K242() is Element of bool K238()
K238() is set
bool K238() is non empty set
K137() is set
bool K137() is non empty set
K180() is TopStruct
the carrier of K180() is set
bool K242() is non empty set
K318() is set
{} is Relation-like non-empty empty-yielding empty finite finite-yielding V30() set
the Relation-like non-empty empty-yielding empty finite finite-yielding V30() set is Relation-like non-empty empty-yielding empty finite finite-yielding V30() set
{{}} is non empty finite V30() set
K283({{}}) is M25({{}})
[:K283({{}}),{{}}:] is Relation-like set
bool [:K283({{}}),{{}}:] is non empty set
K107(K283({{}}),{{}}) is set
1 is non empty set
{{},1} is non empty finite set
S is non empty V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
T is non empty V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
the carrier of S is non empty set
the carrier of T is non empty set
[: the carrier of S, the carrier of T:] is Relation-like non empty set
bool [: the carrier of S, the carrier of T:] is non empty set
the Relation-like the carrier of S -defined the carrier of T -valued Function-like non empty V22( the carrier of S) quasi_total infs-preserving meet-preserving filtered-infs-preserving monotone Element of bool [: the carrier of S, the carrier of T:] is Relation-like the carrier of S -defined the carrier of T -valued Function-like non empty V22( the carrier of S) quasi_total infs-preserving meet-preserving filtered-infs-preserving monotone Element of bool [: the carrier of S, the carrier of T:]
[: the carrier of T, the carrier of S:] is Relation-like non empty set
bool [: the carrier of T, the carrier of S:] is non empty set
x is Relation-like the carrier of T -defined the carrier of S -valued Function-like non empty V22( the carrier of T) quasi_total Element of bool [: the carrier of T, the carrier of S:]
[ the Relation-like the carrier of S -defined the carrier of T -valued Function-like non empty V22( the carrier of S) quasi_total infs-preserving meet-preserving filtered-infs-preserving monotone Element of bool [: the carrier of S, the carrier of T:],x] is Connection of S,T
{ the Relation-like the carrier of S -defined the carrier of T -valued Function-like non empty V22( the carrier of S) quasi_total infs-preserving meet-preserving filtered-infs-preserving monotone Element of bool [: the carrier of S, the carrier of T:],x} is non empty with_non-empty_elements non empty-membered finite set
{ the Relation-like the carrier of S -defined the carrier of T -valued Function-like non empty V22( the carrier of S) quasi_total infs-preserving meet-preserving filtered-infs-preserving monotone Element of bool [: the carrier of S, the carrier of T:]} is non empty with_non-empty_elements non empty-membered finite set
{{ the Relation-like the carrier of S -defined the carrier of T -valued Function-like non empty V22( the carrier of S) quasi_total infs-preserving meet-preserving filtered-infs-preserving monotone Element of bool [: the carrier of S, the carrier of T:],x},{ the Relation-like the carrier of S -defined the carrier of T -valued Function-like non empty V22( the carrier of S) quasi_total infs-preserving meet-preserving filtered-infs-preserving monotone Element of bool [: the carrier of S, the carrier of T:]}} is non empty with_non-empty_elements non empty-membered finite V30() set
S is non empty RelStr
the carrier of S is non empty set
the InternalRel of S is Relation-like the carrier of S -defined the carrier of S -valued Element of bool [: the carrier of S, the carrier of S:]
[: the carrier of S, the carrier of S:] is Relation-like non empty set
bool [: the carrier of S, the carrier of S:] is non empty set
RelStr(# the carrier of S, the InternalRel of S #) is non empty strict RelStr
g is non empty RelStr
the carrier of g is non empty set
the InternalRel of g is Relation-like the carrier of g -defined the carrier of g -valued Element of bool [: the carrier of g, the carrier of g:]
[: the carrier of g, the carrier of g:] is Relation-like non empty set
bool [: the carrier of g, the carrier of g:] is non empty set
RelStr(# the carrier of g, the InternalRel of g #) is non empty strict RelStr
T is non empty RelStr
the carrier of T is non empty set
the InternalRel of T is Relation-like the carrier of T -defined the carrier of T -valued Element of bool [: the carrier of T, the carrier of T:]
[: the carrier of T, the carrier of T:] is Relation-like non empty set
bool [: the carrier of T, the carrier of T:] is non empty set
RelStr(# the carrier of T, the InternalRel of T #) is non empty strict RelStr
x is non empty RelStr
the carrier of x is non empty set
the InternalRel of x is Relation-like the carrier of x -defined the carrier of x -valued Element of bool [: the carrier of x, the carrier of x:]
[: the carrier of x, the carrier of x:] is Relation-like non empty set
bool [: the carrier of x, the carrier of x:] is non empty set
RelStr(# the carrier of x, the InternalRel of x #) is non empty strict RelStr
x is Connection of S,T
x is Connection of g,x
[: the carrier of S, the carrier of T:] is Relation-like non empty set
bool [: the carrier of S, the carrier of T:] is non empty set
[: the carrier of T, the carrier of S:] is Relation-like non empty set
bool [: the carrier of T, the carrier of S:] is non empty set
x is Relation-like the carrier of S -defined the carrier of T -valued Function-like non empty V22( the carrier of S) quasi_total Element of bool [: the carrier of S, the carrier of T:]
y is Relation-like the carrier of T -defined the carrier of S -valued Function-like non empty V22( the carrier of T) quasi_total Element of bool [: the carrier of T, the carrier of S:]
[x,y] is Connection of S,T
{x,y} is non empty with_non-empty_elements non empty-membered finite set
{x} is non empty with_non-empty_elements non empty-membered finite set
{{x,y},{x}} is non empty with_non-empty_elements non empty-membered finite V30() set
[: the carrier of g, the carrier of x:] is Relation-like non empty set
bool [: the carrier of g, the carrier of x:] is non empty set
[: the carrier of x, the carrier of g:] is Relation-like non empty set
bool [: the carrier of x, the carrier of g:] is non empty set
a is Relation-like the carrier of g -defined the carrier of x -valued Function-like non empty V22( the carrier of g) quasi_total Element of bool [: the carrier of g, the carrier of x:]
b is Relation-like the carrier of x -defined the carrier of g -valued Function-like non empty V22( the carrier of x) quasi_total Element of bool [: the carrier of x, the carrier of g:]
[a,b] is Connection of g,x
{a,b} is non empty with_non-empty_elements non empty-membered finite set
{a} is non empty with_non-empty_elements non empty-membered finite set
{{a,b},{a}} is non empty with_non-empty_elements non empty-membered finite V30() set
a is Element of the carrier of x
b . a is Element of the carrier of g
b2 is Element of the carrier of g
a . b2 is Element of the carrier of x
f is Element of the carrier of T
g is Element of the carrier of S
x . g is Element of the carrier of T
y . f is Element of the carrier of S
S is non empty V87() reflexive transitive antisymmetric with_suprema with_infima RelStr
the carrier of S is non empty set
T is non empty V87() reflexive transitive antisymmetric with_suprema with_infima RelStr
the carrier of T is non empty set
[: the carrier of S, the carrier of T:] is Relation-like non empty set
bool [: the carrier of S, the carrier of T:] is non empty set
g is Relation-like the carrier of S -defined the carrier of T -valued Function-like non empty V22( the carrier of S) quasi_total Element of bool [: the carrier of S, the carrier of T:]
[: the carrier of T, the carrier of S:] is Relation-like non empty set
bool [: the carrier of T, the carrier of S:] is non empty set
x is Relation-like the carrier of T -defined the carrier of S -valued Function-like non empty V22( the carrier of T) quasi_total Element of bool [: the carrier of T, the carrier of S:]
[g,x] is Connection of S,T
{g,x} is non empty with_non-empty_elements non empty-membered finite set
{g} is non empty with_non-empty_elements non empty-membered finite set
{{g,x},{g}} is non empty with_non-empty_elements non empty-membered finite V30() set
x is Relation-like the carrier of T -defined the carrier of S -valued Function-like non empty V22( the carrier of T) quasi_total Element of bool [: the carrier of T, the carrier of S:]
[g,x] is Connection of S,T
{g,x} is non empty with_non-empty_elements non empty-membered finite set
{{g,x},{g}} is non empty with_non-empty_elements non empty-membered finite V30() set
x is Element of the carrier of T
x is Element of the carrier of T
x . x is Element of the carrier of S
uparrow x is non empty filtered upper Element of bool the carrier of T
bool the carrier of T is non empty set
{x} is non empty finite Element of bool the carrier of T
uparrow {x} is non empty upper Element of bool the carrier of T
g " (uparrow x) is Element of bool the carrier of S
bool the carrier of S is non empty set
x . x is Element of the carrier of S
x . x is Element of the carrier of S
"/\" ((g " (uparrow x)),S) is Element of the carrier of S
x . x is Element of the carrier of S
T is non empty V87() reflexive transitive antisymmetric with_suprema with_infima RelStr
the carrier of T is non empty set
S is non empty V87() reflexive transitive antisymmetric with_suprema with_infima RelStr
the carrier of S is non empty set
[: the carrier of T, the carrier of S:] is Relation-like non empty set
bool [: the carrier of T, the carrier of S:] is non empty set
g is Relation-like the carrier of T -defined the carrier of S -valued Function-like non empty V22( the carrier of T) quasi_total Element of bool [: the carrier of T, the carrier of S:]
[: the carrier of S, the carrier of T:] is Relation-like non empty set
bool [: the carrier of S, the carrier of T:] is non empty set
x is Relation-like the carrier of S -defined the carrier of T -valued Function-like non empty V22( the carrier of S) quasi_total Element of bool [: the carrier of S, the carrier of T:]
[x,g] is Connection of S,T
{x,g} is non empty with_non-empty_elements non empty-membered finite set
{x} is non empty with_non-empty_elements non empty-membered finite set
{{x,g},{x}} is non empty with_non-empty_elements non empty-membered finite V30() set
x is Relation-like the carrier of S -defined the carrier of T -valued Function-like non empty V22( the carrier of S) quasi_total Element of bool [: the carrier of S, the carrier of T:]
[x,g] is Connection of S,T
{x,g} is non empty with_non-empty_elements non empty-membered finite set
{x} is non empty with_non-empty_elements non empty-membered finite set
{{x,g},{x}} is non empty with_non-empty_elements non empty-membered finite V30() set
x is Element of the carrier of S
x is Element of the carrier of S
x . x is Element of the carrier of T
downarrow x is non empty directed lower Element of bool the carrier of S
bool the carrier of S is non empty set
{x} is non empty finite Element of bool the carrier of S
downarrow {x} is non empty lower Element of bool the carrier of S
g " (downarrow x) is Element of bool the carrier of T
bool the carrier of T is non empty set
x . x is Element of the carrier of T
x . x is Element of the carrier of T
"\/" ((g " (downarrow x)),T) is Element of the carrier of T
x . x is Element of the carrier of T
S is non empty V87() reflexive transitive antisymmetric with_suprema with_infima RelStr
T is non empty V87() reflexive transitive antisymmetric with_suprema with_infima RelStr
the carrier of S is non empty set
the carrier of T is non empty set
[: the carrier of S, the carrier of T:] is Relation-like non empty set
bool [: the carrier of S, the carrier of T:] is non empty set
x is Relation-like the carrier of S -defined the carrier of T -valued Function-like non empty V22( the carrier of S) quasi_total Element of bool [: the carrier of S, the carrier of T:]
g is non empty V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
the carrier of g is non empty set
x is non empty V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
the carrier of x is non empty set
[: the carrier of g, the carrier of x:] is Relation-like non empty set
bool [: the carrier of g, the carrier of x:] is non empty set
x is Relation-like the carrier of g -defined the carrier of x -valued Function-like non empty V22( the carrier of g) quasi_total infs-preserving meet-preserving filtered-infs-preserving monotone Element of bool [: the carrier of g, the carrier of x:]
(g,x,x) is Relation-like the carrier of x -defined the carrier of g -valued Function-like non empty V22( the carrier of x) quasi_total Element of bool [: the carrier of x, the carrier of g:]
[: the carrier of x, the carrier of g:] is Relation-like non empty set
bool [: the carrier of x, the carrier of g:] is non empty set
[x,(g,x,x)] is Connection of g,x
{x,(g,x,x)} is non empty with_non-empty_elements non empty-membered finite set
{x} is non empty with_non-empty_elements non empty-membered finite set
{{x,(g,x,x)},{x}} is non empty with_non-empty_elements non empty-membered finite V30() set
(S,T,x) is Relation-like the carrier of T -defined the carrier of S -valued Function-like non empty V22( the carrier of T) quasi_total Element of bool [: the carrier of T, the carrier of S:]
[: the carrier of T, the carrier of S:] is Relation-like non empty set
bool [: the carrier of T, the carrier of S:] is non empty set
S is non empty V87() reflexive transitive antisymmetric with_suprema with_infima RelStr
T is non empty V87() reflexive transitive antisymmetric with_suprema with_infima RelStr
the carrier of S is non empty set
the carrier of T is non empty set
[: the carrier of S, the carrier of T:] is Relation-like non empty set
bool [: the carrier of S, the carrier of T:] is non empty set
x is Relation-like the carrier of S -defined the carrier of T -valued Function-like non empty V22( the carrier of S) quasi_total Element of bool [: the carrier of S, the carrier of T:]
g is non empty V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
the carrier of g is non empty set
x is non empty V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
the carrier of x is non empty set
[: the carrier of g, the carrier of x:] is Relation-like non empty set
bool [: the carrier of g, the carrier of x:] is non empty set
x is Relation-like the carrier of g -defined the carrier of x -valued Function-like non empty V22( the carrier of g) quasi_total sups-preserving join-preserving directed-sups-preserving monotone Element of bool [: the carrier of g, the carrier of x:]
(x,g,x) is Relation-like the carrier of x -defined the carrier of g -valued Function-like non empty V22( the carrier of x) quasi_total Element of bool [: the carrier of x, the carrier of g:]
[: the carrier of x, the carrier of g:] is Relation-like non empty set
bool [: the carrier of x, the carrier of g:] is non empty set
[(x,g,x),x] is Connection of x,g
{(x,g,x),x} is non empty with_non-empty_elements non empty-membered finite set
{(x,g,x)} is non empty with_non-empty_elements non empty-membered finite set
{{(x,g,x),x},{(x,g,x)}} is non empty with_non-empty_elements non empty-membered finite V30() set
(T,S,x) is Relation-like the carrier of T -defined the carrier of S -valued Function-like non empty V22( the carrier of T) quasi_total Element of bool [: the carrier of T, the carrier of S:]
[: the carrier of T, the carrier of S:] is Relation-like non empty set
bool [: the carrier of T, the carrier of S:] is non empty set
S is non empty V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
the carrier of S is non empty set
T is non empty V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
the carrier of T is non empty set
[: the carrier of S, the carrier of T:] is Relation-like non empty set
bool [: the carrier of S, the carrier of T:] is non empty set
g is Relation-like the carrier of S -defined the carrier of T -valued Function-like non empty V22( the carrier of S) quasi_total infs-preserving meet-preserving filtered-infs-preserving monotone Element of bool [: the carrier of S, the carrier of T:]
(S,T,g) is Relation-like the carrier of T -defined the carrier of S -valued Function-like non empty V22( the carrier of T) quasi_total Element of bool [: the carrier of T, the carrier of S:]
[: the carrier of T, the carrier of S:] is Relation-like non empty set
bool [: the carrier of T, the carrier of S:] is non empty set
T is non empty V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
the carrier of T is non empty set
S is non empty V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
the carrier of S is non empty set
[: the carrier of T, the carrier of S:] is Relation-like non empty set
bool [: the carrier of T, the carrier of S:] is non empty set
g is Relation-like the carrier of T -defined the carrier of S -valued Function-like non empty V22( the carrier of T) quasi_total sups-preserving join-preserving directed-sups-preserving monotone Element of bool [: the carrier of T, the carrier of S:]
(S,T,g) is Relation-like the carrier of S -defined the carrier of T -valued Function-like non empty V22( the carrier of S) quasi_total Element of bool [: the carrier of S, the carrier of T:]
[: the carrier of S, the carrier of T:] is Relation-like non empty set
bool [: the carrier of S, the carrier of T:] is non empty set
S is non empty V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
the carrier of S is non empty set
T is non empty V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
the carrier of T is non empty set
[: the carrier of S, the carrier of T:] is Relation-like non empty set
bool [: the carrier of S, the carrier of T:] is non empty set
g is Relation-like the carrier of S -defined the carrier of T -valued Function-like non empty V22( the carrier of S) quasi_total infs-preserving meet-preserving filtered-infs-preserving monotone Element of bool [: the carrier of S, the carrier of T:]
(S,T,g) is Relation-like the carrier of T -defined the carrier of S -valued Function-like non empty V22( the carrier of T) quasi_total sups-preserving join-preserving directed-sups-preserving monotone lower_adjoint Element of bool [: the carrier of T, the carrier of S:]
[: the carrier of T, the carrier of S:] is Relation-like non empty set
bool [: the carrier of T, the carrier of S:] is non empty set
x is Element of the carrier of T
(S,T,g) . x is Element of the carrier of S
uparrow x is non empty filtered upper Element of bool the carrier of T
bool the carrier of T is non empty set
{x} is non empty finite Element of bool the carrier of T
uparrow {x} is non empty upper Element of bool the carrier of T
g " (uparrow x) is Element of bool the carrier of S
bool the carrier of S is non empty set
"/\" ((g " (uparrow x)),S) is Element of the carrier of S
[g,(S,T,g)] is Connection of S,T
{g,(S,T,g)} is non empty with_non-empty_elements non empty-membered finite set
{g} is non empty with_non-empty_elements non empty-membered finite set
{{g,(S,T,g)},{g}} is non empty with_non-empty_elements non empty-membered finite V30() set
T is non empty V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
the carrier of T is non empty set
S is non empty V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
the carrier of S is non empty set
[: the carrier of T, the carrier of S:] is Relation-like non empty set
bool [: the carrier of T, the carrier of S:] is non empty set
g is Relation-like the carrier of T -defined the carrier of S -valued Function-like non empty V22( the carrier of T) quasi_total sups-preserving join-preserving directed-sups-preserving monotone Element of bool [: the carrier of T, the carrier of S:]
(S,T,g) is Relation-like the carrier of S -defined the carrier of T -valued Function-like non empty V22( the carrier of S) quasi_total infs-preserving meet-preserving filtered-infs-preserving monotone upper_adjoint Element of bool [: the carrier of S, the carrier of T:]
[: the carrier of S, the carrier of T:] is Relation-like non empty set
bool [: the carrier of S, the carrier of T:] is non empty set
x is Element of the carrier of S
(S,T,g) . x is Element of the carrier of T
downarrow x is non empty directed lower Element of bool the carrier of S
bool the carrier of S is non empty set
{x} is non empty finite Element of bool the carrier of S
downarrow {x} is non empty lower Element of bool the carrier of S
g " (downarrow x) is Element of bool the carrier of T
bool the carrier of T is non empty set
"\/" ((g " (downarrow x)),T) is Element of the carrier of T
[(S,T,g),g] is Connection of S,T
{(S,T,g),g} is non empty with_non-empty_elements non empty-membered finite set
{(S,T,g)} is non empty with_non-empty_elements non empty-membered finite set
{{(S,T,g),g},{(S,T,g)}} is non empty with_non-empty_elements non empty-membered finite V30() set
S is RelStr
the carrier of S is set
T is RelStr
the carrier of T is set
[: the carrier of S, the carrier of T:] is Relation-like set
bool [: the carrier of S, the carrier of T:] is non empty set
g is Relation-like the carrier of S -defined the carrier of T -valued Function-like quasi_total Element of bool [: the carrier of S, the carrier of T:]
S ~ is strict RelStr
the InternalRel of S is Relation-like the carrier of S -defined the carrier of S -valued Element of bool [: the carrier of S, the carrier of S:]
[: the carrier of S, the carrier of S:] is Relation-like set
bool [: the carrier of S, the carrier of S:] is non empty set
the InternalRel of S ~ is Relation-like the carrier of S -defined the carrier of S -valued Element of bool [: the carrier of S, the carrier of S:]
RelStr(# the carrier of S,( the InternalRel of S ~) #) is strict RelStr
the carrier of (S ~) is set
T ~ is strict RelStr
the InternalRel of T is Relation-like the carrier of T -defined the carrier of T -valued Element of bool [: the carrier of T, the carrier of T:]
[: the carrier of T, the carrier of T:] is Relation-like set
bool [: the carrier of T, the carrier of T:] is non empty set
the InternalRel of T ~ is Relation-like the carrier of T -defined the carrier of T -valued Element of bool [: the carrier of T, the carrier of T:]
RelStr(# the carrier of T,( the InternalRel of T ~) #) is strict RelStr
the carrier of (T ~) is set
[: the carrier of (S ~), the carrier of (T ~):] is Relation-like set
bool [: the carrier of (S ~), the carrier of (T ~):] is non empty set
S is non empty V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
the carrier of S is non empty set
T is non empty V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
the carrier of T is non empty set
[: the carrier of S, the carrier of T:] is Relation-like non empty set
bool [: the carrier of S, the carrier of T:] is non empty set
T ~ is non empty strict V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
the InternalRel of T is Relation-like the carrier of T -defined the carrier of T -valued V22( the carrier of T) quasi_total V31() V34() V38() Element of bool [: the carrier of T, the carrier of T:]
[: the carrier of T, the carrier of T:] is Relation-like non empty set
bool [: the carrier of T, the carrier of T:] is non empty set
the InternalRel of T ~ is Relation-like the carrier of T -defined the carrier of T -valued Element of bool [: the carrier of T, the carrier of T:]
RelStr(# the carrier of T,( the InternalRel of T ~) #) is non empty strict RelStr
S ~ is non empty strict V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
the InternalRel of S is Relation-like the carrier of S -defined the carrier of S -valued V22( the carrier of S) quasi_total V31() V34() V38() Element of bool [: the carrier of S, the carrier of S:]
[: the carrier of S, the carrier of S:] is Relation-like non empty set
bool [: the carrier of S, the carrier of S:] is non empty set
the InternalRel of S ~ is Relation-like the carrier of S -defined the carrier of S -valued Element of bool [: the carrier of S, the carrier of S:]
RelStr(# the carrier of S,( the InternalRel of S ~) #) is non empty strict RelStr
g is Relation-like the carrier of S -defined the carrier of T -valued Function-like non empty V22( the carrier of S) quasi_total infs-preserving meet-preserving filtered-infs-preserving monotone Element of bool [: the carrier of S, the carrier of T:]
(S,T,g) is Relation-like the carrier of (S ~) -defined the carrier of (T ~) -valued Function-like non empty V22( the carrier of (S ~)) quasi_total Element of bool [: the carrier of (S ~), the carrier of (T ~):]
the carrier of (S ~) is non empty set
the carrier of (T ~) is non empty set
[: the carrier of (S ~), the carrier of (T ~):] is Relation-like non empty set
bool [: the carrier of (S ~), the carrier of (T ~):] is non empty set
(S,T,g) is Relation-like the carrier of T -defined the carrier of S -valued Function-like non empty V22( the carrier of T) quasi_total sups-preserving join-preserving directed-sups-preserving monotone lower_adjoint Element of bool [: the carrier of T, the carrier of S:]
[: the carrier of T, the carrier of S:] is Relation-like non empty set
bool [: the carrier of T, the carrier of S:] is non empty set
[g,(S,T,g)] is Connection of S,T
{g,(S,T,g)} is non empty with_non-empty_elements non empty-membered finite set
{g} is non empty with_non-empty_elements non empty-membered finite set
{{g,(S,T,g)},{g}} is non empty with_non-empty_elements non empty-membered finite V30() set
(T,S,(S,T,g)) is Relation-like the carrier of (T ~) -defined the carrier of (S ~) -valued Function-like non empty V22( the carrier of (T ~)) quasi_total Element of bool [: the carrier of (T ~), the carrier of (S ~):]
[: the carrier of (T ~), the carrier of (S ~):] is Relation-like non empty set
bool [: the carrier of (T ~), the carrier of (S ~):] is non empty set
[(T,S,(S,T,g)),(S,T,g)] is Connection of T ~ ,S ~
{(T,S,(S,T,g)),(S,T,g)} is non empty with_non-empty_elements non empty-membered finite set
{(T,S,(S,T,g))} is non empty with_non-empty_elements non empty-membered finite set
{{(T,S,(S,T,g)),(S,T,g)},{(T,S,(S,T,g))}} is non empty with_non-empty_elements non empty-membered finite V30() set
S is non empty V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
the carrier of S is non empty set
T is non empty V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
the carrier of T is non empty set
[: the carrier of S, the carrier of T:] is Relation-like non empty set
bool [: the carrier of S, the carrier of T:] is non empty set
S ~ is non empty strict V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
the InternalRel of S is Relation-like the carrier of S -defined the carrier of S -valued V22( the carrier of S) quasi_total V31() V34() V38() Element of bool [: the carrier of S, the carrier of S:]
[: the carrier of S, the carrier of S:] is Relation-like non empty set
bool [: the carrier of S, the carrier of S:] is non empty set
the InternalRel of S ~ is Relation-like the carrier of S -defined the carrier of S -valued Element of bool [: the carrier of S, the carrier of S:]
RelStr(# the carrier of S,( the InternalRel of S ~) #) is non empty strict RelStr
T ~ is non empty strict V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
the InternalRel of T is Relation-like the carrier of T -defined the carrier of T -valued V22( the carrier of T) quasi_total V31() V34() V38() Element of bool [: the carrier of T, the carrier of T:]
[: the carrier of T, the carrier of T:] is Relation-like non empty set
bool [: the carrier of T, the carrier of T:] is non empty set
the InternalRel of T ~ is Relation-like the carrier of T -defined the carrier of T -valued Element of bool [: the carrier of T, the carrier of T:]
RelStr(# the carrier of T,( the InternalRel of T ~) #) is non empty strict RelStr
g is Relation-like the carrier of S -defined the carrier of T -valued Function-like non empty V22( the carrier of S) quasi_total sups-preserving join-preserving directed-sups-preserving monotone Element of bool [: the carrier of S, the carrier of T:]
(S,T,g) is Relation-like the carrier of (S ~) -defined the carrier of (T ~) -valued Function-like non empty V22( the carrier of (S ~)) quasi_total Element of bool [: the carrier of (S ~), the carrier of (T ~):]
the carrier of (S ~) is non empty set
the carrier of (T ~) is non empty set
[: the carrier of (S ~), the carrier of (T ~):] is Relation-like non empty set
bool [: the carrier of (S ~), the carrier of (T ~):] is non empty set
(T,S,g) is Relation-like the carrier of T -defined the carrier of S -valued Function-like non empty V22( the carrier of T) quasi_total infs-preserving meet-preserving filtered-infs-preserving monotone upper_adjoint Element of bool [: the carrier of T, the carrier of S:]
[: the carrier of T, the carrier of S:] is Relation-like non empty set
bool [: the carrier of T, the carrier of S:] is non empty set
[(T,S,g),g] is Connection of T,S
{(T,S,g),g} is non empty with_non-empty_elements non empty-membered finite set
{(T,S,g)} is non empty with_non-empty_elements non empty-membered finite set
{{(T,S,g),g},{(T,S,g)}} is non empty with_non-empty_elements non empty-membered finite V30() set
(T,S,(T,S,g)) is Relation-like the carrier of (T ~) -defined the carrier of (S ~) -valued Function-like non empty V22( the carrier of (T ~)) quasi_total sups-preserving join-preserving directed-sups-preserving monotone lower_adjoint Element of bool [: the carrier of (T ~), the carrier of (S ~):]
[: the carrier of (T ~), the carrier of (S ~):] is Relation-like non empty set
bool [: the carrier of (T ~), the carrier of (S ~):] is non empty set
[(S,T,g),(T,S,(T,S,g))] is Connection of S ~ ,T ~
{(S,T,g),(T,S,(T,S,g))} is non empty with_non-empty_elements non empty-membered finite set
{(S,T,g)} is non empty with_non-empty_elements non empty-membered finite set
{{(S,T,g),(T,S,(T,S,g))},{(S,T,g)}} is non empty with_non-empty_elements non empty-membered finite V30() set
S is non empty V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
the carrier of S is non empty set
T is non empty V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
the carrier of T is non empty set
[: the carrier of S, the carrier of T:] is Relation-like non empty set
bool [: the carrier of S, the carrier of T:] is non empty set
T ~ is non empty strict V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
the InternalRel of T is Relation-like the carrier of T -defined the carrier of T -valued V22( the carrier of T) quasi_total V31() V34() V38() Element of bool [: the carrier of T, the carrier of T:]
[: the carrier of T, the carrier of T:] is Relation-like non empty set
bool [: the carrier of T, the carrier of T:] is non empty set
the InternalRel of T ~ is Relation-like the carrier of T -defined the carrier of T -valued Element of bool [: the carrier of T, the carrier of T:]
RelStr(# the carrier of T,( the InternalRel of T ~) #) is non empty strict RelStr
the carrier of (T ~) is non empty set
S ~ is non empty strict V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
the InternalRel of S is Relation-like the carrier of S -defined the carrier of S -valued V22( the carrier of S) quasi_total V31() V34() V38() Element of bool [: the carrier of S, the carrier of S:]
[: the carrier of S, the carrier of S:] is Relation-like non empty set
bool [: the carrier of S, the carrier of S:] is non empty set
the InternalRel of S ~ is Relation-like the carrier of S -defined the carrier of S -valued Element of bool [: the carrier of S, the carrier of S:]
RelStr(# the carrier of S,( the InternalRel of S ~) #) is non empty strict RelStr
the carrier of (S ~) is non empty set
g is Relation-like the carrier of S -defined the carrier of T -valued Function-like non empty V22( the carrier of S) quasi_total infs-preserving meet-preserving filtered-infs-preserving monotone Element of bool [: the carrier of S, the carrier of T:]
(S,T,g) is Relation-like the carrier of T -defined the carrier of S -valued Function-like non empty V22( the carrier of T) quasi_total sups-preserving join-preserving directed-sups-preserving monotone lower_adjoint Element of bool [: the carrier of T, the carrier of S:]
[: the carrier of T, the carrier of S:] is Relation-like non empty set
bool [: the carrier of T, the carrier of S:] is non empty set
(S,T,g) is Relation-like the carrier of (S ~) -defined the carrier of (T ~) -valued Function-like non empty V22( the carrier of (S ~)) quasi_total sups-preserving join-preserving directed-sups-preserving monotone lower_adjoint Element of bool [: the carrier of (S ~), the carrier of (T ~):]
[: the carrier of (S ~), the carrier of (T ~):] is Relation-like non empty set
bool [: the carrier of (S ~), the carrier of (T ~):] is non empty set
((T ~),(S ~),(S,T,g)) is Relation-like the carrier of (T ~) -defined the carrier of (S ~) -valued Function-like non empty V22( the carrier of (T ~)) quasi_total infs-preserving meet-preserving filtered-infs-preserving monotone upper_adjoint Element of bool [: the carrier of (T ~), the carrier of (S ~):]
[: the carrier of (T ~), the carrier of (S ~):] is Relation-like non empty set
bool [: the carrier of (T ~), the carrier of (S ~):] is non empty set
[g,(S,T,g)] is Connection of S,T
{g,(S,T,g)} is non empty with_non-empty_elements non empty-membered finite set
{g} is non empty with_non-empty_elements non empty-membered finite set
{{g,(S,T,g)},{g}} is non empty with_non-empty_elements non empty-membered finite V30() set
(T,S,(S,T,g)) is Relation-like the carrier of (T ~) -defined the carrier of (S ~) -valued Function-like non empty V22( the carrier of (T ~)) quasi_total infs-preserving meet-preserving filtered-infs-preserving monotone upper_adjoint Element of bool [: the carrier of (T ~), the carrier of (S ~):]
[(T,S,(S,T,g)),(S,T,g)] is Connection of T ~ ,S ~
{(T,S,(S,T,g)),(S,T,g)} is non empty with_non-empty_elements non empty-membered finite set
{(T,S,(S,T,g))} is non empty with_non-empty_elements non empty-membered finite set
{{(T,S,(S,T,g)),(S,T,g)},{(T,S,(S,T,g))}} is non empty with_non-empty_elements non empty-membered finite V30() set
S is non empty V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
the carrier of S is non empty set
T is non empty V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
the carrier of T is non empty set
[: the carrier of S, the carrier of T:] is Relation-like non empty set
bool [: the carrier of S, the carrier of T:] is non empty set
T ~ is non empty strict V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
the InternalRel of T is Relation-like the carrier of T -defined the carrier of T -valued V22( the carrier of T) quasi_total V31() V34() V38() Element of bool [: the carrier of T, the carrier of T:]
[: the carrier of T, the carrier of T:] is Relation-like non empty set
bool [: the carrier of T, the carrier of T:] is non empty set
the InternalRel of T ~ is Relation-like the carrier of T -defined the carrier of T -valued Element of bool [: the carrier of T, the carrier of T:]
RelStr(# the carrier of T,( the InternalRel of T ~) #) is non empty strict RelStr
the carrier of (T ~) is non empty set
S ~ is non empty strict V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
the InternalRel of S is Relation-like the carrier of S -defined the carrier of S -valued V22( the carrier of S) quasi_total V31() V34() V38() Element of bool [: the carrier of S, the carrier of S:]
[: the carrier of S, the carrier of S:] is Relation-like non empty set
bool [: the carrier of S, the carrier of S:] is non empty set
the InternalRel of S ~ is Relation-like the carrier of S -defined the carrier of S -valued Element of bool [: the carrier of S, the carrier of S:]
RelStr(# the carrier of S,( the InternalRel of S ~) #) is non empty strict RelStr
the carrier of (S ~) is non empty set
g is Relation-like the carrier of S -defined the carrier of T -valued Function-like non empty V22( the carrier of S) quasi_total sups-preserving join-preserving directed-sups-preserving monotone Element of bool [: the carrier of S, the carrier of T:]
(S,T,g) is Relation-like the carrier of (S ~) -defined the carrier of (T ~) -valued Function-like non empty V22( the carrier of (S ~)) quasi_total infs-preserving meet-preserving filtered-infs-preserving monotone upper_adjoint Element of bool [: the carrier of (S ~), the carrier of (T ~):]
[: the carrier of (S ~), the carrier of (T ~):] is Relation-like non empty set
bool [: the carrier of (S ~), the carrier of (T ~):] is non empty set
((S ~),(T ~),(S,T,g)) is Relation-like the carrier of (T ~) -defined the carrier of (S ~) -valued Function-like non empty V22( the carrier of (T ~)) quasi_total sups-preserving join-preserving directed-sups-preserving monotone lower_adjoint Element of bool [: the carrier of (T ~), the carrier of (S ~):]
[: the carrier of (T ~), the carrier of (S ~):] is Relation-like non empty set
bool [: the carrier of (T ~), the carrier of (S ~):] is non empty set
(T,S,g) is Relation-like the carrier of T -defined the carrier of S -valued Function-like non empty V22( the carrier of T) quasi_total infs-preserving meet-preserving filtered-infs-preserving monotone upper_adjoint Element of bool [: the carrier of T, the carrier of S:]
[: the carrier of T, the carrier of S:] is Relation-like non empty set
bool [: the carrier of T, the carrier of S:] is non empty set
[(T,S,g),g] is Connection of T,S
{(T,S,g),g} is non empty with_non-empty_elements non empty-membered finite set
{(T,S,g)} is non empty with_non-empty_elements non empty-membered finite set
{{(T,S,g),g},{(T,S,g)}} is non empty with_non-empty_elements non empty-membered finite V30() set
(T,S,(T,S,g)) is Relation-like the carrier of (T ~) -defined the carrier of (S ~) -valued Function-like non empty V22( the carrier of (T ~)) quasi_total sups-preserving join-preserving directed-sups-preserving monotone lower_adjoint Element of bool [: the carrier of (T ~), the carrier of (S ~):]
[(S,T,g),(T,S,(T,S,g))] is Connection of S ~ ,T ~
{(S,T,g),(T,S,(T,S,g))} is non empty with_non-empty_elements non empty-membered finite set
{(S,T,g)} is non empty with_non-empty_elements non empty-membered finite set
{{(S,T,g),(T,S,(T,S,g))},{(S,T,g)}} is non empty with_non-empty_elements non empty-membered finite V30() set
S is non empty RelStr
id S is Relation-like the carrier of S -defined the carrier of S -valued Function-like one-to-one non empty V22( the carrier of S) quasi_total isomorphic monotone projection Element of bool [: the carrier of S, the carrier of S:]
the carrier of S is non empty set
[: the carrier of S, the carrier of S:] is Relation-like non empty set
bool [: the carrier of S, the carrier of S:] is non empty set
id the carrier of S is Relation-like the carrier of S -defined the carrier of S -valued non empty V22( the carrier of S) quasi_total Element of bool [: the carrier of S, the carrier of S:]
[(id S),(id S)] is Connection of S,S
{(id S),(id S)} is non empty with_non-empty_elements non empty-membered finite set
{(id S)} is non empty with_non-empty_elements non empty-membered finite set
{{(id S),(id S)},{(id S)}} is non empty with_non-empty_elements non empty-membered finite V30() set
T is Relation-like the carrier of S -defined the carrier of S -valued Function-like one-to-one non empty V22( the carrier of S) quasi_total isomorphic monotone projection Element of bool [: the carrier of S, the carrier of S:]
g is Relation-like the carrier of S -defined the carrier of S -valued Function-like one-to-one non empty V22( the carrier of S) quasi_total isomorphic monotone projection Element of bool [: the carrier of S, the carrier of S:]
[T,g] is Connection of S,S
{T,g} is non empty with_non-empty_elements non empty-membered finite set
{T} is non empty with_non-empty_elements non empty-membered finite set
{{T,g},{T}} is non empty with_non-empty_elements non empty-membered finite V30() set
x is Element of the carrier of S
x is Element of the carrier of S
T . x is Element of the carrier of S
g . x is Element of the carrier of S
S is non empty V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
the carrier of S is non empty set
id S is Relation-like the carrier of S -defined the carrier of S -valued Function-like one-to-one non empty V22( the carrier of S) quasi_total infs-preserving sups-preserving meet-preserving join-preserving filtered-infs-preserving directed-sups-preserving isomorphic monotone projection closure kernel Element of bool [: the carrier of S, the carrier of S:]
[: the carrier of S, the carrier of S:] is Relation-like non empty set
bool [: the carrier of S, the carrier of S:] is non empty set
id the carrier of S is Relation-like the carrier of S -defined the carrier of S -valued non empty V22( the carrier of S) quasi_total Element of bool [: the carrier of S, the carrier of S:]
(S,S,(id S)) is Relation-like the carrier of S -defined the carrier of S -valued Function-like non empty V22( the carrier of S) quasi_total sups-preserving join-preserving directed-sups-preserving monotone lower_adjoint Element of bool [: the carrier of S, the carrier of S:]
(S,S,(id S)) is Relation-like the carrier of S -defined the carrier of S -valued Function-like non empty V22( the carrier of S) quasi_total infs-preserving meet-preserving filtered-infs-preserving monotone upper_adjoint Element of bool [: the carrier of S, the carrier of S:]
[(id S),(id S)] is Connection of S,S
{(id S),(id S)} is non empty with_non-empty_elements non empty-membered finite set
{(id S)} is non empty with_non-empty_elements non empty-membered finite set
{{(id S),(id S)},{(id S)}} is non empty with_non-empty_elements non empty-membered finite V30() set
S is non empty V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
the carrier of S is non empty set
T is non empty V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
the carrier of T is non empty set
[: the carrier of S, the carrier of T:] is Relation-like non empty set
bool [: the carrier of S, the carrier of T:] is non empty set
g is non empty V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
the carrier of g is non empty set
[: the carrier of T, the carrier of g:] is Relation-like non empty set
bool [: the carrier of T, the carrier of g:] is non empty set
x is Relation-like the carrier of S -defined the carrier of T -valued Function-like non empty V22( the carrier of S) quasi_total infs-preserving meet-preserving filtered-infs-preserving monotone Element of bool [: the carrier of S, the carrier of T:]
(S,T,x) is Relation-like the carrier of T -defined the carrier of S -valued Function-like non empty V22( the carrier of T) quasi_total sups-preserving join-preserving directed-sups-preserving monotone lower_adjoint Element of bool [: the carrier of T, the carrier of S:]
[: the carrier of T, the carrier of S:] is Relation-like non empty set
bool [: the carrier of T, the carrier of S:] is non empty set
x is Relation-like the carrier of T -defined the carrier of g -valued Function-like non empty V22( the carrier of T) quasi_total infs-preserving meet-preserving filtered-infs-preserving monotone Element of bool [: the carrier of T, the carrier of g:]
x * x is Relation-like the carrier of S -defined the carrier of g -valued Function-like non empty V22( the carrier of S) quasi_total Element of bool [: the carrier of S, the carrier of g:]
[: the carrier of S, the carrier of g:] is Relation-like non empty set
bool [: the carrier of S, the carrier of g:] is non empty set
(S,g,(x * x)) is Relation-like the carrier of g -defined the carrier of S -valued Function-like non empty V22( the carrier of g) quasi_total Element of bool [: the carrier of g, the carrier of S:]
[: the carrier of g, the carrier of S:] is Relation-like non empty set
bool [: the carrier of g, the carrier of S:] is non empty set
(T,g,x) is Relation-like the carrier of g -defined the carrier of T -valued Function-like non empty V22( the carrier of g) quasi_total sups-preserving join-preserving directed-sups-preserving monotone lower_adjoint Element of bool [: the carrier of g, the carrier of T:]
[: the carrier of g, the carrier of T:] is Relation-like non empty set
bool [: the carrier of g, the carrier of T:] is non empty set
(S,T,x) * (T,g,x) is Relation-like the carrier of g -defined the carrier of S -valued Function-like non empty V22( the carrier of g) quasi_total Element of bool [: the carrier of g, the carrier of S:]
[x,(S,T,x)] is Connection of S,T
{x,(S,T,x)} is non empty with_non-empty_elements non empty-membered finite set
{x} is non empty with_non-empty_elements non empty-membered finite set
{{x,(S,T,x)},{x}} is non empty with_non-empty_elements non empty-membered finite V30() set
[x,(T,g,x)] is Connection of T,g
{x,(T,g,x)} is non empty with_non-empty_elements non empty-membered finite set
{x} is non empty with_non-empty_elements non empty-membered finite set
{{x,(T,g,x)},{x}} is non empty with_non-empty_elements non empty-membered finite V30() set
[(x * x),((S,T,x) * (T,g,x))] is Connection of S,g
{(x * x),((S,T,x) * (T,g,x))} is non empty with_non-empty_elements non empty-membered finite set
{(x * x)} is non empty with_non-empty_elements non empty-membered finite set
{{(x * x),((S,T,x) * (T,g,x))},{(x * x)}} is non empty with_non-empty_elements non empty-membered finite V30() set
S is non empty V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
the carrier of S is non empty set
T is non empty V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
the carrier of T is non empty set
[: the carrier of S, the carrier of T:] is Relation-like non empty set
bool [: the carrier of S, the carrier of T:] is non empty set
g is non empty V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
the carrier of g is non empty set
[: the carrier of T, the carrier of g:] is Relation-like non empty set
bool [: the carrier of T, the carrier of g:] is non empty set
x is Relation-like the carrier of S -defined the carrier of T -valued Function-like non empty V22( the carrier of S) quasi_total sups-preserving join-preserving directed-sups-preserving monotone Element of bool [: the carrier of S, the carrier of T:]
(T,S,x) is Relation-like the carrier of T -defined the carrier of S -valued Function-like non empty V22( the carrier of T) quasi_total infs-preserving meet-preserving filtered-infs-preserving monotone upper_adjoint Element of bool [: the carrier of T, the carrier of S:]
[: the carrier of T, the carrier of S:] is Relation-like non empty set
bool [: the carrier of T, the carrier of S:] is non empty set
x is Relation-like the carrier of T -defined the carrier of g -valued Function-like non empty V22( the carrier of T) quasi_total sups-preserving join-preserving directed-sups-preserving monotone Element of bool [: the carrier of T, the carrier of g:]
x * x is Relation-like the carrier of S -defined the carrier of g -valued Function-like non empty V22( the carrier of S) quasi_total Element of bool [: the carrier of S, the carrier of g:]
[: the carrier of S, the carrier of g:] is Relation-like non empty set
bool [: the carrier of S, the carrier of g:] is non empty set
(g,S,(x * x)) is Relation-like the carrier of g -defined the carrier of S -valued Function-like non empty V22( the carrier of g) quasi_total Element of bool [: the carrier of g, the carrier of S:]
[: the carrier of g, the carrier of S:] is Relation-like non empty set
bool [: the carrier of g, the carrier of S:] is non empty set
(g,T,x) is Relation-like the carrier of g -defined the carrier of T -valued Function-like non empty V22( the carrier of g) quasi_total infs-preserving meet-preserving filtered-infs-preserving monotone upper_adjoint Element of bool [: the carrier of g, the carrier of T:]
[: the carrier of g, the carrier of T:] is Relation-like non empty set
bool [: the carrier of g, the carrier of T:] is non empty set
(T,S,x) * (g,T,x) is Relation-like the carrier of g -defined the carrier of S -valued Function-like non empty V22( the carrier of g) quasi_total Element of bool [: the carrier of g, the carrier of S:]
[(T,S,x),x] is Connection of T,S
{(T,S,x),x} is non empty with_non-empty_elements non empty-membered finite set
{(T,S,x)} is non empty with_non-empty_elements non empty-membered finite set
{{(T,S,x),x},{(T,S,x)}} is non empty with_non-empty_elements non empty-membered finite V30() set
[(g,T,x),x] is Connection of g,T
{(g,T,x),x} is non empty with_non-empty_elements non empty-membered finite set
{(g,T,x)} is non empty with_non-empty_elements non empty-membered finite set
{{(g,T,x),x},{(g,T,x)}} is non empty with_non-empty_elements non empty-membered finite V30() set
[((T,S,x) * (g,T,x)),(x * x)] is Connection of g,S
{((T,S,x) * (g,T,x)),(x * x)} is non empty with_non-empty_elements non empty-membered finite set
{((T,S,x) * (g,T,x))} is non empty with_non-empty_elements non empty-membered finite set
{{((T,S,x) * (g,T,x)),(x * x)},{((T,S,x) * (g,T,x))}} is non empty with_non-empty_elements non empty-membered finite V30() set
S is non empty V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
the carrier of S is non empty set
T is non empty V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
the carrier of T is non empty set
[: the carrier of S, the carrier of T:] is Relation-like non empty set
bool [: the carrier of S, the carrier of T:] is non empty set
g is Relation-like the carrier of S -defined the carrier of T -valued Function-like non empty V22( the carrier of S) quasi_total infs-preserving meet-preserving filtered-infs-preserving monotone Element of bool [: the carrier of S, the carrier of T:]
(S,T,g) is Relation-like the carrier of T -defined the carrier of S -valued Function-like non empty V22( the carrier of T) quasi_total sups-preserving join-preserving directed-sups-preserving monotone lower_adjoint Element of bool [: the carrier of T, the carrier of S:]
[: the carrier of T, the carrier of S:] is Relation-like non empty set
bool [: the carrier of T, the carrier of S:] is non empty set
(S,T,(S,T,g)) is Relation-like the carrier of S -defined the carrier of T -valued Function-like non empty V22( the carrier of S) quasi_total infs-preserving meet-preserving filtered-infs-preserving monotone upper_adjoint Element of bool [: the carrier of S, the carrier of T:]
[g,(S,T,g)] is Connection of S,T
{g,(S,T,g)} is non empty with_non-empty_elements non empty-membered finite set
{g} is non empty with_non-empty_elements non empty-membered finite set
{{g,(S,T,g)},{g}} is non empty with_non-empty_elements non empty-membered finite V30() set
S is non empty V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
the carrier of S is non empty set
T is non empty V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
the carrier of T is non empty set
[: the carrier of S, the carrier of T:] is Relation-like non empty set
bool [: the carrier of S, the carrier of T:] is non empty set
g is Relation-like the carrier of S -defined the carrier of T -valued Function-like non empty V22( the carrier of S) quasi_total sups-preserving join-preserving directed-sups-preserving monotone Element of bool [: the carrier of S, the carrier of T:]
(T,S,g) is Relation-like the carrier of T -defined the carrier of S -valued Function-like non empty V22( the carrier of T) quasi_total infs-preserving meet-preserving filtered-infs-preserving monotone upper_adjoint Element of bool [: the carrier of T, the carrier of S:]
[: the carrier of T, the carrier of S:] is Relation-like non empty set
bool [: the carrier of T, the carrier of S:] is non empty set
(T,S,(T,S,g)) is Relation-like the carrier of S -defined the carrier of T -valued Function-like non empty V22( the carrier of S) quasi_total sups-preserving join-preserving directed-sups-preserving monotone lower_adjoint Element of bool [: the carrier of S, the carrier of T:]
[(T,S,g),g] is Connection of T,S
{(T,S,g),g} is non empty with_non-empty_elements non empty-membered finite set
{(T,S,g)} is non empty with_non-empty_elements non empty-membered finite set
{{(T,S,g),g},{(T,S,g)}} is non empty with_non-empty_elements non empty-membered finite V30() set
S is non empty AltCatStr
the Arrows of S is Relation-like [: the carrier of S, the carrier of S:] -defined Function-like V22([: the carrier of S, the carrier of S:]) set
the carrier of S is non empty set
[: the carrier of S, the carrier of S:] is Relation-like non empty set
x is set
T is set
g is set
the Arrows of S . (T,g) is set
[T,g] is set
{T,g} is non empty finite set
{T} is non empty finite set
{{T,g},{T}} is non empty with_non-empty_elements non empty-membered finite V30() set
the Arrows of S . [T,g] is set
dom the Arrows of S is Relation-like the carrier of S -defined the carrier of S -valued Element of bool [: the carrier of S, the carrier of S:]
bool [: the carrier of S, the carrier of S:] is non empty set
x is Element of the carrier of S
x is Element of the carrier of S
<^x,x^> is set
S is non empty set
T is Element of S
g is non empty set
[:g,g:] is Relation-like non empty set
bool [:g,g:] is non empty set
the Relation-like g -defined g -valued well_founded well-ordering V22(g) quasi_total V31() V34() V36() V38() upper-bounded Element of bool [:g,g:] is Relation-like g -defined g -valued well_founded well-ordering V22(g) quasi_total V31() V34() V36() V38() upper-bounded Element of bool [:g,g:]
RelStr(# g, the Relation-like g -defined g -valued well_founded well-ordering V22(g) quasi_total V31() V34() V36() V38() upper-bounded Element of bool [:g,g:] #) is non empty strict V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() satisfying_axiom_K algebraic connected up-complete /\-complete satisfying_axiom_of_approximation continuous with_suprema with_infima complete RelStr
x is non empty V87() reflexive transitive antisymmetric with_suprema with_infima RelStr
the carrier of x is non empty set
x is non empty V87() reflexive transitive antisymmetric with_suprema with_infima RelStr
the carrier of x is non empty set
[: the carrier of x, the carrier of x:] is Relation-like non empty set
bool [: the carrier of x, the carrier of x:] is non empty set
x is non empty V87() reflexive transitive antisymmetric with_suprema with_infima RelStr
the carrier of x is non empty set
[: the carrier of x, the carrier of x:] is Relation-like non empty set
bool [: the carrier of x, the carrier of x:] is non empty set
y is Relation-like the carrier of x -defined the carrier of x -valued Function-like non empty V22( the carrier of x) quasi_total Element of bool [: the carrier of x, the carrier of x:]
a is Relation-like the carrier of x -defined the carrier of x -valued Function-like non empty V22( the carrier of x) quasi_total Element of bool [: the carrier of x, the carrier of x:]
a * y is Relation-like the carrier of x -defined the carrier of x -valued Function-like non empty V22( the carrier of x) quasi_total Element of bool [: the carrier of x, the carrier of x:]
[: the carrier of x, the carrier of x:] is Relation-like non empty set
bool [: the carrier of x, the carrier of x:] is non empty set
x is non empty V87() reflexive transitive antisymmetric with_suprema with_infima RelStr
id x is Relation-like the carrier of x -defined the carrier of x -valued Function-like one-to-one non empty V22( the carrier of x) quasi_total infs-preserving sups-preserving meet-preserving join-preserving filtered-infs-preserving directed-sups-preserving isomorphic monotone projection closure kernel Element of bool [: the carrier of x, the carrier of x:]
the carrier of x is non empty set
[: the carrier of x, the carrier of x:] is Relation-like non empty set
bool [: the carrier of x, the carrier of x:] is non empty set
id the carrier of x is Relation-like the carrier of x -defined the carrier of x -valued non empty V22( the carrier of x) quasi_total Element of bool [: the carrier of x, the carrier of x:]
g is non empty transitive strict semi-functional V189() with_units reflexive para-functional set-id-inheriting concrete carrier-underlaid lattice-wise AltCatStr
the carrier of g is non empty set
x is non empty transitive strict semi-functional V189() with_units reflexive para-functional set-id-inheriting concrete carrier-underlaid lattice-wise AltCatStr
the carrier of x is non empty set
x is Element of the carrier of g
latt x is non empty V87() reflexive transitive antisymmetric with_suprema with_infima RelStr
the carrier of (latt x) is non empty set
x is Element of the carrier of g
latt x is non empty V87() reflexive transitive antisymmetric with_suprema with_infima RelStr
the carrier of (latt x) is non empty set
[: the carrier of (latt x), the carrier of (latt x):] is Relation-like non empty set
bool [: the carrier of (latt x), the carrier of (latt x):] is non empty set
x is Relation-like the carrier of (latt x) -defined the carrier of (latt x) -valued Function-like non empty V22( the carrier of (latt x)) quasi_total monotone Element of bool [: the carrier of (latt x), the carrier of (latt x):]
<^x,x^> is set
y is non empty V87() reflexive transitive antisymmetric with_suprema with_infima RelStr
the carrier of y is non empty set
a is non empty V87() reflexive transitive antisymmetric with_suprema with_infima RelStr
the carrier of a is non empty set
[: the carrier of y, the carrier of a:] is Relation-like non empty set
bool [: the carrier of y, the carrier of a:] is non empty set
b is Relation-like the carrier of y -defined the carrier of a -valued Function-like non empty V22( the carrier of y) quasi_total Element of bool [: the carrier of y, the carrier of a:]
x is Element of the carrier of x
latt x is non empty V87() reflexive transitive antisymmetric with_suprema with_infima RelStr
the carrier of (latt x) is non empty set
x is Element of the carrier of x
latt x is non empty V87() reflexive transitive antisymmetric with_suprema with_infima RelStr
the carrier of (latt x) is non empty set
[: the carrier of (latt x), the carrier of (latt x):] is Relation-like non empty set
bool [: the carrier of (latt x), the carrier of (latt x):] is non empty set
x is Relation-like the carrier of (latt x) -defined the carrier of (latt x) -valued Function-like non empty V22( the carrier of (latt x)) quasi_total monotone Element of bool [: the carrier of (latt x), the carrier of (latt x):]
<^x,x^> is set
y is non empty V87() reflexive transitive antisymmetric with_suprema with_infima RelStr
the carrier of y is non empty set
a is non empty V87() reflexive transitive antisymmetric with_suprema with_infima RelStr
the carrier of a is non empty set
[: the carrier of y, the carrier of a:] is Relation-like non empty set
bool [: the carrier of y, the carrier of a:] is non empty set
b is Relation-like the carrier of y -defined the carrier of a -valued Function-like non empty V22( the carrier of y) quasi_total Element of bool [: the carrier of y, the carrier of a:]
S is non empty non empty-membered set
(S) is non empty transitive strict semi-functional V189() with_units reflexive para-functional set-id-inheriting concrete carrier-underlaid lattice-wise AltCatStr
the Arrows of (S) is Relation-like [: the carrier of (S), the carrier of (S):] -defined Function-like V22([: the carrier of (S), the carrier of (S):]) set
the carrier of (S) is non empty set
[: the carrier of (S), the carrier of (S):] is Relation-like non empty set
T is non empty V87() reflexive transitive antisymmetric with_suprema with_infima RelStr
the carrier of T is non empty set
g is non empty V87() reflexive transitive antisymmetric with_suprema with_infima RelStr
the carrier of g is non empty set
[: the carrier of T, the carrier of g:] is Relation-like non empty set
bool [: the carrier of T, the carrier of g:] is non empty set
the Arrows of (S) . (T,g) is set
[T,g] is set
{T,g} is non empty finite set
{T} is non empty finite set
{{T,g},{T}} is non empty with_non-empty_elements non empty-membered finite V30() set
the Arrows of (S) . [T,g] is set
x is Relation-like the carrier of T -defined the carrier of g -valued Function-like non empty V22( the carrier of T) quasi_total Element of bool [: the carrier of T, the carrier of g:]
x is Element of the carrier of (S)
x is Element of the carrier of (S)
<^x,x^> is set
a is non empty set
latt x is non empty V87() reflexive transitive antisymmetric with_suprema with_infima RelStr
the carrier of (latt x) is non empty set
latt x is non empty V87() reflexive transitive antisymmetric with_suprema with_infima RelStr
the carrier of (latt x) is non empty set
y is Relation-like Function-like Element of <^x,x^>
@ y is Relation-like the carrier of (latt x) -defined the carrier of (latt x) -valued Function-like non empty V22( the carrier of (latt x)) quasi_total monotone Element of bool [: the carrier of (latt x), the carrier of (latt x):]
[: the carrier of (latt x), the carrier of (latt x):] is Relation-like non empty set
bool [: the carrier of (latt x), the carrier of (latt x):] is non empty set
a is non empty set
x is Element of the carrier of (S)
latt x is non empty V87() reflexive transitive antisymmetric with_suprema with_infima RelStr
x is Element of the carrier of (S)
latt x is non empty V87() reflexive transitive antisymmetric with_suprema with_infima RelStr
<^x,x^> is set
y is non empty set
S is non empty set
T is Element of S
g is non empty set
[:g,g:] is Relation-like non empty set
bool [:g,g:] is non empty set
the Relation-like g -defined g -valued well_founded well-ordering V22(g) quasi_total V31() V34() V36() V38() upper-bounded Element of bool [:g,g:] is Relation-like g -defined g -valued well_founded well-ordering V22(g) quasi_total V31() V34() V36() V38() upper-bounded Element of bool [:g,g:]
RelStr(# g, the Relation-like g -defined g -valued well_founded well-ordering V22(g) quasi_total V31() V34() V36() V38() upper-bounded Element of bool [:g,g:] #) is non empty strict V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() satisfying_axiom_K algebraic connected up-complete /\-complete satisfying_axiom_of_approximation continuous with_suprema with_infima complete RelStr
x is non empty V87() reflexive transitive antisymmetric with_suprema with_infima RelStr
the carrier of x is non empty set
x is non empty V87() reflexive transitive antisymmetric with_suprema with_infima RelStr
the carrier of x is non empty set
[: the carrier of x, the carrier of x:] is Relation-like non empty set
bool [: the carrier of x, the carrier of x:] is non empty set
x is non empty V87() reflexive transitive antisymmetric with_suprema with_infima RelStr
the carrier of x is non empty set
[: the carrier of x, the carrier of x:] is Relation-like non empty set
bool [: the carrier of x, the carrier of x:] is non empty set
y is Relation-like the carrier of x -defined the carrier of x -valued Function-like non empty V22( the carrier of x) quasi_total Element of bool [: the carrier of x, the carrier of x:]
a is Relation-like the carrier of x -defined the carrier of x -valued Function-like non empty V22( the carrier of x) quasi_total Element of bool [: the carrier of x, the carrier of x:]
a * y is Relation-like the carrier of x -defined the carrier of x -valued Function-like non empty V22( the carrier of x) quasi_total Element of bool [: the carrier of x, the carrier of x:]
[: the carrier of x, the carrier of x:] is Relation-like non empty set
bool [: the carrier of x, the carrier of x:] is non empty set
x is non empty V87() reflexive transitive antisymmetric with_suprema with_infima RelStr
id x is Relation-like the carrier of x -defined the carrier of x -valued Function-like one-to-one non empty V22( the carrier of x) quasi_total infs-preserving sups-preserving meet-preserving join-preserving filtered-infs-preserving directed-sups-preserving isomorphic monotone projection closure kernel Element of bool [: the carrier of x, the carrier of x:]
the carrier of x is non empty set
[: the carrier of x, the carrier of x:] is Relation-like non empty set
bool [: the carrier of x, the carrier of x:] is non empty set
id the carrier of x is Relation-like the carrier of x -defined the carrier of x -valued non empty V22( the carrier of x) quasi_total Element of bool [: the carrier of x, the carrier of x:]
g is non empty transitive strict semi-functional V189() with_units reflexive para-functional set-id-inheriting concrete carrier-underlaid lattice-wise AltCatStr
the carrier of g is non empty set
x is non empty transitive strict semi-functional V189() with_units reflexive para-functional set-id-inheriting concrete carrier-underlaid lattice-wise AltCatStr
the carrier of x is non empty set
x is Element of the carrier of g
latt x is non empty V87() reflexive transitive antisymmetric with_suprema with_infima RelStr
the carrier of (latt x) is non empty set
x is Element of the carrier of g
latt x is non empty V87() reflexive transitive antisymmetric with_suprema with_infima RelStr
the carrier of (latt x) is non empty set
[: the carrier of (latt x), the carrier of (latt x):] is Relation-like non empty set
bool [: the carrier of (latt x), the carrier of (latt x):] is non empty set
x is Relation-like the carrier of (latt x) -defined the carrier of (latt x) -valued Function-like non empty V22( the carrier of (latt x)) quasi_total monotone Element of bool [: the carrier of (latt x), the carrier of (latt x):]
<^x,x^> is set
y is non empty V87() reflexive transitive antisymmetric with_suprema with_infima RelStr
the carrier of y is non empty set
a is non empty V87() reflexive transitive antisymmetric with_suprema with_infima RelStr
the carrier of a is non empty set
[: the carrier of y, the carrier of a:] is Relation-like non empty set
bool [: the carrier of y, the carrier of a:] is non empty set
b is Relation-like the carrier of y -defined the carrier of a -valued Function-like non empty V22( the carrier of y) quasi_total Element of bool [: the carrier of y, the carrier of a:]
x is Element of the carrier of x
latt x is non empty V87() reflexive transitive antisymmetric with_suprema with_infima RelStr
the carrier of (latt x) is non empty set
x is Element of the carrier of x
latt x is non empty V87() reflexive transitive antisymmetric with_suprema with_infima RelStr
the carrier of (latt x) is non empty set
[: the carrier of (latt x), the carrier of (latt x):] is Relation-like non empty set
bool [: the carrier of (latt x), the carrier of (latt x):] is non empty set
x is Relation-like the carrier of (latt x) -defined the carrier of (latt x) -valued Function-like non empty V22( the carrier of (latt x)) quasi_total monotone Element of bool [: the carrier of (latt x), the carrier of (latt x):]
<^x,x^> is set
y is non empty V87() reflexive transitive antisymmetric with_suprema with_infima RelStr
the carrier of y is non empty set
a is non empty V87() reflexive transitive antisymmetric with_suprema with_infima RelStr
the carrier of a is non empty set
[: the carrier of y, the carrier of a:] is Relation-like non empty set
bool [: the carrier of y, the carrier of a:] is non empty set
b is Relation-like the carrier of y -defined the carrier of a -valued Function-like non empty V22( the carrier of y) quasi_total Element of bool [: the carrier of y, the carrier of a:]
S is non empty non empty-membered set
(S) is non empty transitive strict semi-functional V189() with_units reflexive para-functional set-id-inheriting concrete carrier-underlaid lattice-wise AltCatStr
the Arrows of (S) is Relation-like [: the carrier of (S), the carrier of (S):] -defined Function-like V22([: the carrier of (S), the carrier of (S):]) set
the carrier of (S) is non empty set
[: the carrier of (S), the carrier of (S):] is Relation-like non empty set
T is non empty V87() reflexive transitive antisymmetric with_suprema with_infima RelStr
the carrier of T is non empty set
g is non empty V87() reflexive transitive antisymmetric with_suprema with_infima RelStr
the carrier of g is non empty set
[: the carrier of T, the carrier of g:] is Relation-like non empty set
bool [: the carrier of T, the carrier of g:] is non empty set
the Arrows of (S) . (T,g) is set
[T,g] is set
{T,g} is non empty finite set
{T} is non empty finite set
{{T,g},{T}} is non empty with_non-empty_elements non empty-membered finite V30() set
the Arrows of (S) . [T,g] is set
x is Relation-like the carrier of T -defined the carrier of g -valued Function-like non empty V22( the carrier of T) quasi_total Element of bool [: the carrier of T, the carrier of g:]
x is Element of the carrier of (S)
x is Element of the carrier of (S)
<^x,x^> is set
a is non empty set
latt x is non empty V87() reflexive transitive antisymmetric with_suprema with_infima RelStr
the carrier of (latt x) is non empty set
latt x is non empty V87() reflexive transitive antisymmetric with_suprema with_infima RelStr
the carrier of (latt x) is non empty set
y is Relation-like Function-like Element of <^x,x^>
@ y is Relation-like the carrier of (latt x) -defined the carrier of (latt x) -valued Function-like non empty V22( the carrier of (latt x)) quasi_total monotone Element of bool [: the carrier of (latt x), the carrier of (latt x):]
[: the carrier of (latt x), the carrier of (latt x):] is Relation-like non empty set
bool [: the carrier of (latt x), the carrier of (latt x):] is non empty set
a is non empty set
x is Element of the carrier of (S)
latt x is non empty V87() reflexive transitive antisymmetric with_suprema with_infima RelStr
x is Element of the carrier of (S)
latt x is non empty V87() reflexive transitive antisymmetric with_suprema with_infima RelStr
<^x,x^> is set
y is non empty set
S is non empty non empty-membered set
(S) is non empty transitive strict semi-functional V189() with_units reflexive para-functional set-id-inheriting concrete carrier-underlaid lattice-wise AltCatStr
the carrier of (S) is non empty set
T is Element of the carrier of (S)
latt T is non empty V87() reflexive transitive antisymmetric with_suprema with_infima RelStr
g is non empty set
(S) is non empty transitive strict semi-functional V189() with_units reflexive para-functional set-id-inheriting concrete carrier-underlaid lattice-wise AltCatStr
the carrier of (S) is non empty set
T is Element of the carrier of (S)
latt T is non empty V87() reflexive transitive antisymmetric with_suprema with_infima RelStr
g is non empty set
S is non empty non empty-membered set
(S) is non empty transitive strict semi-functional V189() with_units reflexive para-functional set-id-inheriting concrete carrier-underlaid lattice-wise with_complete_lattices AltCatStr
the carrier of (S) is non empty set
T is non empty V87() reflexive transitive antisymmetric with_suprema with_infima RelStr
the carrier of T is non empty set
g is non empty V87() reflexive transitive antisymmetric with_suprema with_infima RelStr
the carrier of g is non empty set
x is non empty set
S is non empty non empty-membered set
(S) is non empty transitive strict semi-functional V189() with_units reflexive para-functional set-id-inheriting concrete carrier-underlaid lattice-wise with_complete_lattices AltCatStr
the carrier of (S) is non empty set
T is Element of the carrier of (S)
g is Element of the carrier of (S)
<^T,g^> is set
latt T is non empty V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
the carrier of (latt T) is non empty set
latt g is non empty V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
the carrier of (latt g) is non empty set
[: the carrier of (latt T), the carrier of (latt g):] is Relation-like non empty set
bool [: the carrier of (latt T), the carrier of (latt g):] is non empty set
x is set
x is Relation-like Function-like Element of <^T,g^>
@ x is Relation-like the carrier of (latt T) -defined the carrier of (latt g) -valued Function-like non empty V22( the carrier of (latt T)) quasi_total monotone Element of bool [: the carrier of (latt T), the carrier of (latt g):]
latt T is non empty V87() reflexive transitive antisymmetric with_suprema with_infima RelStr
the carrier of (latt T) is non empty set
latt g is non empty V87() reflexive transitive antisymmetric with_suprema with_infima RelStr
the carrier of (latt g) is non empty set
[: the carrier of (latt T), the carrier of (latt g):] is Relation-like non empty set
bool [: the carrier of (latt T), the carrier of (latt g):] is non empty set
x is non empty set
x is non empty set
S is non empty non empty-membered set
(S) is non empty transitive strict semi-functional V189() with_units reflexive para-functional set-id-inheriting concrete carrier-underlaid lattice-wise with_complete_lattices AltCatStr
the carrier of (S) is non empty set
T is non empty V87() reflexive transitive antisymmetric with_suprema with_infima RelStr
the carrier of T is non empty set
g is non empty V87() reflexive transitive antisymmetric with_suprema with_infima RelStr
the carrier of g is non empty set
x is non empty set
S is non empty non empty-membered set
(S) is non empty transitive strict semi-functional V189() with_units reflexive para-functional set-id-inheriting concrete carrier-underlaid lattice-wise with_complete_lattices AltCatStr
the carrier of (S) is non empty set
T is Element of the carrier of (S)
g is Element of the carrier of (S)
<^T,g^> is set
latt T is non empty V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
the carrier of (latt T) is non empty set
latt g is non empty V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
the carrier of (latt g) is non empty set
[: the carrier of (latt T), the carrier of (latt g):] is Relation-like non empty set
bool [: the carrier of (latt T), the carrier of (latt g):] is non empty set
x is set
x is Relation-like Function-like Element of <^T,g^>
@ x is Relation-like the carrier of (latt T) -defined the carrier of (latt g) -valued Function-like non empty V22( the carrier of (latt T)) quasi_total monotone Element of bool [: the carrier of (latt T), the carrier of (latt g):]
latt T is non empty V87() reflexive transitive antisymmetric with_suprema with_infima RelStr
the carrier of (latt T) is non empty set
latt g is non empty V87() reflexive transitive antisymmetric with_suprema with_infima RelStr
the carrier of (latt g) is non empty set
[: the carrier of (latt T), the carrier of (latt g):] is Relation-like non empty set
bool [: the carrier of (latt T), the carrier of (latt g):] is non empty set
x is non empty set
x is non empty set
S is non empty non empty-membered set
(S) is non empty transitive strict semi-functional V189() with_units reflexive para-functional set-id-inheriting concrete carrier-underlaid lattice-wise with_complete_lattices AltCatStr
the carrier of (S) is non empty set
(S) is non empty transitive strict semi-functional V189() with_units reflexive para-functional set-id-inheriting concrete carrier-underlaid lattice-wise with_complete_lattices AltCatStr
the carrier of (S) is non empty set
T is set
g is non empty V87() reflexive transitive antisymmetric with_suprema with_infima RelStr
x is non empty set
the carrier of g is non empty set
x is non empty set
T is set
g is non empty V87() reflexive transitive antisymmetric with_suprema with_infima RelStr
x is non empty set
the carrier of g is non empty set
x is non empty set
S is non empty non empty-membered set
(S) is non empty transitive strict semi-functional V189() with_units reflexive para-functional set-id-inheriting concrete carrier-underlaid lattice-wise with_complete_lattices AltCatStr
(S) is non empty transitive strict semi-functional V189() with_units reflexive para-functional set-id-inheriting concrete carrier-underlaid lattice-wise with_complete_lattices AltCatStr
the Arrows of (S) is Relation-like [: the carrier of (S), the carrier of (S):] -defined Function-like V22([: the carrier of (S), the carrier of (S):]) set
the carrier of (S) is non empty set
[: the carrier of (S), the carrier of (S):] is Relation-like non empty set
x is non empty V87() reflexive transitive antisymmetric with_suprema with_infima RelStr
the carrier of x is non empty set
x is non empty V87() reflexive transitive antisymmetric with_suprema with_infima RelStr
the carrier of x is non empty set
[: the carrier of x, the carrier of x:] is Relation-like non empty set
bool [: the carrier of x, the carrier of x:] is non empty set
x is non empty V87() reflexive transitive antisymmetric with_suprema with_infima RelStr
the carrier of x is non empty set
y is non empty V87() reflexive transitive antisymmetric with_suprema with_infima RelStr
the carrier of y is non empty set
[: the carrier of x, the carrier of y:] is Relation-like non empty set
bool [: the carrier of x, the carrier of y:] is non empty set
x is Relation-like the carrier of x -defined the carrier of x -valued Function-like non empty V22( the carrier of x) quasi_total Element of bool [: the carrier of x, the carrier of x:]
the Arrows of (S) . (x,x) is set
[x,x] is set
{x,x} is non empty finite set
{x} is non empty finite set
{{x,x},{x}} is non empty with_non-empty_elements non empty-membered finite V30() set
the Arrows of (S) . [x,x] is set
a is Relation-like the carrier of x -defined the carrier of y -valued Function-like non empty V22( the carrier of x) quasi_total Element of bool [: the carrier of x, the carrier of y:]
the Arrows of (S) . (x,y) is set
[x,y] is set
{x,y} is non empty finite set
{x} is non empty finite set
{{x,y},{x}} is non empty with_non-empty_elements non empty-membered finite V30() set
the Arrows of (S) . [x,y] is set
the Arrows of (S) is Relation-like [: the carrier of (S), the carrier of (S):] -defined Function-like V22([: the carrier of (S), the carrier of (S):]) set
the carrier of (S) is non empty set
[: the carrier of (S), the carrier of (S):] is Relation-like non empty set
x is non empty V87() reflexive transitive antisymmetric with_suprema with_infima RelStr
the carrier of x is non empty set
x is non empty V87() reflexive transitive antisymmetric with_suprema with_infima RelStr
the carrier of x is non empty set
[: the carrier of x, the carrier of x:] is Relation-like non empty set
bool [: the carrier of x, the carrier of x:] is non empty set
x is non empty V87() reflexive transitive antisymmetric with_suprema with_infima RelStr
the carrier of x is non empty set
y is non empty V87() reflexive transitive antisymmetric with_suprema with_infima RelStr
the carrier of y is non empty set
[: the carrier of x, the carrier of y:] is Relation-like non empty set
bool [: the carrier of x, the carrier of y:] is non empty set
x is Relation-like the carrier of x -defined the carrier of x -valued Function-like non empty V22( the carrier of x) quasi_total Element of bool [: the carrier of x, the carrier of x:]
the Arrows of (S) . (x,x) is set
[x,x] is set
{x,x} is non empty finite set
{x} is non empty finite set
{{x,x},{x}} is non empty with_non-empty_elements non empty-membered finite V30() set
the Arrows of (S) . [x,x] is set
a is Relation-like the carrier of x -defined the carrier of y -valued Function-like non empty V22( the carrier of x) quasi_total Element of bool [: the carrier of x, the carrier of y:]
the Arrows of (S) . (x,y) is set
[x,y] is set
{x,y} is non empty finite set
{x} is non empty finite set
{{x,y},{x}} is non empty with_non-empty_elements non empty-membered finite V30() set
the Arrows of (S) . [x,y] is set
x is non empty V87() reflexive transitive antisymmetric with_suprema with_infima RelStr
x is non empty V87() reflexive transitive antisymmetric with_suprema with_infima RelStr
the carrier of x is non empty set
x is non empty V87() reflexive transitive antisymmetric with_suprema with_infima RelStr
the carrier of x is non empty set
[: the carrier of x, the carrier of x:] is Relation-like non empty set
bool [: the carrier of x, the carrier of x:] is non empty set
x is Relation-like the carrier of x -defined the carrier of x -valued Function-like non empty V22( the carrier of x) quasi_total Element of bool [: the carrier of x, the carrier of x:]
(x,x,x) is Relation-like the carrier of x -defined the carrier of x -valued Function-like non empty V22( the carrier of x) quasi_total Element of bool [: the carrier of x, the carrier of x:]
[: the carrier of x, the carrier of x:] is Relation-like non empty set
bool [: the carrier of x, the carrier of x:] is non empty set
x is non empty V87() reflexive transitive antisymmetric with_suprema with_infima RelStr
the carrier of x is non empty set
id x is Relation-like the carrier of x -defined the carrier of x -valued Function-like one-to-one non empty V22( the carrier of x) quasi_total infs-preserving sups-preserving meet-preserving join-preserving filtered-infs-preserving directed-sups-preserving isomorphic monotone projection closure kernel Element of bool [: the carrier of x, the carrier of x:]
[: the carrier of x, the carrier of x:] is Relation-like non empty set
bool [: the carrier of x, the carrier of x:] is non empty set
id the carrier of x is Relation-like the carrier of x -defined the carrier of x -valued non empty V22( the carrier of x) quasi_total Element of bool [: the carrier of x, the carrier of x:]
(x,x,(id x)) is Relation-like the carrier of x -defined the carrier of x -valued Function-like non empty V22( the carrier of x) quasi_total Element of bool [: the carrier of x, the carrier of x:]
id H1(x) is Relation-like the carrier of H1(x) -defined the carrier of H1(x) -valued Function-like one-to-one non empty V22( the carrier of H1(x)) quasi_total infs-preserving sups-preserving meet-preserving join-preserving filtered-infs-preserving directed-sups-preserving isomorphic monotone projection closure kernel Element of bool [: the carrier of H1(x), the carrier of H1(x):]
the carrier of H1(x) is non empty set
[: the carrier of H1(x), the carrier of H1(x):] is Relation-like non empty set
bool [: the carrier of H1(x), the carrier of H1(x):] is non empty set
x is non empty V87() reflexive transitive antisymmetric with_suprema with_infima RelStr
the carrier of x is non empty set
x is non empty V87() reflexive transitive antisymmetric with_suprema with_infima RelStr
the carrier of x is non empty set
[: the carrier of x, the carrier of x:] is Relation-like non empty set
bool [: the carrier of x, the carrier of x:] is non empty set
x is non empty V87() reflexive transitive antisymmetric with_suprema with_infima RelStr
the carrier of x is non empty set
[: the carrier of x, the carrier of x:] is Relation-like non empty set
bool [: the carrier of x, the carrier of x:] is non empty set
x is Relation-like the carrier of x -defined the carrier of x -valued Function-like non empty V22( the carrier of x) quasi_total Element of bool [: the carrier of x, the carrier of x:]
y is Relation-like the carrier of x -defined the carrier of x -valued Function-like non empty V22( the carrier of x) quasi_total Element of bool [: the carrier of x, the carrier of x:]
y * x is Relation-like the carrier of x -defined the carrier of x -valued Function-like non empty V22( the carrier of x) quasi_total Element of bool [: the carrier of x, the carrier of x:]
[: the carrier of x, the carrier of x:] is Relation-like non empty set
bool [: the carrier of x, the carrier of x:] is non empty set
(x,x,(y * x)) is Relation-like the carrier of x -defined the carrier of x -valued Function-like non empty V22( the carrier of x) quasi_total Element of bool [: the carrier of x, the carrier of x:]
[: the carrier of x, the carrier of x:] is Relation-like non empty set
bool [: the carrier of x, the carrier of x:] is non empty set
(x,x,y) is Relation-like the carrier of x -defined the carrier of x -valued Function-like non empty V22( the carrier of x) quasi_total Element of bool [: the carrier of x, the carrier of x:]
[: the carrier of x, the carrier of x:] is Relation-like non empty set
bool [: the carrier of x, the carrier of x:] is non empty set
(x,x,x) is Relation-like the carrier of x -defined the carrier of x -valued Function-like non empty V22( the carrier of x) quasi_total Element of bool [: the carrier of x, the carrier of x:]
[: the carrier of x, the carrier of x:] is Relation-like non empty set
bool [: the carrier of x, the carrier of x:] is non empty set
(x,x,x) * (x,x,y) is Relation-like the carrier of x -defined the carrier of x -valued Function-like non empty V22( the carrier of x) quasi_total Element of bool [: the carrier of x, the carrier of x:]
x is reflexive V226((S),(S)) strict Contravariant id-preserving comp-reversing contravariant Functor of (S),(S)
x is reflexive V226((S),(S)) strict Contravariant id-preserving comp-reversing contravariant Functor of (S),(S)
x is Element of the carrier of (S)
x . x is Element of the carrier of (S)
the ObjectMap of x is Relation-like [: the carrier of (S), the carrier of (S):] -defined [: the carrier of (S), the carrier of (S):] -valued Function-like non empty V22([: the carrier of (S), the carrier of (S):]) quasi_total Element of bool [:[: the carrier of (S), the carrier of (S):],[: the carrier of (S), the carrier of (S):]:]
[:[: the carrier of (S), the carrier of (S):],[: the carrier of (S), the carrier of (S):]:] is Relation-like non empty set
bool [:[: the carrier of (S), the carrier of (S):],[: the carrier of (S), the carrier of (S):]:] is non empty set
the ObjectMap of x . (x,x) is Element of [: the carrier of (S), the carrier of (S):]
[x,x] is set
{x,x} is non empty finite set
{x} is non empty finite set
{{x,x},{x}} is non empty with_non-empty_elements non empty-membered finite V30() set
the ObjectMap of x . [x,x] is set
K57(( the ObjectMap of x . (x,x))) is set
latt x is non empty V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
x . x is Element of the carrier of (S)
the ObjectMap of x is Relation-like [: the carrier of (S), the carrier of (S):] -defined [: the carrier of (S), the carrier of (S):] -valued Function-like non empty V22([: the carrier of (S), the carrier of (S):]) quasi_total Element of bool [:[: the carrier of (S), the carrier of (S):],[: the carrier of (S), the carrier of (S):]:]
the ObjectMap of x . (x,x) is Element of [: the carrier of (S), the carrier of (S):]
the ObjectMap of x . [x,x] is set
K57(( the ObjectMap of x . (x,x))) is set
x is Element of the carrier of (S)
x is Element of the carrier of (S)
<^x,x^> is set
y is Relation-like Function-like Element of <^x,x^>
x . y is Relation-like Function-like Element of <^(x . x),(x . x)^>
x . x is Element of the carrier of (S)
the ObjectMap of x . (x,x) is Element of [: the carrier of (S), the carrier of (S):]
[x,x] is set
{x,x} is non empty finite set
{x} is non empty finite set
{{x,x},{x}} is non empty with_non-empty_elements non empty-membered finite V30() set
the ObjectMap of x . [x,x] is set
K57(( the ObjectMap of x . (x,x))) is set
x . x is Element of the carrier of (S)
the ObjectMap of x . (x,x) is Element of [: the carrier of (S), the carrier of (S):]
[x,x] is set
{x,x} is non empty finite set
{x} is non empty finite set
{{x,x},{x}} is non empty with_non-empty_elements non empty-membered finite V30() set
the ObjectMap of x . [x,x] is set
K57(( the ObjectMap of x . (x,x))) is set
<^(x . x),(x . x)^> is set
latt x is non empty V87() reflexive transitive antisymmetric with_suprema with_infima RelStr
latt x is non empty V87() reflexive transitive antisymmetric with_suprema with_infima RelStr
@ y is Relation-like the carrier of (latt x) -defined the carrier of (latt x) -valued Function-like non empty V22( the carrier of (latt x)) quasi_total monotone Element of bool [: the carrier of (latt x), the carrier of (latt x):]
the carrier of (latt x) is non empty set
the carrier of (latt x) is non empty set
[: the carrier of (latt x), the carrier of (latt x):] is Relation-like non empty set
bool [: the carrier of (latt x), the carrier of (latt x):] is non empty set
((latt x),(latt x),(@ y)) is Relation-like the carrier of (latt x) -defined the carrier of (latt x) -valued Function-like non empty V22( the carrier of (latt x)) quasi_total Element of bool [: the carrier of (latt x), the carrier of (latt x):]
[: the carrier of (latt x), the carrier of (latt x):] is Relation-like non empty set
bool [: the carrier of (latt x), the carrier of (latt x):] is non empty set
x . y is Relation-like Function-like Element of <^(x . x),(x . x)^>
x . x is Element of the carrier of (S)
the ObjectMap of x . (x,x) is Element of [: the carrier of (S), the carrier of (S):]
the ObjectMap of x . [x,x] is set
K57(( the ObjectMap of x . (x,x))) is set
x . x is Element of the carrier of (S)
the ObjectMap of x . (x,x) is Element of [: the carrier of (S), the carrier of (S):]
the ObjectMap of x . [x,x] is set
K57(( the ObjectMap of x . (x,x))) is set
<^(x . x),(x . x)^> is set
x is non empty V87() reflexive transitive antisymmetric with_suprema with_infima RelStr
x is non empty V87() reflexive transitive antisymmetric with_suprema with_infima RelStr
the carrier of x is non empty set
x is non empty V87() reflexive transitive antisymmetric with_suprema with_infima RelStr
the carrier of x is non empty set
[: the carrier of x, the carrier of x:] is Relation-like non empty set
bool [: the carrier of x, the carrier of x:] is non empty set
x is Relation-like the carrier of x -defined the carrier of x -valued Function-like non empty V22( the carrier of x) quasi_total Element of bool [: the carrier of x, the carrier of x:]
(x,x,x) is Relation-like the carrier of x -defined the carrier of x -valued Function-like non empty V22( the carrier of x) quasi_total Element of bool [: the carrier of x, the carrier of x:]
[: the carrier of x, the carrier of x:] is Relation-like non empty set
bool [: the carrier of x, the carrier of x:] is non empty set
x is non empty V87() reflexive transitive antisymmetric with_suprema with_infima RelStr
the carrier of x is non empty set
id x is Relation-like the carrier of x -defined the carrier of x -valued Function-like one-to-one non empty V22( the carrier of x) quasi_total infs-preserving sups-preserving meet-preserving join-preserving filtered-infs-preserving directed-sups-preserving isomorphic monotone projection closure kernel Element of bool [: the carrier of x, the carrier of x:]
[: the carrier of x, the carrier of x:] is Relation-like non empty set
bool [: the carrier of x, the carrier of x:] is non empty set
id the carrier of x is Relation-like the carrier of x -defined the carrier of x -valued non empty V22( the carrier of x) quasi_total Element of bool [: the carrier of x, the carrier of x:]
(x,x,(id x)) is Relation-like the carrier of x -defined the carrier of x -valued Function-like non empty V22( the carrier of x) quasi_total Element of bool [: the carrier of x, the carrier of x:]
id H1(x) is Relation-like the carrier of H1(x) -defined the carrier of H1(x) -valued Function-like one-to-one non empty V22( the carrier of H1(x)) quasi_total infs-preserving sups-preserving meet-preserving join-preserving filtered-infs-preserving directed-sups-preserving isomorphic monotone projection closure kernel Element of bool [: the carrier of H1(x), the carrier of H1(x):]
the carrier of H1(x) is non empty set
[: the carrier of H1(x), the carrier of H1(x):] is Relation-like non empty set
bool [: the carrier of H1(x), the carrier of H1(x):] is non empty set
x is non empty V87() reflexive transitive antisymmetric with_suprema with_infima RelStr
the carrier of x is non empty set
x is non empty V87() reflexive transitive antisymmetric with_suprema with_infima RelStr
the carrier of x is non empty set
[: the carrier of x, the carrier of x:] is Relation-like non empty set
bool [: the carrier of x, the carrier of x:] is non empty set
x is non empty V87() reflexive transitive antisymmetric with_suprema with_infima RelStr
the carrier of x is non empty set
[: the carrier of x, the carrier of x:] is Relation-like non empty set
bool [: the carrier of x, the carrier of x:] is non empty set
x is Relation-like the carrier of x -defined the carrier of x -valued Function-like non empty V22( the carrier of x) quasi_total Element of bool [: the carrier of x, the carrier of x:]
y is Relation-like the carrier of x -defined the carrier of x -valued Function-like non empty V22( the carrier of x) quasi_total Element of bool [: the carrier of x, the carrier of x:]
y * x is Relation-like the carrier of x -defined the carrier of x -valued Function-like non empty V22( the carrier of x) quasi_total Element of bool [: the carrier of x, the carrier of x:]
[: the carrier of x, the carrier of x:] is Relation-like non empty set
bool [: the carrier of x, the carrier of x:] is non empty set
(x,x,(y * x)) is Relation-like the carrier of x -defined the carrier of x -valued Function-like non empty V22( the carrier of x) quasi_total Element of bool [: the carrier of x, the carrier of x:]
[: the carrier of x, the carrier of x:] is Relation-like non empty set
bool [: the carrier of x, the carrier of x:] is non empty set
(x,x,y) is Relation-like the carrier of x -defined the carrier of x -valued Function-like non empty V22( the carrier of x) quasi_total Element of bool [: the carrier of x, the carrier of x:]
[: the carrier of x, the carrier of x:] is Relation-like non empty set
bool [: the carrier of x, the carrier of x:] is non empty set
(x,x,x) is Relation-like the carrier of x -defined the carrier of x -valued Function-like non empty V22( the carrier of x) quasi_total Element of bool [: the carrier of x, the carrier of x:]
[: the carrier of x, the carrier of x:] is Relation-like non empty set
bool [: the carrier of x, the carrier of x:] is non empty set
(x,x,x) * (x,x,y) is Relation-like the carrier of x -defined the carrier of x -valued Function-like non empty V22( the carrier of x) quasi_total Element of bool [: the carrier of x, the carrier of x:]
x is reflexive V226((S),(S)) strict Contravariant id-preserving comp-reversing contravariant Functor of (S),(S)
x is reflexive V226((S),(S)) strict Contravariant id-preserving comp-reversing contravariant Functor of (S),(S)
x is Element of the carrier of (S)
x . x is Element of the carrier of (S)
the ObjectMap of x is Relation-like [: the carrier of (S), the carrier of (S):] -defined [: the carrier of (S), the carrier of (S):] -valued Function-like non empty V22([: the carrier of (S), the carrier of (S):]) quasi_total Element of bool [:[: the carrier of (S), the carrier of (S):],[: the carrier of (S), the carrier of (S):]:]
[:[: the carrier of (S), the carrier of (S):],[: the carrier of (S), the carrier of (S):]:] is Relation-like non empty set
bool [:[: the carrier of (S), the carrier of (S):],[: the carrier of (S), the carrier of (S):]:] is non empty set
the ObjectMap of x . (x,x) is Element of [: the carrier of (S), the carrier of (S):]
[x,x] is set
{x,x} is non empty finite set
{x} is non empty finite set
{{x,x},{x}} is non empty with_non-empty_elements non empty-membered finite V30() set
the ObjectMap of x . [x,x] is set
K57(( the ObjectMap of x . (x,x))) is set
latt x is non empty V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
x . x is Element of the carrier of (S)
the ObjectMap of x is Relation-like [: the carrier of (S), the carrier of (S):] -defined [: the carrier of (S), the carrier of (S):] -valued Function-like non empty V22([: the carrier of (S), the carrier of (S):]) quasi_total Element of bool [:[: the carrier of (S), the carrier of (S):],[: the carrier of (S), the carrier of (S):]:]
the ObjectMap of x . (x,x) is Element of [: the carrier of (S), the carrier of (S):]
the ObjectMap of x . [x,x] is set
K57(( the ObjectMap of x . (x,x))) is set
x is Element of the carrier of (S)
x is Element of the carrier of (S)
<^x,x^> is set
y is Relation-like Function-like Element of <^x,x^>
x . y is Relation-like Function-like Element of <^(x . x),(x . x)^>
x . x is Element of the carrier of (S)
the ObjectMap of x . (x,x) is Element of [: the carrier of (S), the carrier of (S):]
[x,x] is set
{x,x} is non empty finite set
{x} is non empty finite set
{{x,x},{x}} is non empty with_non-empty_elements non empty-membered finite V30() set
the ObjectMap of x . [x,x] is set
K57(( the ObjectMap of x . (x,x))) is set
x . x is Element of the carrier of (S)
the ObjectMap of x . (x,x) is Element of [: the carrier of (S), the carrier of (S):]
[x,x] is set
{x,x} is non empty finite set
{x} is non empty finite set
{{x,x},{x}} is non empty with_non-empty_elements non empty-membered finite V30() set
the ObjectMap of x . [x,x] is set
K57(( the ObjectMap of x . (x,x))) is set
<^(x . x),(x . x)^> is set
latt x is non empty V87() reflexive transitive antisymmetric with_suprema with_infima RelStr
latt x is non empty V87() reflexive transitive antisymmetric with_suprema with_infima RelStr
@ y is Relation-like the carrier of (latt x) -defined the carrier of (latt x) -valued Function-like non empty V22( the carrier of (latt x)) quasi_total monotone Element of bool [: the carrier of (latt x), the carrier of (latt x):]
the carrier of (latt x) is non empty set
the carrier of (latt x) is non empty set
[: the carrier of (latt x), the carrier of (latt x):] is Relation-like non empty set
bool [: the carrier of (latt x), the carrier of (latt x):] is non empty set
((latt x),(latt x),(@ y)) is Relation-like the carrier of (latt x) -defined the carrier of (latt x) -valued Function-like non empty V22( the carrier of (latt x)) quasi_total Element of bool [: the carrier of (latt x), the carrier of (latt x):]
[: the carrier of (latt x), the carrier of (latt x):] is Relation-like non empty set
bool [: the carrier of (latt x), the carrier of (latt x):] is non empty set
x . y is Relation-like Function-like Element of <^(x . x),(x . x)^>
x . x is Element of the carrier of (S)
the ObjectMap of x . (x,x) is Element of [: the carrier of (S), the carrier of (S):]
the ObjectMap of x . [x,x] is set
K57(( the ObjectMap of x . (x,x))) is set
x . x is Element of the carrier of (S)
the ObjectMap of x . (x,x) is Element of [: the carrier of (S), the carrier of (S):]
the ObjectMap of x . [x,x] is set
K57(( the ObjectMap of x . (x,x))) is set
<^(x . x),(x . x)^> is set
S is non empty non empty-membered set
(S) is non empty transitive strict semi-functional V189() with_units reflexive para-functional set-id-inheriting concrete carrier-underlaid lattice-wise with_complete_lattices AltCatStr
(S) is non empty transitive strict semi-functional V189() with_units reflexive para-functional set-id-inheriting concrete carrier-underlaid lattice-wise with_complete_lattices AltCatStr
(S) is reflexive V226((S),(S)) strict Contravariant id-preserving comp-reversing contravariant Functor of (S),(S)
the carrier of (S) is non empty set
x is Element of the carrier of (S)
(S) . x is Element of the carrier of (S)
the carrier of (S) is non empty set
[: the carrier of (S), the carrier of (S):] is Relation-like non empty set
the ObjectMap of (S) is Relation-like [: the carrier of (S), the carrier of (S):] -defined [: the carrier of (S), the carrier of (S):] -valued Function-like non empty V22([: the carrier of (S), the carrier of (S):]) quasi_total Element of bool [:[: the carrier of (S), the carrier of (S):],[: the carrier of (S), the carrier of (S):]:]
[: the carrier of (S), the carrier of (S):] is Relation-like non empty set
[:[: the carrier of (S), the carrier of (S):],[: the carrier of (S), the carrier of (S):]:] is Relation-like non empty set
bool [:[: the carrier of (S), the carrier of (S):],[: the carrier of (S), the carrier of (S):]:] is non empty set
the ObjectMap of (S) . (x,x) is Element of [: the carrier of (S), the carrier of (S):]
[x,x] is set
{x,x} is non empty finite set
{x} is non empty finite set
{{x,x},{x}} is non empty with_non-empty_elements non empty-membered finite V30() set
the ObjectMap of (S) . [x,x] is set
K57(( the ObjectMap of (S) . (x,x))) is set
latt x is non empty V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
x is Element of the carrier of (S)
x is Element of the carrier of (S)
<^x,x^> is set
x is Relation-like Function-like Element of <^x,x^>
(S) . x is Relation-like Function-like Element of <^((S) . x),((S) . x)^>
(S) . x is Element of the carrier of (S)
the carrier of (S) is non empty set
[: the carrier of (S), the carrier of (S):] is Relation-like non empty set
the ObjectMap of (S) is Relation-like [: the carrier of (S), the carrier of (S):] -defined [: the carrier of (S), the carrier of (S):] -valued Function-like non empty V22([: the carrier of (S), the carrier of (S):]) quasi_total Element of bool [:[: the carrier of (S), the carrier of (S):],[: the carrier of (S), the carrier of (S):]:]
[: the carrier of (S), the carrier of (S):] is Relation-like non empty set
[:[: the carrier of (S), the carrier of (S):],[: the carrier of (S), the carrier of (S):]:] is Relation-like non empty set
bool [:[: the carrier of (S), the carrier of (S):],[: the carrier of (S), the carrier of (S):]:] is non empty set
the ObjectMap of (S) . (x,x) is Element of [: the carrier of (S), the carrier of (S):]
[x,x] is set
{x,x} is non empty finite set
{x} is non empty finite set
{{x,x},{x}} is non empty with_non-empty_elements non empty-membered finite V30() set
the ObjectMap of (S) . [x,x] is set
K57(( the ObjectMap of (S) . (x,x))) is set
(S) . x is Element of the carrier of (S)
the ObjectMap of (S) . (x,x) is Element of [: the carrier of (S), the carrier of (S):]
[x,x] is set
{x,x} is non empty finite set
{x} is non empty finite set
{{x,x},{x}} is non empty with_non-empty_elements non empty-membered finite V30() set
the ObjectMap of (S) . [x,x] is set
K57(( the ObjectMap of (S) . (x,x))) is set
<^((S) . x),((S) . x)^> is set
latt x is non empty V87() reflexive transitive antisymmetric with_suprema with_infima RelStr
latt x is non empty V87() reflexive transitive antisymmetric with_suprema with_infima RelStr
@ x is Relation-like the carrier of (latt x) -defined the carrier of (latt x) -valued Function-like non empty V22( the carrier of (latt x)) quasi_total monotone Element of bool [: the carrier of (latt x), the carrier of (latt x):]
the carrier of (latt x) is non empty set
the carrier of (latt x) is non empty set
[: the carrier of (latt x), the carrier of (latt x):] is Relation-like non empty set
bool [: the carrier of (latt x), the carrier of (latt x):] is non empty set
((latt x),(latt x),(@ x)) is Relation-like the carrier of (latt x) -defined the carrier of (latt x) -valued Function-like non empty V22( the carrier of (latt x)) quasi_total Element of bool [: the carrier of (latt x), the carrier of (latt x):]
[: the carrier of (latt x), the carrier of (latt x):] is Relation-like non empty set
bool [: the carrier of (latt x), the carrier of (latt x):] is non empty set
x is Element of the carrier of (S)
latt x is non empty V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
x is Element of the carrier of (S)
latt x is non empty V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
x is Element of the carrier of (S)
x is Element of the carrier of (S)
<^x,x^> is set
x is Relation-like Function-like Element of <^x,x^>
@ x is Relation-like the carrier of (latt x) -defined the carrier of (latt x) -valued Function-like non empty V22( the carrier of (latt x)) quasi_total monotone Element of bool [: the carrier of (latt x), the carrier of (latt x):]
latt x is non empty V87() reflexive transitive antisymmetric with_suprema with_infima RelStr
the carrier of (latt x) is non empty set
latt x is non empty V87() reflexive transitive antisymmetric with_suprema with_infima RelStr
the carrier of (latt x) is non empty set
[: the carrier of (latt x), the carrier of (latt x):] is Relation-like non empty set
bool [: the carrier of (latt x), the carrier of (latt x):] is non empty set
y is Relation-like Function-like Element of <^x,x^>
@ y is Relation-like the carrier of (latt x) -defined the carrier of (latt x) -valued Function-like non empty V22( the carrier of (latt x)) quasi_total monotone Element of bool [: the carrier of (latt x), the carrier of (latt x):]
latt x is non empty V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
latt x is non empty V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
a is non empty set
a is non empty set
((latt x),(latt x),(@ x)) is Relation-like the carrier of (latt x) -defined the carrier of (latt x) -valued Function-like non empty V22( the carrier of (latt x)) quasi_total Element of bool [: the carrier of (latt x), the carrier of (latt x):]
[: the carrier of (latt x), the carrier of (latt x):] is Relation-like non empty set
bool [: the carrier of (latt x), the carrier of (latt x):] is non empty set
((latt x),(latt x),(@ y)) is Relation-like the carrier of (latt x) -defined the carrier of (latt x) -valued Function-like non empty V22( the carrier of (latt x)) quasi_total Element of bool [: the carrier of (latt x), the carrier of (latt x):]
((latt x),(latt x),((latt x),(latt x),(@ y))) is Relation-like the carrier of (latt x) -defined the carrier of (latt x) -valued Function-like non empty V22( the carrier of (latt x)) quasi_total Element of bool [: the carrier of (latt x), the carrier of (latt x):]
the carrier of (S) is non empty set
x is Element of the carrier of (S)
x is Element of the carrier of (S)
<^x,x^> is set
latt x is non empty V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
y is non empty set
the carrier of (latt x) is non empty set
y is non empty set
latt x is non empty V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
y is non empty set
the carrier of (latt x) is non empty set
y is non empty set
y is Element of the carrier of (S)
a is Element of the carrier of (S)
b is Element of the carrier of (S)
latt b is non empty V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
a is Element of the carrier of (S)
latt a is non empty V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
x is Relation-like Function-like Element of <^x,x^>
@ x is Relation-like the carrier of (latt x) -defined the carrier of (latt x) -valued Function-like non empty V22( the carrier of (latt x)) quasi_total monotone Element of bool [: the carrier of (latt x), the carrier of (latt x):]
latt x is non empty V87() reflexive transitive antisymmetric with_suprema with_infima RelStr
the carrier of (latt x) is non empty set
latt x is non empty V87() reflexive transitive antisymmetric with_suprema with_infima RelStr
the carrier of (latt x) is non empty set
[: the carrier of (latt x), the carrier of (latt x):] is Relation-like non empty set
bool [: the carrier of (latt x), the carrier of (latt x):] is non empty set
b2 is non empty set
((latt x),(latt x),(@ x)) is Relation-like the carrier of (latt x) -defined the carrier of (latt x) -valued Function-like non empty V22( the carrier of (latt x)) quasi_total Element of bool [: the carrier of (latt x), the carrier of (latt x):]
[: the carrier of (latt x), the carrier of (latt x):] is Relation-like non empty set
bool [: the carrier of (latt x), the carrier of (latt x):] is non empty set
<^b,a^> is set
b2 is non empty set
f is non empty set
b2 is Relation-like Function-like Element of <^b,a^>
g is non empty set
f is Relation-like Function-like Element of <^b,a^>
@ f is Relation-like the carrier of (latt b) -defined the carrier of (latt a) -valued Function-like non empty V22( the carrier of (latt b)) quasi_total monotone Element of bool [: the carrier of (latt b), the carrier of (latt a):]
latt b is non empty V87() reflexive transitive antisymmetric with_suprema with_infima RelStr
the carrier of (latt b) is non empty set
latt a is non empty V87() reflexive transitive antisymmetric with_suprema with_infima RelStr
the carrier of (latt a) is non empty set
[: the carrier of (latt b), the carrier of (latt a):] is Relation-like non empty set
bool [: the carrier of (latt b), the carrier of (latt a):] is non empty set
((latt b),(latt a),(@ f)) is Relation-like the carrier of (latt a) -defined the carrier of (latt b) -valued Function-like non empty V22( the carrier of (latt a)) quasi_total Element of bool [: the carrier of (latt a), the carrier of (latt b):]
[: the carrier of (latt a), the carrier of (latt b):] is Relation-like non empty set
bool [: the carrier of (latt a), the carrier of (latt b):] is non empty set
(S) is reflexive V226((S),(S)) strict Contravariant id-preserving comp-reversing contravariant Functor of (S),(S)
the carrier of (S) is non empty set
x is Element of the carrier of (S)
(S) . x is Element of the carrier of (S)
the carrier of (S) is non empty set
[: the carrier of (S), the carrier of (S):] is Relation-like non empty set
the ObjectMap of (S) is Relation-like [: the carrier of (S), the carrier of (S):] -defined [: the carrier of (S), the carrier of (S):] -valued Function-like non empty V22([: the carrier of (S), the carrier of (S):]) quasi_total Element of bool [:[: the carrier of (S), the carrier of (S):],[: the carrier of (S), the carrier of (S):]:]
[: the carrier of (S), the carrier of (S):] is Relation-like non empty set
[:[: the carrier of (S), the carrier of (S):],[: the carrier of (S), the carrier of (S):]:] is Relation-like non empty set
bool [:[: the carrier of (S), the carrier of (S):],[: the carrier of (S), the carrier of (S):]:] is non empty set
the ObjectMap of (S) . (x,x) is Element of [: the carrier of (S), the carrier of (S):]
[x,x] is set
{x,x} is non empty finite set
{x} is non empty finite set
{{x,x},{x}} is non empty with_non-empty_elements non empty-membered finite V30() set
the ObjectMap of (S) . [x,x] is set
K57(( the ObjectMap of (S) . (x,x))) is set
latt x is non empty V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
x is Element of the carrier of (S)
x is Element of the carrier of (S)
<^x,x^> is set
x is Relation-like Function-like Element of <^x,x^>
(S) . x is Relation-like Function-like Element of <^((S) . x),((S) . x)^>
(S) . x is Element of the carrier of (S)
the carrier of (S) is non empty set
[: the carrier of (S), the carrier of (S):] is Relation-like non empty set
the ObjectMap of (S) is Relation-like [: the carrier of (S), the carrier of (S):] -defined [: the carrier of (S), the carrier of (S):] -valued Function-like non empty V22([: the carrier of (S), the carrier of (S):]) quasi_total Element of bool [:[: the carrier of (S), the carrier of (S):],[: the carrier of (S), the carrier of (S):]:]
[: the carrier of (S), the carrier of (S):] is Relation-like non empty set
[:[: the carrier of (S), the carrier of (S):],[: the carrier of (S), the carrier of (S):]:] is Relation-like non empty set
bool [:[: the carrier of (S), the carrier of (S):],[: the carrier of (S), the carrier of (S):]:] is non empty set
the ObjectMap of (S) . (x,x) is Element of [: the carrier of (S), the carrier of (S):]
[x,x] is set
{x,x} is non empty finite set
{x} is non empty finite set
{{x,x},{x}} is non empty with_non-empty_elements non empty-membered finite V30() set
the ObjectMap of (S) . [x,x] is set
K57(( the ObjectMap of (S) . (x,x))) is set
(S) . x is Element of the carrier of (S)
the ObjectMap of (S) . (x,x) is Element of [: the carrier of (S), the carrier of (S):]
[x,x] is set
{x,x} is non empty finite set
{x} is non empty finite set
{{x,x},{x}} is non empty with_non-empty_elements non empty-membered finite V30() set
the ObjectMap of (S) . [x,x] is set
K57(( the ObjectMap of (S) . (x,x))) is set
<^((S) . x),((S) . x)^> is set
latt x is non empty V87() reflexive transitive antisymmetric with_suprema with_infima RelStr
latt x is non empty V87() reflexive transitive antisymmetric with_suprema with_infima RelStr
@ x is Relation-like the carrier of (latt x) -defined the carrier of (latt x) -valued Function-like non empty V22( the carrier of (latt x)) quasi_total monotone Element of bool [: the carrier of (latt x), the carrier of (latt x):]
the carrier of (latt x) is non empty set
the carrier of (latt x) is non empty set
[: the carrier of (latt x), the carrier of (latt x):] is Relation-like non empty set
bool [: the carrier of (latt x), the carrier of (latt x):] is non empty set
((latt x),(latt x),(@ x)) is Relation-like the carrier of (latt x) -defined the carrier of (latt x) -valued Function-like non empty V22( the carrier of (latt x)) quasi_total Element of bool [: the carrier of (latt x), the carrier of (latt x):]
[: the carrier of (latt x), the carrier of (latt x):] is Relation-like non empty set
bool [: the carrier of (latt x), the carrier of (latt x):] is non empty set
x is Element of the carrier of (S)
latt x is non empty V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
x is Element of the carrier of (S)
latt x is non empty V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
x is Element of the carrier of (S)
x is Element of the carrier of (S)
<^x,x^> is set
x is Relation-like Function-like Element of <^x,x^>
@ x is Relation-like the carrier of (latt x) -defined the carrier of (latt x) -valued Function-like non empty V22( the carrier of (latt x)) quasi_total monotone Element of bool [: the carrier of (latt x), the carrier of (latt x):]
latt x is non empty V87() reflexive transitive antisymmetric with_suprema with_infima RelStr
the carrier of (latt x) is non empty set
latt x is non empty V87() reflexive transitive antisymmetric with_suprema with_infima RelStr
the carrier of (latt x) is non empty set
[: the carrier of (latt x), the carrier of (latt x):] is Relation-like non empty set
bool [: the carrier of (latt x), the carrier of (latt x):] is non empty set
y is Relation-like Function-like Element of <^x,x^>
@ y is Relation-like the carrier of (latt x) -defined the carrier of (latt x) -valued Function-like non empty V22( the carrier of (latt x)) quasi_total monotone Element of bool [: the carrier of (latt x), the carrier of (latt x):]
latt x is non empty V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
latt x is non empty V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
a is non empty set
a is non empty set
((latt x),(latt x),(@ x)) is Relation-like the carrier of (latt x) -defined the carrier of (latt x) -valued Function-like non empty V22( the carrier of (latt x)) quasi_total Element of bool [: the carrier of (latt x), the carrier of (latt x):]
[: the carrier of (latt x), the carrier of (latt x):] is Relation-like non empty set
bool [: the carrier of (latt x), the carrier of (latt x):] is non empty set
((latt x),(latt x),(@ y)) is Relation-like the carrier of (latt x) -defined the carrier of (latt x) -valued Function-like non empty V22( the carrier of (latt x)) quasi_total Element of bool [: the carrier of (latt x), the carrier of (latt x):]
((latt x),(latt x),((latt x),(latt x),(@ y))) is Relation-like the carrier of (latt x) -defined the carrier of (latt x) -valued Function-like non empty V22( the carrier of (latt x)) quasi_total Element of bool [: the carrier of (latt x), the carrier of (latt x):]
the carrier of (S) is non empty set
x is Element of the carrier of (S)
x is Element of the carrier of (S)
<^x,x^> is set
latt x is non empty V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
y is non empty set
the carrier of (latt x) is non empty set
y is non empty set
latt x is non empty V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
y is non empty set
the carrier of (latt x) is non empty set
y is non empty set
y is Element of the carrier of (S)
a is Element of the carrier of (S)
b is Element of the carrier of (S)
latt b is non empty V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
a is Element of the carrier of (S)
latt a is non empty V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
x is Relation-like Function-like Element of <^x,x^>
@ x is Relation-like the carrier of (latt x) -defined the carrier of (latt x) -valued Function-like non empty V22( the carrier of (latt x)) quasi_total monotone Element of bool [: the carrier of (latt x), the carrier of (latt x):]
latt x is non empty V87() reflexive transitive antisymmetric with_suprema with_infima RelStr
the carrier of (latt x) is non empty set
latt x is non empty V87() reflexive transitive antisymmetric with_suprema with_infima RelStr
the carrier of (latt x) is non empty set
[: the carrier of (latt x), the carrier of (latt x):] is Relation-like non empty set
bool [: the carrier of (latt x), the carrier of (latt x):] is non empty set
b2 is non empty set
((latt x),(latt x),(@ x)) is Relation-like the carrier of (latt x) -defined the carrier of (latt x) -valued Function-like non empty V22( the carrier of (latt x)) quasi_total Element of bool [: the carrier of (latt x), the carrier of (latt x):]
[: the carrier of (latt x), the carrier of (latt x):] is Relation-like non empty set
bool [: the carrier of (latt x), the carrier of (latt x):] is non empty set
<^b,a^> is set
b2 is non empty set
f is non empty set
b2 is Relation-like Function-like Element of <^b,a^>
g is non empty set
f is Relation-like Function-like Element of <^b,a^>
@ f is Relation-like the carrier of (latt b) -defined the carrier of (latt a) -valued Function-like non empty V22( the carrier of (latt b)) quasi_total monotone Element of bool [: the carrier of (latt b), the carrier of (latt a):]
latt b is non empty V87() reflexive transitive antisymmetric with_suprema with_infima RelStr
the carrier of (latt b) is non empty set
latt a is non empty V87() reflexive transitive antisymmetric with_suprema with_infima RelStr
the carrier of (latt a) is non empty set
[: the carrier of (latt b), the carrier of (latt a):] is Relation-like non empty set
bool [: the carrier of (latt b), the carrier of (latt a):] is non empty set
((latt a),(latt b),(@ f)) is Relation-like the carrier of (latt a) -defined the carrier of (latt b) -valued Function-like non empty V22( the carrier of (latt a)) quasi_total Element of bool [: the carrier of (latt a), the carrier of (latt b):]
[: the carrier of (latt a), the carrier of (latt b):] is Relation-like non empty set
bool [: the carrier of (latt a), the carrier of (latt b):] is non empty set
S is non empty non empty-membered set
(S) is non empty transitive strict semi-functional V189() with_units reflexive para-functional set-id-inheriting concrete carrier-underlaid lattice-wise with_complete_lattices AltCatStr
(S) is non empty transitive strict semi-functional V189() with_units reflexive para-functional set-id-inheriting concrete carrier-underlaid lattice-wise with_complete_lattices AltCatStr
(S) is reflexive V226((S),(S)) strict Contravariant id-preserving comp-reversing contravariant bijective Functor of (S),(S)
(S) " is strict FunctorStr over (S),(S)
(S) is reflexive V226((S),(S)) strict Contravariant id-preserving comp-reversing contravariant bijective Functor of (S),(S)
(S) " is strict FunctorStr over (S),(S)
the carrier of (S) is non empty set
x is Element of the carrier of (S)
(S) . x is Element of the carrier of (S)
the carrier of (S) is non empty set
[: the carrier of (S), the carrier of (S):] is Relation-like non empty set
the ObjectMap of (S) is Relation-like [: the carrier of (S), the carrier of (S):] -defined [: the carrier of (S), the carrier of (S):] -valued Function-like non empty V22([: the carrier of (S), the carrier of (S):]) quasi_total Element of bool [:[: the carrier of (S), the carrier of (S):],[: the carrier of (S), the carrier of (S):]:]
[: the carrier of (S), the carrier of (S):] is Relation-like non empty set
[:[: the carrier of (S), the carrier of (S):],[: the carrier of (S), the carrier of (S):]:] is Relation-like non empty set
bool [:[: the carrier of (S), the carrier of (S):],[: the carrier of (S), the carrier of (S):]:] is non empty set
the ObjectMap of (S) . (x,x) is Element of [: the carrier of (S), the carrier of (S):]
[x,x] is set
{x,x} is non empty finite set
{x} is non empty finite set
{{x,x},{x}} is non empty with_non-empty_elements non empty-membered finite V30() set
the ObjectMap of (S) . [x,x] is set
K57(( the ObjectMap of (S) . (x,x))) is set
(S) . ((S) . x) is Element of the carrier of (S)
the ObjectMap of (S) is Relation-like [: the carrier of (S), the carrier of (S):] -defined [: the carrier of (S), the carrier of (S):] -valued Function-like non empty V22([: the carrier of (S), the carrier of (S):]) quasi_total Element of bool [:[: the carrier of (S), the carrier of (S):],[: the carrier of (S), the carrier of (S):]:]
[:[: the carrier of (S), the carrier of (S):],[: the carrier of (S), the carrier of (S):]:] is Relation-like non empty set
bool [:[: the carrier of (S), the carrier of (S):],[: the carrier of (S), the carrier of (S):]:] is non empty set
the ObjectMap of (S) . (((S) . x),((S) . x)) is Element of [: the carrier of (S), the carrier of (S):]
[((S) . x),((S) . x)] is set
{((S) . x),((S) . x)} is non empty finite set
{((S) . x)} is non empty finite set
{{((S) . x),((S) . x)},{((S) . x)}} is non empty with_non-empty_elements non empty-membered finite V30() set
the ObjectMap of (S) . [((S) . x),((S) . x)] is set
K57(( the ObjectMap of (S) . (((S) . x),((S) . x)))) is set
latt ((S) . x) is non empty V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
latt x is non empty V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
x is Element of the carrier of (S)
x is Element of the carrier of (S)
<^x,x^> is set
(S) . x is Element of the carrier of (S)
the ObjectMap of (S) . (x,x) is Element of [: the carrier of (S), the carrier of (S):]
[x,x] is set
{x,x} is non empty finite set
{x} is non empty finite set
{{x,x},{x}} is non empty with_non-empty_elements non empty-membered finite V30() set
the ObjectMap of (S) . [x,x] is set
K57(( the ObjectMap of (S) . (x,x))) is set
(S) . x is Element of the carrier of (S)
the ObjectMap of (S) . (x,x) is Element of [: the carrier of (S), the carrier of (S):]
[x,x] is set
{x,x} is non empty finite set
{x} is non empty finite set
{{x,x},{x}} is non empty with_non-empty_elements non empty-membered finite V30() set
the ObjectMap of (S) . [x,x] is set
K57(( the ObjectMap of (S) . (x,x))) is set
<^((S) . x),((S) . x)^> is set
x is Relation-like Function-like Element of <^x,x^>
(S) . x is Relation-like Function-like Element of <^((S) . x),((S) . x)^>
latt x is non empty V87() reflexive transitive antisymmetric with_suprema with_infima RelStr
latt x is non empty V87() reflexive transitive antisymmetric with_suprema with_infima RelStr
@ x is Relation-like the carrier of (latt x) -defined the carrier of (latt x) -valued Function-like non empty V22( the carrier of (latt x)) quasi_total monotone Element of bool [: the carrier of (latt x), the carrier of (latt x):]
the carrier of (latt x) is non empty set
the carrier of (latt x) is non empty set
[: the carrier of (latt x), the carrier of (latt x):] is Relation-like non empty set
bool [: the carrier of (latt x), the carrier of (latt x):] is non empty set
((latt x),(latt x),(@ x)) is Relation-like the carrier of (latt x) -defined the carrier of (latt x) -valued Function-like non empty V22( the carrier of (latt x)) quasi_total Element of bool [: the carrier of (latt x), the carrier of (latt x):]
[: the carrier of (latt x), the carrier of (latt x):] is Relation-like non empty set
bool [: the carrier of (latt x), the carrier of (latt x):] is non empty set
@ ((S) . x) is Relation-like the carrier of (latt ((S) . x)) -defined the carrier of (latt ((S) . x)) -valued Function-like non empty V22( the carrier of (latt ((S) . x))) quasi_total monotone Element of bool [: the carrier of (latt ((S) . x)), the carrier of (latt ((S) . x)):]
latt ((S) . x) is non empty V87() reflexive transitive antisymmetric with_suprema with_infima RelStr
the carrier of (latt ((S) . x)) is non empty set
latt ((S) . x) is non empty V87() reflexive transitive antisymmetric with_suprema with_infima RelStr
the carrier of (latt ((S) . x)) is non empty set
[: the carrier of (latt ((S) . x)), the carrier of (latt ((S) . x)):] is Relation-like non empty set
bool [: the carrier of (latt ((S) . x)), the carrier of (latt ((S) . x)):] is non empty set
latt x is non empty V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
latt x is non empty V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
y is non empty set
(S) . ((S) . x) is Relation-like Function-like Element of <^((S) . ((S) . x)),((S) . ((S) . x))^>
(S) . ((S) . x) is Element of the carrier of (S)
the ObjectMap of (S) . (((S) . x),((S) . x)) is Element of [: the carrier of (S), the carrier of (S):]
[((S) . x),((S) . x)] is set
{((S) . x),((S) . x)} is non empty finite set
{((S) . x)} is non empty finite set
{{((S) . x),((S) . x)},{((S) . x)}} is non empty with_non-empty_elements non empty-membered finite V30() set
the ObjectMap of (S) . [((S) . x),((S) . x)] is set
K57(( the ObjectMap of (S) . (((S) . x),((S) . x)))) is set
(S) . ((S) . x) is Element of the carrier of (S)
the ObjectMap of (S) . (((S) . x),((S) . x)) is Element of [: the carrier of (S), the carrier of (S):]
[((S) . x),((S) . x)] is set
{((S) . x),((S) . x)} is non empty finite set
{((S) . x)} is non empty finite set
{{((S) . x),((S) . x)},{((S) . x)}} is non empty with_non-empty_elements non empty-membered finite V30() set
the ObjectMap of (S) . [((S) . x),((S) . x)] is set
K57(( the ObjectMap of (S) . (((S) . x),((S) . x)))) is set
<^((S) . ((S) . x)),((S) . ((S) . x))^> is set
((latt ((S) . x)),(latt ((S) . x)),(@ ((S) . x))) is Relation-like the carrier of (latt ((S) . x)) -defined the carrier of (latt ((S) . x)) -valued Function-like non empty V22( the carrier of (latt ((S) . x))) quasi_total Element of bool [: the carrier of (latt ((S) . x)), the carrier of (latt ((S) . x)):]
[: the carrier of (latt ((S) . x)), the carrier of (latt ((S) . x)):] is Relation-like non empty set
bool [: the carrier of (latt ((S) . x)), the carrier of (latt ((S) . x)):] is non empty set
y is Element of the carrier of (S)
(S) . y is Element of the carrier of (S)
the ObjectMap of (S) . (y,y) is Element of [: the carrier of (S), the carrier of (S):]
[y,y] is set
{y,y} is non empty finite set
{y} is non empty finite set
{{y,y},{y}} is non empty with_non-empty_elements non empty-membered finite V30() set
the ObjectMap of (S) . [y,y] is set
K57(( the ObjectMap of (S) . (y,y))) is set
(S) . ((S) . y) is Element of the carrier of (S)
the ObjectMap of (S) . (((S) . y),((S) . y)) is Element of [: the carrier of (S), the carrier of (S):]
[((S) . y),((S) . y)] is set
{((S) . y),((S) . y)} is non empty finite set
{((S) . y)} is non empty finite set
{{((S) . y),((S) . y)},{((S) . y)}} is non empty with_non-empty_elements non empty-membered finite V30() set
the ObjectMap of (S) . [((S) . y),((S) . y)] is set
K57(( the ObjectMap of (S) . (((S) . y),((S) . y)))) is set
latt ((S) . y) is non empty V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
latt y is non empty V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
y is Element of the carrier of (S)
a is Element of the carrier of (S)
<^y,a^> is set
(S) . a is Element of the carrier of (S)
the ObjectMap of (S) . (a,a) is Element of [: the carrier of (S), the carrier of (S):]
[a,a] is set
{a,a} is non empty finite set
{a} is non empty finite set
{{a,a},{a}} is non empty with_non-empty_elements non empty-membered finite V30() set
the ObjectMap of (S) . [a,a] is set
K57(( the ObjectMap of (S) . (a,a))) is set
(S) . y is Element of the carrier of (S)
the ObjectMap of (S) . (y,y) is Element of [: the carrier of (S), the carrier of (S):]
[y,y] is set
{y,y} is non empty finite set
{y} is non empty finite set
{{y,y},{y}} is non empty with_non-empty_elements non empty-membered finite V30() set
the ObjectMap of (S) . [y,y] is set
K57(( the ObjectMap of (S) . (y,y))) is set
<^((S) . a),((S) . y)^> is set
b is Relation-like Function-like Element of <^y,a^>
(S) . b is Relation-like Function-like Element of <^((S) . a),((S) . y)^>
latt y is non empty V87() reflexive transitive antisymmetric with_suprema with_infima RelStr
latt a is non empty V87() reflexive transitive antisymmetric with_suprema with_infima RelStr
@ b is Relation-like the carrier of (latt y) -defined the carrier of (latt a) -valued Function-like non empty V22( the carrier of (latt y)) quasi_total monotone Element of bool [: the carrier of (latt y), the carrier of (latt a):]
the carrier of (latt y) is non empty set
the carrier of (latt a) is non empty set
[: the carrier of (latt y), the carrier of (latt a):] is Relation-like non empty set
bool [: the carrier of (latt y), the carrier of (latt a):] is non empty set
((latt y),(latt a),(@ b)) is Relation-like the carrier of (latt a) -defined the carrier of (latt y) -valued Function-like non empty V22( the carrier of (latt a)) quasi_total Element of bool [: the carrier of (latt a), the carrier of (latt y):]
[: the carrier of (latt a), the carrier of (latt y):] is Relation-like non empty set
bool [: the carrier of (latt a), the carrier of (latt y):] is non empty set
@ ((S) . b) is Relation-like the carrier of (latt ((S) . a)) -defined the carrier of (latt ((S) . y)) -valued Function-like non empty V22( the carrier of (latt ((S) . a))) quasi_total monotone Element of bool [: the carrier of (latt ((S) . a)), the carrier of (latt ((S) . y)):]
latt ((S) . a) is non empty V87() reflexive transitive antisymmetric with_suprema with_infima RelStr
the carrier of (latt ((S) . a)) is non empty set
latt ((S) . y) is non empty V87() reflexive transitive antisymmetric with_suprema with_infima RelStr
the carrier of (latt ((S) . y)) is non empty set
[: the carrier of (latt ((S) . a)), the carrier of (latt ((S) . y)):] is Relation-like non empty set
bool [: the carrier of (latt ((S) . a)), the carrier of (latt ((S) . y)):] is non empty set
latt y is non empty V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
latt a is non empty V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
a is non empty set
(S) . ((S) . b) is Relation-like Function-like Element of <^((S) . ((S) . y)),((S) . ((S) . a))^>
(S) . ((S) . y) is Element of the carrier of (S)
the ObjectMap of (S) . (((S) . y),((S) . y)) is Element of [: the carrier of (S), the carrier of (S):]
[((S) . y),((S) . y)] is set
{((S) . y),((S) . y)} is non empty finite set
{((S) . y)} is non empty finite set
{{((S) . y),((S) . y)},{((S) . y)}} is non empty with_non-empty_elements non empty-membered finite V30() set
the ObjectMap of (S) . [((S) . y),((S) . y)] is set
K57(( the ObjectMap of (S) . (((S) . y),((S) . y)))) is set
(S) . ((S) . a) is Element of the carrier of (S)
the ObjectMap of (S) . (((S) . a),((S) . a)) is Element of [: the carrier of (S), the carrier of (S):]
[((S) . a),((S) . a)] is set
{((S) . a),((S) . a)} is non empty finite set
{((S) . a)} is non empty finite set
{{((S) . a),((S) . a)},{((S) . a)}} is non empty with_non-empty_elements non empty-membered finite V30() set
the ObjectMap of (S) . [((S) . a),((S) . a)] is set
K57(( the ObjectMap of (S) . (((S) . a),((S) . a)))) is set
<^((S) . ((S) . y)),((S) . ((S) . a))^> is set
((latt ((S) . y)),(latt ((S) . a)),(@ ((S) . b))) is Relation-like the carrier of (latt ((S) . y)) -defined the carrier of (latt ((S) . a)) -valued Function-like non empty V22( the carrier of (latt ((S) . y))) quasi_total Element of bool [: the carrier of (latt ((S) . y)), the carrier of (latt ((S) . a)):]
[: the carrier of (latt ((S) . y)), the carrier of (latt ((S) . a)):] is Relation-like non empty set
bool [: the carrier of (latt ((S) . y)), the carrier of (latt ((S) . a)):] is non empty set
S is non empty non empty-membered set
(S) is non empty transitive strict semi-functional V189() with_units reflexive para-functional set-id-inheriting concrete carrier-underlaid lattice-wise with_complete_lattices AltCatStr
(S) is non empty transitive strict semi-functional V189() with_units reflexive para-functional set-id-inheriting concrete carrier-underlaid lattice-wise with_complete_lattices AltCatStr
(S) is reflexive V226((S),(S)) strict Contravariant id-preserving comp-reversing contravariant bijective Functor of (S),(S)
(S) is reflexive V226((S),(S)) strict Contravariant id-preserving comp-reversing contravariant bijective Functor of (S),(S)
(S) * (S) is reflexive V226((S),(S)) strict Covariant id-preserving FunctorStr over (S),(S)
id (S) is reflexive V226((S),(S)) strict Covariant id-preserving comp-preserving covariant bijective Functor of (S),(S)
(S) * (S) is reflexive V226((S),(S)) strict Covariant id-preserving FunctorStr over (S),(S)
id (S) is reflexive V226((S),(S)) strict Covariant id-preserving comp-preserving covariant bijective Functor of (S),(S)
(S) " is strict FunctorStr over (S),(S)
(S) " is strict FunctorStr over (S),(S)
S is non empty non empty-membered set
(S) is non empty transitive strict semi-functional V189() with_units reflexive para-functional set-id-inheriting concrete carrier-underlaid lattice-wise with_complete_lattices AltCatStr
(S) is non empty transitive strict semi-functional V189() with_units reflexive para-functional set-id-inheriting concrete carrier-underlaid lattice-wise with_complete_lattices AltCatStr
(S) is reflexive V226((S),(S)) strict Contravariant id-preserving comp-reversing contravariant bijective Functor of (S),(S)
S is non empty V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
the carrier of S is non empty set
T is non empty V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
the carrier of T is non empty set
[: the carrier of S, the carrier of T:] is Relation-like non empty set
bool [: the carrier of S, the carrier of T:] is non empty set
g is Relation-like the carrier of S -defined the carrier of T -valued Function-like non empty V22( the carrier of S) quasi_total infs-preserving meet-preserving filtered-infs-preserving monotone Element of bool [: the carrier of S, the carrier of T:]
(S,T,g) is Relation-like the carrier of T -defined the carrier of S -valued Function-like non empty V22( the carrier of T) quasi_total sups-preserving join-preserving directed-sups-preserving monotone lower_adjoint Element of bool [: the carrier of T, the carrier of S:]
[: the carrier of T, the carrier of S:] is Relation-like non empty set
bool [: the carrier of T, the carrier of S:] is non empty set
x is non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of T
the carrier of x is non empty set
bool the carrier of x is non empty set
the InternalRel of x is Relation-like the carrier of x -defined the carrier of x -valued V22( the carrier of x) quasi_total V31() V34() V38() Element of bool [: the carrier of x, the carrier of x:]
[: the carrier of x, the carrier of x:] is Relation-like non empty set
bool [: the carrier of x, the carrier of x:] is non empty set
RelStr(# the carrier of x, the InternalRel of x #) is non empty strict V87() reflexive transitive antisymmetric upper-bounded up-complete with_suprema with_infima RelStr
the InternalRel of T is Relation-like the carrier of T -defined the carrier of T -valued V22( the carrier of T) quasi_total V31() V34() V38() Element of bool [: the carrier of T, the carrier of T:]
[: the carrier of T, the carrier of T:] is Relation-like non empty set
bool [: the carrier of T, the carrier of T:] is non empty set
RelStr(# the carrier of T, the InternalRel of T #) is non empty strict V87() reflexive transitive antisymmetric upper-bounded up-complete with_suprema with_infima RelStr
x is non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of S
the carrier of x is non empty set
the InternalRel of x is Relation-like the carrier of x -defined the carrier of x -valued V22( the carrier of x) quasi_total V31() V34() V38() Element of bool [: the carrier of x, the carrier of x:]
[: the carrier of x, the carrier of x:] is Relation-like non empty set
bool [: the carrier of x, the carrier of x:] is non empty set
RelStr(# the carrier of x, the InternalRel of x #) is non empty strict V87() reflexive transitive antisymmetric upper-bounded up-complete with_suprema with_infima RelStr
the InternalRel of S is Relation-like the carrier of S -defined the carrier of S -valued V22( the carrier of S) quasi_total V31() V34() V38() Element of bool [: the carrier of S, the carrier of S:]
[: the carrier of S, the carrier of S:] is Relation-like non empty set
bool [: the carrier of S, the carrier of S:] is non empty set
RelStr(# the carrier of S, the InternalRel of S #) is non empty strict V87() reflexive transitive antisymmetric upper-bounded up-complete with_suprema with_infima RelStr
[: the carrier of x, the carrier of x:] is Relation-like non empty set
bool [: the carrier of x, the carrier of x:] is non empty set
[: the carrier of x, the carrier of x:] is Relation-like non empty set
bool [: the carrier of x, the carrier of x:] is non empty set
y is Relation-like the carrier of x -defined the carrier of x -valued Function-like non empty V22( the carrier of x) quasi_total Element of bool [: the carrier of x, the carrier of x:]
x is open Element of bool the carrier of x
y .: x is Element of bool the carrier of x
bool the carrier of x is non empty set
uparrow (y .: x) is upper Element of bool the carrier of x
a is non empty directed Element of bool the carrier of x
"\/" (a,x) is Element of the carrier of x
b is Element of the carrier of x
a is set
y . a is set
x is Relation-like the carrier of x -defined the carrier of x -valued Function-like non empty V22( the carrier of x) quasi_total Element of bool [: the carrier of x, the carrier of x:]
f is Relation-like the carrier of x -defined the carrier of x -valued Function-like non empty V22( the carrier of x) quasi_total Element of bool [: the carrier of x, the carrier of x:]
[f,y] is Connection of x,x
{f,y} is non empty with_non-empty_elements non empty-membered finite set
{f} is non empty with_non-empty_elements non empty-membered finite set
{{f,y},{f}} is non empty with_non-empty_elements non empty-membered finite V30() set
y * f is Relation-like the carrier of x -defined the carrier of x -valued Function-like non empty V22( the carrier of x) quasi_total Element of bool [: the carrier of x, the carrier of x:]
id x is Relation-like the carrier of x -defined the carrier of x -valued Function-like one-to-one non empty V22( the carrier of x) quasi_total continuous infs-preserving sups-preserving meet-preserving join-preserving filtered-infs-preserving directed-sups-preserving isomorphic monotone projection closure kernel Element of bool [: the carrier of x, the carrier of x:]
id the carrier of x is Relation-like the carrier of x -defined the carrier of x -valued non empty V22( the carrier of x) quasi_total Element of bool [: the carrier of x, the carrier of x:]
id x is Relation-like the carrier of x -defined the carrier of x -valued Function-like one-to-one non empty V22( the carrier of x) quasi_total continuous infs-preserving sups-preserving meet-preserving join-preserving filtered-infs-preserving directed-sups-preserving isomorphic monotone projection closure kernel Element of bool [: the carrier of x, the carrier of x:]
id the carrier of x is Relation-like the carrier of x -defined the carrier of x -valued non empty V22( the carrier of x) quasi_total Element of bool [: the carrier of x, the carrier of x:]
f * y is Relation-like the carrier of x -defined the carrier of x -valued Function-like non empty V22( the carrier of x) quasi_total Element of bool [: the carrier of x, the carrier of x:]
b2 is Element of the carrier of x
(id x) . b2 is Element of the carrier of x
(f * y) . b2 is Element of the carrier of x
y . b2 is Element of the carrier of x
f . (y . b2) is Element of the carrier of x
f . b is Element of the carrier of x
f . ("\/" (a,x)) is Element of the carrier of x
f .: a is non empty Element of bool the carrier of x
"\/" ((f .: a),x) is Element of the carrier of x
g is set
f is set
f . f is set
E9 is Element of the carrier of x
(y * f) . E9 is Element of the carrier of x
f . E9 is Element of the carrier of x
y . (f . E9) is Element of the carrier of x
(id x) . E9 is Element of the carrier of x
y . g is set
(S,T,g) .: x is Element of bool the carrier of S
bool the carrier of S is non empty set
uparrow ((S,T,g) .: x) is upper Element of bool the carrier of S
the non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of T is non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of T
the non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of S is non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of S
the carrier of the non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of T is non empty set
the InternalRel of the non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of T is Relation-like the carrier of the non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of T -defined the carrier of the non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of T -valued V22( the carrier of the non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of T) quasi_total V31() V34() V38() Element of bool [: the carrier of the non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of T, the carrier of the non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of T:]
[: the carrier of the non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of T, the carrier of the non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of T:] is Relation-like non empty set
bool [: the carrier of the non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of T, the carrier of the non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of T:] is non empty set
RelStr(# the carrier of the non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of T, the InternalRel of the non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of T #) is non empty strict V87() reflexive transitive antisymmetric upper-bounded up-complete with_suprema with_infima RelStr
the carrier of the non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of S is non empty set
the InternalRel of the non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of S is Relation-like the carrier of the non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of S -defined the carrier of the non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of S -valued V22( the carrier of the non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of S) quasi_total V31() V34() V38() Element of bool [: the carrier of the non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of S, the carrier of the non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of S:]
[: the carrier of the non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of S, the carrier of the non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of S:] is Relation-like non empty set
bool [: the carrier of the non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of S, the carrier of the non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of S:] is non empty set
RelStr(# the carrier of the non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of S, the InternalRel of the non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of S #) is non empty strict V87() reflexive transitive antisymmetric upper-bounded up-complete with_suprema with_infima RelStr
[: the carrier of the non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of S, the carrier of the non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of T:] is Relation-like non empty set
bool [: the carrier of the non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of S, the carrier of the non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of T:] is non empty set
x is Relation-like the carrier of the non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of S -defined the carrier of the non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of T -valued Function-like non empty V22( the carrier of the non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of S) quasi_total Element of bool [: the carrier of the non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of S, the carrier of the non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of T:]
[: the carrier of the non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of T, the carrier of the non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of S:] is Relation-like non empty set
bool [: the carrier of the non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of T, the carrier of the non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of S:] is non empty set
b is Element of bool the carrier of S
g .: b is Element of bool the carrier of T
bool the carrier of T is non empty set
"\/" ((g .: b),T) is Element of the carrier of T
"\/" (b,S) is Element of the carrier of S
g . ("\/" (b,S)) is Element of the carrier of T
bool the carrier of the non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of S is non empty set
a is Element of bool the carrier of the non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of S
x is Relation-like the carrier of the non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of S -defined the carrier of the non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of T -valued Function-like non empty V22( the carrier of the non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of S) quasi_total infs-preserving meet-preserving filtered-infs-preserving monotone Element of bool [: the carrier of the non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of S, the carrier of the non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of T:]
b2 is non empty directed Element of bool the carrier of the non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of S
x .: b2 is non empty Element of bool the carrier of the non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of T
bool the carrier of the non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of T is non empty set
"\/" ((x .: b2), the non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of T) is Element of the carrier of the non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of T
downarrow ("\/" ((x .: b2), the non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of T)) is non empty directed lower property(S) Element of bool the carrier of the non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of T
{("\/" ((x .: b2), the non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of T))} is non empty finite Element of bool the carrier of the non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of T
downarrow {("\/" ((x .: b2), the non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of T))} is non empty lower property(S) Element of bool the carrier of the non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of T
(downarrow ("\/" ((x .: b2), the non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of T))) ` is closed_under_directed_sups Element of bool the carrier of the non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of T
the carrier of the non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of T \ (downarrow ("\/" ((x .: b2), the non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of T))) is set
(S,T,g) .: ((downarrow ("\/" ((x .: b2), the non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of T))) `) is Element of bool the carrier of S
uparrow ((S,T,g) .: ((downarrow ("\/" ((x .: b2), the non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of T))) `)) is upper Element of bool the carrier of S
downarrow ("\/" ((g .: b),T)) is non empty directed lower Element of bool the carrier of T
{("\/" ((g .: b),T))} is non empty finite Element of bool the carrier of T
downarrow {("\/" ((g .: b),T))} is non empty lower Element of bool the carrier of T
[g,(S,T,g)] is Connection of S,T
{g,(S,T,g)} is non empty with_non-empty_elements non empty-membered finite set
{g} is non empty with_non-empty_elements non empty-membered finite set
{{g,(S,T,g)},{g}} is non empty with_non-empty_elements non empty-membered finite V30() set
(S,T,g) * g is Relation-like the carrier of S -defined the carrier of S -valued Function-like non empty V22( the carrier of S) quasi_total Element of bool [: the carrier of S, the carrier of S:]
id S is Relation-like the carrier of S -defined the carrier of S -valued Function-like one-to-one non empty V22( the carrier of S) quasi_total infs-preserving sups-preserving meet-preserving join-preserving filtered-infs-preserving directed-sups-preserving isomorphic monotone projection closure kernel Element of bool [: the carrier of S, the carrier of S:]
id the carrier of S is Relation-like the carrier of S -defined the carrier of S -valued non empty V22( the carrier of S) quasi_total Element of bool [: the carrier of S, the carrier of S:]
id T is Relation-like the carrier of T -defined the carrier of T -valued Function-like one-to-one non empty V22( the carrier of T) quasi_total infs-preserving sups-preserving meet-preserving join-preserving filtered-infs-preserving directed-sups-preserving isomorphic monotone projection closure kernel Element of bool [: the carrier of T, the carrier of T:]
id the carrier of T is Relation-like the carrier of T -defined the carrier of T -valued non empty V22( the carrier of T) quasi_total Element of bool [: the carrier of T, the carrier of T:]
g * (S,T,g) is Relation-like the carrier of T -defined the carrier of T -valued Function-like non empty V22( the carrier of T) quasi_total Element of bool [: the carrier of T, the carrier of T:]
(id S) . ("\/" (b,S)) is Element of the carrier of S
((S,T,g) * g) . ("\/" (b,S)) is Element of the carrier of S
(S,T,g) . (g . ("\/" (b,S))) is Element of the carrier of S
a is Relation-like the carrier of the non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of T -defined the carrier of the non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of S -valued Function-like non empty V22( the carrier of the non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of T) quasi_total Element of bool [: the carrier of the non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of T, the carrier of the non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of S:]
f is Element of the carrier of the non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of S
x . f is Element of the carrier of the non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of T
a . (x . f) is Element of the carrier of the non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of S
"\/" (b2, the non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of S) is Element of the carrier of the non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of S
a .: ((downarrow ("\/" ((x .: b2), the non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of T))) `) is Element of bool the carrier of the non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of S
uparrow (a .: ((downarrow ("\/" ((x .: b2), the non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of T))) `)) is upper Element of bool the carrier of the non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of S
f is set
E9 is Element of the carrier of the non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of S
a is Element of the carrier of the non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of S
b is set
a . b is set
a9 is Element of the carrier of the non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of T
b9 is Element of the carrier of T
(id T) . b9 is Element of the carrier of T
(g * (S,T,g)) . b9 is Element of the carrier of T
(S,T,g) . b9 is Element of the carrier of S
g . ((S,T,g) . b9) is Element of the carrier of T
a . a9 is Element of the carrier of the non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of S
x . (a . a9) is Element of the carrier of the non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of T
x . E9 is Element of the carrier of the non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of T
S is non empty V87() reflexive RelStr
the carrier of S is non empty set
T is non empty V87() reflexive RelStr
the carrier of T is non empty set
[: the carrier of S, the carrier of T:] is Relation-like non empty set
bool [: the carrier of S, the carrier of T:] is non empty set
S is non empty V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
the carrier of S is non empty set
T is non empty V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
the carrier of T is non empty set
[: the carrier of S, the carrier of T:] is Relation-like non empty set
bool [: the carrier of S, the carrier of T:] is non empty set
g is Relation-like the carrier of S -defined the carrier of T -valued Function-like non empty V22( the carrier of S) quasi_total infs-preserving meet-preserving filtered-infs-preserving monotone Element of bool [: the carrier of S, the carrier of T:]
(S,T,g) is Relation-like the carrier of T -defined the carrier of S -valued Function-like non empty V22( the carrier of T) quasi_total sups-preserving join-preserving directed-sups-preserving monotone lower_adjoint Element of bool [: the carrier of T, the carrier of S:]
[: the carrier of T, the carrier of S:] is Relation-like non empty set
bool [: the carrier of T, the carrier of S:] is non empty set
[g,(S,T,g)] is Connection of S,T
{g,(S,T,g)} is non empty with_non-empty_elements non empty-membered finite set
{g} is non empty with_non-empty_elements non empty-membered finite set
{{g,(S,T,g)},{g}} is non empty with_non-empty_elements non empty-membered finite V30() set
x is Element of the carrier of T
x is Element of the carrier of T
(S,T,g) . x is Element of the carrier of S
(S,T,g) . x is Element of the carrier of S
bool the carrier of S is non empty set
x is non empty directed Element of bool the carrier of S
"\/" (x,S) is Element of the carrier of S
g . ("\/" (x,S)) is Element of the carrier of T
g .: x is non empty Element of bool the carrier of T
bool the carrier of T is non empty set
"\/" ((g .: x),T) is Element of the carrier of T
y is Element of the carrier of T
a is set
g . a is set
b is Element of the carrier of S
S is non empty V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
the carrier of S is non empty set
T is non empty V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete satisfying_axiom_of_approximation continuous with_suprema with_infima complete RelStr
the carrier of T is non empty set
[: the carrier of S, the carrier of T:] is Relation-like non empty set
bool [: the carrier of S, the carrier of T:] is non empty set
g is Relation-like the carrier of S -defined the carrier of T -valued Function-like non empty V22( the carrier of S) quasi_total infs-preserving meet-preserving filtered-infs-preserving monotone Element of bool [: the carrier of S, the carrier of T:]
(S,T,g) is Relation-like the carrier of T -defined the carrier of S -valued Function-like non empty V22( the carrier of T) quasi_total sups-preserving join-preserving directed-sups-preserving monotone lower_adjoint Element of bool [: the carrier of T, the carrier of S:]
[: the carrier of T, the carrier of S:] is Relation-like non empty set
bool [: the carrier of T, the carrier of S:] is non empty set
bool the carrier of S is non empty set
x is Element of bool the carrier of S
g .: x is Element of bool the carrier of T
bool the carrier of T is non empty set
"\/" ((g .: x),T) is Element of the carrier of T
"\/" (x,S) is Element of the carrier of S
g . ("\/" (x,S)) is Element of the carrier of T
waybelow (g . ("\/" (x,S))) is non empty directed lower Element of bool the carrier of T
{ b1 where b1 is Element of the carrier of T : b1 is_way_below g . ("\/" (x,S)) } is set
"\/" ((waybelow (g . ("\/" (x,S)))),T) is Element of the carrier of T
x is Element of the carrier of T
(S,T,g) . x is Element of the carrier of S
(S,T,g) . (g . ("\/" (x,S))) is Element of the carrier of S
[g,(S,T,g)] is Connection of S,T
{g,(S,T,g)} is non empty with_non-empty_elements non empty-membered finite set
{g} is non empty with_non-empty_elements non empty-membered finite set
{{g,(S,T,g)},{g}} is non empty with_non-empty_elements non empty-membered finite V30() set
(S,T,g) * g is Relation-like the carrier of S -defined the carrier of S -valued Function-like non empty V22( the carrier of S) quasi_total Element of bool [: the carrier of S, the carrier of S:]
[: the carrier of S, the carrier of S:] is Relation-like non empty set
bool [: the carrier of S, the carrier of S:] is non empty set
id S is Relation-like the carrier of S -defined the carrier of S -valued Function-like one-to-one non empty V22( the carrier of S) quasi_total infs-preserving sups-preserving meet-preserving join-preserving filtered-infs-preserving directed-sups-preserving isomorphic monotone projection closure kernel Element of bool [: the carrier of S, the carrier of S:]
id the carrier of S is Relation-like the carrier of S -defined the carrier of S -valued non empty V22( the carrier of S) quasi_total Element of bool [: the carrier of S, the carrier of S:]
(id S) . ("\/" (x,S)) is Element of the carrier of S
((S,T,g) * g) . ("\/" (x,S)) is Element of the carrier of S
x is Element of the carrier of S
g . x is Element of the carrier of T
S is TopSpace-like TopStruct
the carrier of S is set
T is TopSpace-like TopStruct
the carrier of T is set
[: the carrier of S, the carrier of T:] is Relation-like set
bool [: the carrier of S, the carrier of T:] is non empty set
S is non empty TopSpace-like TopStruct
the carrier of S is non empty set
T is non empty TopSpace-like TopStruct
the carrier of T is non empty set
[: the carrier of S, the carrier of T:] is Relation-like non empty set
bool [: the carrier of S, the carrier of T:] is non empty set
g is Relation-like the carrier of S -defined the carrier of T -valued Function-like non empty V22( the carrier of S) quasi_total Element of bool [: the carrier of S, the carrier of T:]
Image g is TopSpace-like SubSpace of T
corestr g is Relation-like the carrier of S -defined the carrier of (Image g) -valued Function-like quasi_total Element of bool [: the carrier of S, the carrier of (Image g):]
the carrier of (Image g) is set
[: the carrier of S, the carrier of (Image g):] is Relation-like set
bool [: the carrier of S, the carrier of (Image g):] is non empty set
rng g is non empty Element of bool the carrier of T
bool the carrier of T is non empty set
T | (rng g) is non empty strict TopSpace-like SubSpace of T
bool the carrier of S is non empty set
the carrier of (T | (rng g)) is non empty set
bool the carrier of (T | (rng g)) is non empty set
x is Element of bool the carrier of S
(corestr g) .: x is Element of bool the carrier of (Image g)
bool the carrier of (Image g) is non empty set
bool the carrier of S is non empty set
x is open Element of bool the carrier of S
g .: x is Element of bool the carrier of T
the carrier of (T | (rng g)) is non empty set
bool the carrier of (T | (rng g)) is non empty set
S is non empty V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
the carrier of S is non empty set
T is non empty V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
the carrier of T is non empty set
[: the carrier of S, the carrier of T:] is Relation-like non empty set
bool [: the carrier of S, the carrier of T:] is non empty set
g is Relation-like the carrier of S -defined the carrier of T -valued Function-like non empty V22( the carrier of S) quasi_total infs-preserving meet-preserving filtered-infs-preserving monotone Element of bool [: the carrier of S, the carrier of T:]
(S,T,g) is Relation-like the carrier of T -defined the carrier of S -valued Function-like non empty V22( the carrier of T) quasi_total sups-preserving join-preserving directed-sups-preserving monotone lower_adjoint Element of bool [: the carrier of T, the carrier of S:]
[: the carrier of T, the carrier of S:] is Relation-like non empty set
bool [: the carrier of T, the carrier of S:] is non empty set
rng (S,T,g) is non empty Element of bool the carrier of S
bool the carrier of S is non empty set
x is non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of T
the carrier of x is non empty set
bool the carrier of x is non empty set
the InternalRel of x is Relation-like the carrier of x -defined the carrier of x -valued V22( the carrier of x) quasi_total V31() V34() V38() Element of bool [: the carrier of x, the carrier of x:]
[: the carrier of x, the carrier of x:] is Relation-like non empty set
bool [: the carrier of x, the carrier of x:] is non empty set
RelStr(# the carrier of x, the InternalRel of x #) is non empty strict V87() reflexive transitive antisymmetric upper-bounded up-complete with_suprema with_infima RelStr
the InternalRel of T is Relation-like the carrier of T -defined the carrier of T -valued V22( the carrier of T) quasi_total V31() V34() V38() Element of bool [: the carrier of T, the carrier of T:]
[: the carrier of T, the carrier of T:] is Relation-like non empty set
bool [: the carrier of T, the carrier of T:] is non empty set
RelStr(# the carrier of T, the InternalRel of T #) is non empty strict V87() reflexive transitive antisymmetric upper-bounded up-complete with_suprema with_infima RelStr
x is non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of S
the carrier of x is non empty set
the InternalRel of x is Relation-like the carrier of x -defined the carrier of x -valued V22( the carrier of x) quasi_total V31() V34() V38() Element of bool [: the carrier of x, the carrier of x:]
[: the carrier of x, the carrier of x:] is Relation-like non empty set
bool [: the carrier of x, the carrier of x:] is non empty set
RelStr(# the carrier of x, the InternalRel of x #) is non empty strict V87() reflexive transitive antisymmetric upper-bounded up-complete with_suprema with_infima RelStr
the InternalRel of S is Relation-like the carrier of S -defined the carrier of S -valued V22( the carrier of S) quasi_total V31() V34() V38() Element of bool [: the carrier of S, the carrier of S:]
[: the carrier of S, the carrier of S:] is Relation-like non empty set
bool [: the carrier of S, the carrier of S:] is non empty set
RelStr(# the carrier of S, the InternalRel of S #) is non empty strict V87() reflexive transitive antisymmetric upper-bounded up-complete with_suprema with_infima RelStr
[: the carrier of x, the carrier of x:] is Relation-like non empty set
bool [: the carrier of x, the carrier of x:] is non empty set
x is open Element of bool the carrier of x
(S,T,g) .: x is Element of bool the carrier of S
uparrow ((S,T,g) .: x) is upper Element of bool the carrier of S
(rng (S,T,g)) /\ (uparrow ((S,T,g) .: x)) is Element of bool the carrier of S
bool the carrier of x is non empty set
x is Relation-like the carrier of x -defined the carrier of x -valued Function-like non empty V22( the carrier of x) quasi_total Element of bool [: the carrier of x, the carrier of x:]
x .: x is Element of bool the carrier of x
y is Element of bool the carrier of x
rng x is non empty Element of bool the carrier of x
y /\ (rng x) is Element of bool the carrier of x
a is set
b is Element of the carrier of S
a is Element of the carrier of S
b2 is set
(S,T,g) . b2 is set
dom x is non empty Element of bool the carrier of x
f is set
x . f is set
g is Element of the carrier of T
f is Element of the carrier of T
g "\/" f is Element of the carrier of T
(S,T,g) . (g "\/" f) is Element of the carrier of S
a "\/" b is Element of the carrier of S
bool the carrier of T is non empty set
E9 is Element of bool the carrier of T
S is non empty V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
the carrier of S is non empty set
T is non empty V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
the carrier of T is non empty set
[: the carrier of S, the carrier of T:] is Relation-like non empty set
bool [: the carrier of S, the carrier of T:] is non empty set
g is Relation-like the carrier of S -defined the carrier of T -valued Function-like non empty V22( the carrier of S) quasi_total infs-preserving meet-preserving filtered-infs-preserving monotone Element of bool [: the carrier of S, the carrier of T:]
(S,T,g) is Relation-like the carrier of T -defined the carrier of S -valued Function-like non empty V22( the carrier of T) quasi_total sups-preserving join-preserving directed-sups-preserving monotone lower_adjoint Element of bool [: the carrier of T, the carrier of S:]
[: the carrier of T, the carrier of S:] is Relation-like non empty set
bool [: the carrier of T, the carrier of S:] is non empty set
x is non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of T
the carrier of x is non empty set
bool the carrier of x is non empty set
x is non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of S
the carrier of x is non empty set
bool the carrier of x is non empty set
[: the carrier of x, the carrier of x:] is Relation-like non empty set
bool [: the carrier of x, the carrier of x:] is non empty set
x is Relation-like the carrier of x -defined the carrier of x -valued Function-like non empty V22( the carrier of x) quasi_total Element of bool [: the carrier of x, the carrier of x:]
x is open Element of bool the carrier of x
x .: x is Element of bool the carrier of x
rng x is non empty Element of bool the carrier of x
x | (rng x) is non empty strict TopSpace-like SubSpace of x
the carrier of (x | (rng x)) is non empty set
bool the carrier of (x | (rng x)) is non empty set
(S,T,g) .: x is Element of bool the carrier of S
bool the carrier of S is non empty set
uparrow ((S,T,g) .: x) is upper Element of bool the carrier of S
y is open Element of bool the carrier of x
y /\ (rng x) is Element of bool the carrier of x
[#] (x | (rng x)) is non empty non proper closed Element of bool the carrier of (x | (rng x))
([#] (x | (rng x))) /\ y is Element of bool the carrier of x
the topology of x is non empty Element of bool (bool the carrier of x)
bool (bool the carrier of x) is non empty set
a is Element of bool the carrier of (x | (rng x))
the topology of (x | (rng x)) is non empty Element of bool (bool the carrier of (x | (rng x)))
bool (bool the carrier of (x | (rng x))) is non empty set
S is non empty V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
the carrier of S is non empty set
T is non empty V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
the carrier of T is non empty set
[: the carrier of S, the carrier of T:] is Relation-like non empty set
bool [: the carrier of S, the carrier of T:] is non empty set
g is Relation-like the carrier of S -defined the carrier of T -valued Function-like non empty V22( the carrier of S) quasi_total sups-preserving join-preserving directed-sups-preserving monotone Element of bool [: the carrier of S, the carrier of T:]
Image g is non empty strict V87() reflexive transitive antisymmetric full SubRelStr of T
S is non empty V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
the carrier of S is non empty set
T is non empty V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
the carrier of T is non empty set
[: the carrier of S, the carrier of T:] is Relation-like non empty set
bool [: the carrier of S, the carrier of T:] is non empty set
g is Relation-like the carrier of S -defined the carrier of T -valued Function-like non empty V22( the carrier of S) quasi_total infs-preserving meet-preserving filtered-infs-preserving monotone Element of bool [: the carrier of S, the carrier of T:]
(S,T,g) is Relation-like the carrier of T -defined the carrier of S -valued Function-like non empty V22( the carrier of T) quasi_total sups-preserving join-preserving directed-sups-preserving monotone lower_adjoint Element of bool [: the carrier of T, the carrier of S:]
[: the carrier of T, the carrier of S:] is Relation-like non empty set
bool [: the carrier of T, the carrier of S:] is non empty set
Image (S,T,g) is non empty strict V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() full up-complete /\-complete with_suprema with_infima complete SubRelStr of S
x is non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of T
the carrier of x is non empty set
x is non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of S
the carrier of x is non empty set
[: the carrier of x, the carrier of x:] is Relation-like non empty set
bool [: the carrier of x, the carrier of x:] is non empty set
x is non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of Image (S,T,g)
the carrier of x is non empty set
[: the carrier of x, the carrier of x:] is Relation-like non empty set
bool [: the carrier of x, the carrier of x:] is non empty set
x is Relation-like the carrier of x -defined the carrier of x -valued Function-like non empty V22( the carrier of x) quasi_total Element of bool [: the carrier of x, the carrier of x:]
y is Relation-like the carrier of x -defined the carrier of x -valued Function-like non empty V22( the carrier of x) quasi_total Element of bool [: the carrier of x, the carrier of x:]
bool the carrier of x is non empty set
rng x is non empty Element of bool the carrier of x
bool the carrier of x is non empty set
x | (rng x) is non empty strict TopSpace-like SubSpace of x
the carrier of (x | (rng x)) is non empty set
bool the carrier of (x | (rng x)) is non empty set
a is Element of bool the carrier of x
y .: a is Element of bool the carrier of x
bool the carrier of x is non empty set
x .: a is Element of bool the carrier of x
rng (S,T,g) is non empty Element of bool the carrier of S
bool the carrier of S is non empty set
subrelstr (rng (S,T,g)) is non empty strict V87() reflexive transitive antisymmetric full SubRelStr of S
the carrier of (Image (S,T,g)) is non empty set
[#] (x | (rng x)) is non empty non proper closed Element of bool the carrier of (x | (rng x))
the InternalRel of x is Relation-like the carrier of x -defined the carrier of x -valued V22( the carrier of x) quasi_total V31() V34() V38() Element of bool [: the carrier of x, the carrier of x:]
[: the carrier of x, the carrier of x:] is Relation-like non empty set
bool [: the carrier of x, the carrier of x:] is non empty set
RelStr(# the carrier of x, the InternalRel of x #) is non empty strict V87() reflexive transitive antisymmetric upper-bounded up-complete with_suprema with_infima RelStr
the InternalRel of x is Relation-like the carrier of x -defined the carrier of x -valued V22( the carrier of x) quasi_total V31() V34() V38() Element of bool [: the carrier of x, the carrier of x:]
[: the carrier of x, the carrier of x:] is Relation-like non empty set
bool [: the carrier of x, the carrier of x:] is non empty set
RelStr(# the carrier of x, the InternalRel of x #) is non empty strict V87() reflexive transitive antisymmetric upper-bounded up-complete with_suprema with_infima RelStr
the InternalRel of S is Relation-like the carrier of S -defined the carrier of S -valued V22( the carrier of S) quasi_total V31() V34() V38() Element of bool [: the carrier of S, the carrier of S:]
[: the carrier of S, the carrier of S:] is Relation-like non empty set
bool [: the carrier of S, the carrier of S:] is non empty set
RelStr(# the carrier of S, the InternalRel of S #) is non empty strict V87() reflexive transitive antisymmetric upper-bounded up-complete with_suprema with_infima RelStr
b is open Element of bool the carrier of (x | (rng x))
the topology of (x | (rng x)) is non empty Element of bool (bool the carrier of (x | (rng x)))
bool (bool the carrier of (x | (rng x))) is non empty set
the topology of x is non empty Element of bool (bool the carrier of x)
bool (bool the carrier of x) is non empty set
b2 is Element of bool the carrier of x
b2 /\ ([#] (x | (rng x))) is Element of bool the carrier of (x | (rng x))
a is Element of bool the carrier of x
f is Element of the carrier of x
g is Element of the carrier of x
f is Element of the carrier of (Image (S,T,g))
E9 is Element of the carrier of (Image (S,T,g))
a is Element of the carrier of S
b is Element of the carrier of S
a9 is Element of the carrier of x
b9 is Element of the carrier of x
f is non empty directed Element of bool the carrier of x
"\/" (f,x) is Element of the carrier of x
bool the carrier of (Image (S,T,g)) is non empty set
g is non empty Element of bool the carrier of (Image (S,T,g))
f is non empty Element of bool the carrier of S
E9 is non empty Element of bool the carrier of x
"\/" (g,S) is Element of the carrier of S
"\/" (f,S) is Element of the carrier of S
"\/" (g,(Image (S,T,g))) is Element of the carrier of (Image (S,T,g))
"\/" (E9,x) is Element of the carrier of x
S is non empty V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
the carrier of S is non empty set
T is non empty V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
the carrier of T is non empty set
[: the carrier of S, the carrier of T:] is Relation-like non empty set
bool [: the carrier of S, the carrier of T:] is non empty set
g is Relation-like the carrier of S -defined the carrier of T -valued Function-like non empty V22( the carrier of S) quasi_total infs-preserving meet-preserving filtered-infs-preserving monotone Element of bool [: the carrier of S, the carrier of T:]
(S,T,g) is Relation-like the carrier of T -defined the carrier of S -valued Function-like non empty V22( the carrier of T) quasi_total sups-preserving join-preserving directed-sups-preserving monotone lower_adjoint Element of bool [: the carrier of T, the carrier of S:]
[: the carrier of T, the carrier of S:] is Relation-like non empty set
bool [: the carrier of T, the carrier of S:] is non empty set
x is non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of T
the carrier of x is non empty set
x is non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of S
the carrier of x is non empty set
[: the carrier of x, the carrier of x:] is Relation-like non empty set
bool [: the carrier of x, the carrier of x:] is non empty set
[g,(S,T,g)] is Connection of S,T
{g,(S,T,g)} is non empty with_non-empty_elements non empty-membered finite set
{g} is non empty with_non-empty_elements non empty-membered finite set
{{g,(S,T,g)},{g}} is non empty with_non-empty_elements non empty-membered finite V30() set
rng (S,T,g) is non empty Element of bool the carrier of S
bool the carrier of S is non empty set
the InternalRel of x is Relation-like the carrier of x -defined the carrier of x -valued V22( the carrier of x) quasi_total V31() V34() V38() Element of bool [: the carrier of x, the carrier of x:]
[: the carrier of x, the carrier of x:] is Relation-like non empty set
bool [: the carrier of x, the carrier of x:] is non empty set
RelStr(# the carrier of x, the InternalRel of x #) is non empty strict V87() reflexive transitive antisymmetric upper-bounded up-complete with_suprema with_infima RelStr
the InternalRel of S is Relation-like the carrier of S -defined the carrier of S -valued V22( the carrier of S) quasi_total V31() V34() V38() Element of bool [: the carrier of S, the carrier of S:]
[: the carrier of S, the carrier of S:] is Relation-like non empty set
bool [: the carrier of S, the carrier of S:] is non empty set
RelStr(# the carrier of S, the InternalRel of S #) is non empty strict V87() reflexive transitive antisymmetric upper-bounded up-complete with_suprema with_infima RelStr
the InternalRel of x is Relation-like the carrier of x -defined the carrier of x -valued V22( the carrier of x) quasi_total V31() V34() V38() Element of bool [: the carrier of x, the carrier of x:]
[: the carrier of x, the carrier of x:] is Relation-like non empty set
bool [: the carrier of x, the carrier of x:] is non empty set
RelStr(# the carrier of x, the InternalRel of x #) is non empty strict V87() reflexive transitive antisymmetric upper-bounded up-complete with_suprema with_infima RelStr
the InternalRel of T is Relation-like the carrier of T -defined the carrier of T -valued V22( the carrier of T) quasi_total V31() V34() V38() Element of bool [: the carrier of T, the carrier of T:]
[: the carrier of T, the carrier of T:] is Relation-like non empty set
bool [: the carrier of T, the carrier of T:] is non empty set
RelStr(# the carrier of T, the InternalRel of T #) is non empty strict V87() reflexive transitive antisymmetric upper-bounded up-complete with_suprema with_infima RelStr
[#] x is non empty non proper closed directed filtered lower upper inaccessible_by_directed_joins closed_under_directed_sups property(S) Element of bool the carrier of x
bool the carrier of x is non empty set
x is Relation-like the carrier of x -defined the carrier of x -valued Function-like non empty V22( the carrier of x) quasi_total Element of bool [: the carrier of x, the carrier of x:]
rng x is non empty Element of bool the carrier of x
x | (rng x) is non empty strict TopSpace-like SubSpace of x
the topology of x is non empty Element of bool (bool the carrier of x)
bool (bool the carrier of x) is non empty set
TopStruct(# the carrier of x, the topology of x #) is non empty strict TopSpace-like TopStruct
bool the carrier of x is non empty set
x is open Element of bool the carrier of x
(S,T,g) .: x is Element of bool the carrier of S
uparrow ((S,T,g) .: x) is upper Element of bool the carrier of S
x is Element of bool the carrier of x
x .: x is Element of bool the carrier of x
the carrier of (x | (rng x)) is non empty set
bool the carrier of (x | (rng x)) is non empty set
bool the carrier of x is non empty set
x is non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of T
the carrier of x is non empty set
bool the carrier of x is non empty set
the InternalRel of x is Relation-like the carrier of x -defined the carrier of x -valued V22( the carrier of x) quasi_total V31() V34() V38() Element of bool [: the carrier of x, the carrier of x:]
[: the carrier of x, the carrier of x:] is Relation-like non empty set
bool [: the carrier of x, the carrier of x:] is non empty set
RelStr(# the carrier of x, the InternalRel of x #) is non empty strict V87() reflexive transitive antisymmetric upper-bounded up-complete with_suprema with_infima RelStr
y is non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of S
the carrier of y is non empty set
the InternalRel of y is Relation-like the carrier of y -defined the carrier of y -valued V22( the carrier of y) quasi_total V31() V34() V38() Element of bool [: the carrier of y, the carrier of y:]
[: the carrier of y, the carrier of y:] is Relation-like non empty set
bool [: the carrier of y, the carrier of y:] is non empty set
RelStr(# the carrier of y, the InternalRel of y #) is non empty strict V87() reflexive transitive antisymmetric upper-bounded up-complete with_suprema with_infima RelStr
a is open Element of bool the carrier of x
b is Element of bool the carrier of x
[: the carrier of x, the carrier of y:] is Relation-like non empty set
bool [: the carrier of x, the carrier of y:] is non empty set
a is open Element of bool the carrier of x
x .: a is Element of bool the carrier of x
b2 is Relation-like the carrier of x -defined the carrier of y -valued Function-like non empty V22( the carrier of x) quasi_total Element of bool [: the carrier of x, the carrier of y:]
b2 .: a is Element of bool the carrier of y
bool the carrier of y is non empty set
uparrow (b2 .: a) is upper Element of bool the carrier of y
(S,T,g) .: a is Element of bool the carrier of S
uparrow ((S,T,g) .: a) is upper Element of bool the carrier of S
S is non empty V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
the carrier of S is non empty set
[: the carrier of S, the carrier of S:] is Relation-like non empty set
bool [: the carrier of S, the carrier of S:] is non empty set
T is Relation-like the carrier of S -defined the carrier of S -valued Function-like non empty V22( the carrier of S) quasi_total projection Element of bool [: the carrier of S, the carrier of S:]
Image T is non empty strict V87() reflexive transitive antisymmetric full SubRelStr of S
S is non empty V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
the carrier of S is non empty set
[: the carrier of S, the carrier of S:] is Relation-like non empty set
bool [: the carrier of S, the carrier of S:] is non empty set
T is Relation-like the carrier of S -defined the carrier of S -valued Function-like non empty V22( the carrier of S) quasi_total projection kernel Element of bool [: the carrier of S, the carrier of S:]
Image T is non empty strict V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() full up-complete /\-complete with_suprema with_infima complete SubRelStr of S
corestr T is Relation-like the carrier of S -defined the carrier of (Image T) -valued Function-like non empty V22( the carrier of S) quasi_total onto Element of bool [: the carrier of S, the carrier of (Image T):]
the carrier of (Image T) is non empty set
[: the carrier of S, the carrier of (Image T):] is Relation-like non empty set
bool [: the carrier of S, the carrier of (Image T):] is non empty set
the carrier of (Image T) |` T is Relation-like the carrier of S -defined the carrier of S -valued the carrier of (Image T) -valued the carrier of S -valued Element of bool [: the carrier of S, the carrier of S:]
inclusion T is Relation-like the carrier of (Image T) -defined the carrier of S -valued Function-like one-to-one non empty V22( the carrier of (Image T)) quasi_total monotone Element of bool [: the carrier of (Image T), the carrier of S:]
[: the carrier of (Image T), the carrier of S:] is Relation-like non empty set
bool [: the carrier of (Image T), the carrier of S:] is non empty set
id (Image T) is Relation-like the carrier of (Image T) -defined the carrier of (Image T) -valued Function-like one-to-one non empty V22( the carrier of (Image T)) quasi_total infs-preserving sups-preserving meet-preserving join-preserving filtered-infs-preserving directed-sups-preserving isomorphic monotone projection closure kernel Element of bool [: the carrier of (Image T), the carrier of (Image T):]
[: the carrier of (Image T), the carrier of (Image T):] is Relation-like non empty set
bool [: the carrier of (Image T), the carrier of (Image T):] is non empty set
id the carrier of (Image T) is Relation-like the carrier of (Image T) -defined the carrier of (Image T) -valued non empty V22( the carrier of (Image T)) quasi_total Element of bool [: the carrier of (Image T), the carrier of (Image T):]
(S,(Image T),(corestr T)) is Relation-like the carrier of (Image T) -defined the carrier of S -valued Function-like non empty V22( the carrier of (Image T)) quasi_total Element of bool [: the carrier of (Image T), the carrier of S:]
(S,(Image T),(inclusion T)) is Relation-like the carrier of S -defined the carrier of (Image T) -valued Function-like non empty V22( the carrier of S) quasi_total Element of bool [: the carrier of S, the carrier of (Image T):]
[(corestr T),(inclusion T)] is Connection of S, Image T
{(corestr T),(inclusion T)} is non empty with_non-empty_elements non empty-membered finite set
{(corestr T)} is non empty with_non-empty_elements non empty-membered finite set
{{(corestr T),(inclusion T)},{(corestr T)}} is non empty with_non-empty_elements non empty-membered finite V30() set
S is non empty V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
the carrier of S is non empty set
[: the carrier of S, the carrier of S:] is Relation-like non empty set
bool [: the carrier of S, the carrier of S:] is non empty set
T is Relation-like the carrier of S -defined the carrier of S -valued Function-like non empty V22( the carrier of S) quasi_total projection kernel Element of bool [: the carrier of S, the carrier of S:]
Image T is non empty strict V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() full up-complete /\-complete with_suprema with_infima complete SubRelStr of S
corestr T is Relation-like the carrier of S -defined the carrier of (Image T) -valued Function-like non empty V22( the carrier of S) quasi_total onto Element of bool [: the carrier of S, the carrier of (Image T):]
the carrier of (Image T) is non empty set
[: the carrier of S, the carrier of (Image T):] is Relation-like non empty set
bool [: the carrier of S, the carrier of (Image T):] is non empty set
the carrier of (Image T) |` T is Relation-like the carrier of S -defined the carrier of S -valued the carrier of (Image T) -valued the carrier of S -valued Element of bool [: the carrier of S, the carrier of S:]
inclusion T is Relation-like the carrier of (Image T) -defined the carrier of S -valued Function-like one-to-one non empty V22( the carrier of (Image T)) quasi_total monotone Element of bool [: the carrier of (Image T), the carrier of S:]
[: the carrier of (Image T), the carrier of S:] is Relation-like non empty set
bool [: the carrier of (Image T), the carrier of S:] is non empty set
id (Image T) is Relation-like the carrier of (Image T) -defined the carrier of (Image T) -valued Function-like one-to-one non empty V22( the carrier of (Image T)) quasi_total infs-preserving sups-preserving meet-preserving join-preserving filtered-infs-preserving directed-sups-preserving isomorphic monotone projection closure kernel Element of bool [: the carrier of (Image T), the carrier of (Image T):]
[: the carrier of (Image T), the carrier of (Image T):] is Relation-like non empty set
bool [: the carrier of (Image T), the carrier of (Image T):] is non empty set
id the carrier of (Image T) is Relation-like the carrier of (Image T) -defined the carrier of (Image T) -valued non empty V22( the carrier of (Image T)) quasi_total Element of bool [: the carrier of (Image T), the carrier of (Image T):]
[(corestr T),(inclusion T)] is Connection of S, Image T
{(corestr T),(inclusion T)} is non empty with_non-empty_elements non empty-membered finite set
{(corestr T)} is non empty with_non-empty_elements non empty-membered finite set
{{(corestr T),(inclusion T)},{(corestr T)}} is non empty with_non-empty_elements non empty-membered finite V30() set
(inclusion T) * (corestr T) is Relation-like the carrier of S -defined the carrier of S -valued Function-like non empty V22( the carrier of S) quasi_total Element of bool [: the carrier of S, the carrier of S:]
bool the carrier of S is non empty set
x is Element of bool the carrier of S
(corestr T) .: x is Element of bool the carrier of (Image T)
bool the carrier of (Image T) is non empty set
"\/" (((corestr T) .: x),(Image T)) is Element of the carrier of (Image T)
"\/" (x,S) is Element of the carrier of S
(corestr T) . ("\/" (x,S)) is Element of the carrier of (Image T)
T .: x is Element of bool the carrier of S
"\/" ((T .: x),S) is Element of the carrier of S
T . ("\/" (x,S)) is Element of the carrier of S
(inclusion T) . ((corestr T) . ("\/" (x,S))) is Element of the carrier of S
(inclusion T) .: ((corestr T) .: x) is Element of bool the carrier of S
"\/" (((corestr T) .: x),S) is Element of the carrier of S
"\/" (((inclusion T) .: ((corestr T) .: x)),S) is Element of the carrier of S
S is non empty V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
the carrier of S is non empty set
[: the carrier of S, the carrier of S:] is Relation-like non empty set
bool [: the carrier of S, the carrier of S:] is non empty set
bool the carrier of S is non empty set
T is Relation-like the carrier of S -defined the carrier of S -valued Function-like non empty V22( the carrier of S) quasi_total projection kernel Element of bool [: the carrier of S, the carrier of S:]
Image T is non empty strict V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() full up-complete /\-complete with_suprema with_infima complete SubRelStr of S
corestr T is Relation-like the carrier of S -defined the carrier of (Image T) -valued Function-like non empty V22( the carrier of S) quasi_total onto Element of bool [: the carrier of S, the carrier of (Image T):]
the carrier of (Image T) is non empty set
[: the carrier of S, the carrier of (Image T):] is Relation-like non empty set
bool [: the carrier of S, the carrier of (Image T):] is non empty set
the carrier of (Image T) |` T is Relation-like the carrier of S -defined the carrier of S -valued the carrier of (Image T) -valued the carrier of S -valued Element of bool [: the carrier of S, the carrier of S:]
inclusion T is Relation-like the carrier of (Image T) -defined the carrier of S -valued Function-like one-to-one non empty V22( the carrier of (Image T)) quasi_total monotone Element of bool [: the carrier of (Image T), the carrier of S:]
[: the carrier of (Image T), the carrier of S:] is Relation-like non empty set
bool [: the carrier of (Image T), the carrier of S:] is non empty set
id (Image T) is Relation-like the carrier of (Image T) -defined the carrier of (Image T) -valued Function-like one-to-one non empty V22( the carrier of (Image T)) quasi_total infs-preserving sups-preserving meet-preserving join-preserving filtered-infs-preserving directed-sups-preserving isomorphic monotone projection closure kernel Element of bool [: the carrier of (Image T), the carrier of (Image T):]
[: the carrier of (Image T), the carrier of (Image T):] is Relation-like non empty set
bool [: the carrier of (Image T), the carrier of (Image T):] is non empty set
id the carrier of (Image T) is Relation-like the carrier of (Image T) -defined the carrier of (Image T) -valued non empty V22( the carrier of (Image T)) quasi_total Element of bool [: the carrier of (Image T), the carrier of (Image T):]
[(corestr T),(inclusion T)] is Connection of S, Image T
{(corestr T),(inclusion T)} is non empty with_non-empty_elements non empty-membered finite set
{(corestr T)} is non empty with_non-empty_elements non empty-membered finite set
{{(corestr T),(inclusion T)},{(corestr T)}} is non empty with_non-empty_elements non empty-membered finite V30() set
(S,(Image T),(corestr T)) is Relation-like the carrier of (Image T) -defined the carrier of S -valued Function-like non empty V22( the carrier of (Image T)) quasi_total Element of bool [: the carrier of (Image T), the carrier of S:]
g is non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of Image T
the carrier of g is non empty set
the InternalRel of g is Relation-like the carrier of g -defined the carrier of g -valued V22( the carrier of g) quasi_total V31() V34() V38() Element of bool [: the carrier of g, the carrier of g:]
[: the carrier of g, the carrier of g:] is Relation-like non empty set
bool [: the carrier of g, the carrier of g:] is non empty set
RelStr(# the carrier of g, the InternalRel of g #) is non empty strict V87() reflexive transitive antisymmetric upper-bounded up-complete with_suprema with_infima RelStr
x is Element of bool the carrier of S
bool the carrier of g is non empty set
bool the carrier of (Image T) is non empty set
x is open Element of bool the carrier of g
x is Element of bool the carrier of (Image T)
(inclusion T) .: x is Element of bool the carrier of S
uparrow x is upper Element of bool the carrier of S
x is non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of S
the carrier of x is non empty set
bool the carrier of x is non empty set
x is non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of Image T
the carrier of x is non empty set
bool the carrier of x is non empty set
the InternalRel of x is Relation-like the carrier of x -defined the carrier of x -valued V22( the carrier of x) quasi_total V31() V34() V38() Element of bool [: the carrier of x, the carrier of x:]
[: the carrier of x, the carrier of x:] is Relation-like non empty set
bool [: the carrier of x, the carrier of x:] is non empty set
RelStr(# the carrier of x, the InternalRel of x #) is non empty strict V87() reflexive transitive antisymmetric upper-bounded up-complete with_suprema with_infima RelStr
x is open Element of bool the carrier of x
x is Element of bool the carrier of (Image T)
y is Element of bool the carrier of S
uparrow y is upper Element of bool the carrier of S
x is non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of S
the carrier of x is non empty set
bool the carrier of x is non empty set
(S,(Image T),(corestr T)) .: x is Element of bool the carrier of S
uparrow ((S,(Image T),(corestr T)) .: x) is upper Element of bool the carrier of S
S is non empty V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
the carrier of S is non empty set
T is non empty V87() reflexive transitive antisymmetric full join-inheriting sups-inheriting directed-sups-inheriting with_suprema SubRelStr of S
the carrier of T is non empty set
g is Element of the carrier of S
x is Element of the carrier of S
x is Element of the carrier of T
x is Element of the carrier of T
bool the carrier of S is non empty set
bool the carrier of T is non empty set
x is non empty directed Element of bool the carrier of T
"\/" (x,T) is Element of the carrier of T
"\/" (x,S) is Element of the carrier of S
y is non empty directed Element of bool the carrier of S
"\/" (y,S) is Element of the carrier of S
a is Element of the carrier of S
b is Element of the carrier of T
S is non empty V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
the carrier of S is non empty set
[: the carrier of S, the carrier of S:] is Relation-like non empty set
bool [: the carrier of S, the carrier of S:] is non empty set
T is Relation-like the carrier of S -defined the carrier of S -valued Function-like non empty V22( the carrier of S) quasi_total projection kernel Element of bool [: the carrier of S, the carrier of S:]
Image T is non empty strict V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() full up-complete /\-complete with_suprema with_infima complete SubRelStr of S
the carrier of (Image T) is non empty set
corestr T is Relation-like the carrier of S -defined the carrier of (Image T) -valued Function-like non empty V22( the carrier of S) quasi_total onto Element of bool [: the carrier of S, the carrier of (Image T):]
[: the carrier of S, the carrier of (Image T):] is Relation-like non empty set
bool [: the carrier of S, the carrier of (Image T):] is non empty set
the carrier of (Image T) |` T is Relation-like the carrier of S -defined the carrier of S -valued the carrier of (Image T) -valued the carrier of S -valued Element of bool [: the carrier of S, the carrier of S:]
(S,(Image T),(corestr T)) is Relation-like the carrier of (Image T) -defined the carrier of S -valued Function-like non empty V22( the carrier of (Image T)) quasi_total Element of bool [: the carrier of (Image T), the carrier of S:]
[: the carrier of (Image T), the carrier of S:] is Relation-like non empty set
bool [: the carrier of (Image T), the carrier of S:] is non empty set
x is Element of the carrier of S
x is Element of the carrier of S
x is Element of the carrier of (Image T)
x is Element of the carrier of (Image T)
inclusion T is Relation-like the carrier of (Image T) -defined the carrier of S -valued Function-like one-to-one non empty V22( the carrier of (Image T)) quasi_total monotone Element of bool [: the carrier of (Image T), the carrier of S:]
id (Image T) is Relation-like the carrier of (Image T) -defined the carrier of (Image T) -valued Function-like one-to-one non empty V22( the carrier of (Image T)) quasi_total infs-preserving sups-preserving meet-preserving join-preserving filtered-infs-preserving directed-sups-preserving isomorphic monotone projection closure kernel Element of bool [: the carrier of (Image T), the carrier of (Image T):]
[: the carrier of (Image T), the carrier of (Image T):] is Relation-like non empty set
bool [: the carrier of (Image T), the carrier of (Image T):] is non empty set
id the carrier of (Image T) is Relation-like the carrier of (Image T) -defined the carrier of (Image T) -valued non empty V22( the carrier of (Image T)) quasi_total Element of bool [: the carrier of (Image T), the carrier of (Image T):]
(S,(Image T),(corestr T)) . x is Element of the carrier of S
(S,(Image T),(corestr T)) . x is Element of the carrier of S
S is non empty V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
the carrier of S is non empty set
[: the carrier of S, the carrier of S:] is Relation-like non empty set
bool [: the carrier of S, the carrier of S:] is non empty set
T is Relation-like the carrier of S -defined the carrier of S -valued Function-like non empty V22( the carrier of S) quasi_total projection kernel Element of bool [: the carrier of S, the carrier of S:]
Image T is non empty strict V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() full up-complete /\-complete with_suprema with_infima complete SubRelStr of S
the carrier of (Image T) is non empty set
corestr T is Relation-like the carrier of S -defined the carrier of (Image T) -valued Function-like non empty V22( the carrier of S) quasi_total onto Element of bool [: the carrier of S, the carrier of (Image T):]
[: the carrier of S, the carrier of (Image T):] is Relation-like non empty set
bool [: the carrier of S, the carrier of (Image T):] is non empty set
the carrier of (Image T) |` T is Relation-like the carrier of S -defined the carrier of S -valued the carrier of (Image T) -valued the carrier of S -valued Element of bool [: the carrier of S, the carrier of S:]
(S,(Image T),(corestr T)) is Relation-like the carrier of (Image T) -defined the carrier of S -valued Function-like non empty V22( the carrier of (Image T)) quasi_total Element of bool [: the carrier of (Image T), the carrier of S:]
[: the carrier of (Image T), the carrier of S:] is Relation-like non empty set
bool [: the carrier of (Image T), the carrier of S:] is non empty set
x is Element of the carrier of (Image T)
x is Element of the carrier of (Image T)
(S,(Image T),(corestr T)) . x is Element of the carrier of S
(S,(Image T),(corestr T)) . x is Element of the carrier of S
inclusion T is Relation-like the carrier of (Image T) -defined the carrier of S -valued Function-like one-to-one non empty V22( the carrier of (Image T)) quasi_total monotone Element of bool [: the carrier of (Image T), the carrier of S:]
id (Image T) is Relation-like the carrier of (Image T) -defined the carrier of (Image T) -valued Function-like one-to-one non empty V22( the carrier of (Image T)) quasi_total infs-preserving sups-preserving meet-preserving join-preserving filtered-infs-preserving directed-sups-preserving isomorphic monotone projection closure kernel Element of bool [: the carrier of (Image T), the carrier of (Image T):]
[: the carrier of (Image T), the carrier of (Image T):] is Relation-like non empty set
bool [: the carrier of (Image T), the carrier of (Image T):] is non empty set
id the carrier of (Image T) is Relation-like the carrier of (Image T) -defined the carrier of (Image T) -valued non empty V22( the carrier of (Image T)) quasi_total Element of bool [: the carrier of (Image T), the carrier of (Image T):]
S is non empty V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
the carrier of S is non empty set
[: the carrier of S, the carrier of S:] is Relation-like non empty set
bool [: the carrier of S, the carrier of S:] is non empty set
T is Relation-like the carrier of S -defined the carrier of S -valued Function-like non empty V22( the carrier of S) quasi_total projection closure Element of bool [: the carrier of S, the carrier of S:]
Image T is non empty strict V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() full meet-inheriting infs-inheriting filtered-infs-inheriting up-complete /\-complete with_suprema with_infima complete SubRelStr of S
corestr T is Relation-like the carrier of S -defined the carrier of (Image T) -valued Function-like non empty V22( the carrier of S) quasi_total onto Element of bool [: the carrier of S, the carrier of (Image T):]
the carrier of (Image T) is non empty set
[: the carrier of S, the carrier of (Image T):] is Relation-like non empty set
bool [: the carrier of S, the carrier of (Image T):] is non empty set
the carrier of (Image T) |` T is Relation-like the carrier of S -defined the carrier of S -valued the carrier of (Image T) -valued the carrier of S -valued Element of bool [: the carrier of S, the carrier of S:]
inclusion T is Relation-like the carrier of (Image T) -defined the carrier of S -valued Function-like one-to-one non empty V22( the carrier of (Image T)) quasi_total monotone Element of bool [: the carrier of (Image T), the carrier of S:]
[: the carrier of (Image T), the carrier of S:] is Relation-like non empty set
bool [: the carrier of (Image T), the carrier of S:] is non empty set
id (Image T) is Relation-like the carrier of (Image T) -defined the carrier of (Image T) -valued Function-like one-to-one non empty V22( the carrier of (Image T)) quasi_total infs-preserving sups-preserving meet-preserving join-preserving filtered-infs-preserving directed-sups-preserving isomorphic monotone projection closure kernel Element of bool [: the carrier of (Image T), the carrier of (Image T):]
[: the carrier of (Image T), the carrier of (Image T):] is Relation-like non empty set
bool [: the carrier of (Image T), the carrier of (Image T):] is non empty set
id the carrier of (Image T) is Relation-like the carrier of (Image T) -defined the carrier of (Image T) -valued non empty V22( the carrier of (Image T)) quasi_total Element of bool [: the carrier of (Image T), the carrier of (Image T):]
((Image T),S,(corestr T)) is Relation-like the carrier of (Image T) -defined the carrier of S -valued Function-like non empty V22( the carrier of (Image T)) quasi_total Element of bool [: the carrier of (Image T), the carrier of S:]
((Image T),S,(inclusion T)) is Relation-like the carrier of S -defined the carrier of (Image T) -valued Function-like non empty V22( the carrier of S) quasi_total Element of bool [: the carrier of S, the carrier of (Image T):]
[(inclusion T),(corestr T)] is Connection of Image T,S
{(inclusion T),(corestr T)} is non empty with_non-empty_elements non empty-membered finite set
{(inclusion T)} is non empty with_non-empty_elements non empty-membered finite set
{{(inclusion T),(corestr T)},{(inclusion T)}} is non empty with_non-empty_elements non empty-membered finite V30() set
S is non empty V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
the carrier of S is non empty set
[: the carrier of S, the carrier of S:] is Relation-like non empty set
bool [: the carrier of S, the carrier of S:] is non empty set
T is Relation-like the carrier of S -defined the carrier of S -valued Function-like non empty V22( the carrier of S) quasi_total projection closure Element of bool [: the carrier of S, the carrier of S:]
Image T is non empty strict V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() full meet-inheriting infs-inheriting filtered-infs-inheriting up-complete /\-complete with_suprema with_infima complete SubRelStr of S
inclusion T is Relation-like the carrier of (Image T) -defined the carrier of S -valued Function-like one-to-one non empty V22( the carrier of (Image T)) quasi_total monotone Element of bool [: the carrier of (Image T), the carrier of S:]
the carrier of (Image T) is non empty set
[: the carrier of (Image T), the carrier of S:] is Relation-like non empty set
bool [: the carrier of (Image T), the carrier of S:] is non empty set
id (Image T) is Relation-like the carrier of (Image T) -defined the carrier of (Image T) -valued Function-like one-to-one non empty V22( the carrier of (Image T)) quasi_total infs-preserving sups-preserving meet-preserving join-preserving filtered-infs-preserving directed-sups-preserving isomorphic monotone projection closure kernel Element of bool [: the carrier of (Image T), the carrier of (Image T):]
[: the carrier of (Image T), the carrier of (Image T):] is Relation-like non empty set
bool [: the carrier of (Image T), the carrier of (Image T):] is non empty set
id the carrier of (Image T) is Relation-like the carrier of (Image T) -defined the carrier of (Image T) -valued non empty V22( the carrier of (Image T)) quasi_total Element of bool [: the carrier of (Image T), the carrier of (Image T):]
bool the carrier of (Image T) is non empty set
x is Element of bool the carrier of (Image T)
bool the carrier of S is non empty set
(inclusion T) .: x is Element of bool the carrier of S
x is non empty directed Element of bool the carrier of S
"\/" (((inclusion T) .: x),S) is Element of the carrier of S
"\/" (x,(Image T)) is Element of the carrier of (Image T)
(inclusion T) . ("\/" (x,(Image T))) is Element of the carrier of S
bool the carrier of (Image T) is non empty set
x is directed Element of bool the carrier of (Image T)
"\/" (x,S) is Element of the carrier of S
(inclusion T) .: x is Element of bool the carrier of S
bool the carrier of S is non empty set
"\/" (((inclusion T) .: x),S) is Element of the carrier of S
"\/" (x,(Image T)) is Element of the carrier of (Image T)
(inclusion T) . ("\/" (x,(Image T))) is Element of the carrier of S
S is non empty V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
the carrier of S is non empty set
[: the carrier of S, the carrier of S:] is Relation-like non empty set
bool [: the carrier of S, the carrier of S:] is non empty set
T is Relation-like the carrier of S -defined the carrier of S -valued Function-like non empty V22( the carrier of S) quasi_total projection closure Element of bool [: the carrier of S, the carrier of S:]
Image T is non empty strict V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() full meet-inheriting infs-inheriting filtered-infs-inheriting up-complete /\-complete with_suprema with_infima complete SubRelStr of S
the carrier of (Image T) is non empty set
inclusion T is Relation-like the carrier of (Image T) -defined the carrier of S -valued Function-like one-to-one non empty V22( the carrier of (Image T)) quasi_total monotone Element of bool [: the carrier of (Image T), the carrier of S:]
[: the carrier of (Image T), the carrier of S:] is Relation-like non empty set
bool [: the carrier of (Image T), the carrier of S:] is non empty set
id (Image T) is Relation-like the carrier of (Image T) -defined the carrier of (Image T) -valued Function-like one-to-one non empty V22( the carrier of (Image T)) quasi_total infs-preserving sups-preserving meet-preserving join-preserving filtered-infs-preserving directed-sups-preserving isomorphic monotone projection closure kernel Element of bool [: the carrier of (Image T), the carrier of (Image T):]
[: the carrier of (Image T), the carrier of (Image T):] is Relation-like non empty set
bool [: the carrier of (Image T), the carrier of (Image T):] is non empty set
id the carrier of (Image T) is Relation-like the carrier of (Image T) -defined the carrier of (Image T) -valued non empty V22( the carrier of (Image T)) quasi_total Element of bool [: the carrier of (Image T), the carrier of (Image T):]
((Image T),S,(inclusion T)) is Relation-like the carrier of S -defined the carrier of (Image T) -valued Function-like non empty V22( the carrier of S) quasi_total Element of bool [: the carrier of S, the carrier of (Image T):]
[: the carrier of S, the carrier of (Image T):] is Relation-like non empty set
bool [: the carrier of S, the carrier of (Image T):] is non empty set
corestr T is Relation-like the carrier of S -defined the carrier of (Image T) -valued Function-like non empty V22( the carrier of S) quasi_total onto Element of bool [: the carrier of S, the carrier of (Image T):]
the carrier of (Image T) |` T is Relation-like the carrier of S -defined the carrier of S -valued the carrier of (Image T) -valued the carrier of S -valued Element of bool [: the carrier of S, the carrier of S:]
x is non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of S
the carrier of x is non empty set
g is non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of Image T
the carrier of g is non empty set
[: the carrier of x, the carrier of g:] is Relation-like non empty set
bool [: the carrier of x, the carrier of g:] is non empty set
x is Relation-like the carrier of x -defined the carrier of g -valued Function-like non empty V22( the carrier of x) quasi_total Element of bool [: the carrier of x, the carrier of g:]
the non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of Image T is non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of Image T
the non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of S is non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of S
the carrier of the non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of Image T is non empty set
the InternalRel of the non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of Image T is Relation-like the carrier of the non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of Image T -defined the carrier of the non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of Image T -valued V22( the carrier of the non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of Image T) quasi_total V31() V34() V38() Element of bool [: the carrier of the non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of Image T, the carrier of the non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of Image T:]
[: the carrier of the non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of Image T, the carrier of the non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of Image T:] is Relation-like non empty set
bool [: the carrier of the non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of Image T, the carrier of the non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of Image T:] is non empty set
RelStr(# the carrier of the non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of Image T, the InternalRel of the non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of Image T #) is non empty strict V87() reflexive transitive antisymmetric upper-bounded up-complete with_suprema with_infima RelStr
the InternalRel of (Image T) is Relation-like the carrier of (Image T) -defined the carrier of (Image T) -valued V22( the carrier of (Image T)) quasi_total V31() V34() V38() Element of bool [: the carrier of (Image T), the carrier of (Image T):]
RelStr(# the carrier of (Image T), the InternalRel of (Image T) #) is non empty strict V87() reflexive transitive antisymmetric upper-bounded up-complete with_suprema with_infima RelStr
the carrier of the non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of S is non empty set
the InternalRel of the non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of S is Relation-like the carrier of the non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of S -defined the carrier of the non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of S -valued V22( the carrier of the non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of S) quasi_total V31() V34() V38() Element of bool [: the carrier of the non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of S, the carrier of the non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of S:]
[: the carrier of the non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of S, the carrier of the non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of S:] is Relation-like non empty set
bool [: the carrier of the non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of S, the carrier of the non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of S:] is non empty set
RelStr(# the carrier of the non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of S, the InternalRel of the non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of S #) is non empty strict V87() reflexive transitive antisymmetric upper-bounded up-complete with_suprema with_infima RelStr
the InternalRel of S is Relation-like the carrier of S -defined the carrier of S -valued V22( the carrier of S) quasi_total V31() V34() V38() Element of bool [: the carrier of S, the carrier of S:]
RelStr(# the carrier of S, the InternalRel of S #) is non empty strict V87() reflexive transitive antisymmetric upper-bounded up-complete with_suprema with_infima RelStr
[: the carrier of the non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of S, the carrier of the non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of Image T:] is Relation-like non empty set
bool [: the carrier of the non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of S, the carrier of the non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of Image T:] is non empty set
x is Relation-like the carrier of the non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of S -defined the carrier of the non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of Image T -valued Function-like non empty V22( the carrier of the non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of S) quasi_total Element of bool [: the carrier of the non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of S, the carrier of the non empty TopSpace-like V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete Scott with_suprema with_infima complete TopAugmentation of Image T:]
S is non empty V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
the carrier of S is non empty set
[: the carrier of S, the carrier of S:] is Relation-like non empty set
bool [: the carrier of S, the carrier of S:] is non empty set
T is Relation-like the carrier of S -defined the carrier of S -valued Function-like non empty V22( the carrier of S) quasi_total projection closure Element of bool [: the carrier of S, the carrier of S:]
Image T is non empty strict V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() full meet-inheriting infs-inheriting filtered-infs-inheriting up-complete /\-complete with_suprema with_infima complete SubRelStr of S
corestr T is Relation-like the carrier of S -defined the carrier of (Image T) -valued Function-like non empty V22( the carrier of S) quasi_total onto Element of bool [: the carrier of S, the carrier of (Image T):]
the carrier of (Image T) is non empty set
[: the carrier of S, the carrier of (Image T):] is Relation-like non empty set
bool [: the carrier of S, the carrier of (Image T):] is non empty set
the carrier of (Image T) |` T is Relation-like the carrier of S -defined the carrier of S -valued the carrier of (Image T) -valued the carrier of S -valued Element of bool [: the carrier of S, the carrier of S:]
inclusion T is Relation-like the carrier of (Image T) -defined the carrier of S -valued Function-like one-to-one non empty V22( the carrier of (Image T)) quasi_total monotone Element of bool [: the carrier of (Image T), the carrier of S:]
[: the carrier of (Image T), the carrier of S:] is Relation-like non empty set
bool [: the carrier of (Image T), the carrier of S:] is non empty set
id (Image T) is Relation-like the carrier of (Image T) -defined the carrier of (Image T) -valued Function-like one-to-one non empty V22( the carrier of (Image T)) quasi_total infs-preserving sups-preserving meet-preserving join-preserving filtered-infs-preserving directed-sups-preserving isomorphic monotone projection closure kernel Element of bool [: the carrier of (Image T), the carrier of (Image T):]
[: the carrier of (Image T), the carrier of (Image T):] is Relation-like non empty set
bool [: the carrier of (Image T), the carrier of (Image T):] is non empty set
id the carrier of (Image T) is Relation-like the carrier of (Image T) -defined the carrier of (Image T) -valued non empty V22( the carrier of (Image T)) quasi_total Element of bool [: the carrier of (Image T), the carrier of (Image T):]
((Image T),S,(inclusion T)) is Relation-like the carrier of S -defined the carrier of (Image T) -valued Function-like non empty V22( the carrier of S) quasi_total Element of bool [: the carrier of S, the carrier of (Image T):]
S is non empty V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete satisfying_axiom_of_approximation continuous with_suprema with_infima complete RelStr
the carrier of S is non empty set
[: the carrier of S, the carrier of S:] is Relation-like non empty set
bool [: the carrier of S, the carrier of S:] is non empty set
T is Relation-like the carrier of S -defined the carrier of S -valued Function-like non empty V22( the carrier of S) quasi_total projection closure Element of bool [: the carrier of S, the carrier of S:]
Image T is non empty strict V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() full meet-inheriting infs-inheriting filtered-infs-inheriting up-complete /\-complete with_suprema with_infima complete SubRelStr of S
corestr T is Relation-like the carrier of S -defined the carrier of (Image T) -valued Function-like non empty V22( the carrier of S) quasi_total onto Element of bool [: the carrier of S, the carrier of (Image T):]
the carrier of (Image T) is non empty set
[: the carrier of S, the carrier of (Image T):] is Relation-like non empty set
bool [: the carrier of S, the carrier of (Image T):] is non empty set
the carrier of (Image T) |` T is Relation-like the carrier of S -defined the carrier of S -valued the carrier of (Image T) -valued the carrier of S -valued Element of bool [: the carrier of S, the carrier of S:]
inclusion T is Relation-like the carrier of (Image T) -defined the carrier of S -valued Function-like one-to-one non empty V22( the carrier of (Image T)) quasi_total monotone Element of bool [: the carrier of (Image T), the carrier of S:]
[: the carrier of (Image T), the carrier of S:] is Relation-like non empty set
bool [: the carrier of (Image T), the carrier of S:] is non empty set
id (Image T) is Relation-like the carrier of (Image T) -defined the carrier of (Image T) -valued Function-like one-to-one non empty V22( the carrier of (Image T)) quasi_total infs-preserving sups-preserving meet-preserving join-preserving filtered-infs-preserving directed-sups-preserving isomorphic monotone projection closure kernel Element of bool [: the carrier of (Image T), the carrier of (Image T):]
[: the carrier of (Image T), the carrier of (Image T):] is Relation-like non empty set
bool [: the carrier of (Image T), the carrier of (Image T):] is non empty set
id the carrier of (Image T) is Relation-like the carrier of (Image T) -defined the carrier of (Image T) -valued non empty V22( the carrier of (Image T)) quasi_total Element of bool [: the carrier of (Image T), the carrier of (Image T):]
((Image T),S,(inclusion T)) is Relation-like the carrier of S -defined the carrier of (Image T) -valued Function-like non empty V22( the carrier of S) quasi_total Element of bool [: the carrier of S, the carrier of (Image T):]
S is non empty set
(S) is non empty transitive strict semi-functional V189() with_units reflexive para-functional set-id-inheriting concrete carrier-underlaid lattice-wise AltCatStr
the carrier of (S) is non empty set
g is Element of the carrier of (S)
x is Element of the carrier of (S)
x is Element of the carrier of (S)
<^g,x^> is set
<^x,x^> is set
latt g is non empty V87() reflexive transitive antisymmetric with_suprema with_infima RelStr
latt x is non empty V87() reflexive transitive antisymmetric with_suprema with_infima RelStr
latt x is non empty V87() reflexive transitive antisymmetric with_suprema with_infima RelStr
x is Relation-like Function-like Element of <^g,x^>
@ x is Relation-like the carrier of (latt g) -defined the carrier of (latt x) -valued Function-like non empty V22( the carrier of (latt g)) quasi_total monotone Element of bool [: the carrier of (latt g), the carrier of (latt x):]
the carrier of (latt g) is non empty set
the carrier of (latt x) is non empty set
[: the carrier of (latt g), the carrier of (latt x):] is Relation-like non empty set
bool [: the carrier of (latt g), the carrier of (latt x):] is non empty set
x is Relation-like Function-like Element of <^x,x^>
@ x is Relation-like the carrier of (latt x) -defined the carrier of (latt x) -valued Function-like non empty V22( the carrier of (latt x)) quasi_total monotone Element of bool [: the carrier of (latt x), the carrier of (latt x):]
the carrier of (latt x) is non empty set
[: the carrier of (latt x), the carrier of (latt x):] is Relation-like non empty set
bool [: the carrier of (latt x), the carrier of (latt x):] is non empty set
x * x is Relation-like Function-like Element of <^g,x^>
<^g,x^> is set
@ (x * x) is Relation-like the carrier of (latt g) -defined the carrier of (latt x) -valued Function-like non empty V22( the carrier of (latt g)) quasi_total monotone Element of bool [: the carrier of (latt g), the carrier of (latt x):]
[: the carrier of (latt g), the carrier of (latt x):] is Relation-like non empty set
bool [: the carrier of (latt g), the carrier of (latt x):] is non empty set
(@ x) * (@ x) is Relation-like the carrier of (latt g) -defined the carrier of (latt x) -valued Function-like non empty V22( the carrier of (latt g)) quasi_total Element of bool [: the carrier of (latt g), the carrier of (latt x):]
g is Element of the carrier of (S)
idm g is Relation-like Function-like Element of <^g,g^>
<^g,g^> is non empty set
latt g is non empty V87() reflexive transitive antisymmetric with_suprema with_infima RelStr
@ (idm g) is Relation-like the carrier of (latt g) -defined the carrier of (latt g) -valued Function-like non empty V22( the carrier of (latt g)) quasi_total monotone Element of bool [: the carrier of (latt g), the carrier of (latt g):]
the carrier of (latt g) is non empty set
[: the carrier of (latt g), the carrier of (latt g):] is Relation-like non empty set
bool [: the carrier of (latt g), the carrier of (latt g):] is non empty set
id (latt g) is Relation-like the carrier of (latt g) -defined the carrier of (latt g) -valued Function-like one-to-one non empty V22( the carrier of (latt g)) quasi_total infs-preserving sups-preserving meet-preserving join-preserving filtered-infs-preserving directed-sups-preserving isomorphic monotone projection closure kernel Element of bool [: the carrier of (latt g), the carrier of (latt g):]
id the carrier of (latt g) is Relation-like the carrier of (latt g) -defined the carrier of (latt g) -valued non empty V22( the carrier of (latt g)) quasi_total Element of bool [: the carrier of (latt g), the carrier of (latt g):]
g is non empty transitive strict semi-functional V189() with_units reflexive id-inheriting para-functional set-id-inheriting concrete carrier-underlaid lattice-wise SubCatStr of (S)
the carrier of g is non empty set
g is non empty transitive strict semi-functional V189() with_units reflexive id-inheriting para-functional set-id-inheriting concrete carrier-underlaid lattice-wise SubCatStr of (S)
the carrier of g is non empty set
x is non empty transitive strict semi-functional V189() with_units reflexive id-inheriting para-functional set-id-inheriting concrete carrier-underlaid lattice-wise SubCatStr of (S)
the carrier of x is non empty set
x is Element of the carrier of (S)
x is Element of the carrier of (S)
x is Element of the carrier of (S)
x is Element of the carrier of (S)
the Arrows of g is Relation-like [: the carrier of g, the carrier of g:] -defined Function-like V22([: the carrier of g, the carrier of g:]) set
[: the carrier of g, the carrier of g:] is Relation-like non empty set
the Comp of g is Relation-like [: the carrier of g, the carrier of g, the carrier of g:] -defined Function-like V22([: the carrier of g, the carrier of g, the carrier of g:]) M34([: the carrier of g, the carrier of g, the carrier of g:],{| the Arrows of g, the Arrows of g|},{| the Arrows of g|})
[: the carrier of g, the carrier of g, the carrier of g:] is non empty set
{| the Arrows of g, the Arrows of g|} is Relation-like [: the carrier of g, the carrier of g, the carrier of g:] -defined Function-like V22([: the carrier of g, the carrier of g, the carrier of g:]) set
{| the Arrows of g|} is Relation-like [: the carrier of g, the carrier of g, the carrier of g:] -defined Function-like V22([: the carrier of g, the carrier of g, the carrier of g:]) set
AltCatStr(# the carrier of g, the Arrows of g, the Comp of g #) is strict AltCatStr
the Arrows of x is Relation-like [: the carrier of x, the carrier of x:] -defined Function-like V22([: the carrier of x, the carrier of x:]) set
[: the carrier of x, the carrier of x:] is Relation-like non empty set
the Comp of x is Relation-like [: the carrier of x, the carrier of x, the carrier of x:] -defined Function-like V22([: the carrier of x, the carrier of x, the carrier of x:]) M34([: the carrier of x, the carrier of x, the carrier of x:],{| the Arrows of x, the Arrows of x|},{| the Arrows of x|})
[: the carrier of x, the carrier of x, the carrier of x:] is non empty set
{| the Arrows of x, the Arrows of x|} is Relation-like [: the carrier of x, the carrier of x, the carrier of x:] -defined Function-like V22([: the carrier of x, the carrier of x, the carrier of x:]) set
{| the Arrows of x|} is Relation-like [: the carrier of x, the carrier of x, the carrier of x:] -defined Function-like V22([: the carrier of x, the carrier of x, the carrier of x:]) set
AltCatStr(# the carrier of x, the Arrows of x, the Comp of x #) is strict AltCatStr
S is non empty non empty-membered set
(S) is non empty transitive strict semi-functional V189() with_units reflexive para-functional set-id-inheriting concrete carrier-underlaid lattice-wise with_complete_lattices AltCatStr
the carrier of (S) is non empty set
g is Element of the carrier of (S)
x is Element of the carrier of (S)
x is Element of the carrier of (S)
<^g,x^> is set
<^x,x^> is set
latt x is non empty V87() reflexive transitive antisymmetric with_suprema with_infima RelStr
latt g is non empty V87() reflexive transitive antisymmetric with_suprema with_infima RelStr
latt x is non empty V87() reflexive transitive antisymmetric with_suprema with_infima RelStr
x is Relation-like Function-like Element of <^g,x^>
@ x is Relation-like the carrier of (latt g) -defined the carrier of (latt x) -valued Function-like non empty V22( the carrier of (latt g)) quasi_total monotone Element of bool [: the carrier of (latt g), the carrier of (latt x):]
the carrier of (latt g) is non empty set
the carrier of (latt x) is non empty set
[: the carrier of (latt g), the carrier of (latt x):] is Relation-like non empty set
bool [: the carrier of (latt g), the carrier of (latt x):] is non empty set
((latt x),(latt g),(@ x)) is Relation-like the carrier of (latt x) -defined the carrier of (latt g) -valued Function-like non empty V22( the carrier of (latt x)) quasi_total Element of bool [: the carrier of (latt x), the carrier of (latt g):]
[: the carrier of (latt x), the carrier of (latt g):] is Relation-like non empty set
bool [: the carrier of (latt x), the carrier of (latt g):] is non empty set
x is Relation-like Function-like Element of <^x,x^>
@ x is Relation-like the carrier of (latt x) -defined the carrier of (latt x) -valued Function-like non empty V22( the carrier of (latt x)) quasi_total monotone Element of bool [: the carrier of (latt x), the carrier of (latt x):]
the carrier of (latt x) is non empty set
[: the carrier of (latt x), the carrier of (latt x):] is Relation-like non empty set
bool [: the carrier of (latt x), the carrier of (latt x):] is non empty set
((latt x),(latt x),(@ x)) is Relation-like the carrier of (latt x) -defined the carrier of (latt x) -valued Function-like non empty V22( the carrier of (latt x)) quasi_total Element of bool [: the carrier of (latt x), the carrier of (latt x):]
[: the carrier of (latt x), the carrier of (latt x):] is Relation-like non empty set
bool [: the carrier of (latt x), the carrier of (latt x):] is non empty set
x * x is Relation-like Function-like Element of <^g,x^>
<^g,x^> is set
@ (x * x) is Relation-like the carrier of (latt g) -defined the carrier of (latt x) -valued Function-like non empty V22( the carrier of (latt g)) quasi_total monotone Element of bool [: the carrier of (latt g), the carrier of (latt x):]
[: the carrier of (latt g), the carrier of (latt x):] is Relation-like non empty set
bool [: the carrier of (latt g), the carrier of (latt x):] is non empty set
((latt x),(latt g),(@ (x * x))) is Relation-like the carrier of (latt x) -defined the carrier of (latt g) -valued Function-like non empty V22( the carrier of (latt x)) quasi_total Element of bool [: the carrier of (latt x), the carrier of (latt g):]
[: the carrier of (latt x), the carrier of (latt g):] is Relation-like non empty set
bool [: the carrier of (latt x), the carrier of (latt g):] is non empty set
latt x is non empty V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
the carrier of (latt x) is non empty set
latt x is non empty V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
the carrier of (latt x) is non empty set
[: the carrier of (latt x), the carrier of (latt x):] is Relation-like non empty set
bool [: the carrier of (latt x), the carrier of (latt x):] is non empty set
y is non empty set
latt g is non empty V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
the carrier of (latt g) is non empty set
[: the carrier of (latt g), the carrier of (latt x):] is Relation-like non empty set
bool [: the carrier of (latt g), the carrier of (latt x):] is non empty set
y is non empty set
(@ x) * (@ x) is Relation-like the carrier of (latt g) -defined the carrier of (latt x) -valued Function-like non empty V22( the carrier of (latt g)) quasi_total Element of bool [: the carrier of (latt g), the carrier of (latt x):]
((latt x),(latt g),(@ x)) * ((latt x),(latt x),(@ x)) is Relation-like the carrier of (latt x) -defined the carrier of (latt g) -valued Function-like non empty V22( the carrier of (latt x)) quasi_total Element of bool [: the carrier of (latt x), the carrier of (latt g):]
g is Element of the carrier of (S)
idm g is Relation-like Function-like Element of <^g,g^>
<^g,g^> is non empty set
latt g is non empty V87() reflexive transitive antisymmetric with_suprema with_infima RelStr
@ (idm g) is Relation-like the carrier of (latt g) -defined the carrier of (latt g) -valued Function-like non empty V22( the carrier of (latt g)) quasi_total monotone Element of bool [: the carrier of (latt g), the carrier of (latt g):]
the carrier of (latt g) is non empty set
[: the carrier of (latt g), the carrier of (latt g):] is Relation-like non empty set
bool [: the carrier of (latt g), the carrier of (latt g):] is non empty set
((latt g),(latt g),(@ (idm g))) is Relation-like the carrier of (latt g) -defined the carrier of (latt g) -valued Function-like non empty V22( the carrier of (latt g)) quasi_total Element of bool [: the carrier of (latt g), the carrier of (latt g):]
latt g is non empty V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
id (latt g) is Relation-like the carrier of (latt g) -defined the carrier of (latt g) -valued Function-like one-to-one non empty V22( the carrier of (latt g)) quasi_total infs-preserving sups-preserving meet-preserving join-preserving filtered-infs-preserving directed-sups-preserving isomorphic monotone projection closure kernel Element of bool [: the carrier of (latt g), the carrier of (latt g):]
the carrier of (latt g) is non empty set
[: the carrier of (latt g), the carrier of (latt g):] is Relation-like non empty set
bool [: the carrier of (latt g), the carrier of (latt g):] is non empty set
id the carrier of (latt g) is Relation-like the carrier of (latt g) -defined the carrier of (latt g) -valued non empty V22( the carrier of (latt g)) quasi_total Element of bool [: the carrier of (latt g), the carrier of (latt g):]
((latt g),(latt g),(id (latt g))) is Relation-like the carrier of (latt g) -defined the carrier of (latt g) -valued Function-like non empty V22( the carrier of (latt g)) quasi_total infs-preserving meet-preserving filtered-infs-preserving monotone upper_adjoint Element of bool [: the carrier of (latt g), the carrier of (latt g):]
g is non empty transitive strict semi-functional V189() with_units reflexive id-inheriting para-functional set-id-inheriting concrete carrier-underlaid lattice-wise with_complete_lattices SubCatStr of (S)
the carrier of g is non empty set
g is non empty transitive strict semi-functional V189() with_units reflexive id-inheriting para-functional set-id-inheriting concrete carrier-underlaid lattice-wise with_complete_lattices SubCatStr of (S)
the carrier of g is non empty set
x is non empty transitive strict semi-functional V189() with_units reflexive id-inheriting para-functional set-id-inheriting concrete carrier-underlaid lattice-wise with_complete_lattices SubCatStr of (S)
the carrier of x is non empty set
x is Element of the carrier of (S)
x is Element of the carrier of (S)
x is Element of the carrier of (S)
x is Element of the carrier of (S)
the Arrows of g is Relation-like [: the carrier of g, the carrier of g:] -defined Function-like V22([: the carrier of g, the carrier of g:]) set
[: the carrier of g, the carrier of g:] is Relation-like non empty set
the Comp of g is Relation-like [: the carrier of g, the carrier of g, the carrier of g:] -defined Function-like V22([: the carrier of g, the carrier of g, the carrier of g:]) M34([: the carrier of g, the carrier of g, the carrier of g:],{| the Arrows of g, the Arrows of g|},{| the Arrows of g|})
[: the carrier of g, the carrier of g, the carrier of g:] is non empty set
{| the Arrows of g, the Arrows of g|} is Relation-like [: the carrier of g, the carrier of g, the carrier of g:] -defined Function-like V22([: the carrier of g, the carrier of g, the carrier of g:]) set
{| the Arrows of g|} is Relation-like [: the carrier of g, the carrier of g, the carrier of g:] -defined Function-like V22([: the carrier of g, the carrier of g, the carrier of g:]) set
AltCatStr(# the carrier of g, the Arrows of g, the Comp of g #) is strict AltCatStr
the Arrows of x is Relation-like [: the carrier of x, the carrier of x:] -defined Function-like V22([: the carrier of x, the carrier of x:]) set
[: the carrier of x, the carrier of x:] is Relation-like non empty set
the Comp of x is Relation-like [: the carrier of x, the carrier of x, the carrier of x:] -defined Function-like V22([: the carrier of x, the carrier of x, the carrier of x:]) M34([: the carrier of x, the carrier of x, the carrier of x:],{| the Arrows of x, the Arrows of x|},{| the Arrows of x|})
[: the carrier of x, the carrier of x, the carrier of x:] is non empty set
{| the Arrows of x, the Arrows of x|} is Relation-like [: the carrier of x, the carrier of x, the carrier of x:] -defined Function-like V22([: the carrier of x, the carrier of x, the carrier of x:]) set
{| the Arrows of x|} is Relation-like [: the carrier of x, the carrier of x, the carrier of x:] -defined Function-like V22([: the carrier of x, the carrier of x, the carrier of x:]) set
AltCatStr(# the carrier of x, the Arrows of x, the Comp of x #) is strict AltCatStr
S is non empty RelStr
the carrier of S is non empty set
bool the carrier of S is non empty set
T is non empty V87() reflexive antisymmetric RelStr
the carrier of T is non empty set
g is Element of the carrier of T
S --> g is Relation-like the carrier of S -defined the carrier of T -valued Function-like non empty V22( the carrier of S) quasi_total Element of bool [: the carrier of S, the carrier of T:]
[: the carrier of S, the carrier of T:] is Relation-like non empty set
bool [: the carrier of S, the carrier of T:] is non empty set
the carrier of S --> g is Relation-like the carrier of S -defined the carrier of T -valued Function-like non empty V22( the carrier of S) quasi_total Element of bool [: the carrier of S, the carrier of T:]
x is non empty Element of bool the carrier of S
(S --> g) .: x is non empty Element of bool the carrier of T
bool the carrier of T is non empty set
{g} is non empty finite Element of bool the carrier of T
the Element of x is Element of x
(S --> g) . the Element of x is Element of the carrier of T
"\/" (x,S) is Element of the carrier of S
(S --> g) . ("\/" (x,S)) is Element of the carrier of T
"/\" (x,S) is Element of the carrier of S
(S --> g) . ("/\" (x,S)) is Element of the carrier of T
"/\" ({g},T) is Element of the carrier of T
"\/" ({g},T) is Element of the carrier of T
S is non empty RelStr
T is non empty V87() reflexive antisymmetric lower-bounded RelStr
Bottom T is Element of the carrier of T
the carrier of T is non empty set
"\/" ({},T) is Element of the carrier of T
S --> (Bottom T) is Relation-like the carrier of S -defined the carrier of T -valued Function-like non empty V22( the carrier of S) quasi_total Element of bool [: the carrier of S, the carrier of T:]
the carrier of S is non empty set
[: the carrier of S, the carrier of T:] is Relation-like non empty set
bool [: the carrier of S, the carrier of T:] is non empty set
the carrier of S --> (Bottom T) is Relation-like the carrier of S -defined the carrier of T -valued Function-like non empty V22( the carrier of S) quasi_total Element of bool [: the carrier of S, the carrier of T:]
bool the carrier of S is non empty set
g is Element of bool the carrier of S
(S --> (Bottom T)) .: g is Element of bool the carrier of T
bool the carrier of T is non empty set
"\/" (((S --> (Bottom T)) .: g),T) is Element of the carrier of T
"\/" (g,S) is Element of the carrier of S
(S --> (Bottom T)) . ("\/" (g,S)) is Element of the carrier of T
( the carrier of S --> (Bottom T)) . ("\/" (g,S)) is Element of the carrier of T
{(Bottom T)} is non empty finite Element of bool the carrier of T
S is non empty RelStr
T is non empty V87() reflexive antisymmetric upper-bounded RelStr
Top T is Element of the carrier of T
the carrier of T is non empty set
"/\" ({},T) is Element of the carrier of T
S --> (Top T) is Relation-like the carrier of S -defined the carrier of T -valued Function-like non empty V22( the carrier of S) quasi_total Element of bool [: the carrier of S, the carrier of T:]
the carrier of S is non empty set
[: the carrier of S, the carrier of T:] is Relation-like non empty set
bool [: the carrier of S, the carrier of T:] is non empty set
the carrier of S --> (Top T) is Relation-like the carrier of S -defined the carrier of T -valued Function-like non empty V22( the carrier of S) quasi_total Element of bool [: the carrier of S, the carrier of T:]
bool the carrier of S is non empty set
g is Element of bool the carrier of S
(S --> (Top T)) .: g is Element of bool the carrier of T
bool the carrier of T is non empty set
"/\" (((S --> (Top T)) .: g),T) is Element of the carrier of T
"/\" (g,S) is Element of the carrier of S
(S --> (Top T)) . ("/\" (g,S)) is Element of the carrier of T
( the carrier of S --> (Top T)) . ("/\" (g,S)) is Element of the carrier of T
{(Top T)} is non empty finite Element of bool the carrier of T
S is non empty RelStr
T is non empty V87() reflexive antisymmetric upper-bounded RelStr
Top T is Element of the carrier of T
the carrier of T is non empty set
"/\" ({},T) is Element of the carrier of T
S --> (Top T) is Relation-like the carrier of S -defined the carrier of T -valued Function-like non empty V22( the carrier of S) quasi_total Element of bool [: the carrier of S, the carrier of T:]
the carrier of S is non empty set
[: the carrier of S, the carrier of T:] is Relation-like non empty set
bool [: the carrier of S, the carrier of T:] is non empty set
the carrier of S --> (Top T) is Relation-like the carrier of S -defined the carrier of T -valued Function-like non empty V22( the carrier of S) quasi_total Element of bool [: the carrier of S, the carrier of T:]
bool the carrier of S is non empty set
g is Element of bool the carrier of S
S is non empty RelStr
T is non empty V87() reflexive antisymmetric lower-bounded RelStr
Bottom T is Element of the carrier of T
the carrier of T is non empty set
"\/" ({},T) is Element of the carrier of T
S --> (Bottom T) is Relation-like the carrier of S -defined the carrier of T -valued Function-like non empty V22( the carrier of S) quasi_total Element of bool [: the carrier of S, the carrier of T:]
the carrier of S is non empty set
[: the carrier of S, the carrier of T:] is Relation-like non empty set
bool [: the carrier of S, the carrier of T:] is non empty set
the carrier of S --> (Bottom T) is Relation-like the carrier of S -defined the carrier of T -valued Function-like non empty V22( the carrier of S) quasi_total Element of bool [: the carrier of S, the carrier of T:]
bool the carrier of S is non empty set
g is Element of bool the carrier of S
S is non empty RelStr
the carrier of S is non empty set
T is non empty V87() reflexive antisymmetric upper-bounded RelStr
the carrier of T is non empty set
[: the carrier of S, the carrier of T:] is Relation-like non empty set
bool [: the carrier of S, the carrier of T:] is non empty set
Top T is Element of the carrier of T
"/\" ({},T) is Element of the carrier of T
S --> (Top T) is Relation-like the carrier of S -defined the carrier of T -valued Function-like non empty V22( the carrier of S) quasi_total infs-preserving meet-preserving filtered-infs-preserving directed-sups-preserving Element of bool [: the carrier of S, the carrier of T:]
the carrier of S --> (Top T) is Relation-like the carrier of S -defined the carrier of T -valued Function-like non empty V22( the carrier of S) quasi_total Element of bool [: the carrier of S, the carrier of T:]
S is non empty RelStr
the carrier of S is non empty set
T is non empty V87() reflexive antisymmetric lower-bounded RelStr
the carrier of T is non empty set
[: the carrier of S, the carrier of T:] is Relation-like non empty set
bool [: the carrier of S, the carrier of T:] is non empty set
Bottom T is Element of the carrier of T
"\/" ({},T) is Element of the carrier of T
S --> (Bottom T) is Relation-like the carrier of S -defined the carrier of T -valued Function-like non empty V22( the carrier of S) quasi_total sups-preserving join-preserving filtered-infs-preserving directed-sups-preserving Element of bool [: the carrier of S, the carrier of T:]
the carrier of S --> (Bottom T) is Relation-like the carrier of S -defined the carrier of T -valued Function-like non empty V22( the carrier of S) quasi_total Element of bool [: the carrier of S, the carrier of T:]
S is non empty non empty-membered set
(S) is non empty transitive strict semi-functional V189() with_units reflexive id-inheriting para-functional set-id-inheriting concrete carrier-underlaid lattice-wise with_complete_lattices SubCatStr of (S)
(S) is non empty transitive strict semi-functional V189() with_units reflexive para-functional set-id-inheriting concrete carrier-underlaid lattice-wise with_complete_lattices AltCatStr
the carrier of (S) is non empty set
T is non empty V87() reflexive transitive antisymmetric with_suprema with_infima RelStr
the carrier of T is non empty set
the carrier of (S) is non empty set
S is non empty non empty-membered set
(S) is non empty transitive strict semi-functional V189() with_units reflexive id-inheriting para-functional set-id-inheriting concrete carrier-underlaid lattice-wise with_complete_lattices SubCatStr of (S)
(S) is non empty transitive strict semi-functional V189() with_units reflexive para-functional set-id-inheriting concrete carrier-underlaid lattice-wise with_complete_lattices AltCatStr
the carrier of (S) is non empty set
T is Element of the carrier of (S)
g is Element of the carrier of (S)
<^T,g^> is set
latt T is non empty V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
the carrier of (latt T) is non empty set
latt g is non empty V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
the carrier of (latt g) is non empty set
[: the carrier of (latt T), the carrier of (latt g):] is Relation-like non empty set
bool [: the carrier of (latt T), the carrier of (latt g):] is non empty set
x is set
the carrier of (S) is non empty set
x is Element of the carrier of (S)
x is Element of the carrier of (S)
<^x,x^> is set
x is Relation-like Function-like Element of <^x,x^>
@ x is Relation-like the carrier of (latt x) -defined the carrier of (latt x) -valued Function-like non empty V22( the carrier of (latt x)) quasi_total monotone Element of bool [: the carrier of (latt x), the carrier of (latt x):]
latt x is non empty V87() reflexive transitive antisymmetric with_suprema with_infima RelStr
the carrier of (latt x) is non empty set
latt x is non empty V87() reflexive transitive antisymmetric with_suprema with_infima RelStr
the carrier of (latt x) is non empty set
[: the carrier of (latt x), the carrier of (latt x):] is Relation-like non empty set
bool [: the carrier of (latt x), the carrier of (latt x):] is non empty set
x is Relation-like the carrier of (latt T) -defined the carrier of (latt g) -valued Function-like non empty V22( the carrier of (latt T)) quasi_total infs-preserving meet-preserving filtered-infs-preserving directed-sups-preserving monotone Element of bool [: the carrier of (latt T), the carrier of (latt g):]
y is Relation-like Function-like Element of <^x,x^>
@ y is Relation-like the carrier of (latt x) -defined the carrier of (latt x) -valued Function-like non empty V22( the carrier of (latt x)) quasi_total monotone Element of bool [: the carrier of (latt x), the carrier of (latt x):]
S is non empty non empty-membered set
(S) is non empty transitive strict semi-functional V189() with_units reflexive id-inheriting para-functional set-id-inheriting concrete carrier-underlaid lattice-wise with_complete_lattices SubCatStr of (S)
(S) is non empty transitive strict semi-functional V189() with_units reflexive para-functional set-id-inheriting concrete carrier-underlaid lattice-wise with_complete_lattices AltCatStr
the carrier of (S) is non empty set
T is non empty V87() reflexive transitive antisymmetric with_suprema with_infima RelStr
the carrier of T is non empty set
the carrier of (S) is non empty set
S is non empty non empty-membered set
(S) is non empty transitive strict semi-functional V189() with_units reflexive id-inheriting para-functional set-id-inheriting concrete carrier-underlaid lattice-wise with_complete_lattices SubCatStr of (S)
(S) is non empty transitive strict semi-functional V189() with_units reflexive para-functional set-id-inheriting concrete carrier-underlaid lattice-wise with_complete_lattices AltCatStr
the carrier of (S) is non empty set
T is Element of the carrier of (S)
g is Element of the carrier of (S)
<^T,g^> is set
latt T is non empty V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
the carrier of (latt T) is non empty set
latt g is non empty V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
the carrier of (latt g) is non empty set
[: the carrier of (latt T), the carrier of (latt g):] is Relation-like non empty set
bool [: the carrier of (latt T), the carrier of (latt g):] is non empty set
x is set
the carrier of (S) is non empty set
x is Element of the carrier of (S)
x is Element of the carrier of (S)
<^x,x^> is set
x is Relation-like Function-like Element of <^x,x^>
@ x is Relation-like the carrier of (latt x) -defined the carrier of (latt x) -valued Function-like non empty V22( the carrier of (latt x)) quasi_total monotone Element of bool [: the carrier of (latt x), the carrier of (latt x):]
latt x is non empty V87() reflexive transitive antisymmetric with_suprema with_infima RelStr
the carrier of (latt x) is non empty set
latt x is non empty V87() reflexive transitive antisymmetric with_suprema with_infima RelStr
the carrier of (latt x) is non empty set
[: the carrier of (latt x), the carrier of (latt x):] is Relation-like non empty set
bool [: the carrier of (latt x), the carrier of (latt x):] is non empty set
((latt x),(latt x),(@ x)) is Relation-like the carrier of (latt x) -defined the carrier of (latt x) -valued Function-like non empty V22( the carrier of (latt x)) quasi_total Element of bool [: the carrier of (latt x), the carrier of (latt x):]
[: the carrier of (latt x), the carrier of (latt x):] is Relation-like non empty set
bool [: the carrier of (latt x), the carrier of (latt x):] is non empty set
latt x is non empty V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
the carrier of (latt x) is non empty set
latt x is non empty V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
the carrier of (latt x) is non empty set
[: the carrier of (latt x), the carrier of (latt x):] is Relation-like non empty set
bool [: the carrier of (latt x), the carrier of (latt x):] is non empty set
x is Relation-like the carrier of (latt T) -defined the carrier of (latt g) -valued Function-like non empty V22( the carrier of (latt T)) quasi_total sups-preserving join-preserving directed-sups-preserving monotone Element of bool [: the carrier of (latt T), the carrier of (latt g):]
((latt g),(latt T),x) is Relation-like the carrier of (latt g) -defined the carrier of (latt T) -valued Function-like non empty V22( the carrier of (latt g)) quasi_total infs-preserving meet-preserving filtered-infs-preserving monotone upper_adjoint Element of bool [: the carrier of (latt g), the carrier of (latt T):]
[: the carrier of (latt g), the carrier of (latt T):] is Relation-like non empty set
bool [: the carrier of (latt g), the carrier of (latt T):] is non empty set
y is Relation-like Function-like Element of <^x,x^>
@ y is Relation-like the carrier of (latt x) -defined the carrier of (latt x) -valued Function-like non empty V22( the carrier of (latt x)) quasi_total monotone Element of bool [: the carrier of (latt x), the carrier of (latt x):]
S is non empty non empty-membered set
(S) is non empty transitive strict semi-functional V189() with_units reflexive id-inheriting para-functional set-id-inheriting concrete carrier-underlaid lattice-wise with_complete_lattices SubCatStr of (S)
(S) is non empty transitive strict semi-functional V189() with_units reflexive para-functional set-id-inheriting concrete carrier-underlaid lattice-wise with_complete_lattices AltCatStr
S -UPS_category is non empty transitive strict semi-functional V189() with_units reflexive para-functional set-id-inheriting concrete carrier-underlaid lattice-wise with_complete_lattices with_all_isomorphisms AltCatStr
Intersect ((S),(S -UPS_category)) is strict AltCatStr
T is non empty set
[:T,T:] is Relation-like non empty set
bool [:T,T:] is non empty set
the Relation-like T -defined T -valued well_founded well-ordering V22(T) quasi_total V31() V34() V36() V38() upper-bounded Element of bool [:T,T:] is Relation-like T -defined T -valued well_founded well-ordering V22(T) quasi_total V31() V34() V36() V38() upper-bounded Element of bool [:T,T:]
the carrier of (S) is non empty set
the carrier of (S -UPS_category) is non empty set
x is Element of the carrier of (S)
idm x is Relation-like Function-like Element of <^x,x^>
<^x,x^> is non empty set
latt x is non empty V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
id (latt x) is Relation-like the carrier of (latt x) -defined the carrier of (latt x) -valued Function-like one-to-one non empty V22( the carrier of (latt x)) quasi_total infs-preserving sups-preserving meet-preserving join-preserving filtered-infs-preserving directed-sups-preserving isomorphic monotone projection closure kernel Element of bool [: the carrier of (latt x), the carrier of (latt x):]
the carrier of (latt x) is non empty set
[: the carrier of (latt x), the carrier of (latt x):] is Relation-like non empty set
bool [: the carrier of (latt x), the carrier of (latt x):] is non empty set
id the carrier of (latt x) is Relation-like the carrier of (latt x) -defined the carrier of (latt x) -valued non empty V22( the carrier of (latt x)) quasi_total Element of bool [: the carrier of (latt x), the carrier of (latt x):]
x is Element of the carrier of (S -UPS_category)
idm x is Relation-like Function-like Element of <^x,x^>
<^x,x^> is non empty set
the carrier of (Intersect ((S),(S -UPS_category))) is set
the carrier of (S) /\ the carrier of (S -UPS_category) is set
RelStr(# T, the Relation-like T -defined T -valued well_founded well-ordering V22(T) quasi_total V31() V34() V36() V38() upper-bounded Element of bool [:T,T:] #) is non empty strict V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() satisfying_axiom_K algebraic connected up-complete /\-complete satisfying_axiom_of_approximation continuous with_suprema with_infima complete RelStr
the Arrows of (S) is Relation-like [: the carrier of (S), the carrier of (S):] -defined Function-like V22([: the carrier of (S), the carrier of (S):]) set
the carrier of (S) is non empty set
[: the carrier of (S), the carrier of (S):] is Relation-like non empty set
x is non empty transitive semi-functional V189() with_units reflexive id-inheriting para-functional set-id-inheriting concrete carrier-underlaid lattice-wise with_complete_lattices SubCatStr of (S)
the carrier of x is non empty set
x is set
y is Element of the carrier of x
a is non empty V87() reflexive transitive antisymmetric with_suprema with_infima RelStr
the carrier of a is non empty set
x is set
y is Element of the carrier of (S)
a is non empty V87() reflexive transitive antisymmetric with_suprema with_infima RelStr
the carrier of a is non empty set
x is Element of the carrier of (S)
y is Element of the carrier of (S)
<^x,y^> is set
the Arrows of (S) . (x,y) is set
[x,y] is set
{x,y} is non empty finite set
{x} is non empty finite set
{{x,y},{x}} is non empty with_non-empty_elements non empty-membered finite V30() set
the Arrows of (S) . [x,y] is set
x is Element of the carrier of x
y is Element of the carrier of x
[: the carrier of (S), the carrier of (S):] is Relation-like non empty set
the Arrows of (S) is Relation-like [: the carrier of (S), the carrier of (S):] -defined Function-like V22([: the carrier of (S), the carrier of (S):]) set
dom the Arrows of (S) is Relation-like the carrier of (S) -defined the carrier of (S) -valued Element of bool [: the carrier of (S), the carrier of (S):]
bool [: the carrier of (S), the carrier of (S):] is non empty set
[: the carrier of (S -UPS_category), the carrier of (S -UPS_category):] is Relation-like non empty set
the Arrows of (S -UPS_category) is Relation-like [: the carrier of (S -UPS_category), the carrier of (S -UPS_category):] -defined Function-like V22([: the carrier of (S -UPS_category), the carrier of (S -UPS_category):]) set
dom the Arrows of (S -UPS_category) is Relation-like the carrier of (S -UPS_category) -defined the carrier of (S -UPS_category) -valued Element of bool [: the carrier of (S -UPS_category), the carrier of (S -UPS_category):]
bool [: the carrier of (S -UPS_category), the carrier of (S -UPS_category):] is non empty set
(dom the Arrows of (S)) /\ (dom the Arrows of (S -UPS_category)) is Relation-like the carrier of (S) -defined the carrier of (S -UPS_category) -defined the carrier of (S) -valued the carrier of (S -UPS_category) -valued Element of bool [: the carrier of (S -UPS_category), the carrier of (S -UPS_category):]
[:( the carrier of (S) /\ the carrier of (S -UPS_category)),( the carrier of (S) /\ the carrier of (S -UPS_category)):] is Relation-like set
<^x,y^> is set
the Arrows of x is Relation-like [: the carrier of x, the carrier of x:] -defined Function-like V22([: the carrier of x, the carrier of x:]) set
[: the carrier of x, the carrier of x:] is Relation-like non empty set
the Arrows of x . (x,y) is set
[x,y] is set
{x,y} is non empty finite set
{x} is non empty finite set
{{x,y},{x}} is non empty with_non-empty_elements non empty-membered finite V30() set
the Arrows of x . [x,y] is set
Intersect ( the Arrows of (S), the Arrows of (S -UPS_category)) is Relation-like Function-like set
[x,y] is Element of [: the carrier of x, the carrier of x:]
(Intersect ( the Arrows of (S), the Arrows of (S -UPS_category))) . [x,y] is set
the Arrows of (S) . (x,y) is set
the Arrows of (S) . [x,y] is set
the Arrows of (S -UPS_category) . [x,y] is set
( the Arrows of (S) . (x,y)) /\ ( the Arrows of (S -UPS_category) . [x,y]) is set
a is Element of the carrier of (S)
b2 is Element of the carrier of (S)
<^a,b2^> is set
the Arrows of (S -UPS_category) . (x,y) is set
the Arrows of (S -UPS_category) . [x,y] is set
<^a,b2^> /\ ( the Arrows of (S -UPS_category) . (x,y)) is set
f is Element of the carrier of (S -UPS_category)
g is Element of the carrier of (S -UPS_category)
<^f,g^> is non empty set
<^a,b2^> /\ <^f,g^> is set
f is set
latt f is non empty V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
the carrier of (latt f) is non empty set
latt g is non empty V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
the carrier of (latt g) is non empty set
[: the carrier of (latt f), the carrier of (latt g):] is Relation-like non empty set
bool [: the carrier of (latt f), the carrier of (latt g):] is non empty set
latt a is non empty V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
the carrier of (latt a) is non empty set
latt b2 is non empty V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
the carrier of (latt b2) is non empty set
[: the carrier of (latt a), the carrier of (latt b2):] is Relation-like non empty set
bool [: the carrier of (latt a), the carrier of (latt b2):] is non empty set
a is Element of the carrier of (S)
b is Element of the carrier of (S)
<^a,b^> is set
the Arrows of (S) . (x,y) is set
the Arrows of (S) . [x,y] is set
S is non empty non empty-membered set
(S) is non empty transitive strict semi-functional V189() with_units reflexive id-inheriting para-functional set-id-inheriting concrete carrier-underlaid lattice-wise with_complete_lattices SubCatStr of (S)
(S) is non empty transitive strict semi-functional V189() with_units reflexive para-functional set-id-inheriting concrete carrier-underlaid lattice-wise with_complete_lattices AltCatStr
the carrier of (S) is non empty set
T is non empty set
[:T,T:] is Relation-like non empty set
bool [:T,T:] is non empty set
the Relation-like T -defined T -valued well_founded well-ordering V22(T) quasi_total V31() V34() V36() V38() upper-bounded Element of bool [:T,T:] is Relation-like T -defined T -valued well_founded well-ordering V22(T) quasi_total V31() V34() V36() V38() upper-bounded Element of bool [:T,T:]
RelStr(# T, the Relation-like T -defined T -valued well_founded well-ordering V22(T) quasi_total V31() V34() V36() V38() upper-bounded Element of bool [:T,T:] #) is non empty strict V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() satisfying_axiom_K algebraic connected up-complete /\-complete satisfying_axiom_of_approximation continuous with_suprema with_infima complete RelStr
the carrier of (S) is non empty set
x is Element of the carrier of (S)
latt x is non empty V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
x is non empty transitive strict semi-functional V189() with_units reflexive full id-inheriting para-functional set-id-inheriting concrete carrier-underlaid lattice-wise with_complete_lattices SubCatStr of (S)
the carrier of x is non empty set
x is non empty transitive strict semi-functional V189() with_units reflexive full id-inheriting para-functional set-id-inheriting concrete carrier-underlaid lattice-wise with_complete_lattices SubCatStr of (S)
the carrier of x is non empty set
the Arrows of x is Relation-like [: the carrier of x, the carrier of x:] -defined Function-like V22([: the carrier of x, the carrier of x:]) set
[: the carrier of x, the carrier of x:] is Relation-like non empty set
the Comp of x is Relation-like [: the carrier of x, the carrier of x, the carrier of x:] -defined Function-like V22([: the carrier of x, the carrier of x, the carrier of x:]) M34([: the carrier of x, the carrier of x, the carrier of x:],{| the Arrows of x, the Arrows of x|},{| the Arrows of x|})
[: the carrier of x, the carrier of x, the carrier of x:] is non empty set
{| the Arrows of x, the Arrows of x|} is Relation-like [: the carrier of x, the carrier of x, the carrier of x:] -defined Function-like V22([: the carrier of x, the carrier of x, the carrier of x:]) set
{| the Arrows of x|} is Relation-like [: the carrier of x, the carrier of x, the carrier of x:] -defined Function-like V22([: the carrier of x, the carrier of x, the carrier of x:]) set
AltCatStr(# the carrier of x, the Arrows of x, the Comp of x #) is strict AltCatStr
the Arrows of x is Relation-like [: the carrier of x, the carrier of x:] -defined Function-like V22([: the carrier of x, the carrier of x:]) set
[: the carrier of x, the carrier of x:] is Relation-like non empty set
the Comp of x is Relation-like [: the carrier of x, the carrier of x, the carrier of x:] -defined Function-like V22([: the carrier of x, the carrier of x, the carrier of x:]) M34([: the carrier of x, the carrier of x, the carrier of x:],{| the Arrows of x, the Arrows of x|},{| the Arrows of x|})
[: the carrier of x, the carrier of x, the carrier of x:] is non empty set
{| the Arrows of x, the Arrows of x|} is Relation-like [: the carrier of x, the carrier of x, the carrier of x:] -defined Function-like V22([: the carrier of x, the carrier of x, the carrier of x:]) set
{| the Arrows of x|} is Relation-like [: the carrier of x, the carrier of x, the carrier of x:] -defined Function-like V22([: the carrier of x, the carrier of x, the carrier of x:]) set
AltCatStr(# the carrier of x, the Arrows of x, the Comp of x #) is strict AltCatStr
S is non empty non empty-membered set
(S) is non empty transitive strict semi-functional V189() with_units reflexive full id-inheriting para-functional set-id-inheriting concrete carrier-underlaid lattice-wise with_complete_lattices SubCatStr of (S)
(S) is non empty transitive strict semi-functional V189() with_units reflexive id-inheriting para-functional set-id-inheriting concrete carrier-underlaid lattice-wise with_complete_lattices SubCatStr of (S)
(S) is non empty transitive strict semi-functional V189() with_units reflexive para-functional set-id-inheriting concrete carrier-underlaid lattice-wise with_complete_lattices AltCatStr
S is non empty non empty-membered set
(S) is non empty transitive strict semi-functional V189() with_units reflexive full id-inheriting para-functional set-id-inheriting concrete carrier-underlaid lattice-wise with_complete_lattices SubCatStr of (S)
(S) is non empty transitive strict semi-functional V189() with_units reflexive id-inheriting para-functional set-id-inheriting concrete carrier-underlaid lattice-wise with_complete_lattices SubCatStr of (S)
(S) is non empty transitive strict semi-functional V189() with_units reflexive para-functional set-id-inheriting concrete carrier-underlaid lattice-wise with_complete_lattices AltCatStr
the carrier of (S) is non empty set
the carrier of (S) is non empty set
the carrier of (S) is non empty set
T is non empty V87() reflexive transitive antisymmetric with_suprema with_infima RelStr
the carrier of T is non empty set
g is Element of the carrier of (S)
latt g is non empty V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
x is non empty set
g is Element of the carrier of (S)
latt g is non empty V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
S is non empty non empty-membered set
(S) is non empty transitive strict semi-functional V189() with_units reflexive full id-inheriting para-functional set-id-inheriting concrete carrier-underlaid lattice-wise with_complete_lattices SubCatStr of (S)
(S) is non empty transitive strict semi-functional V189() with_units reflexive id-inheriting para-functional set-id-inheriting concrete carrier-underlaid lattice-wise with_complete_lattices SubCatStr of (S)
(S) is non empty transitive strict semi-functional V189() with_units reflexive para-functional set-id-inheriting concrete carrier-underlaid lattice-wise with_complete_lattices AltCatStr
the carrier of (S) is non empty set
T is Element of the carrier of (S)
g is Element of the carrier of (S)
<^T,g^> is set
latt T is non empty V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
the carrier of (latt T) is non empty set
latt g is non empty V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
the carrier of (latt g) is non empty set
[: the carrier of (latt T), the carrier of (latt g):] is Relation-like non empty set
bool [: the carrier of (latt T), the carrier of (latt g):] is non empty set
x is set
the carrier of (S) is non empty set
x is Element of the carrier of (S)
x is Element of the carrier of (S)
<^x,x^> is set
S is non empty non empty-membered set
(S) is non empty transitive strict semi-functional V189() with_units reflexive id-inheriting para-functional set-id-inheriting concrete carrier-underlaid lattice-wise with_complete_lattices SubCatStr of (S)
(S) is non empty transitive strict semi-functional V189() with_units reflexive para-functional set-id-inheriting concrete carrier-underlaid lattice-wise with_complete_lattices AltCatStr
the carrier of (S) is non empty set
T is non empty set
[:T,T:] is Relation-like non empty set
bool [:T,T:] is non empty set
the Relation-like T -defined T -valued well_founded well-ordering V22(T) quasi_total V31() V34() V36() V38() upper-bounded Element of bool [:T,T:] is Relation-like T -defined T -valued well_founded well-ordering V22(T) quasi_total V31() V34() V36() V38() upper-bounded Element of bool [:T,T:]
RelStr(# T, the Relation-like T -defined T -valued well_founded well-ordering V22(T) quasi_total V31() V34() V36() V38() upper-bounded Element of bool [:T,T:] #) is non empty strict V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() satisfying_axiom_K algebraic connected up-complete /\-complete satisfying_axiom_of_approximation continuous with_suprema with_infima complete RelStr
the carrier of (S) is non empty set
x is Element of the carrier of (S)
latt x is non empty V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
x is non empty transitive strict semi-functional V189() with_units reflexive full id-inheriting para-functional set-id-inheriting concrete carrier-underlaid lattice-wise with_complete_lattices SubCatStr of (S)
the carrier of x is non empty set
x is non empty transitive strict semi-functional V189() with_units reflexive full id-inheriting para-functional set-id-inheriting concrete carrier-underlaid lattice-wise with_complete_lattices SubCatStr of (S)
the carrier of x is non empty set
the Arrows of x is Relation-like [: the carrier of x, the carrier of x:] -defined Function-like V22([: the carrier of x, the carrier of x:]) set
[: the carrier of x, the carrier of x:] is Relation-like non empty set
the Comp of x is Relation-like [: the carrier of x, the carrier of x, the carrier of x:] -defined Function-like V22([: the carrier of x, the carrier of x, the carrier of x:]) M34([: the carrier of x, the carrier of x, the carrier of x:],{| the Arrows of x, the Arrows of x|},{| the Arrows of x|})
[: the carrier of x, the carrier of x, the carrier of x:] is non empty set
{| the Arrows of x, the Arrows of x|} is Relation-like [: the carrier of x, the carrier of x, the carrier of x:] -defined Function-like V22([: the carrier of x, the carrier of x, the carrier of x:]) set
{| the Arrows of x|} is Relation-like [: the carrier of x, the carrier of x, the carrier of x:] -defined Function-like V22([: the carrier of x, the carrier of x, the carrier of x:]) set
AltCatStr(# the carrier of x, the Arrows of x, the Comp of x #) is strict AltCatStr
the Arrows of x is Relation-like [: the carrier of x, the carrier of x:] -defined Function-like V22([: the carrier of x, the carrier of x:]) set
[: the carrier of x, the carrier of x:] is Relation-like non empty set
the Comp of x is Relation-like [: the carrier of x, the carrier of x, the carrier of x:] -defined Function-like V22([: the carrier of x, the carrier of x, the carrier of x:]) M34([: the carrier of x, the carrier of x, the carrier of x:],{| the Arrows of x, the Arrows of x|},{| the Arrows of x|})
[: the carrier of x, the carrier of x, the carrier of x:] is non empty set
{| the Arrows of x, the Arrows of x|} is Relation-like [: the carrier of x, the carrier of x, the carrier of x:] -defined Function-like V22([: the carrier of x, the carrier of x, the carrier of x:]) set
{| the Arrows of x|} is Relation-like [: the carrier of x, the carrier of x, the carrier of x:] -defined Function-like V22([: the carrier of x, the carrier of x, the carrier of x:]) set
AltCatStr(# the carrier of x, the Arrows of x, the Comp of x #) is strict AltCatStr
S is non empty non empty-membered set
(S) is non empty transitive strict semi-functional V189() with_units reflexive full id-inheriting para-functional set-id-inheriting concrete carrier-underlaid lattice-wise with_complete_lattices SubCatStr of (S)
(S) is non empty transitive strict semi-functional V189() with_units reflexive id-inheriting para-functional set-id-inheriting concrete carrier-underlaid lattice-wise with_complete_lattices SubCatStr of (S)
(S) is non empty transitive strict semi-functional V189() with_units reflexive para-functional set-id-inheriting concrete carrier-underlaid lattice-wise with_complete_lattices AltCatStr
the carrier of (S) is non empty set
the carrier of (S) is non empty set
the carrier of (S) is non empty set
T is non empty V87() reflexive transitive antisymmetric with_suprema with_infima RelStr
the carrier of T is non empty set
g is Element of the carrier of (S)
latt g is non empty V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
x is non empty set
g is Element of the carrier of (S)
latt g is non empty V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
S is non empty non empty-membered set
(S) is non empty transitive strict semi-functional V189() with_units reflexive full id-inheriting para-functional set-id-inheriting concrete carrier-underlaid lattice-wise with_complete_lattices SubCatStr of (S)
(S) is non empty transitive strict semi-functional V189() with_units reflexive id-inheriting para-functional set-id-inheriting concrete carrier-underlaid lattice-wise with_complete_lattices SubCatStr of (S)
(S) is non empty transitive strict semi-functional V189() with_units reflexive para-functional set-id-inheriting concrete carrier-underlaid lattice-wise with_complete_lattices AltCatStr
the carrier of (S) is non empty set
T is Element of the carrier of (S)
g is Element of the carrier of (S)
<^T,g^> is set
latt T is non empty V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
the carrier of (latt T) is non empty set
latt g is non empty V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
the carrier of (latt g) is non empty set
[: the carrier of (latt T), the carrier of (latt g):] is Relation-like non empty set
bool [: the carrier of (latt T), the carrier of (latt g):] is non empty set
x is set
the carrier of (S) is non empty set
x is Element of the carrier of (S)
x is Element of the carrier of (S)
<^x,x^> is set
x is Relation-like the carrier of (latt T) -defined the carrier of (latt g) -valued Function-like non empty V22( the carrier of (latt T)) quasi_total sups-preserving join-preserving directed-sups-preserving monotone Element of bool [: the carrier of (latt T), the carrier of (latt g):]
((latt g),(latt T),x) is Relation-like the carrier of (latt g) -defined the carrier of (latt T) -valued Function-like non empty V22( the carrier of (latt g)) quasi_total infs-preserving meet-preserving filtered-infs-preserving monotone upper_adjoint Element of bool [: the carrier of (latt g), the carrier of (latt T):]
[: the carrier of (latt g), the carrier of (latt T):] is Relation-like non empty set
bool [: the carrier of (latt g), the carrier of (latt T):] is non empty set
S is non empty non empty-membered set
(S) is non empty transitive strict semi-functional V189() with_units reflexive para-functional set-id-inheriting concrete carrier-underlaid lattice-wise with_complete_lattices AltCatStr
(S) is non empty transitive strict semi-functional V189() with_units reflexive para-functional set-id-inheriting concrete carrier-underlaid lattice-wise with_complete_lattices AltCatStr
(S) is reflexive V226((S),(S)) strict Contravariant id-preserving comp-reversing contravariant bijective Functor of (S),(S)
(S) is non empty transitive strict semi-functional V189() with_units reflexive id-inheriting para-functional set-id-inheriting concrete carrier-underlaid lattice-wise with_complete_lattices SubCatStr of (S)
(S) is non empty transitive strict semi-functional V189() with_units reflexive id-inheriting para-functional set-id-inheriting concrete carrier-underlaid lattice-wise with_complete_lattices SubCatStr of (S)
the carrier of (S) is non empty set
the carrier of (S) is non empty set
the carrier of (S) is non empty set
x is Element of the carrier of (S)
(S) . x is Element of the carrier of (S)
the carrier of (S) is non empty set
[: the carrier of (S), the carrier of (S):] is Relation-like non empty set
the ObjectMap of (S) is Relation-like [: the carrier of (S), the carrier of (S):] -defined [: the carrier of (S), the carrier of (S):] -valued Function-like non empty V22([: the carrier of (S), the carrier of (S):]) quasi_total Element of bool [:[: the carrier of (S), the carrier of (S):],[: the carrier of (S), the carrier of (S):]:]
[: the carrier of (S), the carrier of (S):] is Relation-like non empty set
[:[: the carrier of (S), the carrier of (S):],[: the carrier of (S), the carrier of (S):]:] is Relation-like non empty set
bool [:[: the carrier of (S), the carrier of (S):],[: the carrier of (S), the carrier of (S):]:] is non empty set
the ObjectMap of (S) . (x,x) is Element of [: the carrier of (S), the carrier of (S):]
[x,x] is set
{x,x} is non empty finite set
{x} is non empty finite set
{{x,x},{x}} is non empty with_non-empty_elements non empty-membered finite V30() set
the ObjectMap of (S) . [x,x] is set
K57(( the ObjectMap of (S) . (x,x))) is set
x is Element of the carrier of (S)
(S) . x is Element of the carrier of (S)
the ObjectMap of (S) . (x,x) is Element of [: the carrier of (S), the carrier of (S):]
[x,x] is set
{x,x} is non empty finite set
{x} is non empty finite set
{{x,x},{x}} is non empty with_non-empty_elements non empty-membered finite V30() set
the ObjectMap of (S) . [x,x] is set
K57(( the ObjectMap of (S) . (x,x))) is set
x is Element of the carrier of (S)
x is Element of the carrier of (S)
<^x,x^> is set
y is Element of the carrier of (S)
a is Element of the carrier of (S)
b is Element of the carrier of (S)
(S) . x is Element of the carrier of (S)
the carrier of (S) is non empty set
[: the carrier of (S), the carrier of (S):] is Relation-like non empty set
the ObjectMap of (S) is Relation-like [: the carrier of (S), the carrier of (S):] -defined [: the carrier of (S), the carrier of (S):] -valued Function-like non empty V22([: the carrier of (S), the carrier of (S):]) quasi_total Element of bool [:[: the carrier of (S), the carrier of (S):],[: the carrier of (S), the carrier of (S):]:]
[: the carrier of (S), the carrier of (S):] is Relation-like non empty set
[:[: the carrier of (S), the carrier of (S):],[: the carrier of (S), the carrier of (S):]:] is Relation-like non empty set
bool [:[: the carrier of (S), the carrier of (S):],[: the carrier of (S), the carrier of (S):]:] is non empty set
the ObjectMap of (S) . (x,x) is Element of [: the carrier of (S), the carrier of (S):]
[x,x] is set
{x,x} is non empty finite set
{x} is non empty finite set
{{x,x},{x}} is non empty with_non-empty_elements non empty-membered finite V30() set
the ObjectMap of (S) . [x,x] is set
K57(( the ObjectMap of (S) . (x,x))) is set
a is Element of the carrier of (S)
(S) . x is Element of the carrier of (S)
the ObjectMap of (S) . (x,x) is Element of [: the carrier of (S), the carrier of (S):]
[x,x] is set
{x,x} is non empty finite set
{x} is non empty finite set
{{x,x},{x}} is non empty with_non-empty_elements non empty-membered finite V30() set
the ObjectMap of (S) . [x,x] is set
K57(( the ObjectMap of (S) . (x,x))) is set
<^((S) . x),((S) . x)^> is set
b2 is Relation-like Function-like Element of <^x,x^>
@ b2 is Relation-like the carrier of (latt x) -defined the carrier of (latt x) -valued Function-like non empty V22( the carrier of (latt x)) quasi_total monotone Element of bool [: the carrier of (latt x), the carrier of (latt x):]
latt x is non empty V87() reflexive transitive antisymmetric with_suprema with_infima RelStr
the carrier of (latt x) is non empty set
latt x is non empty V87() reflexive transitive antisymmetric with_suprema with_infima RelStr
the carrier of (latt x) is non empty set
[: the carrier of (latt x), the carrier of (latt x):] is Relation-like non empty set
bool [: the carrier of (latt x), the carrier of (latt x):] is non empty set
(S) . b2 is Relation-like Function-like Element of <^((S) . x),((S) . x)^>
@ ((S) . b2) is Relation-like the carrier of (latt ((S) . x)) -defined the carrier of (latt ((S) . x)) -valued Function-like non empty V22( the carrier of (latt ((S) . x))) quasi_total monotone Element of bool [: the carrier of (latt ((S) . x)), the carrier of (latt ((S) . x)):]
latt ((S) . x) is non empty V87() reflexive transitive antisymmetric with_suprema with_infima RelStr
the carrier of (latt ((S) . x)) is non empty set
latt ((S) . x) is non empty V87() reflexive transitive antisymmetric with_suprema with_infima RelStr
the carrier of (latt ((S) . x)) is non empty set
[: the carrier of (latt ((S) . x)), the carrier of (latt ((S) . x)):] is Relation-like non empty set
bool [: the carrier of (latt ((S) . x)), the carrier of (latt ((S) . x)):] is non empty set
latt x is non empty V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
latt x is non empty V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
((latt x),(latt x),(@ b2)) is Relation-like the carrier of (latt x) -defined the carrier of (latt x) -valued Function-like non empty V22( the carrier of (latt x)) quasi_total Element of bool [: the carrier of (latt x), the carrier of (latt x):]
[: the carrier of (latt x), the carrier of (latt x):] is Relation-like non empty set
bool [: the carrier of (latt x), the carrier of (latt x):] is non empty set
latt y is non empty V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
the carrier of (latt y) is non empty set
latt a is non empty V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
the carrier of (latt a) is non empty set
[: the carrier of (latt y), the carrier of (latt a):] is Relation-like non empty set
bool [: the carrier of (latt y), the carrier of (latt a):] is non empty set
g is non empty set
f is Relation-like the carrier of (latt y) -defined the carrier of (latt a) -valued Function-like non empty V22( the carrier of (latt y)) quasi_total infs-preserving meet-preserving filtered-infs-preserving monotone Element of bool [: the carrier of (latt y), the carrier of (latt a):]
((latt y),(latt a),f) is Relation-like the carrier of (latt a) -defined the carrier of (latt y) -valued Function-like non empty V22( the carrier of (latt a)) quasi_total sups-preserving join-preserving directed-sups-preserving monotone lower_adjoint Element of bool [: the carrier of (latt a), the carrier of (latt y):]
[: the carrier of (latt a), the carrier of (latt y):] is Relation-like non empty set
bool [: the carrier of (latt a), the carrier of (latt y):] is non empty set
((latt y),(latt a),((latt y),(latt a),f)) is Relation-like the carrier of (latt y) -defined the carrier of (latt a) -valued Function-like non empty V22( the carrier of (latt y)) quasi_total infs-preserving meet-preserving filtered-infs-preserving monotone upper_adjoint Element of bool [: the carrier of (latt y), the carrier of (latt a):]
<^y,a^> is set
<^a,b^> is set
S is non empty non empty-membered set
(S) is non empty transitive strict semi-functional V189() with_units reflexive para-functional set-id-inheriting concrete carrier-underlaid lattice-wise with_complete_lattices AltCatStr
(S) is non empty transitive strict semi-functional V189() with_units reflexive para-functional set-id-inheriting concrete carrier-underlaid lattice-wise with_complete_lattices AltCatStr
(S) is reflexive V226((S),(S)) strict Contravariant id-preserving comp-reversing contravariant bijective Functor of (S),(S)
(S) is non empty transitive strict semi-functional V189() with_units reflexive id-inheriting para-functional set-id-inheriting concrete carrier-underlaid lattice-wise with_complete_lattices SubCatStr of (S)
(S) is non empty transitive strict semi-functional V189() with_units reflexive id-inheriting para-functional set-id-inheriting concrete carrier-underlaid lattice-wise with_complete_lattices SubCatStr of (S)
(S) is reflexive V226((S),(S)) strict Contravariant id-preserving comp-reversing contravariant bijective Functor of (S),(S)
(S) " is strict FunctorStr over (S),(S)
S is non empty non empty-membered set
(S) is non empty transitive strict semi-functional V189() with_units reflexive para-functional set-id-inheriting concrete carrier-underlaid lattice-wise with_complete_lattices AltCatStr
(S) is non empty transitive strict semi-functional V189() with_units reflexive para-functional set-id-inheriting concrete carrier-underlaid lattice-wise with_complete_lattices AltCatStr
(S) is reflexive V226((S),(S)) strict Contravariant id-preserving comp-reversing contravariant bijective Functor of (S),(S)
(S) is non empty transitive strict semi-functional V189() with_units reflexive full id-inheriting para-functional set-id-inheriting concrete carrier-underlaid lattice-wise with_complete_lattices SubCatStr of (S)
(S) is non empty transitive strict semi-functional V189() with_units reflexive id-inheriting para-functional set-id-inheriting concrete carrier-underlaid lattice-wise with_complete_lattices SubCatStr of (S)
(S) is non empty transitive strict semi-functional V189() with_units reflexive full id-inheriting para-functional set-id-inheriting concrete carrier-underlaid lattice-wise with_complete_lattices SubCatStr of (S)
(S) is non empty transitive strict semi-functional V189() with_units reflexive id-inheriting para-functional set-id-inheriting concrete carrier-underlaid lattice-wise with_complete_lattices SubCatStr of (S)
the carrier of (S) is non empty set
x is non empty transitive semi-functional V189() with_units reflexive id-inheriting para-functional set-id-inheriting concrete carrier-underlaid lattice-wise with_complete_lattices SubCatStr of (S)
the carrier of x is non empty set
x is non empty transitive semi-functional V189() with_units reflexive id-inheriting para-functional set-id-inheriting concrete carrier-underlaid lattice-wise with_complete_lattices SubCatStr of (S)
the carrier of x is non empty set
x is Element of the carrier of (S)
(S) . x is Element of the carrier of (S)
the carrier of (S) is non empty set
[: the carrier of (S), the carrier of (S):] is Relation-like non empty set
the ObjectMap of (S) is Relation-like [: the carrier of (S), the carrier of (S):] -defined [: the carrier of (S), the carrier of (S):] -valued Function-like non empty V22([: the carrier of (S), the carrier of (S):]) quasi_total Element of bool [:[: the carrier of (S), the carrier of (S):],[: the carrier of (S), the carrier of (S):]:]
[: the carrier of (S), the carrier of (S):] is Relation-like non empty set
[:[: the carrier of (S), the carrier of (S):],[: the carrier of (S), the carrier of (S):]:] is Relation-like non empty set
bool [:[: the carrier of (S), the carrier of (S):],[: the carrier of (S), the carrier of (S):]:] is non empty set
the ObjectMap of (S) . (x,x) is Element of [: the carrier of (S), the carrier of (S):]
[x,x] is set
{x,x} is non empty finite set
{x} is non empty finite set
{{x,x},{x}} is non empty with_non-empty_elements non empty-membered finite V30() set
the ObjectMap of (S) . [x,x] is set
K57(( the ObjectMap of (S) . (x,x))) is set
latt x is non empty V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
the carrier of (latt x) is non empty set
y is non empty set
x is Element of the carrier of (S)
y is Element of the carrier of (S)
<^x,y^> is set
a is Element of the carrier of x
b is Element of the carrier of x
a is Element of the carrier of x
(S) . x is Element of the carrier of (S)
the carrier of (S) is non empty set
[: the carrier of (S), the carrier of (S):] is Relation-like non empty set
the ObjectMap of (S) is Relation-like [: the carrier of (S), the carrier of (S):] -defined [: the carrier of (S), the carrier of (S):] -valued Function-like non empty V22([: the carrier of (S), the carrier of (S):]) quasi_total Element of bool [:[: the carrier of (S), the carrier of (S):],[: the carrier of (S), the carrier of (S):]:]
[: the carrier of (S), the carrier of (S):] is Relation-like non empty set
[:[: the carrier of (S), the carrier of (S):],[: the carrier of (S), the carrier of (S):]:] is Relation-like non empty set
bool [:[: the carrier of (S), the carrier of (S):],[: the carrier of (S), the carrier of (S):]:] is non empty set
the ObjectMap of (S) . (x,x) is Element of [: the carrier of (S), the carrier of (S):]
[x,x] is set
{x,x} is non empty finite set
{x} is non empty finite set
{{x,x},{x}} is non empty with_non-empty_elements non empty-membered finite V30() set
the ObjectMap of (S) . [x,x] is set
K57(( the ObjectMap of (S) . (x,x))) is set
b2 is Element of the carrier of x
(S) . y is Element of the carrier of (S)
the ObjectMap of (S) . (y,y) is Element of [: the carrier of (S), the carrier of (S):]
[y,y] is set
{y,y} is non empty finite set
{y} is non empty finite set
{{y,y},{y}} is non empty with_non-empty_elements non empty-membered finite V30() set
the ObjectMap of (S) . [y,y] is set
K57(( the ObjectMap of (S) . (y,y))) is set
f is Relation-like Function-like Element of <^x,y^>
@ f is Relation-like the carrier of (latt x) -defined the carrier of (latt y) -valued Function-like non empty V22( the carrier of (latt x)) quasi_total monotone Element of bool [: the carrier of (latt x), the carrier of (latt y):]
latt x is non empty V87() reflexive transitive antisymmetric with_suprema with_infima RelStr
the carrier of (latt x) is non empty set
latt y is non empty V87() reflexive transitive antisymmetric with_suprema with_infima RelStr
the carrier of (latt y) is non empty set
[: the carrier of (latt x), the carrier of (latt y):] is Relation-like non empty set
bool [: the carrier of (latt x), the carrier of (latt y):] is non empty set
latt x is non empty V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
latt y is non empty V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
(S) . f is Relation-like Function-like Element of <^((S) . y),((S) . x)^>
<^((S) . y),((S) . x)^> is set
((latt x),(latt y),(@ f)) is Relation-like the carrier of (latt y) -defined the carrier of (latt x) -valued Function-like non empty V22( the carrier of (latt y)) quasi_total Element of bool [: the carrier of (latt y), the carrier of (latt x):]
[: the carrier of (latt y), the carrier of (latt x):] is Relation-like non empty set
bool [: the carrier of (latt y), the carrier of (latt x):] is non empty set
latt a is non empty V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
the carrier of (latt a) is non empty set
latt b is non empty V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
the carrier of (latt b) is non empty set
[: the carrier of (latt a), the carrier of (latt b):] is Relation-like non empty set
bool [: the carrier of (latt a), the carrier of (latt b):] is non empty set
f is non empty set
g is Relation-like the carrier of (latt a) -defined the carrier of (latt b) -valued Function-like non empty V22( the carrier of (latt a)) quasi_total infs-preserving meet-preserving filtered-infs-preserving monotone Element of bool [: the carrier of (latt a), the carrier of (latt b):]
((latt a),(latt b),g) is Relation-like the carrier of (latt b) -defined the carrier of (latt a) -valued Function-like non empty V22( the carrier of (latt b)) quasi_total sups-preserving join-preserving directed-sups-preserving monotone lower_adjoint Element of bool [: the carrier of (latt b), the carrier of (latt a):]
[: the carrier of (latt b), the carrier of (latt a):] is Relation-like non empty set
bool [: the carrier of (latt b), the carrier of (latt a):] is non empty set
((latt a),(latt b),((latt a),(latt b),g)) is Relation-like the carrier of (latt a) -defined the carrier of (latt b) -valued Function-like non empty V22( the carrier of (latt a)) quasi_total infs-preserving meet-preserving filtered-infs-preserving monotone upper_adjoint Element of bool [: the carrier of (latt a), the carrier of (latt b):]
<^a,b^> is set
<^b2,a^> is set
latt b2 is non empty V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
the carrier of (latt b2) is non empty set
latt a is non empty V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
the carrier of (latt a) is non empty set
[: the carrier of (latt b2), the carrier of (latt a):] is Relation-like non empty set
bool [: the carrier of (latt b2), the carrier of (latt a):] is non empty set
f is Relation-like the carrier of (latt b2) -defined the carrier of (latt a) -valued Function-like non empty V22( the carrier of (latt b2)) quasi_total sups-preserving join-preserving directed-sups-preserving monotone Element of bool [: the carrier of (latt b2), the carrier of (latt a):]
((latt a),(latt b2),f) is Relation-like the carrier of (latt a) -defined the carrier of (latt b2) -valued Function-like non empty V22( the carrier of (latt a)) quasi_total infs-preserving meet-preserving filtered-infs-preserving monotone upper_adjoint Element of bool [: the carrier of (latt a), the carrier of (latt b2):]
[: the carrier of (latt a), the carrier of (latt b2):] is Relation-like non empty set
bool [: the carrier of (latt a), the carrier of (latt b2):] is non empty set
S is non empty non empty-membered set
(S) is non empty transitive strict semi-functional V189() with_units reflexive para-functional set-id-inheriting concrete carrier-underlaid lattice-wise with_complete_lattices AltCatStr
(S) is non empty transitive strict semi-functional V189() with_units reflexive para-functional set-id-inheriting concrete carrier-underlaid lattice-wise with_complete_lattices AltCatStr
(S) is reflexive V226((S),(S)) strict Contravariant id-preserving comp-reversing contravariant bijective Functor of (S),(S)
(S) is non empty transitive strict semi-functional V189() with_units reflexive full id-inheriting para-functional set-id-inheriting concrete carrier-underlaid lattice-wise with_complete_lattices SubCatStr of (S)
(S) is non empty transitive strict semi-functional V189() with_units reflexive id-inheriting para-functional set-id-inheriting concrete carrier-underlaid lattice-wise with_complete_lattices SubCatStr of (S)
(S) is non empty transitive strict semi-functional V189() with_units reflexive full id-inheriting para-functional set-id-inheriting concrete carrier-underlaid lattice-wise with_complete_lattices SubCatStr of (S)
(S) is non empty transitive strict semi-functional V189() with_units reflexive id-inheriting para-functional set-id-inheriting concrete carrier-underlaid lattice-wise with_complete_lattices SubCatStr of (S)
(S) is reflexive V226((S),(S)) strict Contravariant id-preserving comp-reversing contravariant bijective Functor of (S),(S)
(S) " is strict FunctorStr over (S),(S)
x is non empty transitive semi-functional V189() with_units reflexive id-inheriting para-functional set-id-inheriting concrete carrier-underlaid lattice-wise with_complete_lattices SubCatStr of (S)
x is non empty transitive semi-functional V189() with_units reflexive id-inheriting para-functional set-id-inheriting concrete carrier-underlaid lattice-wise with_complete_lattices SubCatStr of (S)
S is non empty V87() reflexive RelStr
the carrier of S is non empty set
T is non empty V87() reflexive RelStr
the carrier of T is non empty set
[: the carrier of S, the carrier of T:] is Relation-like non empty set
bool [: the carrier of S, the carrier of T:] is non empty set
T is non empty V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
the carrier of T is non empty set
S is non empty V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
the carrier of S is non empty set
[: the carrier of T, the carrier of S:] is Relation-like non empty set
bool [: the carrier of T, the carrier of S:] is non empty set
g is Relation-like the carrier of T -defined the carrier of S -valued Function-like non empty V22( the carrier of T) quasi_total sups-preserving join-preserving directed-sups-preserving monotone Element of bool [: the carrier of T, the carrier of S:]
x is Element of the carrier of T
g . x is Element of the carrier of S
T is non empty V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
the carrier of T is non empty set
S is non empty V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
the carrier of S is non empty set
[: the carrier of T, the carrier of S:] is Relation-like non empty set
bool [: the carrier of T, the carrier of S:] is non empty set
g is Relation-like the carrier of T -defined the carrier of S -valued Function-like non empty V22( the carrier of T) quasi_total sups-preserving join-preserving directed-sups-preserving monotone Element of bool [: the carrier of T, the carrier of S:]
x is Element of the carrier of T
x is Element of the carrier of T
g . x is Element of the carrier of S
g . x is Element of the carrier of S
CompactSublatt T is non empty strict V87() reflexive transitive antisymmetric full join-inheriting with_suprema SubRelStr of T
the carrier of (CompactSublatt T) is non empty set
x is Element of the carrier of T
g . x is Element of the carrier of S
S is non empty RelStr
the carrier of S is non empty set
bool the carrier of S is non empty set
T is non empty RelStr
the carrier of T is non empty set
[: the carrier of S, the carrier of T:] is Relation-like non empty set
bool [: the carrier of S, the carrier of T:] is non empty set
g is non empty RelStr
the carrier of g is non empty set
[: the carrier of T, the carrier of g:] is Relation-like non empty set
bool [: the carrier of T, the carrier of g:] is non empty set
x is Element of bool the carrier of S
x is Relation-like the carrier of S -defined the carrier of T -valued Function-like non empty V22( the carrier of S) quasi_total Element of bool [: the carrier of S, the carrier of T:]
x .: x is Element of bool the carrier of T
bool the carrier of T is non empty set
x is Relation-like the carrier of T -defined the carrier of g -valued Function-like non empty V22( the carrier of T) quasi_total Element of bool [: the carrier of T, the carrier of g:]
x * x is Relation-like the carrier of S -defined the carrier of g -valued Function-like non empty V22( the carrier of S) quasi_total Element of bool [: the carrier of S, the carrier of g:]
[: the carrier of S, the carrier of g:] is Relation-like non empty set
bool [: the carrier of S, the carrier of g:] is non empty set
"\/" ((x .: x),T) is Element of the carrier of T
"\/" (x,S) is Element of the carrier of S
x . ("\/" (x,S)) is Element of the carrier of T
x .: (x .: x) is Element of bool the carrier of g
bool the carrier of g is non empty set
"\/" ((x .: (x .: x)),g) is Element of the carrier of g
x . ("\/" ((x .: x),T)) is Element of the carrier of g
(x * x) .: x is Element of bool the carrier of g
"\/" (((x * x) .: x),g) is Element of the carrier of g
(x * x) . ("\/" (x,S)) is Element of the carrier of g
S is non empty RelStr
the carrier of S is non empty set
T is non empty RelStr
the carrier of T is non empty set
[: the carrier of S, the carrier of T:] is Relation-like non empty set
bool [: the carrier of S, the carrier of T:] is non empty set
S is non empty RelStr
the carrier of S is non empty set
T is non empty RelStr
the carrier of T is non empty set
[: the carrier of S, the carrier of T:] is Relation-like non empty set
bool [: the carrier of S, the carrier of T:] is non empty set
g is non empty RelStr
the carrier of g is non empty set
[: the carrier of T, the carrier of g:] is Relation-like non empty set
bool [: the carrier of T, the carrier of g:] is non empty set
x is Relation-like the carrier of S -defined the carrier of T -valued Function-like non empty V22( the carrier of S) quasi_total Element of bool [: the carrier of S, the carrier of T:]
x is Relation-like the carrier of T -defined the carrier of g -valued Function-like non empty V22( the carrier of T) quasi_total Element of bool [: the carrier of T, the carrier of g:]
x * x is Relation-like the carrier of S -defined the carrier of g -valued Function-like non empty V22( the carrier of S) quasi_total Element of bool [: the carrier of S, the carrier of g:]
[: the carrier of S, the carrier of g:] is Relation-like non empty set
bool [: the carrier of S, the carrier of g:] is non empty set
bool the carrier of S is non empty set
bool the carrier of T is non empty set
x is finite Element of bool the carrier of S
x .: x is finite Element of bool the carrier of T
S is non empty antisymmetric lower-bounded RelStr
the carrier of S is non empty set
T is non empty antisymmetric lower-bounded RelStr
the carrier of T is non empty set
[: the carrier of S, the carrier of T:] is Relation-like non empty set
bool [: the carrier of S, the carrier of T:] is non empty set
g is Relation-like the carrier of S -defined the carrier of T -valued Function-like non empty V22( the carrier of S) quasi_total Element of bool [: the carrier of S, the carrier of T:]
Bottom S is Element of the carrier of S
"\/" ({},S) is Element of the carrier of S
g . (Bottom S) is Element of the carrier of T
Bottom T is Element of the carrier of T
"\/" ({},T) is Element of the carrier of T
{} S is Relation-like non-empty empty-yielding empty proper finite finite-yielding V30() strongly_connected directed filtered lower upper Element of bool the carrier of S
bool the carrier of S is non empty set
g .: ({} S) is Relation-like non-empty empty-yielding empty proper finite finite-yielding V30() strongly_connected directed filtered lower upper Element of bool the carrier of T
bool the carrier of T is non empty set
"\/" ((g .: ({} S)),T) is Element of the carrier of T
"\/" (({} S),S) is Element of the carrier of S
g . ("\/" (({} S),S)) is Element of the carrier of T
{} S is Relation-like non-empty empty-yielding empty proper finite finite-yielding V30() strongly_connected directed filtered lower upper Element of bool the carrier of S
bool the carrier of S is non empty set
g .: ({} S) is Relation-like non-empty empty-yielding empty proper finite finite-yielding V30() strongly_connected directed filtered lower upper Element of bool the carrier of T
bool the carrier of T is non empty set
"\/" ((g .: ({} S)),T) is Element of the carrier of T
"\/" (({} S),S) is Element of the carrier of S
g . ("\/" (({} S),S)) is Element of the carrier of T
S is non empty RelStr
S is non empty RelStr
the carrier of S is non empty set
T is non empty RelStr
the carrier of T is non empty set
[: the carrier of S, the carrier of T:] is Relation-like non empty set
bool [: the carrier of S, the carrier of T:] is non empty set
g is Relation-like the carrier of S -defined the carrier of T -valued Function-like non empty V22( the carrier of S) quasi_total Element of bool [: the carrier of S, the carrier of T:]
{} S is Relation-like non-empty empty-yielding empty proper finite finite-yielding V30() strongly_connected directed filtered lower upper Element of bool the carrier of S
bool the carrier of S is non empty set
S is non empty antisymmetric lower-bounded RelStr
T is SubRelStr of S
{} T is Relation-like non-empty empty-yielding empty finite finite-yielding V30() strongly_connected directed filtered lower upper Element of bool the carrier of T
the carrier of T is set
bool the carrier of T is non empty set
"\/" (({} T),S) is Element of the carrier of S
the carrier of S is non empty set
Bottom S is Element of the carrier of S
"\/" ({},S) is Element of the carrier of S
g is Element of the carrier of S
x is Element of the carrier of S
{g,x} is non empty finite Element of bool the carrier of S
bool the carrier of S is non empty set
"\/" ({g,x},S) is Element of the carrier of S
x is finite Element of bool the carrier of T
"\/" (x,S) is Element of the carrier of S
S is non empty RelStr
T is SubRelStr of S
the carrier of T is set
bool the carrier of T is non empty set
g is finite Element of bool the carrier of T
"\/" (g,S) is Element of the carrier of S
the carrier of S is non empty set
S is non empty V87() reflexive transitive antisymmetric lower-bounded RelStr
the carrier of S is non empty set
T is non empty V87() reflexive transitive antisymmetric lower-bounded RelStr
the carrier of T is non empty set
[: the carrier of S, the carrier of T:] is Relation-like non empty set
bool [: the carrier of S, the carrier of T:] is non empty set
the Relation-like the carrier of S -defined the carrier of T -valued Function-like non empty V22( the carrier of S) quasi_total sups-preserving join-preserving directed-sups-preserving monotone (S,T) Element of bool [: the carrier of S, the carrier of T:] is Relation-like the carrier of S -defined the carrier of T -valued Function-like non empty V22( the carrier of S) quasi_total sups-preserving join-preserving directed-sups-preserving monotone (S,T) Element of bool [: the carrier of S, the carrier of T:]
S is non empty antisymmetric lower-bounded RelStr
T is antisymmetric full SubRelStr of S
Bottom S is Element of the carrier of S
the carrier of S is non empty set
"\/" ({},S) is Element of the carrier of S
the carrier of T is set
g is Element of the carrier of T
x is Element of the carrier of T
x is Element of the carrier of S
S is non empty antisymmetric lower-bounded RelStr
the non empty antisymmetric lower-bounded full join-inheriting sups-inheriting directed-sups-inheriting (S) (S) SubRelStr of S is non empty antisymmetric lower-bounded full join-inheriting sups-inheriting directed-sups-inheriting (S) (S) SubRelStr of S
S is non empty antisymmetric lower-bounded RelStr
Bottom S is Element of the carrier of S
the carrier of S is non empty set
"\/" ({},S) is Element of the carrier of S
T is non empty antisymmetric lower-bounded full (S) SubRelStr of S
Bottom T is Element of the carrier of T
the carrier of T is non empty set
"\/" ({},T) is Element of the carrier of T
x is Element of the carrier of S
g is Element of the carrier of T
S is non empty V87() reflexive transitive antisymmetric lower-bounded with_suprema RelStr
T is V87() reflexive transitive antisymmetric full SubRelStr of S
the carrier of T is set
bool the carrier of T is non empty set
x is finite Element of bool the carrier of T
"\/" (x,S) is Element of the carrier of S
the carrier of S is non empty set
g is non empty V87() reflexive transitive antisymmetric lower-bounded full join-inheriting with_suprema (S) SubRelStr of S
the carrier of g is non empty set
bool the carrier of g is non empty set
Bottom S is Element of the carrier of S
"\/" ({},S) is Element of the carrier of S
Bottom g is Element of the carrier of g
"\/" ({},g) is Element of the carrier of g
x is finite Element of bool the carrier of g
"\/" (x,S) is Element of the carrier of S
"\/" (x,g) is Element of the carrier of g
x is set
x is set
{x} is non empty finite set
x \/ {x} is non empty set
y is finite Element of bool the carrier of g
"\/" (y,S) is Element of the carrier of S
"\/" (y,g) is Element of the carrier of g
a is finite Element of bool the carrier of g
"\/" (a,S) is Element of the carrier of S
"\/" (a,g) is Element of the carrier of g
b is Element of the carrier of g
a is Element of the carrier of S
{a} is non empty finite Element of bool the carrier of S
bool the carrier of S is non empty set
{b} is non empty finite Element of bool the carrier of g
"\/" ({a},S) is Element of the carrier of S
("\/" (a,S)) "\/" ("\/" ({a},S)) is Element of the carrier of S
("\/" (a,S)) "\/" a is Element of the carrier of S
("\/" (a,g)) "\/" b is Element of the carrier of g
"\/" ({b},g) is Element of the carrier of g
("\/" (a,g)) "\/" ("\/" ({b},g)) is Element of the carrier of g
x is Element of bool the carrier of g
"\/" (x,g) is Element of the carrier of g
S is non empty RelStr
the carrier of S is non empty set
T is non empty RelStr
the carrier of T is non empty set
[: the carrier of S, the carrier of T:] is Relation-like non empty set
bool [: the carrier of S, the carrier of T:] is non empty set
g is Relation-like the carrier of S -defined the carrier of T -valued Function-like non empty V22( the carrier of S) quasi_total Element of bool [: the carrier of S, the carrier of T:]
bool the carrier of S is non empty set
x is Element of the carrier of S
x is Element of the carrier of S
{x,x} is non empty finite Element of bool the carrier of S
{} S is Relation-like non-empty empty-yielding empty proper finite finite-yielding V30() strongly_connected directed filtered lower upper Element of bool the carrier of S
S is non empty V87() reflexive transitive antisymmetric lower-bounded with_suprema RelStr
the carrier of S is non empty set
T is non empty V87() reflexive transitive antisymmetric lower-bounded with_suprema RelStr
the carrier of T is non empty set
[: the carrier of S, the carrier of T:] is Relation-like non empty set
bool [: the carrier of S, the carrier of T:] is non empty set
g is Relation-like the carrier of S -defined the carrier of T -valued Function-like non empty V22( the carrier of S) quasi_total Element of bool [: the carrier of S, the carrier of T:]
bool the carrier of S is non empty set
x is finite Element of bool the carrier of S
{} S is Relation-like non-empty empty-yielding empty proper finite finite-yielding V30() strongly_connected directed filtered lower upper closed_under_directed_sups property(S) Element of bool the carrier of S
x is finite Element of bool the carrier of S
x is set
x is set
{x} is non empty finite set
x \/ {x} is non empty set
x is finite Element of bool the carrier of S
y is finite Element of bool the carrier of S
g .: y is finite Element of bool the carrier of T
bool the carrier of T is non empty set
a is Element of the carrier of S
g . a is Element of the carrier of T
{(g . a)} is non empty finite Element of bool the carrier of T
"\/" (y,S) is Element of the carrier of S
{("\/" (y,S)),a} is non empty finite Element of bool the carrier of S
{a} is non empty finite Element of bool the carrier of S
"\/" ({a},S) is Element of the carrier of S
g .: x is finite Element of bool the carrier of T
"\/" ((g .: x),T) is Element of the carrier of T
"\/" (x,S) is Element of the carrier of S
g . ("\/" (x,S)) is Element of the carrier of T
dom g is non empty Element of bool the carrier of S
Im (g,a) is set
{a} is non empty finite set
g .: {a} is finite set
(g .: y) \/ {(g . a)} is non empty finite Element of bool the carrier of T
"\/" ({(g . a)},T) is Element of the carrier of T
"\/" ((g .: y),T) is Element of the carrier of T
("\/" ((g .: y),T)) "\/" (g . a) is Element of the carrier of T
g . ("\/" (y,S)) is Element of the carrier of T
(g . ("\/" (y,S))) "\/" (g . a) is Element of the carrier of T
("\/" (y,S)) "\/" a is Element of the carrier of S
g . (("\/" (y,S)) "\/" a) is Element of the carrier of T
S is non empty RelStr
the carrier of S is non empty set
T is non empty RelStr
the carrier of T is non empty set
[: the carrier of S, the carrier of T:] is Relation-like non empty set
bool [: the carrier of S, the carrier of T:] is non empty set
g is Relation-like the carrier of S -defined the carrier of T -valued Function-like non empty V22( the carrier of S) quasi_total Element of bool [: the carrier of S, the carrier of T:]
bool the carrier of S is non empty set
x is finite Element of bool the carrier of S
g is Relation-like the carrier of S -defined the carrier of T -valued Function-like non empty V22( the carrier of S) quasi_total Element of bool [: the carrier of S, the carrier of T:]
S is non empty RelStr
the carrier of S is non empty set
T is non empty V87() reflexive antisymmetric lower-bounded RelStr
the carrier of T is non empty set
[: the carrier of S, the carrier of T:] is Relation-like non empty set
bool [: the carrier of S, the carrier of T:] is non empty set
the Relation-like the carrier of S -defined the carrier of T -valued Function-like non empty V22( the carrier of S) quasi_total sups-preserving join-preserving directed-sups-preserving (S,T) (S,T) Element of bool [: the carrier of S, the carrier of T:] is Relation-like the carrier of S -defined the carrier of T -valued Function-like non empty V22( the carrier of S) quasi_total sups-preserving join-preserving directed-sups-preserving (S,T) (S,T) Element of bool [: the carrier of S, the carrier of T:]
S is non empty V87() reflexive transitive antisymmetric lower-bounded RelStr
CompactSublatt S is non empty strict V87() reflexive transitive antisymmetric full SubRelStr of S
Bottom S is Element of the carrier of S
the carrier of S is non empty set
"\/" ({},S) is Element of the carrier of S
the carrier of (CompactSublatt S) is non empty set
T is Element of the carrier of (CompactSublatt S)
g is Element of the carrier of (CompactSublatt S)
x is Element of the carrier of S
S is RelStr
the carrier of S is set
T is non empty RelStr
the carrier of T is non empty set
[: the carrier of S, the carrier of T:] is Relation-like set
bool [: the carrier of S, the carrier of T:] is non empty set
g is Relation-like the carrier of S -defined the carrier of T -valued Function-like V22( the carrier of S) quasi_total Element of bool [: the carrier of S, the carrier of T:]
x is SubRelStr of S
the carrier of x is set
g .: the carrier of x is Element of bool the carrier of T
bool the carrier of T is non empty set
g | the carrier of x is Relation-like the carrier of S -defined the carrier of x -defined the carrier of S -defined the carrier of T -valued Function-like Element of bool [: the carrier of S, the carrier of T:]
x is SubRelStr of T
the carrier of x is set
[: the carrier of x, the carrier of x:] is Relation-like set
bool [: the carrier of x, the carrier of x:] is non empty set
dom g is Element of bool the carrier of S
bool the carrier of S is non empty set
dom (g | the carrier of x) is Element of bool the carrier of S
rng (g | the carrier of x) is Element of bool the carrier of T
S is non empty V87() reflexive transitive antisymmetric with_suprema with_infima RelStr
the carrier of S is non empty set
T is non empty V87() reflexive transitive antisymmetric with_suprema with_infima RelStr
the carrier of T is non empty set
[: the carrier of S, the carrier of T:] is Relation-like non empty set
bool [: the carrier of S, the carrier of T:] is non empty set
g is Relation-like the carrier of S -defined the carrier of T -valued Function-like non empty V22( the carrier of S) quasi_total join-preserving Element of bool [: the carrier of S, the carrier of T:]
x is non empty V87() reflexive transitive antisymmetric full join-inheriting with_suprema SubRelStr of S
the carrier of x is non empty set
g | the carrier of x is Relation-like the carrier of S -defined the carrier of x -defined the carrier of S -defined the carrier of T -valued Function-like Element of bool [: the carrier of S, the carrier of T:]
x is non empty V87() reflexive transitive antisymmetric full join-inheriting with_suprema SubRelStr of T
the carrier of x is non empty set
[: the carrier of x, the carrier of x:] is Relation-like non empty set
bool [: the carrier of x, the carrier of x:] is non empty set
x is Relation-like the carrier of x -defined the carrier of x -valued Function-like non empty V22( the carrier of x) quasi_total Element of bool [: the carrier of x, the carrier of x:]
x is Element of the carrier of x
y is Element of the carrier of x
x "\/" y is Element of the carrier of x
a is Element of the carrier of S
b is Element of the carrier of S
a "\/" b is Element of the carrier of S
x . (x "\/" y) is Element of the carrier of x
g . (a "\/" b) is Element of the carrier of T
x . x is Element of the carrier of x
g . a is Element of the carrier of T
x . y is Element of the carrier of x
g . b is Element of the carrier of T
(g . a) "\/" (g . b) is Element of the carrier of T
(x . x) "\/" (x . y) is Element of the carrier of x
S is non empty V87() reflexive transitive antisymmetric lower-bounded with_suprema with_infima RelStr
the carrier of S is non empty set
T is non empty V87() reflexive transitive antisymmetric lower-bounded with_suprema with_infima RelStr
the carrier of T is non empty set
[: the carrier of S, the carrier of T:] is Relation-like non empty set
bool [: the carrier of S, the carrier of T:] is non empty set
g is Relation-like the carrier of S -defined the carrier of T -valued Function-like non empty V22( the carrier of S) quasi_total join-preserving (S,T) (S,T) Element of bool [: the carrier of S, the carrier of T:]
x is non empty V87() reflexive transitive antisymmetric lower-bounded full join-inheriting with_suprema (S) (S) SubRelStr of S
the carrier of x is non empty set
g | the carrier of x is Relation-like the carrier of S -defined the carrier of x -defined the carrier of S -defined the carrier of T -valued Function-like Element of bool [: the carrier of S, the carrier of T:]
x is non empty V87() reflexive transitive antisymmetric lower-bounded full join-inheriting with_suprema (T) (T) SubRelStr of T
the carrier of x is non empty set
[: the carrier of x, the carrier of x:] is Relation-like non empty set
bool [: the carrier of x, the carrier of x:] is non empty set
x is Relation-like the carrier of x -defined the carrier of x -valued Function-like non empty V22( the carrier of x) quasi_total Element of bool [: the carrier of x, the carrier of x:]
Bottom x is Element of the carrier of x
"\/" ({},x) is Element of the carrier of x
Bottom S is Element of the carrier of S
"\/" ({},S) is Element of the carrier of S
x . (Bottom x) is Element of the carrier of x
g . (Bottom S) is Element of the carrier of T
Bottom T is Element of the carrier of T
"\/" ({},T) is Element of the carrier of T
Bottom x is Element of the carrier of x
"\/" ({},x) is Element of the carrier of x
S is non empty V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
CompactSublatt S is non empty strict V87() reflexive transitive antisymmetric lower-bounded full join-inheriting with_suprema SubRelStr of S
Bottom S is Element of the carrier of S
the carrier of S is non empty set
"\/" ({},S) is Element of the carrier of S
the carrier of (CompactSublatt S) is non empty set
T is non empty V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
the carrier of T is non empty set
S is non empty V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
the carrier of S is non empty set
[: the carrier of T, the carrier of S:] is Relation-like non empty set
bool [: the carrier of T, the carrier of S:] is non empty set
CompactSublatt T is non empty strict V87() reflexive transitive antisymmetric lower-bounded full join-inheriting with_suprema (T) (T) SubRelStr of T
the carrier of (CompactSublatt T) is non empty set
CompactSublatt S is non empty strict V87() reflexive transitive antisymmetric lower-bounded full join-inheriting with_suprema (S) (S) SubRelStr of S
the carrier of (CompactSublatt S) is non empty set
[: the carrier of (CompactSublatt T), the carrier of (CompactSublatt S):] is Relation-like non empty set
bool [: the carrier of (CompactSublatt T), the carrier of (CompactSublatt S):] is non empty set
g is Relation-like the carrier of T -defined the carrier of S -valued Function-like non empty V22( the carrier of T) quasi_total sups-preserving join-preserving directed-sups-preserving monotone (T,S) (T,S) Element of bool [: the carrier of T, the carrier of S:]
g | the carrier of (CompactSublatt T) is Relation-like the carrier of T -defined the carrier of (CompactSublatt T) -defined the carrier of T -defined the carrier of S -valued Function-like Element of bool [: the carrier of T, the carrier of S:]
g .: the carrier of (CompactSublatt T) is Element of bool the carrier of S
bool the carrier of S is non empty set
x is set
x is set
g . x is set
x is Element of the carrier of T
g . x is Element of the carrier of S
x is Element of the carrier of T
g . x is Element of the carrier of S
(g | the carrier of (CompactSublatt T)) . x is set
T is non empty V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
S is non empty V87() reflexive transitive antisymmetric lower-bounded upper-bounded V94() up-complete /\-complete with_suprema with_infima complete RelStr
the carrier of S is non empty set
the carrier of T is non empty set
[: the carrier of S, the carrier of T:] is Relation-like non empty set
bool [: the carrier of S, the carrier of T:] is non empty set
CompactSublatt T is non empty strict V87() reflexive transitive antisymmetric lower-bounded full join-inheriting with_suprema (T) (T) SubRelStr of T
the carrier of (CompactSublatt T) is non empty set
CompactSublatt S is non empty strict V87() reflexive transitive antisymmetric lower-bounded full join-inheriting with_suprema (S) (S) SubRelStr of S
the carrier of (CompactSublatt S) is non empty set
[: the carrier of (CompactSublatt T), the carrier of (CompactSublatt S):] is Relation-like non empty set
bool [: the carrier of (CompactSublatt T), the carrier of (CompactSublatt S):] is non empty set
g is Relation-like the carrier of S -defined the carrier of T -valued Function-like non empty V22( the carrier of S) quasi_total infs-preserving meet-preserving filtered-infs-preserving monotone Element of bool [: the carrier of S, the carrier of T:]
(S,T,g) is Relation-like the carrier of T -defined the carrier of S -valued Function-like non empty V22( the carrier of T) quasi_total sups-preserving join-preserving directed-sups-preserving monotone lower_adjoint (T,S) (T,S) Element of bool [: the carrier of T, the carrier of S:]
[: the carrier of T, the carrier of S:] is Relation-like non empty set
bool [: the carrier of T, the carrier of S:] is non empty set
(S,T,g) | the carrier of (CompactSublatt T) is Relation-like the carrier of T -defined the carrier of (CompactSublatt T) -defined the carrier of T -defined the carrier of S -valued Function-like Element of bool [: the carrier of T, the carrier of S:]