, the carrier of c:], the carrier of c:] is non empty set
bool [:[: the carrier of c, the carrier of c:], the carrier of c:] is non empty set
the L_join of c is Relation-like [: the carrier of c, the carrier of c:] -defined the carrier of c -valued Function-like V18([: the carrier of c, the carrier of c:], the carrier of c) commutative associative idempotent Element of bool [:[: the carrier of c, the carrier of c:], the carrier of c:]
LattStr(# the carrier of c, the L_meet of c, the L_join of c #) is non empty strict LattStr
the carrier of (c .:) is non empty set
p is Element of the carrier of L
q is Element of the carrier of L
L .: is non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the L_meet of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
[: the carrier of L, the carrier of L:] is non empty set
[:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
bool [:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
the L_join of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
LattStr(# the carrier of L, the L_meet of L, the L_join of L #) is non empty strict LattStr
(L,q) is Element of the carrier of (L .:)
the carrier of (L .:) is non empty set
(L,p) is Element of the carrier of (L .:)
K .: is non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the L_meet of K is Relation-like [: the carrier of K, the carrier of K:] -defined the carrier of K -valued Function-like V18([: the carrier of K, the carrier of K:], the carrier of K) commutative associative idempotent Element of bool [:[: the carrier of K, the carrier of K:], the carrier of K:]
[: the carrier of K, the carrier of K:] is non empty set
[:[: the carrier of K, the carrier of K:], the carrier of K:] is non empty set
bool [:[: the carrier of K, the carrier of K:], the carrier of K:] is non empty set
the L_join of K is Relation-like [: the carrier of K, the carrier of K:] -defined the carrier of K -valued Function-like V18([: the carrier of K, the carrier of K:], the carrier of K) commutative associative idempotent Element of bool [:[: the carrier of K, the carrier of K:], the carrier of K:]
LattStr(# the carrier of K, the L_meet of K, the L_join of K #) is non empty strict LattStr
b is Element of the carrier of K
(K,b) is Element of the carrier of (K .:)
the carrier of (K .:) is non empty set
b9 is Element of the carrier of K
(K,b9) is Element of the carrier of (K .:)
a9 is Element of the carrier of (a .:)
c₉ is Element of the carrier of (a .:)
(a,c₉) is Element of the carrier of a
(a,a9) is Element of the carrier of a
d9 is Element of the carrier of (c .:)
(c,d9) is Element of the carrier of c
d is Element of the carrier of (c .:)
(c,d) is Element of the carrier of c
L is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the carrier of L is non empty set
bool the carrier of L is non empty set
p is Element of bool the carrier of L
L .: is non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the L_meet of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
[: the carrier of L, the carrier of L:] is non empty set
[:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
bool [:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
the L_join of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
LattStr(# the carrier of L, the L_meet of L, the L_join of L #) is non empty strict LattStr
the carrier of (L .:) is non empty set
bool the carrier of (L .:) is non empty set
L is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
L .: is non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the carrier of L is non empty set
the L_meet of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
[: the carrier of L, the carrier of L:] is non empty set
[:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
bool [:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
the L_join of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
LattStr(# the carrier of L, the L_meet of L, the L_join of L #) is non empty strict LattStr
the carrier of (L .:) is non empty set
bool the carrier of (L .:) is non empty set
p is Element of bool the carrier of (L .:)
bool the carrier of L is non empty set
L is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the carrier of L is non empty set
bool the carrier of L is non empty set
p is non empty Element of bool the carrier of L
(L,p) is Element of bool the carrier of (L .:)
L .: is non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the L_meet of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
[: the carrier of L, the carrier of L:] is non empty set
[:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
bool [:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
the L_join of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
LattStr(# the carrier of L, the L_meet of L, the L_join of L #) is non empty strict LattStr
the carrier of (L .:) is non empty set
bool the carrier of (L .:) is non empty set
L is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
L .: is non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the carrier of L is non empty set
the L_meet of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
[: the carrier of L, the carrier of L:] is non empty set
[:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
bool [:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
the L_join of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
LattStr(# the carrier of L, the L_meet of L, the L_join of L #) is non empty strict LattStr
the carrier of (L .:) is non empty set
bool the carrier of (L .:) is non empty set
p is non empty Element of bool the carrier of (L .:)
(L,p) is Element of bool the carrier of L
bool the carrier of L is non empty set
L is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the carrier of L is non empty set
bool the carrier of L is non empty set
L .: is non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the L_meet of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
[: the carrier of L, the carrier of L:] is non empty set
[:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
bool [:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
the L_join of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
LattStr(# the carrier of L, the L_meet of L, the L_join of L #) is non empty strict LattStr
the carrier of (L .:) is non empty set
bool the carrier of (L .:) is non empty set
p is meet-closed Element of bool the carrier of L
(L,p) is Element of bool the carrier of (L .:)
q is Element of the carrier of (L .:)
K is Element of the carrier of (L .:)
q "\/" K is Element of the carrier of (L .:)
the L_join of (L .:) is Relation-like [: the carrier of (L .:), the carrier of (L .:):] -defined the carrier of (L .:) -valued Function-like V18([: the carrier of (L .:), the carrier of (L .:):], the carrier of (L .:)) commutative associative idempotent Element of bool [:[: the carrier of (L .:), the carrier of (L .:):], the carrier of (L .:):]
[: the carrier of (L .:), the carrier of (L .:):] is non empty set
[:[: the carrier of (L .:), the carrier of (L .:):], the carrier of (L .:):] is non empty set
bool [:[: the carrier of (L .:), the carrier of (L .:):], the carrier of (L .:):] is non empty set
the L_join of (L .:) . (q,K) is Element of the carrier of (L .:)
[q,K] is set
{q,K} is non empty set
{q} is non empty set
{{q,K},{q}} is non empty set
the L_join of (L .:) . [q,K] is set
(L,q) is Element of the carrier of L
(L,K) is Element of the carrier of L
(L,q) "/\" (L,K) is Element of the carrier of L
the L_meet of L . ((L,q),(L,K)) is Element of the carrier of L
[(L,q),(L,K)] is set
{(L,q),(L,K)} is non empty set
{(L,q)} is non empty set
{{(L,q),(L,K)},{(L,q)}} is non empty set
the L_meet of L . [(L,q),(L,K)] is set
q is Element of bool the carrier of (L .:)
L is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the carrier of L is non empty set
bool the carrier of L is non empty set
L .: is non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the L_meet of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
[: the carrier of L, the carrier of L:] is non empty set
[:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
bool [:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
the L_join of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
LattStr(# the carrier of L, the L_meet of L, the L_join of L #) is non empty strict LattStr
the carrier of (L .:) is non empty set
bool the carrier of (L .:) is non empty set
p is join-closed Element of bool the carrier of L
(L,p) is Element of bool the carrier of (L .:)
q is Element of the carrier of (L .:)
K is Element of the carrier of (L .:)
q "/\" K is Element of the carrier of (L .:)
the L_meet of (L .:) is Relation-like [: the carrier of (L .:), the carrier of (L .:):] -defined the carrier of (L .:) -valued Function-like V18([: the carrier of (L .:), the carrier of (L .:):], the carrier of (L .:)) commutative associative idempotent Element of bool [:[: the carrier of (L .:), the carrier of (L .:):], the carrier of (L .:):]
[: the carrier of (L .:), the carrier of (L .:):] is non empty set
[:[: the carrier of (L .:), the carrier of (L .:):], the carrier of (L .:):] is non empty set
bool [:[: the carrier of (L .:), the carrier of (L .:):], the carrier of (L .:):] is non empty set
the L_meet of (L .:) . (q,K) is Element of the carrier of (L .:)
[q,K] is set
{q,K} is non empty set
{q} is non empty set
{{q,K},{q}} is non empty set
the L_meet of (L .:) . [q,K] is set
(L,q) is Element of the carrier of L
(L,K) is Element of the carrier of L
(L,q) "\/" (L,K) is Element of the carrier of L
the L_join of L . ((L,q),(L,K)) is Element of the carrier of L
[(L,q),(L,K)] is set
{(L,q),(L,K)} is non empty set
{(L,q)} is non empty set
{{(L,q),(L,K)},{(L,q)}} is non empty set
the L_join of L . [(L,q),(L,K)] is set
q is Element of bool the carrier of (L .:)
L is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
L .: is non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the carrier of L is non empty set
the L_meet of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
[: the carrier of L, the carrier of L:] is non empty set
[:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
bool [:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
the L_join of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
LattStr(# the carrier of L, the L_meet of L, the L_join of L #) is non empty strict LattStr
the carrier of (L .:) is non empty set
bool the carrier of (L .:) is non empty set
bool the carrier of L is non empty set
p is meet-closed Element of bool the carrier of (L .:)
(L,p) is Element of bool the carrier of L
q is Element of the carrier of L
K is Element of the carrier of L
q "\/" K is Element of the carrier of L
the L_join of L . (q,K) is Element of the carrier of L
[q,K] is set
{q,K} is non empty set
{q} is non empty set
{{q,K},{q}} is non empty set
the L_join of L . [q,K] is set
(L,q) is Element of the carrier of (L .:)
(L,K) is Element of the carrier of (L .:)
(L,q) "/\" (L,K) is Element of the carrier of (L .:)
the L_meet of (L .:) is Relation-like [: the carrier of (L .:), the carrier of (L .:):] -defined the carrier of (L .:) -valued Function-like V18([: the carrier of (L .:), the carrier of (L .:):], the carrier of (L .:)) commutative associative idempotent Element of bool [:[: the carrier of (L .:), the carrier of (L .:):], the carrier of (L .:):]
[: the carrier of (L .:), the carrier of (L .:):] is non empty set
[:[: the carrier of (L .:), the carrier of (L .:):], the carrier of (L .:):] is non empty set
bool [:[: the carrier of (L .:), the carrier of (L .:):], the carrier of (L .:):] is non empty set
the L_meet of (L .:) . ((L,q),(L,K)) is Element of the carrier of (L .:)
[(L,q),(L,K)] is set
{(L,q),(L,K)} is non empty set
{(L,q)} is non empty set
{{(L,q),(L,K)},{(L,q)}} is non empty set
the L_meet of (L .:) . [(L,q),(L,K)] is set
q is Element of bool the carrier of L
L is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
L .: is non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the carrier of L is non empty set
the L_meet of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
[: the carrier of L, the carrier of L:] is non empty set
[:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
bool [:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
the L_join of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
LattStr(# the carrier of L, the L_meet of L, the L_join of L #) is non empty strict LattStr
the carrier of (L .:) is non empty set
bool the carrier of (L .:) is non empty set
bool the carrier of L is non empty set
p is join-closed Element of bool the carrier of (L .:)
(L,p) is Element of bool the carrier of L
q is Element of the carrier of L
K is Element of the carrier of L
q "/\" K is Element of the carrier of L
the L_meet of L . (q,K) is Element of the carrier of L
[q,K] is set
{q,K} is non empty set
{q} is non empty set
{{q,K},{q}} is non empty set
the L_meet of L . [q,K] is set
(L,q) is Element of the carrier of (L .:)
(L,K) is Element of the carrier of (L .:)
(L,q) "\/" (L,K) is Element of the carrier of (L .:)
the L_join of (L .:) is Relation-like [: the carrier of (L .:), the carrier of (L .:):] -defined the carrier of (L .:) -valued Function-like V18([: the carrier of (L .:), the carrier of (L .:):], the carrier of (L .:)) commutative associative idempotent Element of bool [:[: the carrier of (L .:), the carrier of (L .:):], the carrier of (L .:):]
[: the carrier of (L .:), the carrier of (L .:):] is non empty set
[:[: the carrier of (L .:), the carrier of (L .:):], the carrier of (L .:):] is non empty set
bool [:[: the carrier of (L .:), the carrier of (L .:):], the carrier of (L .:):] is non empty set
the L_join of (L .:) . ((L,q),(L,K)) is Element of the carrier of (L .:)
[(L,q),(L,K)] is set
{(L,q),(L,K)} is non empty set
{(L,q)} is non empty set
{{(L,q),(L,K)},{(L,q)}} is non empty set
the L_join of (L .:) . [(L,q),(L,K)] is set
q is Element of bool the carrier of L
L is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the carrier of L is non empty set
bool the carrier of L is non empty set
L .: is non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the L_meet of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
[: the carrier of L, the carrier of L:] is non empty set
[:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
bool [:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
the L_join of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
LattStr(# the carrier of L, the L_meet of L, the L_join of L #) is non empty strict LattStr
the carrier of (L .:) is non empty set
bool the carrier of (L .:) is non empty set
p is final join-closed Element of bool the carrier of L
(L,p) is meet-closed Element of bool the carrier of (L .:)
q is Element of the carrier of (L .:)
K is Element of the carrier of (L .:)
((L .:),q) is Element of the carrier of ((L .:) .:)
(L .:) .: is non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the L_meet of (L .:) is Relation-like [: the carrier of (L .:), the carrier of (L .:):] -defined the carrier of (L .:) -valued Function-like V18([: the carrier of (L .:), the carrier of (L .:):], the carrier of (L .:)) commutative associative idempotent Element of bool [:[: the carrier of (L .:), the carrier of (L .:):], the carrier of (L .:):]
[: the carrier of (L .:), the carrier of (L .:):] is non empty set
[:[: the carrier of (L .:), the carrier of (L .:):], the carrier of (L .:):] is non empty set
bool [:[: the carrier of (L .:), the carrier of (L .:):], the carrier of (L .:):] is non empty set
the L_join of (L .:) is Relation-like [: the carrier of (L .:), the carrier of (L .:):] -defined the carrier of (L .:) -valued Function-like V18([: the carrier of (L .:), the carrier of (L .:):], the carrier of (L .:)) commutative associative idempotent Element of bool [:[: the carrier of (L .:), the carrier of (L .:):], the carrier of (L .:):]
LattStr(# the carrier of (L .:), the L_meet of (L .:), the L_join of (L .:) #) is non empty strict LattStr
the carrier of ((L .:) .:) is non empty set
((L .:),K) is Element of the carrier of ((L .:) .:)
b is Element of the carrier of L
b9 is Element of the carrier of L
q is Element of bool the carrier of (L .:)
L is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the carrier of L is non empty set
bool the carrier of L is non empty set
L .: is non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the L_meet of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
[: the carrier of L, the carrier of L:] is non empty set
[:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
bool [:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
the L_join of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
LattStr(# the carrier of L, the L_meet of L, the L_join of L #) is non empty strict LattStr
the carrier of (L .:) is non empty set
bool the carrier of (L .:) is non empty set
p is initial meet-closed Element of bool the carrier of L
(L,p) is join-closed Element of bool the carrier of (L .:)
q is Element of the carrier of (L .:)
K is Element of the carrier of (L .:)
((L .:),q) is Element of the carrier of ((L .:) .:)
(L .:) .: is non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the L_meet of (L .:) is Relation-like [: the carrier of (L .:), the carrier of (L .:):] -defined the carrier of (L .:) -valued Function-like V18([: the carrier of (L .:), the carrier of (L .:):], the carrier of (L .:)) commutative associative idempotent Element of bool [:[: the carrier of (L .:), the carrier of (L .:):], the carrier of (L .:):]
[: the carrier of (L .:), the carrier of (L .:):] is non empty set
[:[: the carrier of (L .:), the carrier of (L .:):], the carrier of (L .:):] is non empty set
bool [:[: the carrier of (L .:), the carrier of (L .:):], the carrier of (L .:):] is non empty set
the L_join of (L .:) is Relation-like [: the carrier of (L .:), the carrier of (L .:):] -defined the carrier of (L .:) -valued Function-like V18([: the carrier of (L .:), the carrier of (L .:):], the carrier of (L .:)) commutative associative idempotent Element of bool [:[: the carrier of (L .:), the carrier of (L .:):], the carrier of (L .:):]
LattStr(# the carrier of (L .:), the L_meet of (L .:), the L_join of (L .:) #) is non empty strict LattStr
the carrier of ((L .:) .:) is non empty set
((L .:),K) is Element of the carrier of ((L .:) .:)
b is Element of the carrier of L
b9 is Element of the carrier of L
q is Element of bool the carrier of (L .:)
L is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
L .: is non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the carrier of L is non empty set
the L_meet of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
[: the carrier of L, the carrier of L:] is non empty set
[:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
bool [:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
the L_join of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
LattStr(# the carrier of L, the L_meet of L, the L_join of L #) is non empty strict LattStr
the carrier of (L .:) is non empty set
bool the carrier of (L .:) is non empty set
bool the carrier of L is non empty set
p is final join-closed Element of bool the carrier of (L .:)
(L,p) is meet-closed Element of bool the carrier of L
q is Element of the carrier of L
K is Element of the carrier of L
(L,q) is Element of the carrier of (L .:)
(L,K) is Element of the carrier of (L .:)
b is Element of the carrier of (L .:)
b9 is Element of the carrier of (L .:)
q is Element of bool the carrier of L
L is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
L .: is non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the carrier of L is non empty set
the L_meet of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
[: the carrier of L, the carrier of L:] is non empty set
[:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
bool [:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
the L_join of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
LattStr(# the carrier of L, the L_meet of L, the L_join of L #) is non empty strict LattStr
the carrier of (L .:) is non empty set
bool the carrier of (L .:) is non empty set
bool the carrier of L is non empty set
p is initial meet-closed Element of bool the carrier of (L .:)
((L .:),p) is final join-closed Element of bool the carrier of ((L .:) .:)
(L .:) .: is non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the L_meet of (L .:) is Relation-like [: the carrier of (L .:), the carrier of (L .:):] -defined the carrier of (L .:) -valued Function-like V18([: the carrier of (L .:), the carrier of (L .:):], the carrier of (L .:)) commutative associative idempotent Element of bool [:[: the carrier of (L .:), the carrier of (L .:):], the carrier of (L .:):]
[: the carrier of (L .:), the carrier of (L .:):] is non empty set
[:[: the carrier of (L .:), the carrier of (L .:):], the carrier of (L .:):] is non empty set
bool [:[: the carrier of (L .:), the carrier of (L .:):], the carrier of (L .:):] is non empty set
the L_join of (L .:) is Relation-like [: the carrier of (L .:), the carrier of (L .:):] -defined the carrier of (L .:) -valued Function-like V18([: the carrier of (L .:), the carrier of (L .:):], the carrier of (L .:)) commutative associative idempotent Element of bool [:[: the carrier of (L .:), the carrier of (L .:):], the carrier of (L .:):]
LattStr(# the carrier of (L .:), the L_meet of (L .:), the L_join of (L .:) #) is non empty strict LattStr
the carrier of ((L .:) .:) is non empty set
bool the carrier of ((L .:) .:) is non empty set
(L,p) is join-closed Element of bool the carrier of L
q is Element of the carrier of L
K is Element of the carrier of L
(L,q) is Element of the carrier of (L .:)
(L,K) is Element of the carrier of (L .:)
b is Element of the carrier of (L .:)
b9 is Element of the carrier of (L .:)
q is Element of bool the carrier of L
L is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the carrier of L is non empty set
bool the carrier of L is non empty set
L .: is non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the L_meet of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
[: the carrier of L, the carrier of L:] is non empty set
[:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
bool [:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
the L_join of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
LattStr(# the carrier of L, the L_meet of L, the L_join of L #) is non empty strict LattStr
the carrier of (L .:) is non empty set
bool the carrier of (L .:) is non empty set
p is set
q is non empty initial meet-closed join-closed Element of bool the carrier of L
K is non empty Element of bool the carrier of (L .:)
b9 is Element of the carrier of (L .:)
b is Element of the carrier of (L .:)
b9 "/\" b is Element of the carrier of (L .:)
the L_meet of (L .:) is Relation-like [: the carrier of (L .:), the carrier of (L .:):] -defined the carrier of (L .:) -valued Function-like V18([: the carrier of (L .:), the carrier of (L .:):], the carrier of (L .:)) commutative associative idempotent Element of bool [:[: the carrier of (L .:), the carrier of (L .:):], the carrier of (L .:):]
[: the carrier of (L .:), the carrier of (L .:):] is non empty set
[:[: the carrier of (L .:), the carrier of (L .:):], the carrier of (L .:):] is non empty set
bool [:[: the carrier of (L .:), the carrier of (L .:):], the carrier of (L .:):] is non empty set
the L_meet of (L .:) . (b9,b) is Element of the carrier of (L .:)
[b9,b] is set
{b9,b} is non empty set
{b9} is non empty set
{{b9,b},{b9}} is non empty set
the L_meet of (L .:) . [b9,b] is set
(L,b9) is Element of the carrier of L
(L,b) is Element of the carrier of L
(L,b9) "\/" (L,b) is Element of the carrier of L
the L_join of L . ((L,b9),(L,b)) is Element of the carrier of L
[(L,b9),(L,b)] is set
{(L,b9),(L,b)} is non empty set
{(L,b9)} is non empty set
{{(L,b9),(L,b)},{(L,b9)}} is non empty set
the L_join of L . [(L,b9),(L,b)] is set
q is non empty final meet-closed join-closed Element of bool the carrier of (L .:)
b9 is Element of the carrier of L
(L,b9) is Element of the carrier of (L .:)
b is Element of the carrier of L
(L,b) is Element of the carrier of (L .:)
(L,b9) "/\" (L,b) is Element of the carrier of (L .:)
the L_meet of (L .:) is Relation-like [: the carrier of (L .:), the carrier of (L .:):] -defined the carrier of (L .:) -valued Function-like V18([: the carrier of (L .:), the carrier of (L .:):], the carrier of (L .:)) commutative associative idempotent Element of bool [:[: the carrier of (L .:), the carrier of (L .:):], the carrier of (L .:):]
[: the carrier of (L .:), the carrier of (L .:):] is non empty set
[:[: the carrier of (L .:), the carrier of (L .:):], the carrier of (L .:):] is non empty set
bool [:[: the carrier of (L .:), the carrier of (L .:):], the carrier of (L .:):] is non empty set
the L_meet of (L .:) . ((L,b9),(L,b)) is Element of the carrier of (L .:)
[(L,b9),(L,b)] is set
{(L,b9),(L,b)} is non empty set
{(L,b9)} is non empty set
{{(L,b9),(L,b)},{(L,b9)}} is non empty set
the L_meet of (L .:) . [(L,b9),(L,b)] is set
b9 "\/" b is Element of the carrier of L
the L_join of L . (b9,b) is Element of the carrier of L
[b9,b] is set
{b9,b} is non empty set
{b9} is non empty set
{{b9,b},{b9}} is non empty set
the L_join of L . [b9,b] is set
K is non empty Element of bool the carrier of L
L is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the carrier of L is non empty set
bool the carrier of L is non empty set
p is non empty Element of bool the carrier of L
q is Element of the carrier of L
K is Element of the carrier of L
q "\/" K is Element of the carrier of L
the L_join of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
[: the carrier of L, the carrier of L:] is non empty set
[:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
bool [:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
the L_join of L . (q,K) is Element of the carrier of L
[q,K] is set
{q,K} is non empty set
{q} is non empty set
{{q,K},{q}} is non empty set
the L_join of L . [q,K] is set
q is Element of the carrier of L
K is Element of the carrier of L
K "\/" q is Element of the carrier of L
the L_join of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
[: the carrier of L, the carrier of L:] is non empty set
[:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
bool [:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
the L_join of L . (K,q) is Element of the carrier of L
[K,q] is set
{K,q} is non empty set
{K} is non empty set
{{K,q},{K}} is non empty set
the L_join of L . [K,q] is set
q is Element of the carrier of L
K is Element of the carrier of L
q "\/" K is Element of the carrier of L
the L_join of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
[: the carrier of L, the carrier of L:] is non empty set
[:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
bool [:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
the L_join of L . (q,K) is Element of the carrier of L
[q,K] is set
{q,K} is non empty set
{q} is non empty set
{{q,K},{q}} is non empty set
the L_join of L . [q,K] is set
K "\/" q is Element of the carrier of L
the L_join of L . (K,q) is Element of the carrier of L
[K,q] is set
{K,q} is non empty set
{K} is non empty set
{{K,q},{K}} is non empty set
the L_join of L . [K,q] is set
L is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the carrier of L is non empty set
bool the carrier of L is non empty set
p is Element of the carrier of L
q is Element of the carrier of L
p "/\" q is Element of the carrier of L
the L_meet of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
[: the carrier of L, the carrier of L:] is non empty set
[:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
bool [:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
the L_meet of L . (p,q) is Element of the carrier of L
[p,q] is set
{p,q} is non empty set
{p} is non empty set
{{p,q},{p}} is non empty set
the L_meet of L . [p,q] is set
q "/\" p is Element of the carrier of L
the L_meet of L . (q,p) is Element of the carrier of L
[q,p] is set
{q,p} is non empty set
{q} is non empty set
{{q,p},{q}} is non empty set
the L_meet of L . [q,p] is set
K is non empty initial meet-closed join-closed Element of bool the carrier of L
L is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the carrier of L is non empty set
bool the carrier of L is non empty set
p is non empty initial meet-closed join-closed Element of bool the carrier of L
the Element of p is Element of p
L is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the carrier of L is non empty set
bool the carrier of L is non empty set
Bottom L is Element of the carrier of L
p is non empty initial meet-closed join-closed Element of bool the carrier of L
L .: is non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the L_meet of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
[: the carrier of L, the carrier of L:] is non empty set
[:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
bool [:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
the L_join of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
LattStr(# the carrier of L, the L_meet of L, the L_join of L #) is non empty strict LattStr
Top (L .:) is Element of the carrier of (L .:)
the carrier of (L .:) is non empty set
(L,p) is non empty final meet-closed join-closed Element of bool the carrier of (L .:)
bool the carrier of (L .:) is non empty set
L is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the carrier of L is non empty set
Bottom L is Element of the carrier of L
{(Bottom L)} is non empty Element of bool the carrier of L
bool the carrier of L is non empty set
L .: is non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the L_meet of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
[: the carrier of L, the carrier of L:] is non empty set
[:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
bool [:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
the L_join of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
LattStr(# the carrier of L, the L_meet of L, the L_join of L #) is non empty strict LattStr
Top (L .:) is Element of the carrier of (L .:)
the carrier of (L .:) is non empty set
bool the carrier of (L .:) is non empty set
L is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the carrier of L is non empty set
bool the carrier of L is non empty set
p is Element of the carrier of L
{p} is non empty Element of bool the carrier of L
L .: is non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the L_meet of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
[: the carrier of L, the carrier of L:] is non empty set
[:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
bool [:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
the L_join of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
LattStr(# the carrier of L, the L_meet of L, the L_join of L #) is non empty strict LattStr
the carrier of (L .:) is non empty set
bool the carrier of (L .:) is non empty set
L is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the carrier of L is non empty set
bool the carrier of L is non empty set
L .: is non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the L_meet of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
[: the carrier of L, the carrier of L:] is non empty set
[:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
bool [:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
the L_join of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
LattStr(# the carrier of L, the L_meet of L, the L_join of L #) is non empty strict LattStr
the carrier of (L .:) is non empty set
bool the carrier of (L .:) is non empty set
L is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the carrier of L is non empty set
bool the carrier of L is non empty set
L is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the carrier of L is non empty set
p is Element of the carrier of L
{ b₁ where b₁ is Element of the carrier of L : b₁ [= p } is set
bool the carrier of L is non empty set
K is non empty set
b9 is set
b is Element of the carrier of L
b is Element of the carrier of L
b9 is non empty Element of bool the carrier of L
a is Element of the carrier of L
b "\/" a is Element of the carrier of L
the L_join of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
[: the carrier of L, the carrier of L:] is non empty set
[:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
bool [:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
the L_join of L . (b,a) is Element of the carrier of L
[b,a] is set
{b,a} is non empty set
{b} is non empty set
{{b,a},{b}} is non empty set
the L_join of L . [b,a] is set
a9 is Element of the carrier of L
c₉ is Element of the carrier of L
a9 is Element of the carrier of L
a "\/" b is Element of the carrier of L
the L_join of L . (a,b) is Element of the carrier of L
[a,b] is set
{a,b} is non empty set
{a} is non empty set
{{a,b},{a}} is non empty set
the L_join of L . [a,b] is set
a9 is Element of the carrier of L
L is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the carrier of L is non empty set
p is Element of the carrier of L
q is Element of the carrier of L
(L,q) is non empty initial meet-closed join-closed Element of bool the carrier of L
bool the carrier of L is non empty set
{ b₁ where b₁ is Element of the carrier of L : b₁ [= q } is set
K is Element of the carrier of L
K is Element of the carrier of L
L is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the carrier of L is non empty set
L .: is non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the L_meet of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
[: the carrier of L, the carrier of L:] is non empty set
[:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
bool [:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
the L_join of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
LattStr(# the carrier of L, the L_meet of L, the L_join of L #) is non empty strict LattStr
p is Element of the carrier of L
(L,p) is non empty initial meet-closed join-closed Element of bool the carrier of L
bool the carrier of L is non empty set
{ b₁ where b₁ is Element of the carrier of L : b₁ [= p } is set
(L,p) is Element of the carrier of (L .:)
the carrier of (L .:) is non empty set
<.(L,p).) is non empty final meet-closed join-closed Element of bool the carrier of (L .:)
bool the carrier of (L .:) is non empty set
{ b₁ where b₁ is Element of the carrier of (L .:) : (L,p) [= b₁ } is set
((L .:),(L,p)) is non empty initial meet-closed join-closed Element of bool the carrier of (L .:)
{ b₁ where b₁ is Element of the carrier of (L .:) : b₁ [= (L,p) } is set
<.p.) is non empty final meet-closed join-closed Element of bool the carrier of L
{ b₁ where b₁ is Element of the carrier of L : p [= b₁ } is set
(L,<.(L,p).)) is non empty initial meet-closed join-closed Element of bool the carrier of L
q is Element of the carrier of L
(L,q) is Element of the carrier of (L .:)
(L,((L .:),(L,p))) is non empty meet-closed join-closed Element of bool the carrier of L
q is Element of the carrier of L
(L,q) is Element of the carrier of (L .:)
L is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the carrier of L is non empty set
p is Element of the carrier of L
(L,p) is non empty initial meet-closed join-closed Element of bool the carrier of L
bool the carrier of L is non empty set
{ b₁ where b₁ is Element of the carrier of L : b₁ [= p } is set
q is Element of the carrier of L
p "/\" q is Element of the carrier of L
the L_meet of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
[: the carrier of L, the carrier of L:] is non empty set
[:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
bool [:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
the L_meet of L . (p,q) is Element of the carrier of L
[p,q] is set
{p,q} is non empty set
{p} is non empty set
{{p,q},{p}} is non empty set
the L_meet of L . [p,q] is set
q "/\" p is Element of the carrier of L
the L_meet of L . (q,p) is Element of the carrier of L
[q,p] is set
{q,p} is non empty set
{q} is non empty set
{{q,p},{q}} is non empty set
the L_meet of L . [q,p] is set
L is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the carrier of L is non empty set
(L) is non empty initial meet-closed join-closed Element of bool the carrier of L
bool the carrier of L is non empty set
Top L is Element of the carrier of L
(L,(Top L)) is non empty initial meet-closed join-closed Element of bool the carrier of L
{ b₁ where b₁ is Element of the carrier of L : b₁ [= Top L } is set
L .: is non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the L_meet of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
[: the carrier of L, the carrier of L:] is non empty set
[:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
bool [:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
the L_join of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
LattStr(# the carrier of L, the L_meet of L, the L_join of L #) is non empty strict LattStr
the carrier of (L .:) is non empty set
<.(L .:).) is non empty final meet-closed join-closed Element of bool the carrier of (L .:)
bool the carrier of (L .:) is non empty set
Bottom (L .:) is Element of the carrier of (L .:)
<.(Bottom (L .:)).) is non empty final meet-closed join-closed Element of bool the carrier of (L .:)
{ b₁ where b₁ is Element of the carrier of (L .:) : Bottom (L .:) [= b₁ } is set
(L,(Top L)) is Element of the carrier of (L .:)
L is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the carrier of L is non empty set
bool the carrier of L is non empty set
L is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the carrier of L is non empty set
bool the carrier of L is non empty set
L .: is non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the L_meet of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
[: the carrier of L, the carrier of L:] is non empty set
[:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
bool [:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
the L_join of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
LattStr(# the carrier of L, the L_meet of L, the L_join of L #) is non empty strict LattStr
p is non empty initial meet-closed join-closed Element of bool the carrier of L
(L,p) is non empty final meet-closed join-closed Element of bool the carrier of (L .:)
the carrier of (L .:) is non empty set
bool the carrier of (L .:) is non empty set
q is non empty final meet-closed join-closed Element of bool the carrier of (L .:)
(L,q) is non empty initial meet-closed join-closed Element of bool the carrier of L
q is non empty initial meet-closed join-closed Element of bool the carrier of L
(L,q) is non empty final meet-closed join-closed Element of bool the carrier of (L .:)
L is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the carrier of L is non empty set
bool the carrier of L is non empty set
L .: is non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the L_meet of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
[: the carrier of L, the carrier of L:] is non empty set
[:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
bool [:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
the L_join of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
LattStr(# the carrier of L, the L_meet of L, the L_join of L #) is non empty strict LattStr
p is non empty initial meet-closed join-closed Element of bool the carrier of L
the carrier of (L .:) is non empty set
bool the carrier of (L .:) is non empty set
(L,p) is non empty final meet-closed join-closed Element of bool the carrier of (L .:)
q is non empty final meet-closed join-closed Element of bool the carrier of (L .:)
(L,q) is non empty initial meet-closed join-closed Element of bool the carrier of L
(L,(L,q)) is non empty final meet-closed join-closed Element of bool the carrier of (L .:)
L is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the carrier of L is non empty set
p is Element of the carrier of L
(L,p) is non empty initial meet-closed join-closed Element of bool the carrier of L
bool the carrier of L is non empty set
{ b₁ where b₁ is Element of the carrier of L : b₁ [= p } is set
q is Element of the carrier of L
p "\/" q is Element of the carrier of L
the L_join of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
[: the carrier of L, the carrier of L:] is non empty set
[:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
bool [:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
the L_join of L . (p,q) is Element of the carrier of L
[p,q] is set
{p,q} is non empty set
{p} is non empty set
{{p,q},{p}} is non empty set
the L_join of L . [p,q] is set
L .: is non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the L_meet of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
LattStr(# the carrier of L, the L_meet of L, the L_join of L #) is non empty strict LattStr
(L,p) is Element of the carrier of (L .:)
the carrier of (L .:) is non empty set
(L,q) is Element of the carrier of (L .:)
(L,p) "/\" (L,q) is Element of the carrier of (L .:)
the L_meet of (L .:) is Relation-like [: the carrier of (L .:), the carrier of (L .:):] -defined the carrier of (L .:) -valued Function-like V18([: the carrier of (L .:), the carrier of (L .:):], the carrier of (L .:)) commutative associative idempotent Element of bool [:[: the carrier of (L .:), the carrier of (L .:):], the carrier of (L .:):]
[: the carrier of (L .:), the carrier of (L .:):] is non empty set
[:[: the carrier of (L .:), the carrier of (L .:):], the carrier of (L .:):] is non empty set
bool [:[: the carrier of (L .:), the carrier of (L .:):], the carrier of (L .:):] is non empty set
the L_meet of (L .:) . ((L,p),(L,q)) is Element of the carrier of (L .:)
[(L,p),(L,q)] is set
{(L,p),(L,q)} is non empty set
{(L,p)} is non empty set
{{(L,p),(L,q)},{(L,p)}} is non empty set
the L_meet of (L .:) . [(L,p),(L,q)] is set
<.(L,p).) is non empty final meet-closed join-closed Element of bool the carrier of (L .:)
bool the carrier of (L .:) is non empty set
{ b₁ where b₁ is Element of the carrier of (L .:) : (L,p) [= b₁ } is set
L is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the carrier of L is non empty set
Top L is Element of the carrier of L
bool the carrier of L is non empty set
p is Element of the carrier of L
L .: is non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the L_meet of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
[: the carrier of L, the carrier of L:] is non empty set
[:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
bool [:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
the L_join of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
LattStr(# the carrier of L, the L_meet of L, the L_join of L #) is non empty strict LattStr
Bottom (L .:) is Element of the carrier of (L .:)
the carrier of (L .:) is non empty set
bool the carrier of (L .:) is non empty set
(L,p) is Element of the carrier of (L .:)
q is non empty final meet-closed join-closed Element of bool the carrier of (L .:)
(L,q) is non empty initial meet-closed join-closed Element of bool the carrier of L
(L,(L,q)) is non empty final meet-closed join-closed Element of bool the carrier of (L .:)
L is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the carrier of L is non empty set
bool the carrier of L is non empty set
p is non empty Element of bool the carrier of L
L .: is non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the L_meet of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
[: the carrier of L, the carrier of L:] is non empty set
[:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
bool [:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
the L_join of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
LattStr(# the carrier of L, the L_meet of L, the L_join of L #) is non empty strict LattStr
(L,p) is non empty Element of bool the carrier of (L .:)
the carrier of (L .:) is non empty set
bool the carrier of (L .:) is non empty set
<.(L,p).) is non empty final meet-closed join-closed Element of bool the carrier of (L .:)
(L,<.(L,p).)) is non empty initial meet-closed join-closed Element of bool the carrier of L
q is non empty initial meet-closed join-closed Element of bool the carrier of L
K is non empty initial meet-closed join-closed Element of bool the carrier of L
(L,K) is non empty final meet-closed join-closed Element of bool the carrier of (L .:)
q is non empty initial meet-closed join-closed Element of bool the carrier of L
K is non empty initial meet-closed join-closed Element of bool the carrier of L
L is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the carrier of L is non empty set
bool the carrier of L is non empty set
p is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the carrier of p is non empty set
bool the carrier of p is non empty set
the L_join of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
[: the carrier of L, the carrier of L:] is non empty set
[:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
bool [:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
the L_meet of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
LattStr(# the carrier of L, the L_join of L, the L_meet of L #) is non empty strict LattStr
the L_join of p is Relation-like [: the carrier of p, the carrier of p:] -defined the carrier of p -valued Function-like V18([: the carrier of p, the carrier of p:], the carrier of p) commutative associative idempotent Element of bool [:[: the carrier of p, the carrier of p:], the carrier of p:]
[: the carrier of p, the carrier of p:] is non empty set
[:[: the carrier of p, the carrier of p:], the carrier of p:] is non empty set
bool [:[: the carrier of p, the carrier of p:], the carrier of p:] is non empty set
the L_meet of p is Relation-like [: the carrier of p, the carrier of p:] -defined the carrier of p -valued Function-like V18([: the carrier of p, the carrier of p:], the carrier of p) commutative associative idempotent Element of bool [:[: the carrier of p, the carrier of p:], the carrier of p:]
LattStr(# the carrier of p, the L_join of p, the L_meet of p #) is non empty strict LattStr
q is non empty Element of bool the carrier of L
<.q.) is non empty final meet-closed join-closed Element of bool the carrier of L
(L,q) is non empty initial meet-closed join-closed Element of bool the carrier of L
K is non empty Element of bool the carrier of p
<.K.) is non empty final meet-closed join-closed Element of bool the carrier of p
(p,K) is non empty initial meet-closed join-closed Element of bool the carrier of p
L is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the carrier of L is non empty set
bool the carrier of L is non empty set
L .: is non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the L_meet of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
[: the carrier of L, the carrier of L:] is non empty set
[:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
bool [:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
the L_join of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
LattStr(# the carrier of L, the L_meet of L, the L_join of L #) is non empty strict LattStr
the carrier of (L .:) is non empty set
bool the carrier of (L .:) is non empty set
p is non empty Element of bool the carrier of L
(L,p) is non empty Element of bool the carrier of (L .:)
<.(L,p).) is non empty final meet-closed join-closed Element of bool the carrier of (L .:)
(L,p) is non empty initial meet-closed join-closed Element of bool the carrier of L
<.p.) is non empty final meet-closed join-closed Element of bool the carrier of L
((L .:),(L,p)) is non empty initial meet-closed join-closed Element of bool the carrier of (L .:)
q is non empty Element of bool the carrier of (L .:)
(L,q) is non empty Element of bool the carrier of L
<.(L,q).) is non empty final meet-closed join-closed Element of bool the carrier of L
((L .:),q) is non empty initial meet-closed join-closed Element of bool the carrier of (L .:)
<.q.) is non empty final meet-closed join-closed Element of bool the carrier of (L .:)
(L,(L,q)) is non empty initial meet-closed join-closed Element of bool the carrier of L
K is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the carrier of K is non empty set
bool the carrier of K is non empty set
K .: is non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the L_meet of K is Relation-like [: the carrier of K, the carrier of K:] -defined the carrier of K -valued Function-like V18([: the carrier of K, the carrier of K:], the carrier of K) commutative associative idempotent Element of bool [:[: the carrier of K, the carrier of K:], the carrier of K:]
[: the carrier of K, the carrier of K:] is non empty set
[:[: the carrier of K, the carrier of K:], the carrier of K:] is non empty set
bool [:[: the carrier of K, the carrier of K:], the carrier of K:] is non empty set
the L_join of K is Relation-like [: the carrier of K, the carrier of K:] -defined the carrier of K -valued Function-like V18([: the carrier of K, the carrier of K:], the carrier of K) commutative associative idempotent Element of bool [:[: the carrier of K, the carrier of K:], the carrier of K:]
LattStr(# the carrier of K, the L_meet of K, the L_join of K #) is non empty strict LattStr
b9 is non empty Element of bool the carrier of K
(K,b9) is non empty Element of bool the carrier of (K .:)
the carrier of (K .:) is non empty set
bool the carrier of (K .:) is non empty set
<.(K,b9).) is non empty final meet-closed join-closed Element of bool the carrier of (K .:)
(K,b9) is non empty initial meet-closed join-closed Element of bool the carrier of K
(K,(K,b9)) is non empty final meet-closed join-closed Element of bool the carrier of (K .:)
(K,<.(K,b9).)) is non empty initial meet-closed join-closed Element of bool the carrier of K
(L .:) .: is non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the L_meet of (L .:) is Relation-like [: the carrier of (L .:), the carrier of (L .:):] -defined the carrier of (L .:) -valued Function-like V18([: the carrier of (L .:), the carrier of (L .:):], the carrier of (L .:)) commutative associative idempotent Element of bool [:[: the carrier of (L .:), the carrier of (L .:):], the carrier of (L .:):]
[: the carrier of (L .:), the carrier of (L .:):] is non empty set
[:[: the carrier of (L .:), the carrier of (L .:):], the carrier of (L .:):] is non empty set
bool [:[: the carrier of (L .:), the carrier of (L .:):], the carrier of (L .:):] is non empty set
the L_join of (L .:) is Relation-like [: the carrier of (L .:), the carrier of (L .:):] -defined the carrier of (L .:) -valued Function-like V18([: the carrier of (L .:), the carrier of (L .:):], the carrier of (L .:)) commutative associative idempotent Element of bool [:[: the carrier of (L .:), the carrier of (L .:):], the carrier of (L .:):]
LattStr(# the carrier of (L .:), the L_meet of (L .:), the L_join of (L .:) #) is non empty strict LattStr
((L .:),(L,p)) is non empty Element of bool the carrier of ((L .:) .:)
the carrier of ((L .:) .:) is non empty set
bool the carrier of ((L .:) .:) is non empty set
<.((L .:),(L,p)).) is non empty final meet-closed join-closed Element of bool the carrier of ((L .:) .:)
(L,(L,q)) is non empty Element of bool the carrier of (L .:)
((L .:),(L,(L,q))) is non empty Element of bool the carrier of ((L .:) .:)
<.((L .:),(L,(L,q))).) is non empty final meet-closed join-closed Element of bool the carrier of ((L .:) .:)
L is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the carrier of L is non empty set
bool the carrier of L is non empty set
p is non empty initial meet-closed join-closed Element of bool the carrier of L
(L,p) is non empty initial meet-closed join-closed Element of bool the carrier of L
L is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the carrier of L is non empty set
bool the carrier of L is non empty set
p is non empty Element of bool the carrier of L
(L,p) is non empty initial meet-closed join-closed Element of bool the carrier of L
(L,(L,p)) is non empty initial meet-closed join-closed Element of bool the carrier of L
q is non empty Element of bool the carrier of L
(L,q) is non empty initial meet-closed join-closed Element of bool the carrier of L
K is non empty Element of bool the carrier of L
(L,K) is non empty initial meet-closed join-closed Element of bool the carrier of L
L is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the carrier of L is non empty set
bool the carrier of L is non empty set
p is Element of the carrier of L
(L,p) is non empty initial meet-closed join-closed Element of bool the carrier of L
{ b₁ where b₁ is Element of the carrier of L : b₁ [= p } is set
q is non empty Element of bool the carrier of L
(L,q) is non empty initial meet-closed join-closed Element of bool the carrier of L
L .: is non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the L_meet of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
[: the carrier of L, the carrier of L:] is non empty set
[:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
bool [:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
the L_join of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
LattStr(# the carrier of L, the L_meet of L, the L_join of L #) is non empty strict LattStr
(L,p) is Element of the carrier of (L .:)
the carrier of (L .:) is non empty set
<.(L,p).) is non empty final meet-closed join-closed Element of bool the carrier of (L .:)
bool the carrier of (L .:) is non empty set
{ b₁ where b₁ is Element of the carrier of (L .:) : (L,p) [= b₁ } is set
(L,q) is non empty Element of bool the carrier of (L .:)
<.(L,q).) is non empty final meet-closed join-closed Element of bool the carrier of (L .:)
L is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the carrier of L is non empty set
bool the carrier of L is non empty set
p is Element of the carrier of L
{p} is non empty Element of bool the carrier of L
(L,p) is non empty initial meet-closed join-closed Element of bool the carrier of L
{ b₁ where b₁ is Element of the carrier of L : b₁ [= p } is set
q is non empty Element of bool the carrier of L
(L,q) is non empty initial meet-closed join-closed Element of bool the carrier of L
L .: is non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the L_meet of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
[: the carrier of L, the carrier of L:] is non empty set
[:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
bool [:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
the L_join of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
LattStr(# the carrier of L, the L_meet of L, the L_join of L #) is non empty strict LattStr
(L,p) is Element of the carrier of (L .:)
the carrier of (L .:) is non empty set
<.(L,p).) is non empty final meet-closed join-closed Element of bool the carrier of (L .:)
bool the carrier of (L .:) is non empty set
{ b₁ where b₁ is Element of the carrier of (L .:) : (L,p) [= b₁ } is set
(L,q) is non empty Element of bool the carrier of (L .:)
<.(L,q).) is non empty final meet-closed join-closed Element of bool the carrier of (L .:)
L is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the carrier of L is non empty set
bool the carrier of L is non empty set
Top L is Element of the carrier of L
(L) is non empty initial meet-closed join-closed Element of bool the carrier of L
p is non empty Element of bool the carrier of L
(L,p) is non empty initial meet-closed join-closed Element of bool the carrier of L
L .: is non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the L_meet of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
[: the carrier of L, the carrier of L:] is non empty set
[:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
bool [:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
the L_join of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
LattStr(# the carrier of L, the L_meet of L, the L_join of L #) is non empty strict LattStr
Bottom (L .:) is Element of the carrier of (L .:)
the carrier of (L .:) is non empty set
(L,p) is non empty Element of bool the carrier of (L .:)
bool the carrier of (L .:) is non empty set
<.(L,p).) is non empty final meet-closed join-closed Element of bool the carrier of (L .:)
<.(L .:).) is non empty final meet-closed join-closed Element of bool the carrier of (L .:)
L is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the carrier of L is non empty set
bool the carrier of L is non empty set
Top L is Element of the carrier of L
(L) is non empty initial meet-closed join-closed Element of bool the carrier of L
p is non empty initial meet-closed join-closed Element of bool the carrier of L
(L,p) is non empty initial meet-closed join-closed Element of bool the carrier of L
L is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the carrier of L is non empty set
bool the carrier of L is non empty set
L is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the carrier of L is non empty set
bool the carrier of L is non empty set
L .: is non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the L_meet of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
[: the carrier of L, the carrier of L:] is non empty set
[:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
bool [:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
the L_join of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
LattStr(# the carrier of L, the L_meet of L, the L_join of L #) is non empty strict LattStr
p is non empty initial meet-closed join-closed Element of bool the carrier of L
(L,p) is non empty final meet-closed join-closed Element of bool the carrier of (L .:)
the carrier of (L .:) is non empty set
bool the carrier of (L .:) is non empty set
q is Element of the carrier of (L .:)
K is Element of the carrier of (L .:)
q "\/" K is Element of the carrier of (L .:)
the L_join of (L .:) is Relation-like [: the carrier of (L .:), the carrier of (L .:):] -defined the carrier of (L .:) -valued Function-like V18([: the carrier of (L .:), the carrier of (L .:):], the carrier of (L .:)) commutative associative idempotent Element of bool [:[: the carrier of (L .:), the carrier of (L .:):], the carrier of (L .:):]
[: the carrier of (L .:), the carrier of (L .:):] is non empty set
[:[: the carrier of (L .:), the carrier of (L .:):], the carrier of (L .:):] is non empty set
bool [:[: the carrier of (L .:), the carrier of (L .:):], the carrier of (L .:):] is non empty set
the L_join of (L .:) . (q,K) is Element of the carrier of (L .:)
[q,K] is set
{q,K} is non empty set
{q} is non empty set
{{q,K},{q}} is non empty set
the L_join of (L .:) . [q,K] is set
(L,q) is Element of the carrier of L
(L,K) is Element of the carrier of L
(L,q) "/\" (L,K) is Element of the carrier of L
the L_meet of L . ((L,q),(L,K)) is Element of the carrier of L
[(L,q),(L,K)] is set
{(L,q),(L,K)} is non empty set
{(L,q)} is non empty set
{{(L,q),(L,K)},{(L,q)}} is non empty set
the L_meet of L . [(L,q),(L,K)] is set
q is Element of the carrier of L
K is Element of the carrier of L
q "/\" K is Element of the carrier of L
the L_meet of L . (q,K) is Element of the carrier of L
[q,K] is set
{q,K} is non empty set
{q} is non empty set
{{q,K},{q}} is non empty set
the L_meet of L . [q,K] is set
(L,q) is Element of the carrier of (L .:)
(L,K) is Element of the carrier of (L .:)
(L,q) "\/" (L,K) is Element of the carrier of (L .:)
the L_join of (L .:) is Relation-like [: the carrier of (L .:), the carrier of (L .:):] -defined the carrier of (L .:) -valued Function-like V18([: the carrier of (L .:), the carrier of (L .:):], the carrier of (L .:)) commutative associative idempotent Element of bool [:[: the carrier of (L .:), the carrier of (L .:):], the carrier of (L .:):]
[: the carrier of (L .:), the carrier of (L .:):] is non empty set
[:[: the carrier of (L .:), the carrier of (L .:):], the carrier of (L .:):] is non empty set
bool [:[: the carrier of (L .:), the carrier of (L .:):], the carrier of (L .:):] is non empty set
the L_join of (L .:) . ((L,q),(L,K)) is Element of the carrier of (L .:)
[(L,q),(L,K)] is set
{(L,q),(L,K)} is non empty set
{(L,q)} is non empty set
{{(L,q),(L,K)},{(L,q)}} is non empty set
the L_join of (L .:) . [(L,q),(L,K)] is set
L is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the carrier of L is non empty set
bool the carrier of L is non empty set
p is non empty Element of bool the carrier of L
q is non empty Element of bool the carrier of L
{ (b₁ "\/" b₂) where b₁, b₂ is Element of the carrier of L : ( b₁ in p & b₂ in q ) } is set
b9 is set
b is Element of the carrier of L
a is Element of the carrier of L
b "\/" a is Element of the carrier of L
the L_join of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
[: the carrier of L, the carrier of L:] is non empty set
[:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
bool [:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
the L_join of L . (b,a) is Element of the carrier of L
[b,a] is set
{b,a} is non empty set
{b} is non empty set
{{b,a},{b}} is non empty set
the L_join of L . [b,a] is set
L is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the carrier of L is non empty set
bool the carrier of L is non empty set
p is non empty Element of bool the carrier of L
q is non empty Element of bool the carrier of L
(L,p,q) is Element of bool the carrier of L
{ (b₁ "\/" b₂) where b₁, b₂ is Element of the carrier of L : ( b₁ in p & b₂ in q ) } is set
the Element of p is Element of p
the Element of q is Element of q
the Element of p "\/" the Element of q is Element of the carrier of L
the L_join of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
[: the carrier of L, the carrier of L:] is non empty set
[:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
bool [:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
the L_join of L . ( the Element of p, the Element of q) is Element of the carrier of L
[ the Element of p, the Element of q] is set
{ the Element of p, the Element of q} is non empty set
{ the Element of p} is non empty set
{{ the Element of p, the Element of q},{ the Element of p}} is non empty set
the L_join of L . [ the Element of p, the Element of q] is set
a is non empty set
L is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the carrier of L is non empty set
bool the carrier of L is non empty set
p is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the carrier of p is non empty set
bool the carrier of p is non empty set
the L_join of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
[: the carrier of L, the carrier of L:] is non empty set
[:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
bool [:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
the L_meet of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
LattStr(# the carrier of L, the L_join of L, the L_meet of L #) is non empty strict LattStr
the L_join of p is Relation-like [: the carrier of p, the carrier of p:] -defined the carrier of p -valued Function-like V18([: the carrier of p, the carrier of p:], the carrier of p) commutative associative idempotent Element of bool [:[: the carrier of p, the carrier of p:], the carrier of p:]
[: the carrier of p, the carrier of p:] is non empty set
[:[: the carrier of p, the carrier of p:], the carrier of p:] is non empty set
bool [:[: the carrier of p, the carrier of p:], the carrier of p:] is non empty set
the L_meet of p is Relation-like [: the carrier of p, the carrier of p:] -defined the carrier of p -valued Function-like V18([: the carrier of p, the carrier of p:], the carrier of p) commutative associative idempotent Element of bool [:[: the carrier of p, the carrier of p:], the carrier of p:]
LattStr(# the carrier of p, the L_join of p, the L_meet of p #) is non empty strict LattStr
q is non empty Element of bool the carrier of L
K is non empty Element of bool the carrier of L
q "/\" K is non empty Element of bool the carrier of L
{ (b₁ "/\" b₂) where b₁, b₂ is Element of the carrier of L : ( b₁ in q & b₂ in K ) } is set
b9 is non empty Element of bool the carrier of p
b is non empty Element of bool the carrier of p
b9 "/\" b is non empty Element of bool the carrier of p
{ (b₁ "/\" b₂) where b₁, b₂ is Element of the carrier of p : ( b₁ in b9 & b₂ in b ) } is set
a is set
a9 is Element of the carrier of L
c₉ is Element of the carrier of L
a9 "/\" c₉ is Element of the carrier of L
the L_meet of L . (a9,c₉) is Element of the carrier of L
[a9,c₉] is set
{a9,c₉} is non empty set
{a9} is non empty set
{{a9,c₉},{a9}} is non empty set
the L_meet of L . [a9,c₉] is set
c is Element of the carrier of p
d is Element of the carrier of p
c "/\" d is Element of the carrier of p
the L_meet of p . (c,d) is Element of the carrier of p
[c,d] is set
{c,d} is non empty set
{c} is non empty set
{{c,d},{c}} is non empty set
the L_meet of p . [c,d] is set
a is set
a9 is Element of the carrier of p
c₉ is Element of the carrier of p
a9 "/\" c₉ is Element of the carrier of p
the L_meet of p . (a9,c₉) is Element of the carrier of p
[a9,c₉] is set
{a9,c₉} is non empty set
{a9} is non empty set
{{a9,c₉},{a9}} is non empty set
the L_meet of p . [a9,c₉] is set
c is Element of the carrier of L
d is Element of the carrier of L
c "/\" d is Element of the carrier of L
the L_meet of L . (c,d) is Element of the carrier of L
[c,d] is set
{c,d} is non empty set
{c} is non empty set
{{c,d},{c}} is non empty set
the L_meet of L . [c,d] is set
L is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the carrier of L is non empty set
bool the carrier of L is non empty set
L .: is non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the L_meet of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
[: the carrier of L, the carrier of L:] is non empty set
[:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
bool [:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
the L_join of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
LattStr(# the carrier of L, the L_meet of L, the L_join of L #) is non empty strict LattStr
the carrier of (L .:) is non empty set
bool the carrier of (L .:) is non empty set
p is non empty Element of bool the carrier of L
(L,p) is non empty Element of bool the carrier of (L .:)
q is non empty Element of bool the carrier of L
(L,p,q) is non empty Element of bool the carrier of L
{ (b₁ "\/" b₂) where b₁, b₂ is Element of the carrier of L : ( b₁ in p & b₂ in q ) } is set
(L,q) is non empty Element of bool the carrier of (L .:)
(L,p) "/\" (L,q) is non empty Element of bool the carrier of (L .:)
{ (b₁ "/\" b₂) where b₁, b₂ is Element of the carrier of (L .:) : ( b₁ in (L,p) & b₂ in (L,q) ) } is set
((L .:),(L,p),(L,q)) is non empty Element of bool the carrier of (L .:)
{ (b₁ "\/" b₂) where b₁, b₂ is Element of the carrier of (L .:) : ( b₁ in (L,p) & b₂ in (L,q) ) } is set
p "/\" q is non empty Element of bool the carrier of L
{ (b₁ "/\" b₂) where b₁, b₂ is Element of the carrier of L : ( b₁ in p & b₂ in q ) } is set
K is non empty Element of bool the carrier of (L .:)
(L,K) is non empty Element of bool the carrier of L
b9 is non empty Element of bool the carrier of (L .:)
((L .:),K,b9) is non empty Element of bool the carrier of (L .:)
{ (b₁ "\/" b₂) where b₁, b₂ is Element of the carrier of (L .:) : ( b₁ in K & b₂ in b9 ) } is set
(L,b9) is non empty Element of bool the carrier of L
(L,K) "/\" (L,b9) is non empty Element of bool the carrier of L
{ (b₁ "/\" b₂) where b₁, b₂ is Element of the carrier of L : ( b₁ in (L,K) & b₂ in (L,b9) ) } is set
(L,(L,K),(L,b9)) is non empty Element of bool the carrier of L
{ (b₁ "\/" b₂) where b₁, b₂ is Element of the carrier of L : ( b₁ in (L,K) & b₂ in (L,b9) ) } is set
K "/\" b9 is non empty Element of bool the carrier of (L .:)
{ (b₁ "/\" b₂) where b₁, b₂ is Element of the carrier of (L .:) : ( b₁ in K & b₂ in b9 ) } is set
b is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the carrier of b is non empty set
bool the carrier of b is non empty set
b .: is non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the L_meet of b is Relation-like [: the carrier of b, the carrier of b:] -defined the carrier of b -valued Function-like V18([: the carrier of b, the carrier of b:], the carrier of b) commutative associative idempotent Element of bool [:[: the carrier of b, the carrier of b:], the carrier of b:]
[: the carrier of b, the carrier of b:] is non empty set
[:[: the carrier of b, the carrier of b:], the carrier of b:] is non empty set
bool [:[: the carrier of b, the carrier of b:], the carrier of b:] is non empty set
the L_join of b is Relation-like [: the carrier of b, the carrier of b:] -defined the carrier of b -valued Function-like V18([: the carrier of b, the carrier of b:], the carrier of b) commutative associative idempotent Element of bool [:[: the carrier of b, the carrier of b:], the carrier of b:]
LattStr(# the carrier of b, the L_meet of b, the L_join of b #) is non empty strict LattStr
a is non empty Element of bool the carrier of b
(b,a) is non empty Element of bool the carrier of (b .:)
the carrier of (b .:) is non empty set
bool the carrier of (b .:) is non empty set
a9 is non empty Element of bool the carrier of b
(b,a,a9) is non empty Element of bool the carrier of b
{ (b₁ "\/" b₂) where b₁, b₂ is Element of the carrier of b : ( b₁ in a & b₂ in a9 ) } is set
(b,a9) is non empty Element of bool the carrier of (b .:)
(b,a) "/\" (b,a9) is non empty Element of bool the carrier of (b .:)
{ (b₁ "/\" b₂) where b₁, b₂ is Element of the carrier of (b .:) : ( b₁ in (b,a) & b₂ in (b,a9) ) } is set
c₉ is set
c is Element of the carrier of b
d is Element of the carrier of b
c "\/" d is Element of the carrier of b
the L_join of b . (c,d) is Element of the carrier of b
[c,d] is set
{c,d} is non empty set
{c} is non empty set
{{c,d},{c}} is non empty set
the L_join of b . [c,d] is set
(b,c) is Element of the carrier of (b .:)
(b,d) is Element of the carrier of (b .:)
(b,c) "/\" (b,d) is Element of the carrier of (b .:)
the L_meet of (b .:) is Relation-like [: the carrier of (b .:), the carrier of (b .:):] -defined the carrier of (b .:) -valued Function-like V18([: the carrier of (b .:), the carrier of (b .:):], the carrier of (b .:)) commutative associative idempotent Element of bool [:[: the carrier of (b .:), the carrier of (b .:):], the carrier of (b .:):]
[: the carrier of (b .:), the carrier of (b .:):] is non empty set
[:[: the carrier of (b .:), the carrier of (b .:):], the carrier of (b .:):] is non empty set
bool [:[: the carrier of (b .:), the carrier of (b .:):], the carrier of (b .:):] is non empty set
the L_meet of (b .:) . ((b,c),(b,d)) is Element of the carrier of (b .:)
[(b,c),(b,d)] is set
{(b,c),(b,d)} is non empty set
{(b,c)} is non empty set
{{(b,c),(b,d)},{(b,c)}} is non empty set
the L_meet of (b .:) . [(b,c),(b,d)] is set
c₉ is set
c is Element of the carrier of (b .:)
d is Element of the carrier of (b .:)
c "/\" d is Element of the carrier of (b .:)
the L_meet of (b .:) is Relation-like [: the carrier of (b .:), the carrier of (b .:):] -defined the carrier of (b .:) -valued Function-like V18([: the carrier of (b .:), the carrier of (b .:):], the carrier of (b .:)) commutative associative idempotent Element of bool [:[: the carrier of (b .:), the carrier of (b .:):], the carrier of (b .:):]
[: the carrier of (b .:), the carrier of (b .:):] is non empty set
[:[: the carrier of (b .:), the carrier of (b .:):], the carrier of (b .:):] is non empty set
bool [:[: the carrier of (b .:), the carrier of (b .:):], the carrier of (b .:):] is non empty set
the L_meet of (b .:) . (c,d) is Element of the carrier of (b .:)
[c,d] is set
{c,d} is non empty set
{c} is non empty set
{{c,d},{c}} is non empty set
the L_meet of (b .:) . [c,d] is set
(b,c) is Element of the carrier of b
(b,d) is Element of the carrier of b
(b,c) "\/" (b,d) is Element of the carrier of b
the L_join of b . ((b,c),(b,d)) is Element of the carrier of b
[(b,c),(b,d)] is set
{(b,c),(b,d)} is non empty set
{(b,c)} is non empty set
{{(b,c),(b,d)},{(b,c)}} is non empty set
the L_join of b . [(b,c),(b,d)] is set
(L,(L,K)) is non empty Element of bool the carrier of (L .:)
(L,(L,b9)) is non empty Element of bool the carrier of (L .:)
(L .:) .: is non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the L_meet of (L .:) is Relation-like [: the carrier of (L .:), the carrier of (L .:):] -defined the carrier of (L .:) -valued Function-like V18([: the carrier of (L .:), the carrier of (L .:):], the carrier of (L .:)) commutative associative idempotent Element of bool [:[: the carrier of (L .:), the carrier of (L .:):], the carrier of (L .:):]
[: the carrier of (L .:), the carrier of (L .:):] is non empty set
[:[: the carrier of (L .:), the carrier of (L .:):], the carrier of (L .:):] is non empty set
bool [:[: the carrier of (L .:), the carrier of (L .:):], the carrier of (L .:):] is non empty set
the L_join of (L .:) is Relation-like [: the carrier of (L .:), the carrier of (L .:):] -defined the carrier of (L .:) -valued Function-like V18([: the carrier of (L .:), the carrier of (L .:):], the carrier of (L .:)) commutative associative idempotent Element of bool [:[: the carrier of (L .:), the carrier of (L .:):], the carrier of (L .:):]
LattStr(# the carrier of (L .:), the L_meet of (L .:), the L_join of (L .:) #) is non empty strict LattStr
((L .:),(L,p)) is non empty Element of bool the carrier of ((L .:) .:)
the carrier of ((L .:) .:) is non empty set
bool the carrier of ((L .:) .:) is non empty set
((L .:),(L,q)) is non empty Element of bool the carrier of ((L .:) .:)
((L .:),(L,p)) "/\" ((L .:),(L,q)) is non empty Element of bool the carrier of ((L .:) .:)
{ (b₁ "/\" b₂) where b₁, b₂ is Element of the carrier of ((L .:) .:) : ( b₁ in ((L .:),(L,p)) & b₂ in ((L .:),(L,q)) ) } is set
((L .:),K) is non empty Element of bool the carrier of ((L .:) .:)
((L .:),b9) is non empty Element of bool the carrier of ((L .:) .:)
((L .:),K) "/\" ((L .:),b9) is non empty Element of bool the carrier of ((L .:) .:)
{ (b₁ "/\" b₂) where b₁, b₂ is Element of the carrier of ((L .:) .:) : ( b₁ in ((L .:),K) & b₂ in ((L .:),b9) ) } is set
L is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the carrier of L is non empty set
bool the carrier of L is non empty set
p is Element of the carrier of L
K is non empty Element of bool the carrier of L
q is Element of the carrier of L
b9 is non empty Element of bool the carrier of L
p "\/" q is Element of the carrier of L
the L_join of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
[: the carrier of L, the carrier of L:] is non empty set
[:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
bool [:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
the L_join of L . (p,q) is Element of the carrier of L
[p,q] is set
{p,q} is non empty set
{p} is non empty set
{{p,q},{p}} is non empty set
the L_join of L . [p,q] is set
(L,K,b9) is non empty Element of bool the carrier of L
{ (b₁ "\/" b₂) where b₁, b₂ is Element of the carrier of L : ( b₁ in K & b₂ in b9 ) } is set
q "\/" p is Element of the carrier of L
the L_join of L . (q,p) is Element of the carrier of L
[q,p] is set
{q,p} is non empty set
{q} is non empty set
{{q,p},{q}} is non empty set
the L_join of L . [q,p] is set
L is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the carrier of L is non empty set
bool the carrier of L is non empty set
p is set
q is non empty Element of bool the carrier of L
K is non empty Element of bool the carrier of L
(L,q,K) is non empty Element of bool the carrier of L
{ (b₁ "\/" b₂) where b₁, b₂ is Element of the carrier of L : ( b₁ in q & b₂ in K ) } is set
L is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the carrier of L is non empty set
bool the carrier of L is non empty set
p is non empty Element of bool the carrier of L
q is non empty Element of bool the carrier of L
(L,p,q) is non empty Element of bool the carrier of L
{ (b₁ "\/" b₂) where b₁, b₂ is Element of the carrier of L : ( b₁ in p & b₂ in q ) } is set
(L,q,p) is non empty Element of bool the carrier of L
{ (b₁ "\/" b₂) where b₁, b₂ is Element of the carrier of L : ( b₁ in q & b₂ in p ) } is set
K is Element of the carrier of L
b9 is Element of the carrier of L
b is Element of the carrier of L
b9 "\/" b is Element of the carrier of L
the L_join of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
[: the carrier of L, the carrier of L:] is non empty set
[:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
bool [:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
the L_join of L . (b9,b) is Element of the carrier of L
[b9,b] is set
{b9,b} is non empty set
{b9} is non empty set
{{b9,b},{b9}} is non empty set
the L_join of L . [b9,b] is set
b9 is Element of the carrier of L
b is Element of the carrier of L
b9 "\/" b is Element of the carrier of L
the L_join of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
[: the carrier of L, the carrier of L:] is non empty set
[:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
bool [:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
the L_join of L . (b9,b) is Element of the carrier of L
[b9,b] is set
{b9,b} is non empty set
{b9} is non empty set
{{b9,b},{b9}} is non empty set
the L_join of L . [b9,b] is set
L is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular LattStr
the carrier of L is non empty set
bool the carrier of L is non empty set
p is non empty initial meet-closed join-closed Element of bool the carrier of L
q is non empty initial meet-closed join-closed Element of bool the carrier of L
(L,p,q) is non empty Element of bool the carrier of L
{ (b₁ "\/" b₂) where b₁, b₂ is Element of the carrier of L : ( b₁ in p & b₂ in q ) } is set
L .: is non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the L_meet of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
[: the carrier of L, the carrier of L:] is non empty set
[:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
bool [:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
the L_join of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
LattStr(# the carrier of L, the L_meet of L, the L_join of L #) is non empty strict LattStr
K is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular LattStr
the carrier of K is non empty set
bool the carrier of K is non empty set
(L,p) is non empty final meet-closed join-closed Element of bool the carrier of (L .:)
the carrier of (L .:) is non empty set
bool the carrier of (L .:) is non empty set
(L,q) is non empty final meet-closed join-closed Element of bool the carrier of (L .:)
(L,p) "/\" (L,q) is non empty Element of bool the carrier of (L .:)
{ (b₁ "/\" b₂) where b₁, b₂ is Element of the carrier of (L .:) : ( b₁ in (L,p) & b₂ in (L,q) ) } is set
b9 is non empty final meet-closed join-closed Element of bool the carrier of K
b is non empty final meet-closed join-closed Element of bool the carrier of K
b9 "/\" b is non empty final meet-closed join-closed Element of bool the carrier of K
{ (b₁ "/\" b₂) where b₁, b₂ is Element of the carrier of K : ( b₁ in b9 & b₂ in b ) } is set
(K,(b9 "/\" b)) is non empty initial meet-closed join-closed Element of bool the carrier of (K .:)
K .: is non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the L_meet of K is Relation-like [: the carrier of K, the carrier of K:] -defined the carrier of K -valued Function-like V18([: the carrier of K, the carrier of K:], the carrier of K) commutative associative idempotent Element of bool [:[: the carrier of K, the carrier of K:], the carrier of K:]
[: the carrier of K, the carrier of K:] is non empty set
[:[: the carrier of K, the carrier of K:], the carrier of K:] is non empty set
bool [:[: the carrier of K, the carrier of K:], the carrier of K:] is non empty set
the L_join of K is Relation-like [: the carrier of K, the carrier of K:] -defined the carrier of K -valued Function-like V18([: the carrier of K, the carrier of K:], the carrier of K) commutative associative idempotent Element of bool [:[: the carrier of K, the carrier of K:], the carrier of K:]
LattStr(# the carrier of K, the L_meet of K, the L_join of K #) is non empty strict LattStr
the carrier of (K .:) is non empty set
bool the carrier of (K .:) is non empty set
L is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the carrier of L is non empty set
bool the carrier of L is non empty set
p is non empty Element of bool the carrier of L
(L,p) is non empty initial meet-closed join-closed Element of bool the carrier of L
q is non empty Element of bool the carrier of L
p \/ q is non empty Element of bool the carrier of L
(L,(p \/ q)) is non empty initial meet-closed join-closed Element of bool the carrier of L
(L,p) \/ q is non empty Element of bool the carrier of L
(L,((L,p) \/ q)) is non empty initial meet-closed join-closed Element of bool the carrier of L
(L,q) is non empty initial meet-closed join-closed Element of bool the carrier of L
p \/ (L,q) is non empty Element of bool the carrier of L
(L,(p \/ (L,q))) is non empty initial meet-closed join-closed Element of bool the carrier of L
L .: is non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the L_meet of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
[: the carrier of L, the carrier of L:] is non empty set
[:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
bool [:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
the L_join of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
LattStr(# the carrier of L, the L_meet of L, the L_join of L #) is non empty strict LattStr
(L,(p \/ (L,q))) is non empty Element of bool the carrier of (L .:)
the carrier of (L .:) is non empty set
bool the carrier of (L .:) is non empty set
<.(L,(p \/ (L,q))).) is non empty final meet-closed join-closed Element of bool the carrier of (L .:)
(L,q) is non empty Element of bool the carrier of (L .:)
<.(L,q).) is non empty final meet-closed join-closed Element of bool the carrier of (L .:)
(L,(p \/ q)) is non empty Element of bool the carrier of (L .:)
<.(L,(p \/ q)).) is non empty final meet-closed join-closed Element of bool the carrier of (L .:)
(L,((L,p) \/ q)) is non empty Element of bool the carrier of (L .:)
<.(L,((L,p) \/ q)).) is non empty final meet-closed join-closed Element of bool the carrier of (L .:)
(L,p) is non empty Element of bool the carrier of (L .:)
<.(L,p).) is non empty final meet-closed join-closed Element of bool the carrier of (L .:)
L is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the carrier of L is non empty set
bool the carrier of L is non empty set
p is non empty initial meet-closed join-closed Element of bool the carrier of L
q is non empty initial meet-closed join-closed Element of bool the carrier of L
p \/ q is non empty Element of bool the carrier of L
(L,(p \/ q)) is non empty initial meet-closed join-closed Element of bool the carrier of L
{ b₁ where b₁ is Element of the carrier of L : ex b₂, b₃ being Element of the carrier of L st
( b₁ [= b₂ "\/" b₃ & b₂ in p & b₃ in q ) } is set
(L,q) is non empty final meet-closed join-closed Element of bool the carrier of (L .:)
L .: is non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the L_meet of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
[: the carrier of L, the carrier of L:] is non empty set
[:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
bool [:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
the L_join of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
LattStr(# the carrier of L, the L_meet of L, the L_join of L #) is non empty strict LattStr
the carrier of (L .:) is non empty set
bool the carrier of (L .:) is non empty set
(L,(p \/ q)) is non empty Element of bool the carrier of (L .:)
<.(L,(p \/ q)).) is non empty final meet-closed join-closed Element of bool the carrier of (L .:)
(L,p) is non empty final meet-closed join-closed Element of bool the carrier of (L .:)
{ b₁ where b₁ is Element of the carrier of (L .:) : ex b₂, b₃ being Element of the carrier of (L .:) st
( b₂ "/\" b₃ [= b₁ & b₂ in p & b₃ in q ) } is set
K is set
b9 is Element of the carrier of (L .:)
b is Element of the carrier of (L .:)
a is Element of the carrier of (L .:)
b "/\" a is Element of the carrier of (L .:)
the L_meet of (L .:) is Relation-like [: the carrier of (L .:), the carrier of (L .:):] -defined the carrier of (L .:) -valued Function-like V18([: the carrier of (L .:), the carrier of (L .:):], the carrier of (L .:)) commutative associative idempotent Element of bool [:[: the carrier of (L .:), the carrier of (L .:):], the carrier of (L .:):]
[: the carrier of (L .:), the carrier of (L .:):] is non empty set
[:[: the carrier of (L .:), the carrier of (L .:):], the carrier of (L .:):] is non empty set
bool [:[: the carrier of (L .:), the carrier of (L .:):], the carrier of (L .:):] is non empty set
the L_meet of (L .:) . (b,a) is Element of the carrier of (L .:)
[b,a] is set
{b,a} is non empty set
{b} is non empty set
{{b,a},{b}} is non empty set
the L_meet of (L .:) . [b,a] is set
b is Element of the carrier of (L .:)
a is Element of the carrier of (L .:)
b "/\" a is Element of the carrier of (L .:)
the L_meet of (L .:) is Relation-like [: the carrier of (L .:), the carrier of (L .:):] -defined the carrier of (L .:) -valued Function-like V18([: the carrier of (L .:), the carrier of (L .:):], the carrier of (L .:)) commutative associative idempotent Element of bool [:[: the carrier of (L .:), the carrier of (L .:):], the carrier of (L .:):]
[: the carrier of (L .:), the carrier of (L .:):] is non empty set
[:[: the carrier of (L .:), the carrier of (L .:):], the carrier of (L .:):] is non empty set
bool [:[: the carrier of (L .:), the carrier of (L .:):], the carrier of (L .:):] is non empty set
the L_meet of (L .:) . (b,a) is Element of the carrier of (L .:)
[b,a] is set
{b,a} is non empty set
{b} is non empty set
{{b,a},{b}} is non empty set
the L_meet of (L .:) . [b,a] is set
(L,b) is Element of the carrier of L
(L,a) is Element of the carrier of L
(L,b) "\/" (L,a) is Element of the carrier of L
the L_join of L . ((L,b),(L,a)) is Element of the carrier of L
[(L,b),(L,a)] is set
{(L,b),(L,a)} is non empty set
{(L,b)} is non empty set
{{(L,b),(L,a)},{(L,b)}} is non empty set
the L_join of L . [(L,b),(L,a)] is set
(L,b9) is Element of the carrier of L
(L,(b "/\" a)) is Element of the carrier of L
K is set
b9 is Element of the carrier of L
b is Element of the carrier of L
a is Element of the carrier of L
b "\/" a is Element of the carrier of L
the L_join of L . (b,a) is Element of the carrier of L
[b,a] is set
{b,a} is non empty set
{b} is non empty set
{{b,a},{b}} is non empty set
the L_join of L . [b,a] is set
b is Element of the carrier of L
a is Element of the carrier of L
b "\/" a is Element of the carrier of L
the L_join of L . (b,a) is Element of the carrier of L
[b,a] is set
{b,a} is non empty set
{b} is non empty set
{{b,a},{b}} is non empty set
the L_join of L . [b,a] is set
(L,b) is Element of the carrier of (L .:)
(L,a) is Element of the carrier of (L .:)
(L,b) "/\" (L,a) is Element of the carrier of (L .:)
the L_meet of (L .:) is Relation-like [: the carrier of (L .:), the carrier of (L .:):] -defined the carrier of (L .:) -valued Function-like V18([: the carrier of (L .:), the carrier of (L .:):], the carrier of (L .:)) commutative associative idempotent Element of bool [:[: the carrier of (L .:), the carrier of (L .:):], the carrier of (L .:):]
[: the carrier of (L .:), the carrier of (L .:):] is non empty set
[:[: the carrier of (L .:), the carrier of (L .:):], the carrier of (L .:):] is non empty set
bool [:[: the carrier of (L .:), the carrier of (L .:):], the carrier of (L .:):] is non empty set
the L_meet of (L .:) . ((L,b),(L,a)) is Element of the carrier of (L .:)
[(L,b),(L,a)] is set
{(L,b),(L,a)} is non empty set
{(L,b)} is non empty set
{{(L,b),(L,a)},{(L,b)}} is non empty set
the L_meet of (L .:) . [(L,b),(L,a)] is set
(L,(b "\/" a)) is Element of the carrier of (L .:)
(L,b9) is Element of the carrier of (L .:)
L is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the carrier of L is non empty set
bool the carrier of L is non empty set
p is non empty initial meet-closed join-closed Element of bool the carrier of L
q is non empty initial meet-closed join-closed Element of bool the carrier of L
(L,p,q) is non empty Element of bool the carrier of L
{ (b₁ "\/" b₂) where b₁, b₂ is Element of the carrier of L : ( b₁ in p & b₂ in q ) } is set
L .: is non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the L_meet of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
[: the carrier of L, the carrier of L:] is non empty set
[:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
bool [:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
the L_join of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
LattStr(# the carrier of L, the L_meet of L, the L_join of L #) is non empty strict LattStr
(L,p) is non empty final meet-closed join-closed Element of bool the carrier of (L .:)
the carrier of (L .:) is non empty set
bool the carrier of (L .:) is non empty set
(L,q) is non empty final meet-closed join-closed Element of bool the carrier of (L .:)
(L,p) "/\" (L,q) is non empty Element of bool the carrier of (L .:)
{ (b₁ "/\" b₂) where b₁, b₂ is Element of the carrier of (L .:) : ( b₁ in (L,p) & b₂ in (L,q) ) } is set
L is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the carrier of L is non empty set
bool the carrier of L is non empty set
p is non empty initial meet-closed join-closed Element of bool the carrier of L
q is non empty initial meet-closed join-closed Element of bool the carrier of L
p \/ q is non empty Element of bool the carrier of L
(L,(p \/ q)) is non empty initial meet-closed join-closed Element of bool the carrier of L
(L,p,q) is non empty Element of bool the carrier of L
{ (b₁ "\/" b₂) where b₁, b₂ is Element of the carrier of L : ( b₁ in p & b₂ in q ) } is set
(L,(L,p,q)) is non empty initial meet-closed join-closed Element of bool the carrier of L
L .: is non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the L_meet of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
[: the carrier of L, the carrier of L:] is non empty set
[:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
bool [:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
the L_join of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
LattStr(# the carrier of L, the L_meet of L, the L_join of L #) is non empty strict LattStr
(L,(L,p,q)) is non empty Element of bool the carrier of (L .:)
the carrier of (L .:) is non empty set
bool the carrier of (L .:) is non empty set
<.(L,(L,p,q)).) is non empty final meet-closed join-closed Element of bool the carrier of (L .:)
(L,(p \/ q)) is non empty Element of bool the carrier of (L .:)
<.(L,(p \/ q)).) is non empty final meet-closed join-closed Element of bool the carrier of (L .:)
(L,p) is non empty final meet-closed join-closed Element of bool the carrier of (L .:)
(L,q) is non empty final meet-closed join-closed Element of bool the carrier of (L .:)
(L,p) "/\" (L,q) is non empty Element of bool the carrier of (L .:)
{ (b₁ "/\" b₂) where b₁, b₂ is Element of the carrier of (L .:) : ( b₁ in (L,p) & b₂ in (L,q) ) } is set
L is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
L .: is non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the carrier of L is non empty set
the L_meet of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
[: the carrier of L, the carrier of L:] is non empty set
[:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
bool [:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
the L_join of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
LattStr(# the carrier of L, the L_meet of L, the L_join of L #) is non empty strict LattStr
the carrier of (L .:) is non empty set
p is Element of the carrier of (L .:)
(L,p) is Element of the carrier of L
q is Element of the carrier of L
(L,q) is Element of the carrier of (L .:)
q "\/" (L,p) is Element of the carrier of L
the L_join of L . (q,(L,p)) is Element of the carrier of L
[q,(L,p)] is set
{q,(L,p)} is non empty set
{q} is non empty set
{{q,(L,p)},{q}} is non empty set
the L_join of L . [q,(L,p)] is set
Top L is Element of the carrier of L
(L,q) "/\" p is Element of the carrier of (L .:)
the L_meet of (L .:) is Relation-like [: the carrier of (L .:), the carrier of (L .:):] -defined the carrier of (L .:) -valued Function-like V18([: the carrier of (L .:), the carrier of (L .:):], the carrier of (L .:)) commutative associative idempotent Element of bool [:[: the carrier of (L .:), the carrier of (L .:):], the carrier of (L .:):]
[: the carrier of (L .:), the carrier of (L .:):] is non empty set
[:[: the carrier of (L .:), the carrier of (L .:):], the carrier of (L .:):] is non empty set
bool [:[: the carrier of (L .:), the carrier of (L .:):], the carrier of (L .:):] is non empty set
the L_meet of (L .:) . ((L,q),p) is Element of the carrier of (L .:)
[(L,q),p] is set
{(L,q),p} is non empty set
{(L,q)} is non empty set
{{(L,q),p},{(L,q)}} is non empty set
the L_meet of (L .:) . [(L,q),p] is set
q "/\" (L,p) is Element of the carrier of L
the L_meet of L . (q,(L,p)) is Element of the carrier of L
the L_meet of L . [q,(L,p)] is set
Bottom L is Element of the carrier of L
(L,q) "\/" p is Element of the carrier of (L .:)
the L_join of (L .:) is Relation-like [: the carrier of (L .:), the carrier of (L .:):] -defined the carrier of (L .:) -valued Function-like V18([: the carrier of (L .:), the carrier of (L .:):], the carrier of (L .:)) commutative associative idempotent Element of bool [:[: the carrier of (L .:), the carrier of (L .:):], the carrier of (L .:):]
the L_join of (L .:) . ((L,q),p) is Element of the carrier of (L .:)
the L_join of (L .:) . [(L,q),p] is set
Top (L .:) is Element of the carrier of (L .:)
Bottom (L .:) is Element of the carrier of (L .:)
K is Element of the carrier of (L .:)
the carrier of (L .:) is non empty set
p is Element of the carrier of L
(L,p) is Element of the carrier of (L .:)
q is Element of the carrier of (L .:)
q "\/" (L,p) is Element of the carrier of (L .:)
the L_join of (L .:) is Relation-like [: the carrier of (L .:), the carrier of (L .:):] -defined the carrier of (L .:) -valued Function-like V18([: the carrier of (L .:), the carrier of (L .:):], the carrier of (L .:)) commutative associative idempotent Element of bool [:[: the carrier of (L .:), the carrier of (L .:):], the carrier of (L .:):]
[: the carrier of (L .:), the carrier of (L .:):] is non empty set
[:[: the carrier of (L .:), the carrier of (L .:):], the carrier of (L .:):] is non empty set
bool [:[: the carrier of (L .:), the carrier of (L .:):], the carrier of (L .:):] is non empty set
the L_join of (L .:) . (q,(L,p)) is Element of the carrier of (L .:)
[q,(L,p)] is set
{q,(L,p)} is non empty set
{q} is non empty set
{{q,(L,p)},{q}} is non empty set
the L_join of (L .:) . [q,(L,p)] is set
Top (L .:) is Element of the carrier of (L .:)
(L,q) is Element of the carrier of L
(L,q) "/\" p is Element of the carrier of L
the L_meet of L . ((L,q),p) is Element of the carrier of L
[(L,q),p] is set
{(L,q),p} is non empty set
{(L,q)} is non empty set
{{(L,q),p},{(L,q)}} is non empty set
the L_meet of L . [(L,q),p] is set
Bottom (L .:) is Element of the carrier of (L .:)
Top L is Element of the carrier of L
q "/\" (L,p) is Element of the carrier of (L .:)
the L_meet of (L .:) is Relation-like [: the carrier of (L .:), the carrier of (L .:):] -defined the carrier of (L .:) -valued Function-like V18([: the carrier of (L .:), the carrier of (L .:):], the carrier of (L .:)) commutative associative idempotent Element of bool [:[: the carrier of (L .:), the carrier of (L .:):], the carrier of (L .:):]
the L_meet of (L .:) . (q,(L,p)) is Element of the carrier of (L .:)
the L_meet of (L .:) . [q,(L,p)] is set
(L,q) "\/" p is Element of the carrier of L
the L_join of L . ((L,q),p) is Element of the carrier of L
the L_join of L . [(L,q),p] is set
Bottom L is Element of the carrier of L
K is Element of the carrier of L
L is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
L .: is non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the carrier of L is non empty set
the L_meet of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
[: the carrier of L, the carrier of L:] is non empty set
[:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
bool [:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
the L_join of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
LattStr(# the carrier of L, the L_meet of L, the L_join of L #) is non empty strict LattStr
L is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded upper-bounded bounded complemented Boolean implicative Heyting LattStr
L .: is non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the carrier of L is non empty set
the L_meet of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
[: the carrier of L, the carrier of L:] is non empty set
[:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
bool [:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
the L_join of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
LattStr(# the carrier of L, the L_meet of L, the L_join of L #) is non empty strict LattStr
L is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded upper-bounded bounded complemented Boolean implicative Heyting LattStr
the carrier of L is non empty set
L .: is non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded upper-bounded bounded complemented Boolean implicative Heyting LattStr
the L_meet of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
[: the carrier of L, the carrier of L:] is non empty set
[:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
bool [:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
the L_join of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
LattStr(# the carrier of L, the L_meet of L, the L_join of L #) is non empty strict LattStr
p is Element of the carrier of L
(L,p) is Element of the carrier of (L .:)
the carrier of (L .:) is non empty set
(L,p) ` is Element of the carrier of (L .:)
p ` is Element of the carrier of L
((L,p) `) "/\" (L,p) is Element of the carrier of (L .:)
the L_meet of (L .:) is Relation-like [: the carrier of (L .:), the carrier of (L .:):] -defined the carrier of (L .:) -valued Function-like V18([: the carrier of (L .:), the carrier of (L .:):], the carrier of (L .:)) commutative associative idempotent Element of bool [:[: the carrier of (L .:), the carrier of (L .:):], the carrier of (L .:):]
[: the carrier of (L .:), the carrier of (L .:):] is non empty set
[:[: the carrier of (L .:), the carrier of (L .:):], the carrier of (L .:):] is non empty set
bool [:[: the carrier of (L .:), the carrier of (L .:):], the carrier of (L .:):] is non empty set
the L_meet of (L .:) . (((L,p) `),(L,p)) is Element of the carrier of (L .:)
[((L,p) `),(L,p)] is set
{((L,p) `),(L,p)} is non empty set
{((L,p) `)} is non empty set
{{((L,p) `),(L,p)},{((L,p) `)}} is non empty set
the L_meet of (L .:) . [((L,p) `),(L,p)] is set
Bottom (L .:) is Element of the carrier of (L .:)
(L,((L,p) `)) is Element of the carrier of L
(L,(L,p)) is Element of the carrier of L
(L,((L,p) `)) "\/" (L,(L,p)) is Element of the carrier of L
the L_join of L . ((L,((L,p) `)),(L,(L,p))) is Element of the carrier of L
[(L,((L,p) `)),(L,(L,p))] is set
{(L,((L,p) `)),(L,(L,p))} is non empty set
{(L,((L,p) `))} is non empty set
{{(L,((L,p) `)),(L,(L,p))},{(L,((L,p) `))}} is non empty set
the L_join of L . [(L,((L,p) `)),(L,(L,p))] is set
Top L is Element of the carrier of L
((L,p) `) "\/" (L,p) is Element of the carrier of (L .:)
the L_join of (L .:) is Relation-like [: the carrier of (L .:), the carrier of (L .:):] -defined the carrier of (L .:) -valued Function-like V18([: the carrier of (L .:), the carrier of (L .:):], the carrier of (L .:)) commutative associative idempotent Element of bool [:[: the carrier of (L .:), the carrier of (L .:):], the carrier of (L .:):]
the L_join of (L .:) . (((L,p) `),(L,p)) is Element of the carrier of (L .:)
the L_join of (L .:) . [((L,p) `),(L,p)] is set
Top (L .:) is Element of the carrier of (L .:)
(L,((L,p) `)) "/\" (L,(L,p)) is Element of the carrier of L
the L_meet of L . ((L,((L,p) `)),(L,(L,p))) is Element of the carrier of L
the L_meet of L . [(L,((L,p) `)),(L,(L,p))] is set
Bottom L is Element of the carrier of L
L is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded upper-bounded bounded complemented Boolean implicative Heyting LattStr
the carrier of L is non empty set
L .: is non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded upper-bounded bounded complemented Boolean implicative Heyting LattStr
the L_meet of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
[: the carrier of L, the carrier of L:] is non empty set
[:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
bool [:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
the L_join of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
LattStr(# the carrier of L, the L_meet of L, the L_join of L #) is non empty strict LattStr
the carrier of (L .:) is non empty set
p is Element of the carrier of L
(L,p) is Element of the carrier of (L .:)
(L,p) ` is Element of the carrier of (L .:)
p ` is Element of the carrier of L
q is Element of the carrier of (L .:)
(L,q) is Element of the carrier of L
(L,q) ` is Element of the carrier of L
q ` is Element of the carrier of (L .:)
(L,(L,q)) is Element of the carrier of (L .:)
L is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded upper-bounded bounded complemented Boolean implicative Heyting LattStr
the carrier of L is non empty set
bool the carrier of L is non empty set
p is non empty initial meet-closed join-closed Element of bool the carrier of L
q is non empty initial meet-closed join-closed Element of bool the carrier of L
p \/ q is non empty Element of bool the carrier of L
(L,(p \/ q)) is non empty initial meet-closed join-closed Element of bool the carrier of L
(L,p,q) is non empty initial meet-closed join-closed Element of bool the carrier of L
{ (b<sub>1</sub> "\/" b<sub>2</sub>) where b<sub>1</sub>, b<sub>2</sub> is Element of the carrier of L : ( b<sub>1</sub> in p & b<sub>2</sub> in q ) } is set
L .: is non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded upper-bounded bounded complemented Boolean implicative Heyting LattStr
the L_meet of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
[: the carrier of L, the carrier of L:] is non empty set
[:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
bool [:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
the L_join of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
LattStr(# the carrier of L, the L_meet of L, the L_join of L #) is non empty strict LattStr
the carrier of (L .:) is non empty set
bool the carrier of (L .:) is non empty set
(L,p) is non empty final meet-closed join-closed Element of bool the carrier of (L .:)
(L,q) is non empty final meet-closed join-closed Element of bool the carrier of (L .:)
(L,(p \/ q)) is non empty Element of bool the carrier of (L .:)
<.(L,(p \/ q)).) is non empty final meet-closed join-closed Element of bool the carrier of (L .:)
K is non empty final meet-closed join-closed Element of bool the carrier of (L .:)
b9 is non empty final meet-closed join-closed Element of bool the carrier of (L .:)
K "/\" b9 is non empty final meet-closed join-closed Element of bool the carrier of (L .:)
{ (b<sub>1</sub> "/\" b<sub>2</sub>) where b<sub>1</sub>, b<sub>2</sub> is Element of the carrier of (L .:) : ( b<sub>1</sub> in K & b<sub>2</sub> in b9 ) } is set
L is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded upper-bounded bounded complemented Boolean implicative Heyting LattStr
the carrier of L is non empty set
bool the carrier of L is non empty set
p is non empty initial meet-closed join-closed Element of bool the carrier of L
L .: is non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded upper-bounded bounded complemented Boolean implicative Heyting LattStr
the L_meet of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
[: the carrier of L, the carrier of L:] is non empty set
[:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
bool [:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
the L_join of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
LattStr(# the carrier of L, the L_meet of L, the L_join of L #) is non empty strict LattStr
the carrier of (L .:) is non empty set
bool the carrier of (L .:) is non empty set
(L,p) is non empty final meet-closed join-closed Element of bool the carrier of (L .:)
q is non empty final meet-closed join-closed Element of bool the carrier of (L .:)
K is Element of the carrier of (L .:)
K ` is Element of the carrier of (L .:)
K is Element of the carrier of (L .:)
K ` is Element of the carrier of (L .:)
K is Element of the carrier of L
K ` is Element of the carrier of L
b9 is Element of the carrier of (L .:)
b9 ` is Element of the carrier of (L .:)
(L,K) is Element of the carrier of (L .:)
(L,K) ` is Element of the carrier of (L .:)
b is Element of the carrier of (L .:)
b ` is Element of the carrier of (L .:)
K is Element of the carrier of (L .:)
b9 is Element of the carrier of L
b9 ` is Element of the carrier of L
(L,K) is Element of the carrier of L
(L,K) ` is Element of the carrier of L
K ` is Element of the carrier of (L .:)
K is Element of the carrier of (L .:)
K ` is Element of the carrier of (L .:)
L is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded upper-bounded bounded complemented Boolean implicative Heyting LattStr
the carrier of L is non empty set
bool the carrier of L is non empty set
(L) is non empty initial meet-closed join-closed Element of bool the carrier of L
p is non empty initial meet-closed join-closed Element of bool the carrier of L
L .: is non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded upper-bounded bounded complemented Boolean implicative Heyting LattStr
the L_meet of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
[: the carrier of L, the carrier of L:] is non empty set
[:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
bool [:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
the L_join of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
LattStr(# the carrier of L, the L_meet of L, the L_join of L #) is non empty strict LattStr
the carrier of (L .:) is non empty set
bool the carrier of (L .:) is non empty set
(L,p) is non empty final meet-closed join-closed Element of bool the carrier of (L .:)
<.(L .:).) is non empty final meet-closed join-closed Element of bool the carrier of (L .:)
q is non empty final meet-closed join-closed Element of bool the carrier of (L .:)
L is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded upper-bounded bounded complemented Boolean implicative Heyting LattStr
the carrier of L is non empty set
bool the carrier of L is non empty set
p is non empty initial meet-closed join-closed Element of bool the carrier of L
L .: is non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded upper-bounded bounded complemented Boolean implicative Heyting LattStr
the L_meet of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
[: the carrier of L, the carrier of L:] is non empty set
[:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
bool [:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
the L_join of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
LattStr(# the carrier of L, the L_meet of L, the L_join of L #) is non empty strict LattStr
the carrier of (L .:) is non empty set
bool the carrier of (L .:) is non empty set
(L,p) is non empty final meet-closed join-closed Element of bool the carrier of (L .:)
q is non empty final meet-closed join-closed Element of bool the carrier of (L .:)
K is Element of the carrier of L
K ` is Element of the carrier of L
(L,K) is Element of the carrier of (L .:)
(L,K) ` is Element of the carrier of (L .:)
L is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded upper-bounded bounded complemented Boolean implicative Heyting LattStr
the carrier of L is non empty set
bool the carrier of L is non empty set
p is Element of the carrier of L
q is Element of the carrier of L
L .: is non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded upper-bounded bounded complemented Boolean implicative Heyting LattStr
the L_meet of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
[: the carrier of L, the carrier of L:] is non empty set
[:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
bool [:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
the L_join of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
LattStr(# the carrier of L, the L_meet of L, the L_join of L #) is non empty strict LattStr
the carrier of (L .:) is non empty set
bool the carrier of (L .:) is non empty set
(L,p) is Element of the carrier of (L .:)
(L,q) is Element of the carrier of (L .:)
K is non empty final meet-closed join-closed Element of bool the carrier of (L .:)
(L,K) is non empty initial meet-closed join-closed Element of bool the carrier of L
b9 is non empty initial meet-closed join-closed Element of bool the carrier of L
(L,b9) is non empty final meet-closed join-closed Element of bool the carrier of (L .:)
L is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the carrier of L is non empty set
bool the carrier of L is non empty set
the L_join of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
[: the carrier of L, the carrier of L:] is non empty set
[:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
bool [:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
the L_meet of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
p is non empty meet-closed join-closed Element of bool the carrier of L
the L_join of L || p is Relation-like Function-like set
[:p,p:] is non empty set
the L_join of L | [:p,p:] is Relation-like set
[:[:p,p:],p:] is non empty set
bool [:[:p,p:],p:] is non empty set
the L_meet of L || p is Relation-like Function-like set
the L_meet of L | [:p,p:] is Relation-like set
dom H₂(L) is Relation-like set
[:H₁(L),H₁(L):] is set
H₂(L) || p is Relation-like Function-like set
dom (H₂(L) || p) is set
[:p,p:] is Relation-like the carrier of L -defined the carrier of L -valued non empty Element of bool [: the carrier of L, the carrier of L:]
bool [: the carrier of L, the carrier of L:] is non empty set
rng (H₂(L) || p) is set
q is set
K is set
(H₂(L) || p) . K is set
b9 is Element of p
b is Element of p
[b9,b] is Element of [:p,p:]
{b9,b} is non empty set
{b9} is non empty set
{{b9,b},{b9}} is non empty set
b9 "\/" b is Element of the carrier of L
the L_join of L . (b9,b) is Element of the carrier of L
[b9,b] is set
the L_join of L . [b9,b] is set
dom H₃(L) is Relation-like set
H₃(L) || p is Relation-like Function-like set
dom (H₃(L) || p) is set
rng (H₃(L) || p) is set
q is set
K is set
(H₃(L) || p) . K is set
b9 is Element of p
b is Element of p
[b9,b] is Element of [:p,p:]
{b9,b} is non empty set
{b9} is non empty set
{{b9,b},{b9}} is non empty set
b9 "/\" b is Element of the carrier of L
the L_meet of L . (b9,b) is Element of the carrier of L
[b9,b] is set
the L_meet of L . [b9,b] is set
L is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the carrier of L is non empty set
bool the carrier of L is non empty set
the L_join of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
[: the carrier of L, the carrier of L:] is non empty set
[:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
bool [:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
the L_meet of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
p is non empty meet-closed join-closed Element of bool the carrier of L
[:p,p:] is non empty set
[:[:p,p:],p:] is non empty set
bool [:[:p,p:],p:] is non empty set
the L_join of L || p is Relation-like Function-like set
the L_join of L | [:p,p:] is Relation-like set
the L_meet of L || p is Relation-like Function-like set
the L_meet of L | [:p,p:] is Relation-like set
q is Relation-like [:p,p:] -defined p -valued Function-like V18([:p,p:],p) Element of bool [:[:p,p:],p:]
K is Relation-like [:p,p:] -defined p -valued Function-like V18([:p,p:],p) Element of bool [:[:p,p:],p:]
L is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the carrier of L is non empty set
p is Element of the carrier of L
q is Element of the carrier of L
{ b₁ where b₁ is Element of the carrier of L : ( p [= b₁ & b₁ [= q ) } is set
bool the carrier of L is non empty set
b9 is non empty set
b is set
a is Element of the carrier of L
a is Element of the carrier of L
b is non empty Element of bool the carrier of L
a9 is Element of the carrier of L
a "\/" a9 is Element of the carrier of L
the L_join of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
[: the carrier of L, the carrier of L:] is non empty set
[:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
bool [:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
the L_join of L . (a,a9) is Element of the carrier of L
[a,a9] is set
{a,a9} is non empty set
{a} is non empty set
{{a,a9},{a}} is non empty set
the L_join of L . [a,a9] is set
a "/\" a9 is Element of the carrier of L
the L_meet of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
the L_meet of L . (a,a9) is Element of the carrier of L
the L_meet of L . [a,a9] is set
c₉ is Element of the carrier of L
c₉ is Element of the carrier of L
c is Element of the carrier of L
a is Element of the carrier of L
a9 is Element of the carrier of L
a "/\" a9 is Element of the carrier of L
the L_meet of L . (a,a9) is Element of the carrier of L
[a,a9] is set
{a,a9} is non empty set
{a} is non empty set
{{a,a9},{a}} is non empty set
the L_meet of L . [a,a9] is set
c₉ is Element of the carrier of L
c is Element of the carrier of L
c₉ "\/" c is Element of the carrier of L
the L_join of L . (c₉,c) is Element of the carrier of L
[c₉,c] is set
{c₉,c} is non empty set
{c₉} is non empty set
{{c₉,c},{c₉}} is non empty set
the L_join of L . [c₉,c] is set
L is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the carrier of L is non empty set
p is Element of the carrier of L
q is Element of the carrier of L
(L,p,q) is non empty meet-closed join-closed Element of bool the carrier of L
bool the carrier of L is non empty set
K is Element of the carrier of L
{ b₁ where b₁ is Element of the carrier of L : ( p [= b₁ & b₁ [= q ) } is set
b9 is Element of the carrier of L
b9 is Element of the carrier of L
L is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the carrier of L is non empty set
p is Element of the carrier of L
q is Element of the carrier of L
(L,p,q) is non empty meet-closed join-closed Element of bool the carrier of L
bool the carrier of L is non empty set
L is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the carrier of L is non empty set
p is Element of the carrier of L
(L,p,p) is non empty meet-closed join-closed Element of bool the carrier of L
bool the carrier of L is non empty set
{p} is non empty Element of bool the carrier of L
q is Element of the carrier of L
L is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the carrier of L is non empty set
Top L is Element of the carrier of L
p is Element of the carrier of L
<.p.) is non empty final meet-closed join-closed Element of bool the carrier of L
bool the carrier of L is non empty set
{ b₁ where b₁ is Element of the carrier of L : p [= b₁ } is set
(L,p,(Top L)) is non empty meet-closed join-closed Element of bool the carrier of L
q is Element of the carrier of L
L is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the carrier of L is non empty set
Bottom L is Element of the carrier of L
p is Element of the carrier of L
(L,p) is non empty initial meet-closed join-closed Element of bool the carrier of L
bool the carrier of L is non empty set
{ b₁ where b₁ is Element of the carrier of L : b₁ [= p } is set
(L,(Bottom L),p) is non empty meet-closed join-closed Element of bool the carrier of L
q is Element of the carrier of L
L is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the carrier of L is non empty set
bool the carrier of L is non empty set
p is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the carrier of p is non empty set
bool the carrier of p is non empty set
the L_join of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
[: the carrier of L, the carrier of L:] is non empty set
[:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
bool [:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
the L_meet of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
LattStr(# the carrier of L, the L_join of L, the L_meet of L #) is non empty strict LattStr
the L_join of p is Relation-like [: the carrier of p, the carrier of p:] -defined the carrier of p -valued Function-like V18([: the carrier of p, the carrier of p:], the carrier of p) commutative associative idempotent Element of bool [:[: the carrier of p, the carrier of p:], the carrier of p:]
[: the carrier of p, the carrier of p:] is non empty set
[:[: the carrier of p, the carrier of p:], the carrier of p:] is non empty set
bool [:[: the carrier of p, the carrier of p:], the carrier of p:] is non empty set
the L_meet of p is Relation-like [: the carrier of p, the carrier of p:] -defined the carrier of p -valued Function-like V18([: the carrier of p, the carrier of p:], the carrier of p) commutative associative idempotent Element of bool [:[: the carrier of p, the carrier of p:], the carrier of p:]
LattStr(# the carrier of p, the L_join of p, the L_meet of p #) is non empty strict LattStr
q is non empty final meet-closed join-closed Element of bool the carrier of L
latt q is non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
K is non empty final meet-closed join-closed Element of bool the carrier of p
latt K is non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
[:q,q:] is non empty set
[:[:q,q:],q:] is non empty set
bool [:[:q,q:],q:] is non empty set
the L_join of L || q is Relation-like Function-like set
the L_join of L | [:q,q:] is Relation-like set
the L_meet of L || q is Relation-like Function-like set
the L_meet of L | [:q,q:] is Relation-like set
b9 is Relation-like [:q,q:] -defined q -valued Function-like V18([:q,q:],q) Element of bool [:[:q,q:],q:]
b is Relation-like [:q,q:] -defined q -valued Function-like V18([:q,q:],q) Element of bool [:[:q,q:],q:]
LattStr(# q,b9,b #) is non empty strict LattStr
L is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the carrier of L is non empty set
bool the carrier of L is non empty set
the L_join of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
[: the carrier of L, the carrier of L:] is non empty set
[:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
bool [:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
the L_meet of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
p is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the carrier of p is non empty set
the L_join of p is Relation-like [: the carrier of p, the carrier of p:] -defined the carrier of p -valued Function-like V18([: the carrier of p, the carrier of p:], the carrier of p) commutative associative idempotent Element of bool [:[: the carrier of p, the carrier of p:], the carrier of p:]
[: the carrier of p, the carrier of p:] is non empty set
[:[: the carrier of p, the carrier of p:], the carrier of p:] is non empty set
bool [:[: the carrier of p, the carrier of p:], the carrier of p:] is non empty set
the L_meet of p is Relation-like [: the carrier of p, the carrier of p:] -defined the carrier of p -valued Function-like V18([: the carrier of p, the carrier of p:], the carrier of p) commutative associative idempotent Element of bool [:[: the carrier of p, the carrier of p:], the carrier of p:]
LattStr(# the carrier of p, the L_join of p, the L_meet of p #) is non empty strict LattStr
H₃(L) || H₁(p) is Relation-like Function-like set
the L_meet of L | [: the carrier of p, the carrier of p:] is Relation-like set
H₂(L) || H₁(p) is Relation-like Function-like set
the L_join of L | [: the carrier of p, the carrier of p:] is Relation-like set
K is Element of the carrier of L
q is non empty Element of bool the carrier of L
b9 is Element of the carrier of L
[K,b9] is Element of [: the carrier of L, the carrier of L:]
{K,b9} is non empty set
{K} is non empty set
{{K,b9},{K}} is non empty set
[:q,q:] is Relation-like the carrier of L -defined the carrier of L -valued non empty Element of bool [: the carrier of L, the carrier of L:]
bool [: the carrier of L, the carrier of L:] is non empty set
dom H₃(p) is Relation-like set
H₃(L) . [K,b9] is set
H₃(p) . [K,b9] is set
dom H₂(p) is Relation-like set
H₂(L) . [K,b9] is set
H₂(p) . [K,b9] is set
K "/\" b9 is Element of the carrier of L
the L_meet of L . (K,b9) is Element of the carrier of L
[K,b9] is set
the L_meet of L . [K,b9] is set
K "\/" b9 is Element of the carrier of L
the L_join of L . (K,b9) is Element of the carrier of L
the L_join of L . [K,b9] is set
K is Element of the carrier of L
b9 is Element of the carrier of L
K "/\" b9 is Element of the carrier of L
the L_meet of L . (K,b9) is Element of the carrier of L
[K,b9] is set
{K,b9} is non empty set
{K} is non empty set
{{K,b9},{K}} is non empty set
the L_meet of L . [K,b9] is set
b is Element of the carrier of L
a is Element of the carrier of L
b "\/" a is Element of the carrier of L
the L_join of L . (b,a) is Element of the carrier of L
[b,a] is set
{b,a} is non empty set
{b} is non empty set
{{b,a},{b}} is non empty set
the L_join of L . [b,a] is set
K is non empty meet-closed join-closed Element of bool the carrier of L
[:K,K:] is non empty set
[:[:K,K:],K:] is non empty set
bool [:[:K,K:],K:] is non empty set
the L_join of L || K is Relation-like Function-like set
the L_join of L | [:K,K:] is Relation-like set
the L_meet of L || K is Relation-like Function-like set
the L_meet of L | [:K,K:] is Relation-like set
b9 is Relation-like [:K,K:] -defined K -valued Function-like V18([:K,K:],K) Element of bool [:[:K,K:],K:]
b is Relation-like [:K,K:] -defined K -valued Function-like V18([:K,K:],K) Element of bool [:[:K,K:],K:]
LattStr(# K,b9,b #) is non empty strict LattStr
q is non empty meet-closed join-closed Element of bool the carrier of L
[:q,q:] is non empty set
[:[:q,q:],q:] is non empty set
bool [:[:q,q:],q:] is non empty set
K is Relation-like [:q,q:] -defined q -valued Function-like V18([:q,q:],q) Element of bool [:[:q,q:],q:]
the L_join of L || q is Relation-like Function-like set
the L_join of L | [:q,q:] is Relation-like set
b9 is Relation-like [:q,q:] -defined q -valued Function-like V18([:q,q:],q) Element of bool [:[:q,q:],q:]
the L_meet of L || q is Relation-like Function-like set
the L_meet of L | [:q,q:] is Relation-like set
LattStr(# q,K,b9 #) is non empty strict LattStr
L is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the carrier of L is non empty set
p is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like SubLattice of L
the carrier of p is non empty set
q is Element of the carrier of p
L is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the carrier of L is non empty set
bool the carrier of L is non empty set
p is non empty meet-closed join-closed Element of bool the carrier of L
[:p,p:] is non empty set
[:[:p,p:],p:] is non empty set
bool [:[:p,p:],p:] is non empty set
the L_join of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
[: the carrier of L, the carrier of L:] is non empty set
[:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
bool [:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
the L_join of L || p is Relation-like Function-like set
the L_join of L | [:p,p:] is Relation-like set
the L_meet of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
the L_meet of L || p is Relation-like Function-like set
the L_meet of L | [:p,p:] is Relation-like set
H₂(L) || p is Relation-like Function-like set
H₃(L) || p is Relation-like Function-like set
q is Relation-like [:p,p:] -defined p -valued Function-like V18([:p,p:],p) Element of bool [:[:p,p:],p:]
K is Relation-like [:p,p:] -defined p -valued Function-like V18([:p,p:],p) Element of bool [:[:p,p:],p:]
LattStr(# p,q,K #) is non empty strict LattStr
b9 is non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
b is non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like SubLattice of L
b9 is Relation-like [:p,p:] -defined p -valued Function-like V18([:p,p:],p) Element of bool [:[:p,p:],p:]
b is Relation-like [:p,p:] -defined p -valued Function-like V18([:p,p:],p) Element of bool [:[:p,p:],p:]
q is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like SubLattice of L
LattStr(# p,b9,b #) is non empty strict LattStr
a is Relation-like [:p,p:] -defined p -valued Function-like V18([:p,p:],p) Element of bool [:[:p,p:],p:]
a9 is Relation-like [:p,p:] -defined p -valued Function-like V18([:p,p:],p) Element of bool [:[:p,p:],p:]
K is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like SubLattice of L
LattStr(# p,a,a9 #) is non empty strict LattStr
L is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the carrier of L is non empty set
bool the carrier of L is non empty set
p is non empty meet-closed join-closed Element of bool the carrier of L
(L,p) is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like SubLattice of L
[:p,p:] is non empty set
[:[:p,p:],p:] is non empty set
bool [:[:p,p:],p:] is non empty set
the L_join of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
[: the carrier of L, the carrier of L:] is non empty set
[:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
bool [:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
the L_join of L || p is Relation-like Function-like set
the L_join of L | [:p,p:] is Relation-like set
the L_meet of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
the L_meet of L || p is Relation-like Function-like set
the L_meet of L | [:p,p:] is Relation-like set
q is Relation-like [:p,p:] -defined p -valued Function-like V18([:p,p:],p) Element of bool [:[:p,p:],p:]
K is Relation-like [:p,p:] -defined p -valued Function-like V18([:p,p:],p) Element of bool [:[:p,p:],p:]
LattStr(# p,q,K #) is non empty strict LattStr
L is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
p is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like SubLattice of L
p .: is non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the carrier of p is non empty set
the L_meet of p is Relation-like [: the carrier of p, the carrier of p:] -defined the carrier of p -valued Function-like V18([: the carrier of p, the carrier of p:], the carrier of p) commutative associative idempotent Element of bool [:[: the carrier of p, the carrier of p:], the carrier of p:]
[: the carrier of p, the carrier of p:] is non empty set
[:[: the carrier of p, the carrier of p:], the carrier of p:] is non empty set
bool [:[: the carrier of p, the carrier of p:], the carrier of p:] is non empty set
the L_join of p is Relation-like [: the carrier of p, the carrier of p:] -defined the carrier of p -valued Function-like V18([: the carrier of p, the carrier of p:], the carrier of p) commutative associative idempotent Element of bool [:[: the carrier of p, the carrier of p:], the carrier of p:]
LattStr(# the carrier of p, the L_meet of p, the L_join of p #) is non empty strict LattStr
L .: is non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the carrier of L is non empty set
the L_meet of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
[: the carrier of L, the carrier of L:] is non empty set
[:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
bool [:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
the L_join of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
LattStr(# the carrier of L, the L_meet of L, the L_join of L #) is non empty strict LattStr
bool the carrier of L is non empty set
LattStr(# the carrier of p, the L_join of p, the L_meet of p #) is non empty strict LattStr
q is non empty meet-closed join-closed Element of bool the carrier of L
[:q,q:] is non empty set
[:[:q,q:],q:] is non empty set
bool [:[:q,q:],q:] is non empty set
K is Relation-like [:q,q:] -defined q -valued Function-like V18([:q,q:],q) Element of bool [:[:q,q:],q:]
H₂(L) || q is Relation-like Function-like set
the L_join of L | [:q,q:] is Relation-like set
b9 is Relation-like [:q,q:] -defined q -valued Function-like V18([:q,q:],q) Element of bool [:[:q,q:],q:]
H₃(L) || q is Relation-like Function-like set
the L_meet of L | [:q,q:] is Relation-like set
LattStr(# q,K,b9 #) is non empty strict LattStr
(L,q) is non empty meet-closed join-closed Element of bool the carrier of (L .:)
the carrier of (L .:) is non empty set
bool the carrier of (L .:) is non empty set
[:(L,q),(L,q):] is non empty set
[:[:(L,q),(L,q):],(L,q):] is non empty set
bool [:[:(L,q),(L,q):],(L,q):] is non empty set
the L_join of (L .:) is Relation-like [: the carrier of (L .:), the carrier of (L .:):] -defined the carrier of (L .:) -valued Function-like V18([: the carrier of (L .:), the carrier of (L .:):], the carrier of (L .:)) commutative associative idempotent Element of bool [:[: the carrier of (L .:), the carrier of (L .:):], the carrier of (L .:):]
[: the carrier of (L .:), the carrier of (L .:):] is non empty set
[:[: the carrier of (L .:), the carrier of (L .:):], the carrier of (L .:):] is non empty set
bool [:[: the carrier of (L .:), the carrier of (L .:):], the carrier of (L .:):] is non empty set
the L_join of (L .:) || (L,q) is Relation-like Function-like set
the L_join of (L .:) | [:(L,q),(L,q):] is Relation-like set
the L_meet of (L .:) is Relation-like [: the carrier of (L .:), the carrier of (L .:):] -defined the carrier of (L .:) -valued Function-like V18([: the carrier of (L .:), the carrier of (L .:):], the carrier of (L .:)) commutative associative idempotent Element of bool [:[: the carrier of (L .:), the carrier of (L .:):], the carrier of (L .:):]
the L_meet of (L .:) || (L,q) is Relation-like Function-like set
the L_meet of (L .:) | [:(L,q),(L,q):] is Relation-like set
the carrier of (p .:) is non empty set
the L_join of (p .:) is Relation-like [: the carrier of (p .:), the carrier of (p .:):] -defined the carrier of (p .:) -valued Function-like V18([: the carrier of (p .:), the carrier of (p .:):], the carrier of (p .:)) commutative associative idempotent Element of bool [:[: the carrier of (p .:), the carrier of (p .:):], the carrier of (p .:):]
[: the carrier of (p .:), the carrier of (p .:):] is non empty set
[:[: the carrier of (p .:), the carrier of (p .:):], the carrier of (p .:):] is non empty set
bool [:[: the carrier of (p .:), the carrier of (p .:):], the carrier of (p .:):] is non empty set
the L_meet of (p .:) is Relation-like [: the carrier of (p .:), the carrier of (p .:):] -defined the carrier of (p .:) -valued Function-like V18([: the carrier of (p .:), the carrier of (p .:):], the carrier of (p .:)) commutative associative idempotent Element of bool [:[: the carrier of (p .:), the carrier of (p .:):], the carrier of (p .:):]
LattStr(# the carrier of (p .:), the L_join of (p .:), the L_meet of (p .:) #) is non empty strict LattStr
L is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the carrier of L is non empty set
bool the carrier of L is non empty set
p is non empty final meet-closed join-closed Element of bool the carrier of L
latt p is non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
(L,p) is non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like SubLattice of L
[:p,p:] is non empty set
[:[:p,p:],p:] is non empty set
bool [:[:p,p:],p:] is non empty set
the L_join of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
[: the carrier of L, the carrier of L:] is non empty set
[:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
bool [:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
H₂(L) || p is Relation-like Function-like set
the L_join of L | [:p,p:] is Relation-like set
the L_meet of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
H₃(L) || p is Relation-like Function-like set
the L_meet of L | [:p,p:] is Relation-like set
q is Relation-like [:p,p:] -defined p -valued Function-like V18([:p,p:],p) Element of bool [:[:p,p:],p:]
K is Relation-like [:p,p:] -defined p -valued Function-like V18([:p,p:],p) Element of bool [:[:p,p:],p:]
LattStr(# p,q,K #) is non empty strict LattStr
L is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the carrier of L is non empty set
bool the carrier of L is non empty set
L .: is non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the L_meet of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
[: the carrier of L, the carrier of L:] is non empty set
[:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
bool [:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
the L_join of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
LattStr(# the carrier of L, the L_meet of L, the L_join of L #) is non empty strict LattStr
p is non empty meet-closed join-closed Element of bool the carrier of L
(L,p) is non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like SubLattice of L
(L,p) is non empty meet-closed join-closed Element of bool the carrier of (L .:)
the carrier of (L .:) is non empty set
bool the carrier of (L .:) is non empty set
((L .:),(L,p)) is non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like SubLattice of L .:
((L .:),((L .:),(L,p))) is non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like SubLattice of (L .:) .:
(L .:) .: is non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the L_meet of (L .:) is Relation-like [: the carrier of (L .:), the carrier of (L .:):] -defined the carrier of (L .:) -valued Function-like V18([: the carrier of (L .:), the carrier of (L .:):], the carrier of (L .:)) commutative associative idempotent Element of bool [:[: the carrier of (L .:), the carrier of (L .:):], the carrier of (L .:):]
[: the carrier of (L .:), the carrier of (L .:):] is non empty set
[:[: the carrier of (L .:), the carrier of (L .:):], the carrier of (L .:):] is non empty set
bool [:[: the carrier of (L .:), the carrier of (L .:):], the carrier of (L .:):] is non empty set
the L_join of (L .:) is Relation-like [: the carrier of (L .:), the carrier of (L .:):] -defined the carrier of (L .:) -valued Function-like V18([: the carrier of (L .:), the carrier of (L .:):], the carrier of (L .:)) commutative associative idempotent Element of bool [:[: the carrier of (L .:), the carrier of (L .:):], the carrier of (L .:):]
LattStr(# the carrier of (L .:), the L_meet of (L .:), the L_join of (L .:) #) is non empty strict LattStr
the carrier of ((L .:),(L,p)) is non empty set
the L_meet of ((L .:),(L,p)) is Relation-like [: the carrier of ((L .:),(L,p)), the carrier of ((L .:),(L,p)):] -defined the carrier of ((L .:),(L,p)) -valued Function-like V18([: the carrier of ((L .:),(L,p)), the carrier of ((L .:),(L,p)):], the carrier of ((L .:),(L,p))) commutative associative idempotent Element of bool [:[: the carrier of ((L .:),(L,p)), the carrier of ((L .:),(L,p)):], the carrier of ((L .:),(L,p)):]
[: the carrier of ((L .:),(L,p)), the carrier of ((L .:),(L,p)):] is non empty set
[:[: the carrier of ((L .:),(L,p)), the carrier of ((L .:),(L,p)):], the carrier of ((L .:),(L,p)):] is non empty set
bool [:[: the carrier of ((L .:),(L,p)), the carrier of ((L .:),(L,p)):], the carrier of ((L .:),(L,p)):] is non empty set
the L_join of ((L .:),(L,p)) is Relation-like [: the carrier of ((L .:),(L,p)), the carrier of ((L .:),(L,p)):] -defined the carrier of ((L .:),(L,p)) -valued Function-like V18([: the carrier of ((L .:),(L,p)), the carrier of ((L .:),(L,p)):], the carrier of ((L .:),(L,p))) commutative associative idempotent Element of bool [:[: the carrier of ((L .:),(L,p)), the carrier of ((L .:),(L,p)):], the carrier of ((L .:),(L,p)):]
LattStr(# the carrier of ((L .:),(L,p)), the L_meet of ((L .:),(L,p)), the L_join of ((L .:),(L,p)) #) is non empty strict LattStr
[:p,p:] is non empty set
[:[:p,p:],p:] is non empty set
bool [:[:p,p:],p:] is non empty set
H₂(L) || p is Relation-like Function-like set
the L_join of L | [:p,p:] is Relation-like set
H₃(L) || p is Relation-like Function-like set
the L_meet of L | [:p,p:] is Relation-like set
[:(L,p),(L,p):] is non empty set
[:[:(L,p),(L,p):],(L,p):] is non empty set
bool [:[:(L,p),(L,p):],(L,p):] is non empty set
H₂(L .: ) || (L,p) is Relation-like Function-like set
the L_join of (L .:) | [:(L,p),(L,p):] is Relation-like set
H₃(L .: ) || (L,p) is Relation-like Function-like set
the L_meet of (L .:) | [:(L,p),(L,p):] is Relation-like set
q is Relation-like [:p,p:] -defined p -valued Function-like V18([:p,p:],p) Element of bool [:[:p,p:],p:]
K is Relation-like [:p,p:] -defined p -valued Function-like V18([:p,p:],p) Element of bool [:[:p,p:],p:]
LattStr(# p,q,K #) is non empty strict LattStr
b9 is Relation-like [:(L,p),(L,p):] -defined (L,p) -valued Function-like V18([:(L,p),(L,p):],(L,p)) Element of bool [:[:(L,p),(L,p):],(L,p):]
b is Relation-like [:(L,p),(L,p):] -defined (L,p) -valued Function-like V18([:(L,p),(L,p):],(L,p)) Element of bool [:[:(L,p),(L,p):],(L,p):]
LattStr(# (L,p),b9,b #) is non empty strict LattStr
L is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
(L) is non empty initial meet-closed join-closed Element of bool the carrier of L
the carrier of L is non empty set
bool the carrier of L is non empty set
(L,(L)) is non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like SubLattice of L
the L_join of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
[: the carrier of L, the carrier of L:] is non empty set
[:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
bool [:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
the L_meet of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
LattStr(# the carrier of L, the L_join of L, the L_meet of L #) is non empty strict LattStr
<.L.) is non empty final meet-closed join-closed Element of bool the carrier of L
(L,<.L.)) is non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like SubLattice of L
dom H₃(L) is Relation-like set
[:H₁(L),H₁(L):] is set
dom H₂(L) is Relation-like set
H₂(L) | (dom H₂(L)) is Relation-like set
[:(L),(L):] is non empty set
[:[:(L),(L):],(L):] is non empty set
bool [:[:(L),(L):],(L):] is non empty set
the L_join of L || (L) is Relation-like Function-like set
the L_join of L | [:(L),(L):] is Relation-like set
the L_meet of L || (L) is Relation-like Function-like set
the L_meet of L | [:(L),(L):] is Relation-like set
p is Relation-like [:(L),(L):] -defined (L) -valued Function-like V18([:(L),(L):],(L)) Element of bool [:[:(L),(L):],(L):]
q is Relation-like [:(L),(L):] -defined (L) -valued Function-like V18([:(L),(L):],(L)) Element of bool [:[:(L),(L):],(L):]
LattStr(# (L),p,q #) is non empty strict LattStr
L is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the carrier of L is non empty set
bool the carrier of L is non empty set
the L_join of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
[: the carrier of L, the carrier of L:] is non empty set
[:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
bool [:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
the L_meet of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
p is non empty meet-closed join-closed Element of bool the carrier of L
(L,p) is non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like SubLattice of L
the carrier of (L,p) is non empty set
the L_join of (L,p) is Relation-like [: the carrier of (L,p), the carrier of (L,p):] -defined the carrier of (L,p) -valued Function-like V18([: the carrier of (L,p), the carrier of (L,p):], the carrier of (L,p)) commutative associative idempotent Element of bool [:[: the carrier of (L,p), the carrier of (L,p):], the carrier of (L,p):]
[: the carrier of (L,p), the carrier of (L,p):] is non empty set
[:[: the carrier of (L,p), the carrier of (L,p):], the carrier of (L,p):] is non empty set
bool [:[: the carrier of (L,p), the carrier of (L,p):], the carrier of (L,p):] is non empty set
the L_join of L || p is Relation-like Function-like set
[:p,p:] is non empty set
the L_join of L | [:p,p:] is Relation-like set
the L_meet of (L,p) is Relation-like [: the carrier of (L,p), the carrier of (L,p):] -defined the carrier of (L,p) -valued Function-like V18([: the carrier of (L,p), the carrier of (L,p):], the carrier of (L,p)) commutative associative idempotent Element of bool [:[: the carrier of (L,p), the carrier of (L,p):], the carrier of (L,p):]
the L_meet of L || p is Relation-like Function-like set
the L_meet of L | [:p,p:] is Relation-like set
[:[:p,p:],p:] is non empty set
bool [:[:p,p:],p:] is non empty set
H₂(L) || p is Relation-like Function-like set
H₃(L) || p is Relation-like Function-like set
q is Relation-like [:p,p:] -defined p -valued Function-like V18([:p,p:],p) Element of bool [:[:p,p:],p:]
K is Relation-like [:p,p:] -defined p -valued Function-like V18([:p,p:],p) Element of bool [:[:p,p:],p:]
LattStr(# p,q,K #) is non empty strict LattStr
L is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the carrier of L is non empty set
bool the carrier of L is non empty set
p is non empty meet-closed join-closed Element of bool the carrier of L
(L,p) is non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like SubLattice of L
the carrier of (L,p) is non empty set
q is Element of the carrier of L
K is Element of the carrier of L
q "\/" K is Element of the carrier of L
the L_join of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
[: the carrier of L, the carrier of L:] is non empty set
[:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
bool [:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
the L_join of L . (q,K) is Element of the carrier of L
[q,K] is set
{q,K} is non empty set
{q} is non empty set
{{q,K},{q}} is non empty set
the L_join of L . [q,K] is set
q "/\" K is Element of the carrier of L
the L_meet of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
the L_meet of L . (q,K) is Element of the carrier of L
the L_meet of L . [q,K] is set
b9 is Element of the carrier of (L,p)
b is Element of the carrier of (L,p)
b9 "\/" b is Element of the carrier of (L,p)
the L_join of (L,p) is Relation-like [: the carrier of (L,p), the carrier of (L,p):] -defined the carrier of (L,p) -valued Function-like V18([: the carrier of (L,p), the carrier of (L,p):], the carrier of (L,p)) commutative associative idempotent Element of bool [:[: the carrier of (L,p), the carrier of (L,p):], the carrier of (L,p):]
[: the carrier of (L,p), the carrier of (L,p):] is non empty set
[:[: the carrier of (L,p), the carrier of (L,p):], the carrier of (L,p):] is non empty set
bool [:[: the carrier of (L,p), the carrier of (L,p):], the carrier of (L,p):] is non empty set
the L_join of (L,p) . (b9,b) is Element of the carrier of (L,p)
[b9,b] is set
{b9,b} is non empty set
{b9} is non empty set
{{b9,b},{b9}} is non empty set
the L_join of (L,p) . [b9,b] is set
b9 "/\" b is Element of the carrier of (L,p)
the L_meet of (L,p) is Relation-like [: the carrier of (L,p), the carrier of (L,p):] -defined the carrier of (L,p) -valued Function-like V18([: the carrier of (L,p), the carrier of (L,p):], the carrier of (L,p)) commutative associative idempotent Element of bool [:[: the carrier of (L,p), the carrier of (L,p):], the carrier of (L,p):]
the L_meet of (L,p) . (b9,b) is Element of the carrier of (L,p)
the L_meet of (L,p) . [b9,b] is set
[:p,p:] is non empty set
[:[:p,p:],p:] is non empty set
bool [:[:p,p:],p:] is non empty set
H₂(L) || p is Relation-like Function-like set
the L_join of L | [:p,p:] is Relation-like set
H₃(L) || p is Relation-like Function-like set
the L_meet of L | [:p,p:] is Relation-like set
a is Relation-like [:p,p:] -defined p -valued Function-like V18([:p,p:],p) Element of bool [:[:p,p:],p:]
a9 is Relation-like [:p,p:] -defined p -valued Function-like V18([:p,p:],p) Element of bool [:[:p,p:],p:]
LattStr(# p,a,a9 #) is non empty strict LattStr
[b9,b] is Element of [: the carrier of (L,p), the carrier of (L,p):]
[:p,p:] is Relation-like the carrier of L -defined the carrier of L -valued non empty Element of bool [: the carrier of L, the carrier of L:]
bool [: the carrier of L, the carrier of L:] is non empty set
dom a is Relation-like set
dom a9 is Relation-like set
L is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the carrier of L is non empty set
bool the carrier of L is non empty set
p is non empty meet-closed join-closed Element of bool the carrier of L
(L,p) is non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like SubLattice of L
the carrier of (L,p) is non empty set
q is Element of the carrier of L
K is Element of the carrier of L
b9 is Element of the carrier of (L,p)
b is Element of the carrier of (L,p)
q "\/" K is Element of the carrier of L
the L_join of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
[: the carrier of L, the carrier of L:] is non empty set
[:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
bool [:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
the L_join of L . (q,K) is Element of the carrier of L
[q,K] is set
{q,K} is non empty set
{q} is non empty set
{{q,K},{q}} is non empty set
the L_join of L . [q,K] is set
b9 "\/" b is Element of the carrier of (L,p)
the L_join of (L,p) is Relation-like [: the carrier of (L,p), the carrier of (L,p):] -defined the carrier of (L,p) -valued Function-like V18([: the carrier of (L,p), the carrier of (L,p):], the carrier of (L,p)) commutative associative idempotent Element of bool [:[: the carrier of (L,p), the carrier of (L,p):], the carrier of (L,p):]
[: the carrier of (L,p), the carrier of (L,p):] is non empty set
[:[: the carrier of (L,p), the carrier of (L,p):], the carrier of (L,p):] is non empty set
bool [:[: the carrier of (L,p), the carrier of (L,p):], the carrier of (L,p):] is non empty set
the L_join of (L,p) . (b9,b) is Element of the carrier of (L,p)
[b9,b] is set
{b9,b} is non empty set
{b9} is non empty set
{{b9,b},{b9}} is non empty set
the L_join of (L,p) . [b9,b] is set
b9 "\/" b is Element of the carrier of (L,p)
the L_join of (L,p) is Relation-like [: the carrier of (L,p), the carrier of (L,p):] -defined the carrier of (L,p) -valued Function-like V18([: the carrier of (L,p), the carrier of (L,p):], the carrier of (L,p)) commutative associative idempotent Element of bool [:[: the carrier of (L,p), the carrier of (L,p):], the carrier of (L,p):]
[: the carrier of (L,p), the carrier of (L,p):] is non empty set
[:[: the carrier of (L,p), the carrier of (L,p):], the carrier of (L,p):] is non empty set
bool [:[: the carrier of (L,p), the carrier of (L,p):], the carrier of (L,p):] is non empty set
the L_join of (L,p) . (b9,b) is Element of the carrier of (L,p)
[b9,b] is set
{b9,b} is non empty set
{b9} is non empty set
{{b9,b},{b9}} is non empty set
the L_join of (L,p) . [b9,b] is set
q "\/" K is Element of the carrier of L
the L_join of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
[: the carrier of L, the carrier of L:] is non empty set
[:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
bool [:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
the L_join of L . (q,K) is Element of the carrier of L
[q,K] is set
{q,K} is non empty set
{q} is non empty set
{{q,K},{q}} is non empty set
the L_join of L . [q,K] is set
L is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the carrier of L is non empty set
bool the carrier of L is non empty set
p is non empty initial meet-closed join-closed Element of bool the carrier of L
(L,p) is non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like SubLattice of L
the carrier of (L,p) is non empty set
the Element of the carrier of (L,p) is Element of the carrier of (L,p)
b9 is Element of the carrier of L
K is Element of the carrier of L
b9 "/\" K is Element of the carrier of L
the L_meet of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
[: the carrier of L, the carrier of L:] is non empty set
[:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
bool [:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
the L_meet of L . (b9,K) is Element of the carrier of L
[b9,K] is set
{b9,K} is non empty set
{b9} is non empty set
{{b9,K},{b9}} is non empty set
the L_meet of L . [b9,K] is set
b is Element of the carrier of (L,p)
a is Element of the carrier of (L,p)
b "/\" a is Element of the carrier of (L,p)
the L_meet of (L,p) is Relation-like [: the carrier of (L,p), the carrier of (L,p):] -defined the carrier of (L,p) -valued Function-like V18([: the carrier of (L,p), the carrier of (L,p):], the carrier of (L,p)) commutative associative idempotent Element of bool [:[: the carrier of (L,p), the carrier of (L,p):], the carrier of (L,p):]
[: the carrier of (L,p), the carrier of (L,p):] is non empty set
[:[: the carrier of (L,p), the carrier of (L,p):], the carrier of (L,p):] is non empty set
bool [:[: the carrier of (L,p), the carrier of (L,p):], the carrier of (L,p):] is non empty set
the L_meet of (L,p) . (b,a) is Element of the carrier of (L,p)
[b,a] is set
{b,a} is non empty set
{b} is non empty set
{{b,a},{b}} is non empty set
the L_meet of (L,p) . [b,a] is set
a "/\" b is Element of the carrier of (L,p)
the L_meet of (L,p) . (a,b) is Element of the carrier of (L,p)
[a,b] is set
{a,b} is non empty set
{a} is non empty set
{{a,b},{a}} is non empty set
the L_meet of (L,p) . [a,b] is set
b "/\" a is Element of the carrier of (L,p)
a9 is Element of the carrier of L
b9 "/\" a9 is Element of the carrier of L
the L_meet of L . (b9,a9) is Element of the carrier of L
[b9,a9] is set
{b9,a9} is non empty set
{{b9,a9},{b9}} is non empty set
the L_meet of L . [b9,a9] is set
a "/\" b is Element of the carrier of (L,p)
L is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the carrier of L is non empty set
bool the carrier of L is non empty set
p is non empty meet-closed join-closed Element of bool the carrier of L
(L,p) is non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like SubLattice of L
the carrier of (L,p) is non empty set
q is Element of the carrier of (L,p)
b9 is Element of the carrier of (L,p)
K is Element of the carrier of (L,p)
K "/\" b9 is Element of the carrier of (L,p)
the L_meet of (L,p) is Relation-like [: the carrier of (L,p), the carrier of (L,p):] -defined the carrier of (L,p) -valued Function-like V18([: the carrier of (L,p), the carrier of (L,p):], the carrier of (L,p)) commutative associative idempotent Element of bool [:[: the carrier of (L,p), the carrier of (L,p):], the carrier of (L,p):]
[: the carrier of (L,p), the carrier of (L,p):] is non empty set
[:[: the carrier of (L,p), the carrier of (L,p):], the carrier of (L,p):] is non empty set
bool [:[: the carrier of (L,p), the carrier of (L,p):], the carrier of (L,p):] is non empty set
the L_meet of (L,p) . (K,b9) is Element of the carrier of (L,p)
[K,b9] is set
{K,b9} is non empty set
{K} is non empty set
{{K,b9},{K}} is non empty set
the L_meet of (L,p) . [K,b9] is set
q "\/" (K "/\" b9) is Element of the carrier of (L,p)
the L_join of (L,p) is Relation-like [: the carrier of (L,p), the carrier of (L,p):] -defined the carrier of (L,p) -valued Function-like V18([: the carrier of (L,p), the carrier of (L,p):], the carrier of (L,p)) commutative associative idempotent Element of bool [:[: the carrier of (L,p), the carrier of (L,p):], the carrier of (L,p):]
the L_join of (L,p) . (q,(K "/\" b9)) is Element of the carrier of (L,p)
[q,(K "/\" b9)] is set
{q,(K "/\" b9)} is non empty set
{q} is non empty set
{{q,(K "/\" b9)},{q}} is non empty set
the L_join of (L,p) . [q,(K "/\" b9)] is set
q "\/" K is Element of the carrier of (L,p)
the L_join of (L,p) . (q,K) is Element of the carrier of (L,p)
[q,K] is set
{q,K} is non empty set
{{q,K},{q}} is non empty set
the L_join of (L,p) . [q,K] is set
(q "\/" K) "/\" b9 is Element of the carrier of (L,p)
the L_meet of (L,p) . ((q "\/" K),b9) is Element of the carrier of (L,p)
[(q "\/" K),b9] is set
{(q "\/" K),b9} is non empty set
{(q "\/" K)} is non empty set
{{(q "\/" K),b9},{(q "\/" K)}} is non empty set
the L_meet of (L,p) . [(q "\/" K),b9] is set
K "/\" b9 is Element of the carrier of (L,p)
q "\/" K is Element of the carrier of (L,p)
b is Element of the carrier of L
a9 is Element of the carrier of L
q "\/" (K "/\" b9) is Element of the carrier of (L,p)
the L_join of (L,p) . (q,(K "/\" b9)) is Element of the carrier of (L,p)
[q,(K "/\" b9)] is set
{q,(K "/\" b9)} is non empty set
{{q,(K "/\" b9)},{q}} is non empty set
the L_join of (L,p) . [q,(K "/\" b9)] is set
c₉ is Element of the carrier of L
b "\/" c₉ is Element of the carrier of L
the L_join of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
[: the carrier of L, the carrier of L:] is non empty set
[:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
bool [:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
the L_join of L . (b,c₉) is Element of the carrier of L
[b,c₉] is set
{b,c₉} is non empty set
{b} is non empty set
{{b,c₉},{b}} is non empty set
the L_join of L . [b,c₉] is set
a is Element of the carrier of L
a "/\" a9 is Element of the carrier of L
the L_meet of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
the L_meet of L . (a,a9) is Element of the carrier of L
[a,a9] is set
{a,a9} is non empty set
{a} is non empty set
{{a,a9},{a}} is non empty set
the L_meet of L . [a,a9] is set
b "\/" (a "/\" a9) is Element of the carrier of L
the L_join of L . (b,(a "/\" a9)) is Element of the carrier of L
[b,(a "/\" a9)] is set
{b,(a "/\" a9)} is non empty set
{{b,(a "/\" a9)},{b}} is non empty set
the L_join of L . [b,(a "/\" a9)] is set
b "\/" a is Element of the carrier of L
the L_join of L . (b,a) is Element of the carrier of L
[b,a] is set
{b,a} is non empty set
{{b,a},{b}} is non empty set
the L_join of L . [b,a] is set
(b "\/" a) "/\" a9 is Element of the carrier of L
the L_meet of L . ((b "\/" a),a9) is Element of the carrier of L
[(b "\/" a),a9] is set
{(b "\/" a),a9} is non empty set
{(b "\/" a)} is non empty set
{{(b "\/" a),a9},{(b "\/" a)}} is non empty set
the L_meet of L . [(b "\/" a),a9] is set
c is Element of the carrier of L
c "/\" a9 is Element of the carrier of L
the L_meet of L . (c,a9) is Element of the carrier of L
[c,a9] is set
{c,a9} is non empty set
{c} is non empty set
{{c,a9},{c}} is non empty set
the L_meet of L . [c,a9] is set
(q "\/" K) "/\" b9 is Element of the carrier of (L,p)
the L_meet of (L,p) . ((q "\/" K),b9) is Element of the carrier of (L,p)
[(q "\/" K),b9] is set
{(q "\/" K),b9} is non empty set
{(q "\/" K)} is non empty set
{{(q "\/" K),b9},{(q "\/" K)}} is non empty set
the L_meet of (L,p) . [(q "\/" K),b9] is set
L is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the carrier of L is non empty set
bool the carrier of L is non empty set
p is non empty meet-closed join-closed Element of bool the carrier of L
(L,p) is non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like SubLattice of L
the carrier of (L,p) is non empty set
q is Element of the carrier of (L,p)
K is Element of the carrier of (L,p)
b9 is Element of the carrier of (L,p)
K "\/" b9 is Element of the carrier of (L,p)
the L_join of (L,p) is Relation-like [: the carrier of (L,p), the carrier of (L,p):] -defined the carrier of (L,p) -valued Function-like V18([: the carrier of (L,p), the carrier of (L,p):], the carrier of (L,p)) commutative associative idempotent Element of bool [:[: the carrier of (L,p), the carrier of (L,p):], the carrier of (L,p):]
[: the carrier of (L,p), the carrier of (L,p):] is non empty set
[:[: the carrier of (L,p), the carrier of (L,p):], the carrier of (L,p):] is non empty set
bool [:[: the carrier of (L,p), the carrier of (L,p):], the carrier of (L,p):] is non empty set
the L_join of (L,p) . (K,b9) is Element of the carrier of (L,p)
[K,b9] is set
{K,b9} is non empty set
{K} is non empty set
{{K,b9},{K}} is non empty set
the L_join of (L,p) . [K,b9] is set
q "/\" (K "\/" b9) is Element of the carrier of (L,p)
the L_meet of (L,p) is Relation-like [: the carrier of (L,p), the carrier of (L,p):] -defined the carrier of (L,p) -valued Function-like V18([: the carrier of (L,p), the carrier of (L,p):], the carrier of (L,p)) commutative associative idempotent Element of bool [:[: the carrier of (L,p), the carrier of (L,p):], the carrier of (L,p):]
the L_meet of (L,p) . (q,(K "\/" b9)) is Element of the carrier of (L,p)
[q,(K "\/" b9)] is set
{q,(K "\/" b9)} is non empty set
{q} is non empty set
{{q,(K "\/" b9)},{q}} is non empty set
the L_meet of (L,p) . [q,(K "\/" b9)] is set
q "/\" K is Element of the carrier of (L,p)
the L_meet of (L,p) . (q,K) is Element of the carrier of (L,p)
[q,K] is set
{q,K} is non empty set
{{q,K},{q}} is non empty set
the L_meet of (L,p) . [q,K] is set
q "/\" b9 is Element of the carrier of (L,p)
the L_meet of (L,p) . (q,b9) is Element of the carrier of (L,p)
[q,b9] is set
{q,b9} is non empty set
{{q,b9},{q}} is non empty set
the L_meet of (L,p) . [q,b9] is set
(q "/\" K) "\/" (q "/\" b9) is Element of the carrier of (L,p)
the L_join of (L,p) . ((q "/\" K),(q "/\" b9)) is Element of the carrier of (L,p)
[(q "/\" K),(q "/\" b9)] is set
{(q "/\" K),(q "/\" b9)} is non empty set
{(q "/\" K)} is non empty set
{{(q "/\" K),(q "/\" b9)},{(q "/\" K)}} is non empty set
the L_join of (L,p) . [(q "/\" K),(q "/\" b9)] is set
K "\/" b9 is Element of the carrier of (L,p)
q "/\" K is Element of the carrier of (L,p)
q "/\" b9 is Element of the carrier of (L,p)
q "/\" (K "\/" b9) is Element of the carrier of (L,p)
the L_meet of (L,p) . (q,(K "\/" b9)) is Element of the carrier of (L,p)
[q,(K "\/" b9)] is set
{q,(K "\/" b9)} is non empty set
{{q,(K "\/" b9)},{q}} is non empty set
the L_meet of (L,p) . [q,(K "\/" b9)] is set
b is Element of the carrier of L
c₉ is Element of the carrier of L
b "/\" c₉ is Element of the carrier of L
the L_meet of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
[: the carrier of L, the carrier of L:] is non empty set
[:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
bool [:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
the L_meet of L . (b,c₉) is Element of the carrier of L
[b,c₉] is set
{b,c₉} is non empty set
{b} is non empty set
{{b,c₉},{b}} is non empty set
the L_meet of L . [b,c₉] is set
a is Element of the carrier of L
a9 is Element of the carrier of L
a "\/" a9 is Element of the carrier of L
the L_join of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
the L_join of L . (a,a9) is Element of the carrier of L
[a,a9] is set
{a,a9} is non empty set
{a} is non empty set
{{a,a9},{a}} is non empty set
the L_join of L . [a,a9] is set
b "/\" (a "\/" a9) is Element of the carrier of L
the L_meet of L . (b,(a "\/" a9)) is Element of the carrier of L
[b,(a "\/" a9)] is set
{b,(a "\/" a9)} is non empty set
{{b,(a "\/" a9)},{b}} is non empty set
the L_meet of L . [b,(a "\/" a9)] is set
b "/\" a is Element of the carrier of L
the L_meet of L . (b,a) is Element of the carrier of L
[b,a] is set
{b,a} is non empty set
{{b,a},{b}} is non empty set
the L_meet of L . [b,a] is set
b "/\" a9 is Element of the carrier of L
the L_meet of L . (b,a9) is Element of the carrier of L
[b,a9] is set
{b,a9} is non empty set
{{b,a9},{b}} is non empty set
the L_meet of L . [b,a9] is set
(b "/\" a) "\/" (b "/\" a9) is Element of the carrier of L
the L_join of L . ((b "/\" a),(b "/\" a9)) is Element of the carrier of L
[(b "/\" a),(b "/\" a9)] is set
{(b "/\" a),(b "/\" a9)} is non empty set
{(b "/\" a)} is non empty set
{{(b "/\" a),(b "/\" a9)},{(b "/\" a)}} is non empty set
the L_join of L . [(b "/\" a),(b "/\" a9)] is set
c is Element of the carrier of L
c "\/" (b "/\" a9) is Element of the carrier of L
the L_join of L . (c,(b "/\" a9)) is Element of the carrier of L
[c,(b "/\" a9)] is set
{c,(b "/\" a9)} is non empty set
{c} is non empty set
{{c,(b "/\" a9)},{c}} is non empty set
the L_join of L . [c,(b "/\" a9)] is set
d is Element of the carrier of L
c "\/" d is Element of the carrier of L
the L_join of L . (c,d) is Element of the carrier of L
[c,d] is set
{c,d} is non empty set
{{c,d},{c}} is non empty set
the L_join of L . [c,d] is set
(q "/\" K) "\/" (q "/\" b9) is Element of the carrier of (L,p)
the L_join of (L,p) . ((q "/\" K),(q "/\" b9)) is Element of the carrier of (L,p)
[(q "/\" K),(q "/\" b9)] is set
{(q "/\" K),(q "/\" b9)} is non empty set
{(q "/\" K)} is non empty set
{{(q "/\" K),(q "/\" b9)},{(q "/\" K)}} is non empty set
the L_join of (L,p) . [(q "/\" K),(q "/\" b9)] is set
L is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the carrier of L is non empty set
p is Element of the carrier of L
q is Element of the carrier of L
(L,p,q) is non empty meet-closed join-closed Element of bool the carrier of L
bool the carrier of L is non empty set
(L,(L,p,q)) is non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like SubLattice of L
the carrier of (L,(L,p,q)) is non empty set
b is Element of the carrier of (L,(L,p,q))
a is Element of the carrier of (L,(L,p,q))
a9 is Element of the carrier of L
c₉ is Element of the carrier of L
a9 => c₉ is Element of the carrier of L
(a9 => c₉) "/\" a9 is Element of the carrier of L
the L_meet of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
[: the carrier of L, the carrier of L:] is non empty set
[:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
bool [:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
the L_meet of L . ((a9 => c₉),a9) is Element of the carrier of L
[(a9 => c₉),a9] is set
{(a9 => c₉),a9} is non empty set
{(a9 => c₉)} is non empty set
{{(a9 => c₉),a9},{(a9 => c₉)}} is non empty set
the L_meet of L . [(a9 => c₉),a9] is set
p "\/" ((a9 => c₉) "/\" a9) is Element of the carrier of L
the L_join of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
the L_join of L . (p,((a9 => c₉) "/\" a9)) is Element of the carrier of L
[p,((a9 => c₉) "/\" a9)] is set
{p,((a9 => c₉) "/\" a9)} is non empty set
{p} is non empty set
{{p,((a9 => c₉) "/\" a9)},{p}} is non empty set
the L_join of L . [p,((a9 => c₉) "/\" a9)] is set
p "\/" (a9 => c₉) is Element of the carrier of L
the L_join of L . (p,(a9 => c₉)) is Element of the carrier of L
[p,(a9 => c₉)] is set
{p,(a9 => c₉)} is non empty set
{{p,(a9 => c₉)},{p}} is non empty set
the L_join of L . [p,(a9 => c₉)] is set
(p "\/" (a9 => c₉)) "/\" a9 is Element of the carrier of L
the L_meet of L . ((p "\/" (a9 => c₉)),a9) is Element of the carrier of L
[(p "\/" (a9 => c₉)),a9] is set
{(p "\/" (a9 => c₉)),a9} is non empty set
{(p "\/" (a9 => c₉))} is non empty set
{{(p "\/" (a9 => c₉)),a9},{(p "\/" (a9 => c₉))}} is non empty set
the L_meet of L . [(p "\/" (a9 => c₉)),a9] is set
a9 "/\" (a9 => c₉) is Element of the carrier of L
the L_meet of L . (a9,(a9 => c₉)) is Element of the carrier of L
[a9,(a9 => c₉)] is set
{a9,(a9 => c₉)} is non empty set
{a9} is non empty set
{{a9,(a9 => c₉)},{a9}} is non empty set
the L_meet of L . [a9,(a9 => c₉)] is set
p "\/" (a9 "/\" (a9 => c₉)) is Element of the carrier of L
the L_join of L . (p,(a9 "/\" (a9 => c₉))) is Element of the carrier of L
[p,(a9 "/\" (a9 => c₉))] is set
{p,(a9 "/\" (a9 => c₉))} is non empty set
{{p,(a9 "/\" (a9 => c₉))},{p}} is non empty set
the L_join of L . [p,(a9 "/\" (a9 => c₉))] is set
(p "\/" (a9 "/\" (a9 => c₉))) "/\" q is Element of the carrier of L
the L_meet of L . ((p "\/" (a9 "/\" (a9 => c₉))),q) is Element of the carrier of L
[(p "\/" (a9 "/\" (a9 => c₉))),q] is set
{(p "\/" (a9 "/\" (a9 => c₉))),q} is non empty set
{(p "\/" (a9 "/\" (a9 => c₉)))} is non empty set
{{(p "\/" (a9 "/\" (a9 => c₉))),q},{(p "\/" (a9 "/\" (a9 => c₉)))}} is non empty set
the L_meet of L . [(p "\/" (a9 "/\" (a9 => c₉))),q] is set
(a9 => c₉) "\/" p is Element of the carrier of L
the L_join of L . ((a9 => c₉),p) is Element of the carrier of L
[(a9 => c₉),p] is set
{(a9 => c₉),p} is non empty set
{{(a9 => c₉),p},{(a9 => c₉)}} is non empty set
the L_join of L . [(a9 => c₉),p] is set
((a9 => c₉) "\/" p) "/\" q is Element of the carrier of L
the L_meet of L . (((a9 => c₉) "\/" p),q) is Element of the carrier of L
[((a9 => c₉) "\/" p),q] is set
{((a9 => c₉) "\/" p),q} is non empty set
{((a9 => c₉) "\/" p)} is non empty set
{{((a9 => c₉) "\/" p),q},{((a9 => c₉) "\/" p)}} is non empty set
the L_meet of L . [((a9 => c₉) "\/" p),q] is set
d9 is Element of the carrier of (L,(L,p,q))
b "/\" d9 is Element of the carrier of (L,(L,p,q))
the L_meet of (L,(L,p,q)) is Relation-like [: the carrier of (L,(L,p,q)), the carrier of (L,(L,p,q)):] -defined the carrier of (L,(L,p,q)) -valued Function-like V18([: the carrier of (L,(L,p,q)), the carrier of (L,(L,p,q)):], the carrier of (L,(L,p,q))) commutative associative idempotent Element of bool [:[: the carrier of (L,(L,p,q)), the carrier of (L,(L,p,q)):], the carrier of (L,(L,p,q)):]
[: the carrier of (L,(L,p,q)), the carrier of (L,(L,p,q)):] is non empty set
[:[: the carrier of (L,(L,p,q)), the carrier of (L,(L,p,q)):], the carrier of (L,(L,p,q)):] is non empty set
bool [:[: the carrier of (L,(L,p,q)), the carrier of (L,(L,p,q)):], the carrier of (L,(L,p,q)):] is non empty set
the L_meet of (L,(L,p,q)) . (b,d9) is Element of the carrier of (L,(L,p,q))
[b,d9] is set
{b,d9} is non empty set
{b} is non empty set
{{b,d9},{b}} is non empty set
the L_meet of (L,(L,p,q)) . [b,d9] is set
((p "\/" (a9 => c₉)) "/\" a9) "/\" q is Element of the carrier of L
the L_meet of L . (((p "\/" (a9 => c₉)) "/\" a9),q) is Element of the carrier of L
[((p "\/" (a9 => c₉)) "/\" a9),q] is set
{((p "\/" (a9 => c₉)) "/\" a9),q} is non empty set
{((p "\/" (a9 => c₉)) "/\" a9)} is non empty set
{{((p "\/" (a9 => c₉)) "/\" a9),q},{((p "\/" (a9 => c₉)) "/\" a9)}} is non empty set
the L_meet of L . [((p "\/" (a9 => c₉)) "/\" a9),q] is set
a9 "/\" (((a9 => c₉) "\/" p) "/\" q) is Element of the carrier of L
the L_meet of L . (a9,(((a9 => c₉) "\/" p) "/\" q)) is Element of the carrier of L
[a9,(((a9 => c₉) "\/" p) "/\" q)] is set
{a9,(((a9 => c₉) "\/" p) "/\" q)} is non empty set
{{a9,(((a9 => c₉) "\/" p) "/\" q)},{a9}} is non empty set
the L_meet of L . [a9,(((a9 => c₉) "\/" p) "/\" q)] is set
b "/\" d9 is Element of the carrier of (L,(L,p,q))
e9 is Element of the carrier of (L,(L,p,q))
b "/\" e9 is Element of the carrier of (L,(L,p,q))
the L_meet of (L,(L,p,q)) . (b,e9) is Element of the carrier of (L,(L,p,q))
[b,e9] is set
{b,e9} is non empty set
{{b,e9},{b}} is non empty set
the L_meet of (L,(L,p,q)) . [b,e9] is set
b "/\" e9 is Element of the carrier of (L,(L,p,q))
e is Element of the carrier of L
e "/\" q is Element of the carrier of L
the L_meet of L . (e,q) is Element of the carrier of L
[e,q] is set
{e,q} is non empty set
{e} is non empty set
{{e,q},{e}} is non empty set
the L_meet of L . [e,q] is set
ae is Element of the carrier of L
a9 "/\" e is Element of the carrier of L
the L_meet of L . (a9,e) is Element of the carrier of L
[a9,e] is set
{a9,e} is non empty set
{{a9,e},{a9}} is non empty set
the L_meet of L . [a9,e] is set
e "\/" p is Element of the carrier of L
the L_join of L . (e,p) is Element of the carrier of L
[e,p] is set
{e,p} is non empty set
{{e,p},{e}} is non empty set
the L_join of L . [e,p] is set
L is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the carrier of L is non empty set
p is Element of the carrier of L
(L,p) is non empty initial meet-closed join-closed Element of bool the carrier of L
bool the carrier of L is non empty set
{ b₁ where b₁ is Element of the carrier of L : b₁ [= p } is set
(L,(L,p)) is non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like SubLattice of L
L .: is non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the L_meet of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
[: the carrier of L, the carrier of L:] is non empty set
[:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
bool [:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
the L_join of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
LattStr(# the carrier of L, the L_meet of L, the L_join of L #) is non empty strict LattStr
(L,(L,p)) is non empty final meet-closed join-closed Element of bool the carrier of (L .:)
the carrier of (L .:) is non empty set
bool the carrier of (L .:) is non empty set
((L .:),(L,(L,p))) is non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like SubLattice of L .:
((L .:),((L .:),(L,(L,p)))) is non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like SubLattice of (L .:) .:
(L .:) .: is non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the L_meet of (L .:) is Relation-like [: the carrier of (L .:), the carrier of (L .:):] -defined the carrier of (L .:) -valued Function-like V18([: the carrier of (L .:), the carrier of (L .:):], the carrier of (L .:)) commutative associative idempotent Element of bool [:[: the carrier of (L .:), the carrier of (L .:):], the carrier of (L .:):]
[: the carrier of (L .:), the carrier of (L .:):] is non empty set
[:[: the carrier of (L .:), the carrier of (L .:):], the carrier of (L .:):] is non empty set
bool [:[: the carrier of (L .:), the carrier of (L .:):], the carrier of (L .:):] is non empty set
the L_join of (L .:) is Relation-like [: the carrier of (L .:), the carrier of (L .:):] -defined the carrier of (L .:) -valued Function-like V18([: the carrier of (L .:), the carrier of (L .:):], the carrier of (L .:)) commutative associative idempotent Element of bool [:[: the carrier of (L .:), the carrier of (L .:):], the carrier of (L .:):]
LattStr(# the carrier of (L .:), the L_meet of (L .:), the L_join of (L .:) #) is non empty strict LattStr
the carrier of ((L .:),(L,(L,p))) is non empty set
the L_meet of ((L .:),(L,(L,p))) is Relation-like [: the carrier of ((L .:),(L,(L,p))), the carrier of ((L .:),(L,(L,p))):] -defined the carrier of ((L .:),(L,(L,p))) -valued Function-like V18([: the carrier of ((L .:),(L,(L,p))), the carrier of ((L .:),(L,(L,p))):], the carrier of ((L .:),(L,(L,p)))) commutative associative idempotent Element of bool [:[: the carrier of ((L .:),(L,(L,p))), the carrier of ((L .:),(L,(L,p))):], the carrier of ((L .:),(L,(L,p))):]
[: the carrier of ((L .:),(L,(L,p))), the carrier of ((L .:),(L,(L,p))):] is non empty set
[:[: the carrier of ((L .:),(L,(L,p))), the carrier of ((L .:),(L,(L,p))):], the carrier of ((L .:),(L,(L,p))):] is non empty set
bool [:[: the carrier of ((L .:),(L,(L,p))), the carrier of ((L .:),(L,(L,p))):], the carrier of ((L .:),(L,(L,p))):] is non empty set
the L_join of ((L .:),(L,(L,p))) is Relation-like [: the carrier of ((L .:),(L,(L,p))), the carrier of ((L .:),(L,(L,p))):] -defined the carrier of ((L .:),(L,(L,p))) -valued Function-like V18([: the carrier of ((L .:),(L,(L,p))), the carrier of ((L .:),(L,(L,p))):], the carrier of ((L .:),(L,(L,p)))) commutative associative idempotent Element of bool [:[: the carrier of ((L .:),(L,(L,p))), the carrier of ((L .:),(L,(L,p))):], the carrier of ((L .:),(L,(L,p))):]
LattStr(# the carrier of ((L .:),(L,(L,p))), the L_meet of ((L .:),(L,(L,p))), the L_join of ((L .:),(L,(L,p))) #) is non empty strict LattStr
(L,p) is Element of the carrier of (L .:)
<.(L,p).) is non empty final meet-closed join-closed Element of bool the carrier of (L .:)
{ b₁ where b₁ is Element of the carrier of (L .:) : (L,p) [= b₁ } is set
((L .:),<.(L,p).)) is non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like SubLattice of L .:
((L .:),((L .:),<.(L,p).))) is non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like SubLattice of (L .:) .:
the carrier of ((L .:),<.(L,p).)) is non empty set
the L_meet of ((L .:),<.(L,p).)) is Relation-like [: the carrier of ((L .:),<.(L,p).)), the carrier of ((L .:),<.(L,p).)):] -defined the carrier of ((L .:),<.(L,p).)) -valued Function-like V18([: the carrier of ((L .:),<.(L,p).)), the carrier of ((L .:),<.(L,p).)):], the carrier of ((L .:),<.(L,p).))) commutative associative idempotent Element of bool [:[: the carrier of ((L .:),<.(L,p).)), the carrier of ((L .:),<.(L,p).)):], the carrier of ((L .:),<.(L,p).)):]
[: the carrier of ((L .:),<.(L,p).)), the carrier of ((L .:),<.(L,p).)):] is non empty set
[:[: the carrier of ((L .:),<.(L,p).)), the carrier of ((L .:),<.(L,p).)):], the carrier of ((L .:),<.(L,p).)):] is non empty set
bool [:[: the carrier of ((L .:),<.(L,p).)), the carrier of ((L .:),<.(L,p).)):], the carrier of ((L .:),<.(L,p).)):] is non empty set
the L_join of ((L .:),<.(L,p).)) is Relation-like [: the carrier of ((L .:),<.(L,p).)), the carrier of ((L .:),<.(L,p).)):] -defined the carrier of ((L .:),<.(L,p).)) -valued Function-like V18([: the carrier of ((L .:),<.(L,p).)), the carrier of ((L .:),<.(L,p).)):], the carrier of ((L .:),<.(L,p).))) commutative associative idempotent Element of bool [:[: the carrier of ((L .:),<.(L,p).)), the carrier of ((L .:),<.(L,p).)):], the carrier of ((L .:),<.(L,p).)):]
LattStr(# the carrier of ((L .:),<.(L,p).)), the L_meet of ((L .:),<.(L,p).)), the L_join of ((L .:),<.(L,p).)) #) is non empty strict LattStr
latt <.(L,p).) is non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded LattStr
(latt <.(L,p).)) .: is non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the carrier of (latt <.(L,p).)) is non empty set
the L_meet of (latt <.(L,p).)) is Relation-like [: the carrier of (latt <.(L,p).)), the carrier of (latt <.(L,p).)):] -defined the carrier of (latt <.(L,p).)) -valued Function-like V18([: the carrier of (latt <.(L,p).)), the carrier of (latt <.(L,p).)):], the carrier of (latt <.(L,p).))) commutative associative idempotent Element of bool [:[: the carrier of (latt <.(L,p).)), the carrier of (latt <.(L,p).)):], the carrier of (latt <.(L,p).)):]
[: the carrier of (latt <.(L,p).)), the carrier of (latt <.(L,p).)):] is non empty set
[:[: the carrier of (latt <.(L,p).)), the carrier of (latt <.(L,p).)):], the carrier of (latt <.(L,p).)):] is non empty set
bool [:[: the carrier of (latt <.(L,p).)), the carrier of (latt <.(L,p).)):], the carrier of (latt <.(L,p).)):] is non empty set
the L_join of (latt <.(L,p).)) is Relation-like [: the carrier of (latt <.(L,p).)), the carrier of (latt <.(L,p).)):] -defined the carrier of (latt <.(L,p).)) -valued Function-like V18([: the carrier of (latt <.(L,p).)), the carrier of (latt <.(L,p).)):], the carrier of (latt <.(L,p).))) commutative associative idempotent Element of bool [:[: the carrier of (latt <.(L,p).)), the carrier of (latt <.(L,p).)):], the carrier of (latt <.(L,p).)):]
LattStr(# the carrier of (latt <.(L,p).)), the L_meet of (latt <.(L,p).)), the L_join of (latt <.(L,p).)) #) is non empty strict LattStr
L is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the carrier of L is non empty set
p is Element of the carrier of L
(L,p) is non empty initial meet-closed join-closed Element of bool the carrier of L
bool the carrier of L is non empty set
{ b₁ where b₁ is Element of the carrier of L : b₁ [= p } is set
(L,(L,p)) is non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like upper-bounded SubLattice of L
Top (L,(L,p)) is Element of the carrier of (L,(L,p))
the carrier of (L,(L,p)) is non empty set
L .: is non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the L_meet of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
[: the carrier of L, the carrier of L:] is non empty set
[:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
bool [:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
the L_join of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
LattStr(# the carrier of L, the L_meet of L, the L_join of L #) is non empty strict LattStr
(L,(L,p)) is non empty final meet-closed join-closed Element of bool the carrier of (L .:)
the carrier of (L .:) is non empty set
bool the carrier of (L .:) is non empty set
((L .:),(L,(L,p))) is non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like SubLattice of L .:
((L .:),((L .:),(L,(L,p)))) is non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like SubLattice of (L .:) .:
(L .:) .: is non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the L_meet of (L .:) is Relation-like [: the carrier of (L .:), the carrier of (L .:):] -defined the carrier of (L .:) -valued Function-like V18([: the carrier of (L .:), the carrier of (L .:):], the carrier of (L .:)) commutative associative idempotent Element of bool [:[: the carrier of (L .:), the carrier of (L .:):], the carrier of (L .:):]
[: the carrier of (L .:), the carrier of (L .:):] is non empty set
[:[: the carrier of (L .:), the carrier of (L .:):], the carrier of (L .:):] is non empty set
bool [:[: the carrier of (L .:), the carrier of (L .:):], the carrier of (L .:):] is non empty set
the L_join of (L .:) is Relation-like [: the carrier of (L .:), the carrier of (L .:):] -defined the carrier of (L .:) -valued Function-like V18([: the carrier of (L .:), the carrier of (L .:):], the carrier of (L .:)) commutative associative idempotent Element of bool [:[: the carrier of (L .:), the carrier of (L .:):], the carrier of (L .:):]
LattStr(# the carrier of (L .:), the L_meet of (L .:), the L_join of (L .:) #) is non empty strict LattStr
the carrier of ((L .:),(L,(L,p))) is non empty set
the L_meet of ((L .:),(L,(L,p))) is Relation-like [: the carrier of ((L .:),(L,(L,p))), the carrier of ((L .:),(L,(L,p))):] -defined the carrier of ((L .:),(L,(L,p))) -valued Function-like V18([: the carrier of ((L .:),(L,(L,p))), the carrier of ((L .:),(L,(L,p))):], the carrier of ((L .:),(L,(L,p)))) commutative associative idempotent Element of bool [:[: the carrier of ((L .:),(L,(L,p))), the carrier of ((L .:),(L,(L,p))):], the carrier of ((L .:),(L,(L,p))):]
[: the carrier of ((L .:),(L,(L,p))), the carrier of ((L .:),(L,(L,p))):] is non empty set
[:[: the carrier of ((L .:),(L,(L,p))), the carrier of ((L .:),(L,(L,p))):], the carrier of ((L .:),(L,(L,p))):] is non empty set
bool [:[: the carrier of ((L .:),(L,(L,p))), the carrier of ((L .:),(L,(L,p))):], the carrier of ((L .:),(L,(L,p))):] is non empty set
the L_join of ((L .:),(L,(L,p))) is Relation-like [: the carrier of ((L .:),(L,(L,p))), the carrier of ((L .:),(L,(L,p))):] -defined the carrier of ((L .:),(L,(L,p))) -valued Function-like V18([: the carrier of ((L .:),(L,(L,p))), the carrier of ((L .:),(L,(L,p))):], the carrier of ((L .:),(L,(L,p)))) commutative associative idempotent Element of bool [:[: the carrier of ((L .:),(L,(L,p))), the carrier of ((L .:),(L,(L,p))):], the carrier of ((L .:),(L,(L,p))):]
LattStr(# the carrier of ((L .:),(L,(L,p))), the L_meet of ((L .:),(L,(L,p))), the L_join of ((L .:),(L,(L,p))) #) is non empty strict LattStr
(L,p) is Element of the carrier of (L .:)
<.(L,p).) is non empty final meet-closed join-closed Element of bool the carrier of (L .:)
{ b₁ where b₁ is Element of the carrier of (L .:) : (L,p) [= b₁ } is set
((L .:),<.(L,p).)) is non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like SubLattice of L .:
((L .:),((L .:),<.(L,p).))) is non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like SubLattice of (L .:) .:
the carrier of ((L .:),<.(L,p).)) is non empty set
the L_meet of ((L .:),<.(L,p).)) is Relation-like [: the carrier of ((L .:),<.(L,p).)), the carrier of ((L .:),<.(L,p).)):] -defined the carrier of ((L .:),<.(L,p).)) -valued Function-like V18([: the carrier of ((L .:),<.(L,p).)), the carrier of ((L .:),<.(L,p).)):], the carrier of ((L .:),<.(L,p).))) commutative associative idempotent Element of bool [:[: the carrier of ((L .:),<.(L,p).)), the carrier of ((L .:),<.(L,p).)):], the carrier of ((L .:),<.(L,p).)):]
[: the carrier of ((L .:),<.(L,p).)), the carrier of ((L .:),<.(L,p).)):] is non empty set
[:[: the carrier of ((L .:),<.(L,p).)), the carrier of ((L .:),<.(L,p).)):], the carrier of ((L .:),<.(L,p).)):] is non empty set
bool [:[: the carrier of ((L .:),<.(L,p).)), the carrier of ((L .:),<.(L,p).)):], the carrier of ((L .:),<.(L,p).)):] is non empty set
the L_join of ((L .:),<.(L,p).)) is Relation-like [: the carrier of ((L .:),<.(L,p).)), the carrier of ((L .:),<.(L,p).)):] -defined the carrier of ((L .:),<.(L,p).)) -valued Function-like V18([: the carrier of ((L .:),<.(L,p).)), the carrier of ((L .:),<.(L,p).)):], the carrier of ((L .:),<.(L,p).))) commutative associative idempotent Element of bool [:[: the carrier of ((L .:),<.(L,p).)), the carrier of ((L .:),<.(L,p).)):], the carrier of ((L .:),<.(L,p).)):]
LattStr(# the carrier of ((L .:),<.(L,p).)), the L_meet of ((L .:),<.(L,p).)), the L_join of ((L .:),<.(L,p).)) #) is non empty strict LattStr
latt <.(L,p).) is non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded LattStr
(latt <.(L,p).)) .: is non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the carrier of (latt <.(L,p).)) is non empty set
the L_meet of (latt <.(L,p).)) is Relation-like [: the carrier of (latt <.(L,p).)), the carrier of (latt <.(L,p).)):] -defined the carrier of (latt <.(L,p).)) -valued Function-like V18([: the carrier of (latt <.(L,p).)), the carrier of (latt <.(L,p).)):], the carrier of (latt <.(L,p).))) commutative associative idempotent Element of bool [:[: the carrier of (latt <.(L,p).)), the carrier of (latt <.(L,p).)):], the carrier of (latt <.(L,p).)):]
[: the carrier of (latt <.(L,p).)), the carrier of (latt <.(L,p).)):] is non empty set
[:[: the carrier of (latt <.(L,p).)), the carrier of (latt <.(L,p).)):], the carrier of (latt <.(L,p).)):] is non empty set
bool [:[: the carrier of (latt <.(L,p).)), the carrier of (latt <.(L,p).)):], the carrier of (latt <.(L,p).)):] is non empty set
the L_join of (latt <.(L,p).)) is Relation-like [: the carrier of (latt <.(L,p).)), the carrier of (latt <.(L,p).)):] -defined the carrier of (latt <.(L,p).)) -valued Function-like V18([: the carrier of (latt <.(L,p).)), the carrier of (latt <.(L,p).)):], the carrier of (latt <.(L,p).))) commutative associative idempotent Element of bool [:[: the carrier of (latt <.(L,p).)), the carrier of (latt <.(L,p).)):], the carrier of (latt <.(L,p).)):]
LattStr(# the carrier of (latt <.(L,p).)), the L_meet of (latt <.(L,p).)), the L_join of (latt <.(L,p).)) #) is non empty strict LattStr
Bottom (latt <.(L,p).)) is Element of the carrier of (latt <.(L,p).))
L is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the carrier of L is non empty set
Bottom L is Element of the carrier of L
p is Element of the carrier of L
(L,p) is non empty initial meet-closed join-closed Element of bool the carrier of L
bool the carrier of L is non empty set
{ b₁ where b₁ is Element of the carrier of L : b₁ [= p } is set
(L,(L,p)) is non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like upper-bounded SubLattice of L
Bottom (L,(L,p)) is Element of the carrier of (L,(L,p))
the carrier of (L,(L,p)) is non empty set
L .: is non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the L_meet of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
[: the carrier of L, the carrier of L:] is non empty set
[:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
bool [:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
the L_join of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
LattStr(# the carrier of L, the L_meet of L, the L_join of L #) is non empty strict LattStr
(L,p) is Element of the carrier of (L .:)
the carrier of (L .:) is non empty set
<.(L,p).) is non empty final meet-closed join-closed Element of bool the carrier of (L .:)
bool the carrier of (L .:) is non empty set
{ b₁ where b₁ is Element of the carrier of (L .:) : (L,p) [= b₁ } is set
latt <.(L,p).) is non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded LattStr
(L,(L,p)) is non empty final meet-closed join-closed Element of bool the carrier of (L .:)
((L .:),(L,(L,p))) is non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like SubLattice of L .:
((L .:),((L .:),(L,(L,p)))) is non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like SubLattice of (L .:) .:
(L .:) .: is non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the L_meet of (L .:) is Relation-like [: the carrier of (L .:), the carrier of (L .:):] -defined the carrier of (L .:) -valued Function-like V18([: the carrier of (L .:), the carrier of (L .:):], the carrier of (L .:)) commutative associative idempotent Element of bool [:[: the carrier of (L .:), the carrier of (L .:):], the carrier of (L .:):]
[: the carrier of (L .:), the carrier of (L .:):] is non empty set
[:[: the carrier of (L .:), the carrier of (L .:):], the carrier of (L .:):] is non empty set
bool [:[: the carrier of (L .:), the carrier of (L .:):], the carrier of (L .:):] is non empty set
the L_join of (L .:) is Relation-like [: the carrier of (L .:), the carrier of (L .:):] -defined the carrier of (L .:) -valued Function-like V18([: the carrier of (L .:), the carrier of (L .:):], the carrier of (L .:)) commutative associative idempotent Element of bool [:[: the carrier of (L .:), the carrier of (L .:):], the carrier of (L .:):]
LattStr(# the carrier of (L .:), the L_meet of (L .:), the L_join of (L .:) #) is non empty strict LattStr
the carrier of ((L .:),(L,(L,p))) is non empty set
the L_meet of ((L .:),(L,(L,p))) is Relation-like [: the carrier of ((L .:),(L,(L,p))), the carrier of ((L .:),(L,(L,p))):] -defined the carrier of ((L .:),(L,(L,p))) -valued Function-like V18([: the carrier of ((L .:),(L,(L,p))), the carrier of ((L .:),(L,(L,p))):], the carrier of ((L .:),(L,(L,p)))) commutative associative idempotent Element of bool [:[: the carrier of ((L .:),(L,(L,p))), the carrier of ((L .:),(L,(L,p))):], the carrier of ((L .:),(L,(L,p))):]
[: the carrier of ((L .:),(L,(L,p))), the carrier of ((L .:),(L,(L,p))):] is non empty set
[:[: the carrier of ((L .:),(L,(L,p))), the carrier of ((L .:),(L,(L,p))):], the carrier of ((L .:),(L,(L,p))):] is non empty set
bool [:[: the carrier of ((L .:),(L,(L,p))), the carrier of ((L .:),(L,(L,p))):], the carrier of ((L .:),(L,(L,p))):] is non empty set
the L_join of ((L .:),(L,(L,p))) is Relation-like [: the carrier of ((L .:),(L,(L,p))), the carrier of ((L .:),(L,(L,p))):] -defined the carrier of ((L .:),(L,(L,p))) -valued Function-like V18([: the carrier of ((L .:),(L,(L,p))), the carrier of ((L .:),(L,(L,p))):], the carrier of ((L .:),(L,(L,p)))) commutative associative idempotent Element of bool [:[: the carrier of ((L .:),(L,(L,p))), the carrier of ((L .:),(L,(L,p))):], the carrier of ((L .:),(L,(L,p))):]
LattStr(# the carrier of ((L .:),(L,(L,p))), the L_meet of ((L .:),(L,(L,p))), the L_join of ((L .:),(L,(L,p))) #) is non empty strict LattStr
((L .:),<.(L,p).)) is non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like SubLattice of L .:
((L .:),((L .:),<.(L,p).))) is non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like SubLattice of (L .:) .:
the carrier of ((L .:),<.(L,p).)) is non empty set
the L_meet of ((L .:),<.(L,p).)) is Relation-like [: the carrier of ((L .:),<.(L,p).)), the carrier of ((L .:),<.(L,p).)):] -defined the carrier of ((L .:),<.(L,p).)) -valued Function-like V18([: the carrier of ((L .:),<.(L,p).)), the carrier of ((L .:),<.(L,p).)):], the carrier of ((L .:),<.(L,p).))) commutative associative idempotent Element of bool [:[: the carrier of ((L .:),<.(L,p).)), the carrier of ((L .:),<.(L,p).)):], the carrier of ((L .:),<.(L,p).)):]
[: the carrier of ((L .:),<.(L,p).)), the carrier of ((L .:),<.(L,p).)):] is non empty set
[:[: the carrier of ((L .:),<.(L,p).)), the carrier of ((L .:),<.(L,p).)):], the carrier of ((L .:),<.(L,p).)):] is non empty set
bool [:[: the carrier of ((L .:),<.(L,p).)), the carrier of ((L .:),<.(L,p).)):], the carrier of ((L .:),<.(L,p).)):] is non empty set
the L_join of ((L .:),<.(L,p).)) is Relation-like [: the carrier of ((L .:),<.(L,p).)), the carrier of ((L .:),<.(L,p).)):] -defined the carrier of ((L .:),<.(L,p).)) -valued Function-like V18([: the carrier of ((L .:),<.(L,p).)), the carrier of ((L .:),<.(L,p).)):], the carrier of ((L .:),<.(L,p).))) commutative associative idempotent Element of bool [:[: the carrier of ((L .:),<.(L,p).)), the carrier of ((L .:),<.(L,p).)):], the carrier of ((L .:),<.(L,p).)):]
LattStr(# the carrier of ((L .:),<.(L,p).)), the L_meet of ((L .:),<.(L,p).)), the L_join of ((L .:),<.(L,p).)) #) is non empty strict LattStr
(latt <.(L,p).)) .: is non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the carrier of (latt <.(L,p).)) is non empty set
the L_meet of (latt <.(L,p).)) is Relation-like [: the carrier of (latt <.(L,p).)), the carrier of (latt <.(L,p).)):] -defined the carrier of (latt <.(L,p).)) -valued Function-like V18([: the carrier of (latt <.(L,p).)), the carrier of (latt <.(L,p).)):], the carrier of (latt <.(L,p).))) commutative associative idempotent Element of bool [:[: the carrier of (latt <.(L,p).)), the carrier of (latt <.(L,p).)):], the carrier of (latt <.(L,p).)):]
[: the carrier of (latt <.(L,p).)), the carrier of (latt <.(L,p).)):] is non empty set
[:[: the carrier of (latt <.(L,p).)), the carrier of (latt <.(L,p).)):], the carrier of (latt <.(L,p).)):] is non empty set
bool [:[: the carrier of (latt <.(L,p).)), the carrier of (latt <.(L,p).)):], the carrier of (latt <.(L,p).)):] is non empty set
the L_join of (latt <.(L,p).)) is Relation-like [: the carrier of (latt <.(L,p).)), the carrier of (latt <.(L,p).)):] -defined the carrier of (latt <.(L,p).)) -valued Function-like V18([: the carrier of (latt <.(L,p).)), the carrier of (latt <.(L,p).)):], the carrier of (latt <.(L,p).))) commutative associative idempotent Element of bool [:[: the carrier of (latt <.(L,p).)), the carrier of (latt <.(L,p).)):], the carrier of (latt <.(L,p).)):]
LattStr(# the carrier of (latt <.(L,p).)), the L_meet of (latt <.(L,p).)), the L_join of (latt <.(L,p).)) #) is non empty strict LattStr
Top (latt <.(L,p).)) is Element of the carrier of (latt <.(L,p).))
Top (L .:) is Element of the carrier of (L .:)
L is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the carrier of L is non empty set
p is Element of the carrier of L
(L,p) is non empty initial meet-closed join-closed Element of bool the carrier of L
bool the carrier of L is non empty set
{ b₁ where b₁ is Element of the carrier of L : b₁ [= p } is set
(L,(L,p)) is non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like upper-bounded SubLattice of L
L is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the carrier of L is non empty set
p is Element of the carrier of L
q is Element of the carrier of L
(L,p,q) is non empty meet-closed join-closed Element of bool the carrier of L
bool the carrier of L is non empty set
(L,(L,p,q)) is non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like SubLattice of L
Top (L,(L,p,q)) is Element of the carrier of (L,(L,p,q))
the carrier of (L,(L,p,q)) is non empty set
Bottom (L,(L,p,q)) is Element of the carrier of (L,(L,p,q))
b is Element of the carrier of (L,(L,p,q))
a is Element of the carrier of L
b9 is Element of the carrier of (L,(L,p,q))
b9 "\/" b is Element of the carrier of (L,(L,p,q))
the L_join of (L,(L,p,q)) is Relation-like [: the carrier of (L,(L,p,q)), the carrier of (L,(L,p,q)):] -defined the carrier of (L,(L,p,q)) -valued Function-like V18([: the carrier of (L,(L,p,q)), the carrier of (L,(L,p,q)):], the carrier of (L,(L,p,q))) commutative associative idempotent Element of bool [:[: the carrier of (L,(L,p,q)), the carrier of (L,(L,p,q)):], the carrier of (L,(L,p,q)):]
[: the carrier of (L,(L,p,q)), the carrier of (L,(L,p,q)):] is non empty set
[:[: the carrier of (L,(L,p,q)), the carrier of (L,(L,p,q)):], the carrier of (L,(L,p,q)):] is non empty set
bool [:[: the carrier of (L,(L,p,q)), the carrier of (L,(L,p,q)):], the carrier of (L,(L,p,q)):] is non empty set
the L_join of (L,(L,p,q)) . (b9,b) is Element of the carrier of (L,(L,p,q))
[b9,b] is set
{b9,b} is non empty set
{b9} is non empty set
{{b9,b},{b9}} is non empty set
the L_join of (L,(L,p,q)) . [b9,b] is set
q "\/" a is Element of the carrier of L
the L_join of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
[: the carrier of L, the carrier of L:] is non empty set
[:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
bool [:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
the L_join of L . (q,a) is Element of the carrier of L
[q,a] is set
{q,a} is non empty set
{q} is non empty set
{{q,a},{q}} is non empty set
the L_join of L . [q,a] is set
b "\/" b9 is Element of the carrier of (L,(L,p,q))
the L_join of (L,(L,p,q)) . (b,b9) is Element of the carrier of (L,(L,p,q))
[b,b9] is set
{b,b9} is non empty set
{b} is non empty set
{{b,b9},{b}} is non empty set
the L_join of (L,(L,p,q)) . [b,b9] is set
b is Element of the carrier of (L,(L,p,q))
a is Element of the carrier of L
K is Element of the carrier of (L,(L,p,q))
K "/\" b is Element of the carrier of (L,(L,p,q))
the L_meet of (L,(L,p,q)) is Relation-like [: the carrier of (L,(L,p,q)), the carrier of (L,(L,p,q)):] -defined the carrier of (L,(L,p,q)) -valued Function-like V18([: the carrier of (L,(L,p,q)), the carrier of (L,(L,p,q)):], the carrier of (L,(L,p,q))) commutative associative idempotent Element of bool [:[: the carrier of (L,(L,p,q)), the carrier of (L,(L,p,q)):], the carrier of (L,(L,p,q)):]
the L_meet of (L,(L,p,q)) . (K,b) is Element of the carrier of (L,(L,p,q))
[K,b] is set
{K,b} is non empty set
{K} is non empty set
{{K,b},{K}} is non empty set
the L_meet of (L,(L,p,q)) . [K,b] is set
p "/\" a is Element of the carrier of L
the L_meet of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
the L_meet of L . (p,a) is Element of the carrier of L
[p,a] is set
{p,a} is non empty set
{p} is non empty set
{{p,a},{p}} is non empty set
the L_meet of L . [p,a] is set
b "/\" K is Element of the carrier of (L,(L,p,q))
the L_meet of (L,(L,p,q)) . (b,K) is Element of the carrier of (L,(L,p,q))
[b,K] is set
{b,K} is non empty set
{b} is non empty set
{{b,K},{b}} is non empty set
the L_meet of (L,(L,p,q)) . [b,K] is set
L is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the carrier of L is non empty set
p is Element of the carrier of L
(L,p) is non empty initial meet-closed join-closed Element of bool the carrier of L
bool the carrier of L is non empty set
{ b₁ where b₁ is Element of the carrier of L : b₁ [= p } is set
(L,(L,p)) is non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like upper-bounded SubLattice of L
q is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded upper-bounded bounded LattStr
the carrier of q is non empty set
K is Element of the carrier of q
b9 is Element of the carrier of L
b is Element of the carrier of L
b "\/" b9 is Element of the carrier of L
the L_join of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
[: the carrier of L, the carrier of L:] is non empty set
[:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
bool [:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
the L_join of L . (b,b9) is Element of the carrier of L
[b,b9] is set
{b,b9} is non empty set
{b} is non empty set
{{b,b9},{b}} is non empty set
the L_join of L . [b,b9] is set
Top L is Element of the carrier of L
p "/\" b is Element of the carrier of L
the L_meet of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
the L_meet of L . (p,b) is Element of the carrier of L
[p,b] is set
{p,b} is non empty set
{p} is non empty set
{{p,b},{p}} is non empty set
the L_meet of L . [p,b] is set
a is Element of the carrier of q
a "\/" K is Element of the carrier of q
the L_join of q is Relation-like [: the carrier of q, the carrier of q:] -defined the carrier of q -valued Function-like V18([: the carrier of q, the carrier of q:], the carrier of q) commutative associative idempotent Element of bool [:[: the carrier of q, the carrier of q:], the carrier of q:]
[: the carrier of q, the carrier of q:] is non empty set
[:[: the carrier of q, the carrier of q:], the carrier of q:] is non empty set
bool [:[: the carrier of q, the carrier of q:], the carrier of q:] is non empty set
the L_join of q . (a,K) is Element of the carrier of q
[a,K] is set
{a,K} is non empty set
{a} is non empty set
{{a,K},{a}} is non empty set
the L_join of q . [a,K] is set
(p "/\" b) "\/" b9 is Element of the carrier of L
the L_join of L . ((p "/\" b),b9) is Element of the carrier of L
[(p "/\" b),b9] is set
{(p "/\" b),b9} is non empty set
{(p "/\" b)} is non empty set
{{(p "/\" b),b9},{(p "/\" b)}} is non empty set
the L_join of L . [(p "/\" b),b9] is set
b9 "\/" b is Element of the carrier of L
the L_join of L . (b9,b) is Element of the carrier of L
[b9,b] is set
{b9,b} is non empty set
{b9} is non empty set
{{b9,b},{b9}} is non empty set
the L_join of L . [b9,b] is set
(b9 "\/" b) "/\" p is Element of the carrier of L
the L_meet of L . ((b9 "\/" b),p) is Element of the carrier of L
[(b9 "\/" b),p] is set
{(b9 "\/" b),p} is non empty set
{(b9 "\/" b)} is non empty set
{{(b9 "\/" b),p},{(b9 "\/" b)}} is non empty set
the L_meet of L . [(b9 "\/" b),p] is set
Top q is Element of the carrier of q
K "\/" a is Element of the carrier of q
the L_join of q . (K,a) is Element of the carrier of q
[K,a] is set
{K,a} is non empty set
{K} is non empty set
{{K,a},{K}} is non empty set
the L_join of q . [K,a] is set
a "/\" K is Element of the carrier of q
the L_meet of q is Relation-like [: the carrier of q, the carrier of q:] -defined the carrier of q -valued Function-like V18([: the carrier of q, the carrier of q:], the carrier of q) commutative associative idempotent Element of bool [:[: the carrier of q, the carrier of q:], the carrier of q:]
the L_meet of q . (a,K) is Element of the carrier of q
the L_meet of q . [a,K] is set
Bottom q is Element of the carrier of q
K "/\" a is Element of the carrier of q
the L_meet of q . (K,a) is Element of the carrier of q
the L_meet of q . [K,a] is set
K "\/" a is Element of the carrier of q
b "/\" b9 is Element of the carrier of L
the L_meet of L . (b,b9) is Element of the carrier of L
the L_meet of L . [b,b9] is set
Bottom L is Element of the carrier of L
a "/\" K is Element of the carrier of q
(p "/\" b) "/\" b9 is Element of the carrier of L
the L_meet of L . ((p "/\" b),b9) is Element of the carrier of L
the L_meet of L . [(p "/\" b),b9] is set
p "/\" (Bottom L) is Element of the carrier of L
the L_meet of L . (p,(Bottom L)) is Element of the carrier of L
[p,(Bottom L)] is set
{p,(Bottom L)} is non empty set
{{p,(Bottom L)},{p}} is non empty set
the L_meet of L . [p,(Bottom L)] is set
L is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the carrier of L is non empty set
p is Element of the carrier of L
q is Element of the carrier of L
(L,p,q) is non empty meet-closed join-closed Element of bool the carrier of L
bool the carrier of L is non empty set
(L,(L,p,q)) is non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like SubLattice of L
K is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded upper-bounded bounded LattStr
the carrier of K is non empty set
b9 is Element of the carrier of K
b is Element of the carrier of L
a is Element of the carrier of L
a "/\" b is Element of the carrier of L
the L_meet of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
[: the carrier of L, the carrier of L:] is non empty set
[:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
bool [:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
the L_meet of L . (a,b) is Element of the carrier of L
[a,b] is set
{a,b} is non empty set
{a} is non empty set
{{a,b},{a}} is non empty set
the L_meet of L . [a,b] is set
Bottom L is Element of the carrier of L
a "\/" b is Element of the carrier of L
the L_join of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
the L_join of L . (a,b) is Element of the carrier of L
the L_join of L . [a,b] is set
Top L is Element of the carrier of L
a "/\" q is Element of the carrier of L
the L_meet of L . (a,q) is Element of the carrier of L
[a,q] is set
{a,q} is non empty set
{{a,q},{a}} is non empty set
the L_meet of L . [a,q] is set
p "\/" (a "/\" q) is Element of the carrier of L
the L_join of L . (p,(a "/\" q)) is Element of the carrier of L
[p,(a "/\" q)] is set
{p,(a "/\" q)} is non empty set
{p} is non empty set
{{p,(a "/\" q)},{p}} is non empty set
the L_join of L . [p,(a "/\" q)] is set
a9 is Element of the carrier of K
a9 "\/" b9 is Element of the carrier of K
the L_join of K is Relation-like [: the carrier of K, the carrier of K:] -defined the carrier of K -valued Function-like V18([: the carrier of K, the carrier of K:], the carrier of K) commutative associative idempotent Element of bool [:[: the carrier of K, the carrier of K:], the carrier of K:]
[: the carrier of K, the carrier of K:] is non empty set
[:[: the carrier of K, the carrier of K:], the carrier of K:] is non empty set
bool [:[: the carrier of K, the carrier of K:], the carrier of K:] is non empty set
the L_join of K . (a9,b9) is Element of the carrier of K
[a9,b9] is set
{a9,b9} is non empty set
{a9} is non empty set
{{a9,b9},{a9}} is non empty set
the L_join of K . [a9,b9] is set
(p "\/" (a "/\" q)) "\/" b is Element of the carrier of L
the L_join of L . ((p "\/" (a "/\" q)),b) is Element of the carrier of L
[(p "\/" (a "/\" q)),b] is set
{(p "\/" (a "/\" q)),b} is non empty set
{(p "\/" (a "/\" q))} is non empty set
{{(p "\/" (a "/\" q)),b},{(p "\/" (a "/\" q))}} is non empty set
the L_join of L . [(p "\/" (a "/\" q)),b] is set
b "\/" p is Element of the carrier of L
the L_join of L . (b,p) is Element of the carrier of L
[b,p] is set
{b,p} is non empty set
{b} is non empty set
{{b,p},{b}} is non empty set
the L_join of L . [b,p] is set
(b "\/" p) "\/" (a "/\" q) is Element of the carrier of L
the L_join of L . ((b "\/" p),(a "/\" q)) is Element of the carrier of L
[(b "\/" p),(a "/\" q)] is set
{(b "\/" p),(a "/\" q)} is non empty set
{(b "\/" p)} is non empty set
{{(b "\/" p),(a "/\" q)},{(b "\/" p)}} is non empty set
the L_join of L . [(b "\/" p),(a "/\" q)] is set
b "\/" (a "/\" q) is Element of the carrier of L
the L_join of L . (b,(a "/\" q)) is Element of the carrier of L
[b,(a "/\" q)] is set
{b,(a "/\" q)} is non empty set
{{b,(a "/\" q)},{b}} is non empty set
the L_join of L . [b,(a "/\" q)] is set
(Top L) "/\" q is Element of the carrier of L
the L_meet of L . ((Top L),q) is Element of the carrier of L
[(Top L),q] is set
{(Top L),q} is non empty set
{(Top L)} is non empty set
{{(Top L),q},{(Top L)}} is non empty set
the L_meet of L . [(Top L),q] is set
Top K is Element of the carrier of K
b9 "\/" a9 is Element of the carrier of K
the L_join of K . (b9,a9) is Element of the carrier of K
[b9,a9] is set
{b9,a9} is non empty set
{b9} is non empty set
{{b9,a9},{b9}} is non empty set
the L_join of K . [b9,a9] is set
a9 "/\" b9 is Element of the carrier of K
the L_meet of K is Relation-like [: the carrier of K, the carrier of K:] -defined the carrier of K -valued Function-like V18([: the carrier of K, the carrier of K:], the carrier of K) commutative associative idempotent Element of bool [:[: the carrier of K, the carrier of K:], the carrier of K:]
the L_meet of K . (a9,b9) is Element of the carrier of K
the L_meet of K . [a9,b9] is set
Bottom K is Element of the carrier of K
b9 "/\" a9 is Element of the carrier of K
the L_meet of K . (b9,a9) is Element of the carrier of K
the L_meet of K . [b9,a9] is set
b9 "\/" a9 is Element of the carrier of K
a9 "/\" b9 is Element of the carrier of K
(p "\/" (a "/\" q)) "/\" b is Element of the carrier of L
the L_meet of L . ((p "\/" (a "/\" q)),b) is Element of the carrier of L
the L_meet of L . [(p "\/" (a "/\" q)),b] is set
(a "/\" q) "/\" b is Element of the carrier of L
the L_meet of L . ((a "/\" q),b) is Element of the carrier of L
[(a "/\" q),b] is set
{(a "/\" q),b} is non empty set
{(a "/\" q)} is non empty set
{{(a "/\" q),b},{(a "/\" q)}} is non empty set
the L_meet of L . [(a "/\" q),b] is set
p "\/" ((a "/\" q) "/\" b) is Element of the carrier of L
the L_join of L . (p,((a "/\" q) "/\" b)) is Element of the carrier of L
[p,((a "/\" q) "/\" b)] is set
{p,((a "/\" q) "/\" b)} is non empty set
{{p,((a "/\" q) "/\" b)},{p}} is non empty set
the L_join of L . [p,((a "/\" q) "/\" b)] is set
q "/\" (Bottom L) is Element of the carrier of L
the L_meet of L . (q,(Bottom L)) is Element of the carrier of L
[q,(Bottom L)] is set
{q,(Bottom L)} is non empty set
{q} is non empty set
{{q,(Bottom L)},{q}} is non empty set
the L_meet of L . [q,(Bottom L)] is set
p "\/" (q "/\" (Bottom L)) is Element of the carrier of L
the L_join of L . (p,(q "/\" (Bottom L))) is Element of the carrier of L
[p,(q "/\" (Bottom L))] is set
{p,(q "/\" (Bottom L))} is non empty set
{{p,(q "/\" (Bottom L))},{p}} is non empty set
the L_join of L . [p,(q "/\" (Bottom L))] is set
p "\/" (Bottom L) is Element of the carrier of L
the L_join of L . (p,(Bottom L)) is Element of the carrier of L
[p,(Bottom L)] is set
{p,(Bottom L)} is non empty set
{{p,(Bottom L)},{p}} is non empty set
the L_join of L . [p,(Bottom L)] is set
L is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the carrier of L is non empty set
p is Element of the carrier of L
q is Element of the carrier of L
(L,p,q) is non empty meet-closed join-closed Element of bool the carrier of L
bool the carrier of L is non empty set
(L,(L,p,q)) is non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like SubLattice of L
L is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the carrier of L is non empty set
bool the carrier of L is non empty set
a is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr
the carrier of a is non empty set
bool the carrier of a is non empty set
p is non empty Element of bool the carrier of L
q is Element of the carrier of L
K is Element of the carrier of L
q "\/" K is Element of the carrier of L
the L_join of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like V18([: the carrier of L, the carrier of L:], the carrier of L) commutative associative idempotent Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
[: the carrier of L, the carrier of L:] is non empty set
[:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
bool [:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
the L_join of L . (q,K) is Element of the carrier of L
[q,K] is set
{q,K} is non empty set
{q} is non empty set
{{q,K},{q}} is non empty set
the L_join of L . [q,K] is set
b9 is Element of the carrier of L
b is Element of the carrier of L
b9 "\/" b is Element of the carrier of L
the L_join of L . (b9,b) is Element of the carrier of L
[b9,b] is set
{b9,b} is non empty set
{b9} is non empty set
{{b9,b},{b9}} is non empty set
the L_join of L . [b9,b] is set
a9 is non empty Element of bool the carrier of a