:: LATTICE3 semantic presentation

K118() is set
bool K118() is non empty set

the empty set is empty set
1 is non empty set
D is set
bool D is non empty Element of bool (bool D)
bool D is non empty set
bool (bool D) is non empty set
[:(bool D),(bool D):] is non empty set
[:[:(bool D),(bool D):],(bool D):] is non empty set
bool [:[:(bool D),(bool D):],(bool D):] is non empty set
f is Relation-like [:(bool D),(bool D):] -defined bool D -valued Function-like V18([:(bool D),(bool D):], bool D) Element of bool [:[:(bool D),(bool D):],(bool D):]
R is Relation-like [:(bool D),(bool D):] -defined bool D -valued Function-like V18([:(bool D),(bool D):], bool D) Element of bool [:[:(bool D),(bool D):],(bool D):]
LattStr(# (bool D),f,R #) is non empty strict LattStr
the carrier of LattStr(# (bool D),f,R #) is non empty set
the L_join of LattStr(# (bool D),f,R #) is Relation-like [: the carrier of LattStr(# (bool D),f,R #), the carrier of LattStr(# (bool D),f,R #):] -defined the carrier of LattStr(# (bool D),f,R #) -valued Function-like V18([: the carrier of LattStr(# (bool D),f,R #), the carrier of LattStr(# (bool D),f,R #):], the carrier of LattStr(# (bool D),f,R #)) Element of bool [:[: the carrier of LattStr(# (bool D),f,R #), the carrier of LattStr(# (bool D),f,R #):], the carrier of LattStr(# (bool D),f,R #):]
[: the carrier of LattStr(# (bool D),f,R #), the carrier of LattStr(# (bool D),f,R #):] is non empty set
[:[: the carrier of LattStr(# (bool D),f,R #), the carrier of LattStr(# (bool D),f,R #):], the carrier of LattStr(# (bool D),f,R #):] is non empty set
bool [:[: the carrier of LattStr(# (bool D),f,R #), the carrier of LattStr(# (bool D),f,R #):], the carrier of LattStr(# (bool D),f,R #):] is non empty set
the L_meet of LattStr(# (bool D),f,R #) is Relation-like [: the carrier of LattStr(# (bool D),f,R #), the carrier of LattStr(# (bool D),f,R #):] -defined the carrier of LattStr(# (bool D),f,R #) -valued Function-like V18([: the carrier of LattStr(# (bool D),f,R #), the carrier of LattStr(# (bool D),f,R #):], the carrier of LattStr(# (bool D),f,R #)) Element of bool [:[: the carrier of LattStr(# (bool D),f,R #), the carrier of LattStr(# (bool D),f,R #):], the carrier of LattStr(# (bool D),f,R #):]
R is Element of bool D
A is Element of bool D
the L_join of LattStr(# (bool D),f,R #) . (R,A) is set
R \/ A is Element of bool D
A is Element of bool D
L is Element of bool D
the L_meet of LattStr(# (bool D),f,R #) . (A,L) is set
A /\ L is Element of bool D

the carrier of f is set
the L_join of f is Relation-like [: the carrier of f, the carrier of f:] -defined the carrier of f -valued Function-like V18([: the carrier of f, the carrier of f:], the carrier of f) Element of bool [:[: the carrier of f, the carrier of f:], the carrier of f:]
[: the carrier of f, the carrier of f:] is set
[:[: the carrier of f, the carrier of f:], the carrier of f:] is set
bool [:[: the carrier of f, the carrier of f:], the carrier of f:] is non empty set
the L_meet of f is Relation-like [: the carrier of f, the carrier of f:] -defined the carrier of f -valued Function-like V18([: the carrier of f, the carrier of f:], the carrier of f) Element of bool [:[: the carrier of f, the carrier of f:], the carrier of f:]

the carrier of R is set
the L_join of R is Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like V18([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the L_meet of R is Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like V18([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[:(bool D),(bool D):] is non empty set
[:[:(bool D),(bool D):],(bool D):] is non empty set
bool [:[:(bool D),(bool D):],(bool D):] is non empty set
R is Relation-like [:(bool D),(bool D):] -defined bool D -valued Function-like V18([:(bool D),(bool D):], bool D) Element of bool [:[:(bool D),(bool D):],(bool D):]
X is Element of bool D
Y is Element of bool D
R . (X,Y) is Element of bool D
X \/ Y is Element of bool D
A is Relation-like [:(bool D),(bool D):] -defined bool D -valued Function-like V18([:(bool D),(bool D):], bool D) Element of bool [:[:(bool D),(bool D):],(bool D):]
A . (X,Y) is Element of bool D
A is Relation-like [:(bool D),(bool D):] -defined bool D -valued Function-like V18([:(bool D),(bool D):], bool D) Element of bool [:[:(bool D),(bool D):],(bool D):]
X is Element of bool D
Y is Element of bool D
A . (X,Y) is Element of bool D
X /\ Y is Element of bool D
L is Relation-like [:(bool D),(bool D):] -defined bool D -valued Function-like V18([:(bool D),(bool D):], bool D) Element of bool [:[:(bool D),(bool D):],(bool D):]
L . (X,Y) is Element of bool D
D is set
(D) is strict LattStr
the carrier of (D) is set
bool D is non empty Element of bool (bool D)
bool D is non empty set
bool (bool D) is non empty set
D is set
(D) is non empty strict LattStr
the carrier of (D) is non empty set
f is Element of the carrier of (D)
R is set
R is Element of the carrier of (D)
A is set
f "\/" R is Element of the carrier of (D)
the L_join of (D) is Relation-like [: the carrier of (D), the carrier of (D):] -defined the carrier of (D) -valued Function-like V18([: the carrier of (D), the carrier of (D):], the carrier of (D)) Element of bool [:[: the carrier of (D), the carrier of (D):], the carrier of (D):]
[: the carrier of (D), the carrier of (D):] is non empty set
[:[: the carrier of (D), the carrier of (D):], the carrier of (D):] is non empty set
bool [:[: the carrier of (D), the carrier of (D):], the carrier of (D):] is non empty set
the L_join of (D) . (f,R) is Element of the carrier of (D)
R \/ A is set
bool D is non empty set
A is Element of bool D
L is Element of bool D
H2((D)) . (A,L) is set
f \/ R is set
f "/\" R is Element of the carrier of (D)
the L_meet of (D) is Relation-like [: the carrier of (D), the carrier of (D):] -defined the carrier of (D) -valued Function-like V18([: the carrier of (D), the carrier of (D):], the carrier of (D)) Element of bool [:[: the carrier of (D), the carrier of (D):], the carrier of (D):]
the L_meet of (D) . (f,R) is Element of the carrier of (D)
R /\ A is set
bool D is non empty set
A is Element of bool D
L is Element of bool D
H3((D)) . (A,L) is set
f /\ R is set
D is set
(D) is non empty strict LattStr
the carrier of (D) is non empty set
R is set
(R) is non empty strict LattStr
the carrier of (R) is non empty set
f is Element of the carrier of (D)
R is Element of the carrier of (D)
f "\/" R is Element of the carrier of (D)
the L_join of (D) is Relation-like [: the carrier of (D), the carrier of (D):] -defined the carrier of (D) -valued Function-like V18([: the carrier of (D), the carrier of (D):], the carrier of (D)) Element of bool [:[: the carrier of (D), the carrier of (D):], the carrier of (D):]
[: the carrier of (D), the carrier of (D):] is non empty set
[:[: the carrier of (D), the carrier of (D):], the carrier of (D):] is non empty set
bool [:[: the carrier of (D), the carrier of (D):], the carrier of (D):] is non empty set
the L_join of (D) . (f,R) is Element of the carrier of (D)
f \/ R is set
A is Element of the carrier of (R)
A is Element of the carrier of (R)
A "/\" A is Element of the carrier of (R)
the L_meet of (R) is Relation-like [: the carrier of (R), the carrier of (R):] -defined the carrier of (R) -valued Function-like V18([: the carrier of (R), the carrier of (R):], the carrier of (R)) Element of bool [:[: the carrier of (R), the carrier of (R):], the carrier of (R):]
[: the carrier of (R), the carrier of (R):] is non empty set
[:[: the carrier of (R), the carrier of (R):], the carrier of (R):] is non empty set
bool [:[: the carrier of (R), the carrier of (R):], the carrier of (R):] is non empty set
the L_meet of (R) . (A,A) is Element of the carrier of (R)
A /\ A is set
D is set
(D) is non empty strict LattStr
the carrier of (D) is non empty set
f is Element of the carrier of (D)
R is Element of the carrier of (D)
f "\/" R is Element of the carrier of (D)
the L_join of (D) is Relation-like [: the carrier of (D), the carrier of (D):] -defined the carrier of (D) -valued Function-like V18([: the carrier of (D), the carrier of (D):], the carrier of (D)) Element of bool [:[: the carrier of (D), the carrier of (D):], the carrier of (D):]
[: the carrier of (D), the carrier of (D):] is non empty set
[:[: the carrier of (D), the carrier of (D):], the carrier of (D):] is non empty set
bool [:[: the carrier of (D), the carrier of (D):], the carrier of (D):] is non empty set
the L_join of (D) . (f,R) is Element of the carrier of (D)
f \/ R is set
D is set
(D) is non empty strict LattStr
the carrier of (D) is non empty set
f is Element of the carrier of (D)
R is Element of the carrier of (D)
f "\/" R is Element of the carrier of (D)
the L_join of (D) is Relation-like [: the carrier of (D), the carrier of (D):] -defined the carrier of (D) -valued Function-like V18([: the carrier of (D), the carrier of (D):], the carrier of (D)) Element of bool [:[: the carrier of (D), the carrier of (D):], the carrier of (D):]
[: the carrier of (D), the carrier of (D):] is non empty set
[:[: the carrier of (D), the carrier of (D):], the carrier of (D):] is non empty set
bool [:[: the carrier of (D), the carrier of (D):], the carrier of (D):] is non empty set
the L_join of (D) . (f,R) is Element of the carrier of (D)
f \/ R is set
R "\/" f is Element of the carrier of (D)
the L_join of (D) . (R,f) is Element of the carrier of (D)
R \/ f is set
f is Element of the carrier of (D)
R is Element of the carrier of (D)
R is Element of the carrier of (D)
R "\/" R is Element of the carrier of (D)
the L_join of (D) is Relation-like [: the carrier of (D), the carrier of (D):] -defined the carrier of (D) -valued Function-like V18([: the carrier of (D), the carrier of (D):], the carrier of (D)) Element of bool [:[: the carrier of (D), the carrier of (D):], the carrier of (D):]
[: the carrier of (D), the carrier of (D):] is non empty set
[:[: the carrier of (D), the carrier of (D):], the carrier of (D):] is non empty set
bool [:[: the carrier of (D), the carrier of (D):], the carrier of (D):] is non empty set
the L_join of (D) . (R,R) is Element of the carrier of (D)
R \/ R is set
f "\/" (R "\/" R) is Element of the carrier of (D)
the L_join of (D) . (f,(R "\/" R)) is Element of the carrier of (D)
f \/ (R "\/" R) is set
f "\/" R is Element of the carrier of (D)
the L_join of (D) . (f,R) is Element of the carrier of (D)
f \/ R is set
(f "\/" R) "\/" R is Element of the carrier of (D)
the L_join of (D) . ((f "\/" R),R) is Element of the carrier of (D)
(f "\/" R) \/ R is set
f is Element of the carrier of (D)
R is Element of the carrier of (D)
f "/\" R is Element of the carrier of (D)
the L_meet of (D) is Relation-like [: the carrier of (D), the carrier of (D):] -defined the carrier of (D) -valued Function-like V18([: the carrier of (D), the carrier of (D):], the carrier of (D)) Element of bool [:[: the carrier of (D), the carrier of (D):], the carrier of (D):]
[: the carrier of (D), the carrier of (D):] is non empty set
[:[: the carrier of (D), the carrier of (D):], the carrier of (D):] is non empty set
bool [:[: the carrier of (D), the carrier of (D):], the carrier of (D):] is non empty set
the L_meet of (D) . (f,R) is Element of the carrier of (D)
f /\ R is set
(f "/\" R) "\/" R is Element of the carrier of (D)
the L_join of (D) is Relation-like [: the carrier of (D), the carrier of (D):] -defined the carrier of (D) -valued Function-like V18([: the carrier of (D), the carrier of (D):], the carrier of (D)) Element of bool [:[: the carrier of (D), the carrier of (D):], the carrier of (D):]
the L_join of (D) . ((f "/\" R),R) is Element of the carrier of (D)
(f "/\" R) \/ R is set
f is Element of the carrier of (D)
R is Element of the carrier of (D)
f "/\" R is Element of the carrier of (D)
the L_meet of (D) is Relation-like [: the carrier of (D), the carrier of (D):] -defined the carrier of (D) -valued Function-like V18([: the carrier of (D), the carrier of (D):], the carrier of (D)) Element of bool [:[: the carrier of (D), the carrier of (D):], the carrier of (D):]
[: the carrier of (D), the carrier of (D):] is non empty set
[:[: the carrier of (D), the carrier of (D):], the carrier of (D):] is non empty set
bool [:[: the carrier of (D), the carrier of (D):], the carrier of (D):] is non empty set
the L_meet of (D) . (f,R) is Element of the carrier of (D)
f /\ R is set
R "/\" f is Element of the carrier of (D)
the L_meet of (D) . (R,f) is Element of the carrier of (D)
R /\ f is set
f is Element of the carrier of (D)
R is Element of the carrier of (D)
R is Element of the carrier of (D)
R "/\" R is Element of the carrier of (D)
the L_meet of (D) is Relation-like [: the carrier of (D), the carrier of (D):] -defined the carrier of (D) -valued Function-like V18([: the carrier of (D), the carrier of (D):], the carrier of (D)) Element of bool [:[: the carrier of (D), the carrier of (D):], the carrier of (D):]
[: the carrier of (D), the carrier of (D):] is non empty set
[:[: the carrier of (D), the carrier of (D):], the carrier of (D):] is non empty set
bool [:[: the carrier of (D), the carrier of (D):], the carrier of (D):] is non empty set
the L_meet of (D) . (R,R) is Element of the carrier of (D)
R /\ R is set
f "/\" (R "/\" R) is Element of the carrier of (D)
the L_meet of (D) . (f,(R "/\" R)) is Element of the carrier of (D)
f /\ (R "/\" R) is set
f "/\" R is Element of the carrier of (D)
the L_meet of (D) . (f,R) is Element of the carrier of (D)
f /\ R is set
(f "/\" R) "/\" R is Element of the carrier of (D)
the L_meet of (D) . ((f "/\" R),R) is Element of the carrier of (D)
(f "/\" R) /\ R is set
f is Element of the carrier of (D)
R is Element of the carrier of (D)
f "\/" R is Element of the carrier of (D)
the L_join of (D) is Relation-like [: the carrier of (D), the carrier of (D):] -defined the carrier of (D) -valued Function-like V18([: the carrier of (D), the carrier of (D):], the carrier of (D)) Element of bool [:[: the carrier of (D), the carrier of (D):], the carrier of (D):]
[: the carrier of (D), the carrier of (D):] is non empty set
[:[: the carrier of (D), the carrier of (D):], the carrier of (D):] is non empty set
bool [:[: the carrier of (D), the carrier of (D):], the carrier of (D):] is non empty set
the L_join of (D) . (f,R) is Element of the carrier of (D)
f \/ R is set
f "/\" (f "\/" R) is Element of the carrier of (D)
the L_meet of (D) is Relation-like [: the carrier of (D), the carrier of (D):] -defined the carrier of (D) -valued Function-like V18([: the carrier of (D), the carrier of (D):], the carrier of (D)) Element of bool [:[: the carrier of (D), the carrier of (D):], the carrier of (D):]
the L_meet of (D) . (f,(f "\/" R)) is Element of the carrier of (D)
f /\ (f "\/" R) is set
D is set

Bottom (D) is Element of the carrier of (D)
the carrier of (D) is non empty set
f is Element of the carrier of (D)
R is Element of the carrier of (D)
f "/\" R is Element of the carrier of (D)
the L_meet of (D) is Relation-like [: the carrier of (D), the carrier of (D):] -defined the carrier of (D) -valued Function-like V18([: the carrier of (D), the carrier of (D):], the carrier of (D)) commutative associative idempotent Element of bool [:[: the carrier of (D), the carrier of (D):], the carrier of (D):]
[: the carrier of (D), the carrier of (D):] is non empty set
[:[: the carrier of (D), the carrier of (D):], the carrier of (D):] is non empty set
bool [:[: the carrier of (D), the carrier of (D):], the carrier of (D):] is non empty set
the L_meet of (D) . (f,R) is Element of the carrier of (D)
f /\ R is set
R is Element of the carrier of (D)
R "/\" f is Element of the carrier of (D)
the L_meet of (D) is Relation-like [: the carrier of (D), the carrier of (D):] -defined the carrier of (D) -valued Function-like V18([: the carrier of (D), the carrier of (D):], the carrier of (D)) commutative associative idempotent Element of bool [:[: the carrier of (D), the carrier of (D):], the carrier of (D):]
[: the carrier of (D), the carrier of (D):] is non empty set
[:[: the carrier of (D), the carrier of (D):], the carrier of (D):] is non empty set
bool [:[: the carrier of (D), the carrier of (D):], the carrier of (D):] is non empty set
the L_meet of (D) . (R,f) is Element of the carrier of (D)
R /\ f is set
R is Element of the carrier of (D)
f "/\" R is Element of the carrier of (D)
the L_meet of (D) is Relation-like [: the carrier of (D), the carrier of (D):] -defined the carrier of (D) -valued Function-like V18([: the carrier of (D), the carrier of (D):], the carrier of (D)) commutative associative idempotent Element of bool [:[: the carrier of (D), the carrier of (D):], the carrier of (D):]
[: the carrier of (D), the carrier of (D):] is non empty set
[:[: the carrier of (D), the carrier of (D):], the carrier of (D):] is non empty set
bool [:[: the carrier of (D), the carrier of (D):], the carrier of (D):] is non empty set
the L_meet of (D) . (f,R) is Element of the carrier of (D)
f /\ R is set
R is Element of the carrier of (D)
R "/\" f is Element of the carrier of (D)
the L_meet of (D) . (R,f) is Element of the carrier of (D)
R /\ f is set
D is set

Top (D) is Element of the carrier of (D)
the carrier of (D) is non empty set
bool D is non empty Element of bool (bool D)
bool D is non empty set
bool (bool D) is non empty set
f is Element of the carrier of (D)
R is Element of the carrier of (D)
f "\/" R is Element of the carrier of (D)
the L_join of (D) is Relation-like [: the carrier of (D), the carrier of (D):] -defined the carrier of (D) -valued Function-like V18([: the carrier of (D), the carrier of (D):], the carrier of (D)) commutative associative idempotent Element of bool [:[: the carrier of (D), the carrier of (D):], the carrier of (D):]
[: the carrier of (D), the carrier of (D):] is non empty set
[:[: the carrier of (D), the carrier of (D):], the carrier of (D):] is non empty set
bool [:[: the carrier of (D), the carrier of (D):], the carrier of (D):] is non empty set
the L_join of (D) . (f,R) is Element of the carrier of (D)
f \/ R is set
R is Element of the carrier of (D)
R "\/" f is Element of the carrier of (D)
the L_join of (D) is Relation-like [: the carrier of (D), the carrier of (D):] -defined the carrier of (D) -valued Function-like V18([: the carrier of (D), the carrier of (D):], the carrier of (D)) commutative associative idempotent Element of bool [:[: the carrier of (D), the carrier of (D):], the carrier of (D):]
[: the carrier of (D), the carrier of (D):] is non empty set
[:[: the carrier of (D), the carrier of (D):], the carrier of (D):] is non empty set
bool [:[: the carrier of (D), the carrier of (D):], the carrier of (D):] is non empty set
the L_join of (D) . (R,f) is Element of the carrier of (D)
R \/ f is set
R is Element of the carrier of (D)
f "\/" R is Element of the carrier of (D)
the L_join of (D) is Relation-like [: the carrier of (D), the carrier of (D):] -defined the carrier of (D) -valued Function-like V18([: the carrier of (D), the carrier of (D):], the carrier of (D)) commutative associative idempotent Element of bool [:[: the carrier of (D), the carrier of (D):], the carrier of (D):]
[: the carrier of (D), the carrier of (D):] is non empty set
[:[: the carrier of (D), the carrier of (D):], the carrier of (D):] is non empty set
bool [:[: the carrier of (D), the carrier of (D):], the carrier of (D):] is non empty set
the L_join of (D) . (f,R) is Element of the carrier of (D)
f \/ R is set
R is Element of the carrier of (D)
R "\/" f is Element of the carrier of (D)
the L_join of (D) . (R,f) is Element of the carrier of (D)
R \/ f is set
D is set

the carrier of R is non empty set
R is Element of the carrier of R
bool D is non empty Element of bool (bool D)
bool D is non empty set
bool (bool D) is non empty set
D \ R is Element of bool D
A is Element of the carrier of R
A "\/" R is Element of the carrier of R
the L_join of R is Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like V18([: the carrier of R, the carrier of R:], the carrier of R) commutative associative idempotent Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the L_join of R . (A,R) is Element of the carrier of R
A \/ R is set
D \/ R is set
Top R is Element of the carrier of R
R "\/" A is Element of the carrier of R
the L_join of R . (R,A) is Element of the carrier of R
A "/\" R is Element of the carrier of R
the L_meet of R is Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like V18([: the carrier of R, the carrier of R:], the carrier of R) commutative associative idempotent Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
the L_meet of R . (A,R) is Element of the carrier of R
Bottom R is Element of the carrier of R
R "/\" A is Element of the carrier of R
the L_meet of R . (R,A) is Element of the carrier of R
R "\/" A is Element of the carrier of R
A "/\" R is Element of the carrier of R
A /\ R is set
the carrier of R is non empty set
R is Element of the carrier of R
A is Element of the carrier of R
A is Element of the carrier of R
A "\/" A is Element of the carrier of R
the L_join of R is Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like V18([: the carrier of R, the carrier of R:], the carrier of R) commutative associative idempotent Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the L_join of R . (A,A) is Element of the carrier of R
R "/\" (A "\/" A) is Element of the carrier of R
the L_meet of R is Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like V18([: the carrier of R, the carrier of R:], the carrier of R) commutative associative idempotent Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
the L_meet of R . (R,(A "\/" A)) is Element of the carrier of R
R "/\" A is Element of the carrier of R
the L_meet of R . (R,A) is Element of the carrier of R
R "/\" A is Element of the carrier of R
the L_meet of R . (R,A) is Element of the carrier of R
(R "/\" A) "\/" (R "/\" A) is Element of the carrier of R
the L_join of R . ((R "/\" A),(R "/\" A)) is Element of the carrier of R
A "\/" A is Element of the carrier of R
R "/\" (A "\/" A) is Element of the carrier of R
the L_meet of R . (R,(A "\/" A)) is Element of the carrier of R
R /\ (A "\/" A) is set
A \/ A is set
R /\ (A \/ A) is set
R /\ A is set
R /\ A is set
(R /\ A) \/ (R /\ A) is set
R "/\" A is Element of the carrier of R
(R "/\" A) \/ (R /\ A) is set
R "/\" A is Element of the carrier of R
(R "/\" A) \/ (R "/\" A) is set
(R "/\" A) "\/" (R "/\" A) is Element of the carrier of R
the L_join of R . ((R "/\" A),(R "/\" A)) is Element of the carrier of R
D is set

the carrier of (D) is non empty set
R is Element of the carrier of (D)
R ` is Element of the carrier of (D)
D \ R is Element of bool D
bool D is non empty set
(R `) "/\" R is Element of the carrier of (D)
the L_meet of (D) is Relation-like [: the carrier of (D), the carrier of (D):] -defined the carrier of (D) -valued Function-like V18([: the carrier of (D), the carrier of (D):], the carrier of (D)) commutative associative idempotent Element of bool [:[: the carrier of (D), the carrier of (D):], the carrier of (D):]
[: the carrier of (D), the carrier of (D):] is non empty set
[:[: the carrier of (D), the carrier of (D):], the carrier of (D):] is non empty set
bool [:[: the carrier of (D), the carrier of (D):], the carrier of (D):] is non empty set
the L_meet of (D) . ((R `),R) is Element of the carrier of (D)
(R `) /\ R is set
Bottom (D) is Element of the carrier of (D)
R "\/" (R `) is Element of the carrier of (D)
the L_join of (D) is Relation-like [: the carrier of (D), the carrier of (D):] -defined the carrier of (D) -valued Function-like V18([: the carrier of (D), the carrier of (D):], the carrier of (D)) commutative associative idempotent Element of bool [:[: the carrier of (D), the carrier of (D):], the carrier of (D):]
the L_join of (D) . (R,(R `)) is Element of the carrier of (D)
R \/ (R `) is set
Top (D) is Element of the carrier of (D)
R \ R is Element of bool R
bool R is non empty set
(R `) \ R is Element of bool (R `)
bool (R `) is non empty set
(R \ R) \/ ((R `) \ R) is set

the carrier of D is non empty set
[: the carrier of D, the carrier of D:] is non empty set
bool [: the carrier of D, the carrier of D:] is non empty set
{ [b1,b2] where b1, b2 is Element of the carrier of D : b1 [= b2 } is set
[:H1(D),H1(D):] is set
f is set
R is set
[f,R] is set
{f,R} is non empty set
{f} is non empty set
{{f,R},{f}} is non empty set
R is Element of the carrier of D
A is Element of the carrier of D
[R,A] is Element of the carrier of [:D,D:]
[:D,D:] is strict LattStr
the carrier of [:D,D:] is set
{R,A} is non empty set
{R} is non empty set
{{R,A},{R}} is non empty set
bool [:H1(D),H1(D):] is non empty set
f is Relation-like H1(D) -defined H1(D) -valued Element of bool [:H1(D),H1(D):]
R is set
[R,R] is set
{R,R} is non empty set
{R} is non empty set
{{R,R},{R}} is non empty set
R is Element of the carrier of D
R is set
R is set
[R,R] is set
{R,R} is non empty set
{R} is non empty set
{{R,R},{R}} is non empty set
[R,R] is set
{R,R} is non empty set
{R} is non empty set
{{R,R},{R}} is non empty set
A is Element of the carrier of D
A is Element of the carrier of D
R is set
R is set
A is set
[R,R] is set
{R,R} is non empty set
{R} is non empty set
{{R,R},{R}} is non empty set
[R,A] is set
{R,A} is non empty set
{R} is non empty set
{{R,A},{R}} is non empty set
[R,A] is set
{R,A} is non empty set
{{R,A},{R}} is non empty set
A is Element of the carrier of D
L is Element of the carrier of D
X is Element of the carrier of D
dom f is Element of bool H1(D)
bool H1(D) is non empty set
field f is set
dom f is set
rng f is set
(dom f) \/ (rng f) is set

the carrier of D is non empty set
(D) is Relation-like the carrier of D -defined the carrier of D -valued V14( the carrier of D) reflexive antisymmetric transitive Element of bool [: the carrier of D, the carrier of D:]
[: the carrier of D, the carrier of D:] is non empty set
bool [: the carrier of D, the carrier of D:] is non empty set
RelStr(# the carrier of D,(D) #) is non empty strict V58() reflexive transitive antisymmetric RelStr

the carrier of D is non empty set
(D) is Relation-like the carrier of D -defined the carrier of D -valued V14( the carrier of D) reflexive antisymmetric transitive Element of bool [: the carrier of D, the carrier of D:]
[: the carrier of D, the carrier of D:] is non empty set
bool [: the carrier of D, the carrier of D:] is non empty set
RelStr(# the carrier of D,(D) #) is non empty strict V58() reflexive transitive antisymmetric RelStr

(D) is non empty strict V58() reflexive transitive antisymmetric RelStr
the carrier of D is non empty set
(D) is Relation-like the carrier of D -defined the carrier of D -valued V14( the carrier of D) reflexive antisymmetric transitive Element of bool [: the carrier of D, the carrier of D:]
[: the carrier of D, the carrier of D:] is non empty set
bool [: the carrier of D, the carrier of D:] is non empty set
RelStr(# the carrier of D,(D) #) is non empty strict V58() reflexive transitive antisymmetric RelStr

(f) is non empty strict V58() reflexive transitive antisymmetric RelStr
the carrier of f is non empty set
(f) is Relation-like the carrier of f -defined the carrier of f -valued V14( the carrier of f) reflexive antisymmetric transitive Element of bool [: the carrier of f, the carrier of f:]
[: the carrier of f, the carrier of f:] is non empty set
bool [: the carrier of f, the carrier of f:] is non empty set
RelStr(# the carrier of f,(f) #) is non empty strict V58() reflexive transitive antisymmetric RelStr
the L_join of D is Relation-like [: the carrier of D, the carrier of D:] -defined the carrier of D -valued Function-like V18([: the carrier of D, the carrier of D:], the carrier of D) commutative associative idempotent Element of bool [:[: the carrier of D, the carrier of D:], the carrier of D:]
[:[: the carrier of D, the carrier of D:], the carrier of D:] is non empty set
bool [:[: the carrier of D, the carrier of D:], the carrier of D:] is non empty set
the L_meet of D is Relation-like [: the carrier of D, the carrier of D:] -defined the carrier of D -valued Function-like V18([: the carrier of D, the carrier of D:], the carrier of D) commutative associative idempotent Element of bool [:[: the carrier of D, the carrier of D:], the carrier of D:]
LattStr(# the carrier of D, the L_join of D, the L_meet of D #) is non empty strict LattStr
the L_join of f is Relation-like [: the carrier of f, the carrier of f:] -defined the carrier of f -valued Function-like V18([: the carrier of f, the carrier of f:], the carrier of f) commutative associative idempotent Element of bool [:[: the carrier of f, the carrier of f:], the carrier of f:]
[:[: the carrier of f, the carrier of f:], the carrier of f:] is non empty set
bool [:[: the carrier of f, the carrier of f:], the carrier of f:] is non empty set
the L_meet of f is Relation-like [: the carrier of f, the carrier of f:] -defined the carrier of f -valued Function-like V18([: the carrier of f, the carrier of f:], the carrier of f) commutative associative idempotent Element of bool [:[: the carrier of f, the carrier of f:], the carrier of f:]
LattStr(# the carrier of f, the L_join of f, the L_meet of f #) is non empty strict LattStr
[:H1(D),H1(D):] is set
[:[:H1(D),H1(D):],H1(D):] is set
bool [:[:H1(D),H1(D):],H1(D):] is non empty set
A is Element of the carrier of D
A is Element of the carrier of D
A "\/" A is Element of the carrier of D
the L_join of D . (A,A) is Element of the carrier of D
L is Element of the carrier of f
X is Element of the carrier of f
L "\/" X is Element of the carrier of f
the L_join of f . (L,X) is Element of the carrier of f
A "\/" A is Element of the carrier of D
the L_join of D . (A,A) is Element of the carrier of D
X "\/" L is Element of the carrier of f
the L_join of f . (X,L) is Element of the carrier of f
Y is Element of the carrier of f
[L,Y] is Element of the carrier of [:f,f:]
[:f,f:] is strict LattStr
the carrier of [:f,f:] is set
{L,Y} is non empty set
{L} is non empty set
{{L,Y},{L}} is non empty set
[X,Y] is Element of the carrier of [:f,f:]
{X,Y} is non empty set
{X} is non empty set
{{X,Y},{X}} is non empty set
a is Element of the carrier of D
[A,a] is Element of the carrier of [:D,D:]
[:D,D:] is strict LattStr
the carrier of [:D,D:] is set
{A,a} is non empty set
{A} is non empty set
{{A,a},{A}} is non empty set
[A,a] is Element of the carrier of [:D,D:]
{A,a} is non empty set
{A} is non empty set
{{A,a},{A}} is non empty set
[a,(A "\/" A)] is Element of the carrier of [:D,D:]
{a,(A "\/" A)} is non empty set
{a} is non empty set
{{a,(A "\/" A)},{a}} is non empty set
H2(D) . (A,A) is Element of the carrier of D
R is Relation-like [:H1(D),H1(D):] -defined H1(D) -valued Function-like V18([:H1(D),H1(D):],H1(D)) Element of bool [:[:H1(D),H1(D):],H1(D):]
R . (A,A) is Element of the carrier of D
A is Element of the carrier of D
A is Element of the carrier of D
A "/\" A is Element of the carrier of D
the L_meet of D . (A,A) is Element of the carrier of D
L is Element of the carrier of f
X is Element of the carrier of f
L "/\" X is Element of the carrier of f
the L_meet of f . (L,X) is Element of the carrier of f
A "/\" A is Element of the carrier of D
the L_meet of D . (A,A) is Element of the carrier of D
X "/\" L is Element of the carrier of f
the L_meet of f . (X,L) is Element of the carrier of f
Y is Element of the carrier of f
[Y,L] is Element of the carrier of [:f,f:]
{Y,L} is non empty set
{Y} is non empty set
{{Y,L},{Y}} is non empty set
[Y,X] is Element of the carrier of [:f,f:]
{Y,X} is non empty set
{{Y,X},{Y}} is non empty set
a is Element of the carrier of D
[a,A] is Element of the carrier of [:D,D:]
{a,A} is non empty set
{a} is non empty set
{{a,A},{a}} is non empty set
[a,A] is Element of the carrier of [:D,D:]
{a,A} is non empty set
{{a,A},{a}} is non empty set
[(A "/\" A),a] is Element of the carrier of [:D,D:]
{(A "/\" A),a} is non empty set
{(A "/\" A)} is non empty set
{{(A "/\" A),a},{(A "/\" A)}} is non empty set
H3(D) . (A,A) is Element of the carrier of D
R is Relation-like [:H1(D),H1(D):] -defined H1(D) -valued Function-like V18([:H1(D),H1(D):],H1(D)) Element of bool [:[:H1(D),H1(D):],H1(D):]
R . (A,A) is Element of the carrier of D

the carrier of D is non empty set
f is Element of the carrier of D
(D) is non empty strict V58() reflexive transitive antisymmetric RelStr
(D) is Relation-like the carrier of D -defined the carrier of D -valued V14( the carrier of D) reflexive antisymmetric transitive Element of bool [: the carrier of D, the carrier of D:]
[: the carrier of D, the carrier of D:] is non empty set
bool [: the carrier of D, the carrier of D:] is non empty set
RelStr(# the carrier of D,(D) #) is non empty strict V58() reflexive transitive antisymmetric RelStr
the carrier of (D) is non empty set

(D) is non empty strict V58() reflexive transitive antisymmetric RelStr
the carrier of D is non empty set
(D) is Relation-like the carrier of D -defined the carrier of D -valued V14( the carrier of D) reflexive antisymmetric transitive Element of bool [: the carrier of D, the carrier of D:]
[: the carrier of D, the carrier of D:] is non empty set
bool [: the carrier of D, the carrier of D:] is non empty set
RelStr(# the carrier of D,(D) #) is non empty strict V58() reflexive transitive antisymmetric RelStr
the carrier of (D) is non empty set
f is Element of the carrier of (D)

the carrier of D is non empty set
(D) is non empty strict V58() reflexive transitive antisymmetric RelStr
(D) is Relation-like the carrier of D -defined the carrier of D -valued V14( the carrier of D) reflexive antisymmetric transitive Element of bool [: the carrier of D, the carrier of D:]
[: the carrier of D, the carrier of D:] is non empty set
bool [: the carrier of D, the carrier of D:] is non empty set
RelStr(# the carrier of D,(D) #) is non empty strict V58() reflexive transitive antisymmetric RelStr
f is Element of the carrier of D
(D,f) is Element of the carrier of (D)
the carrier of (D) is non empty set
R is Element of the carrier of D
(D,R) is Element of the carrier of (D)
[f,R] is Element of the carrier of [:D,D:]
[:D,D:] is strict LattStr
the carrier of [:D,D:] is set
{f,R} is non empty set
{f} is non empty set
{{f,R},{f}} is non empty set
D is set
[:D,D:] is set
bool [:D,D:] is non empty set

dom f is Element of bool D
bool D is non empty set

dom (f ~) is Element of bool D
D is RelStr
the carrier of D is set
the InternalRel of D is Relation-like the carrier of D -defined the carrier of D -valued Element of bool [: the carrier of D, the carrier of D:]
[: the carrier of D, the carrier of D:] is set
bool [: the carrier of D, the carrier of D:] is non empty set
the InternalRel of D ~ is Relation-like the carrier of D -defined the carrier of D -valued Element of bool [: the carrier of D, the carrier of D:]
RelStr(# the carrier of D,( the InternalRel of D ~) #) is strict RelStr

(D) is strict RelStr
the carrier of D is set
the InternalRel of D is Relation-like the carrier of D -defined the carrier of D -valued V14( the carrier of D) reflexive antisymmetric transitive Element of bool [: the carrier of D, the carrier of D:]
[: the carrier of D, the carrier of D:] is set
bool [: the carrier of D, the carrier of D:] is non empty set
the InternalRel of D ~ is Relation-like the carrier of D -defined the carrier of D -valued reflexive antisymmetric transitive Element of bool [: the carrier of D, the carrier of D:]
RelStr(# the carrier of D,( the InternalRel of D ~) #) is strict RelStr
( the carrier of D, the InternalRel of D) is Relation-like the carrier of D -defined the carrier of D -valued V14( the carrier of D) reflexive antisymmetric transitive Element of bool [: the carrier of D, the carrier of D:]
RelStr(# the carrier of D,( the carrier of D, the InternalRel of D) #) is strict V58() reflexive transitive antisymmetric RelStr
D is non empty RelStr
(D) is strict RelStr
the carrier of D is non empty set
the InternalRel of D is Relation-like the carrier of D -defined the carrier of D -valued Element of bool [: the carrier of D, the carrier of D:]
[: the carrier of D, the carrier of D:] is non empty set
bool [: the carrier of D, the carrier of D:] is non empty set
the InternalRel of D ~ is Relation-like the carrier of D -defined the carrier of D -valued Element of bool [: the carrier of D, the carrier of D:]
RelStr(# the carrier of D,( the InternalRel of D ~) #) is non empty strict RelStr
D is RelStr
(D) is strict RelStr
the carrier of D is set
the InternalRel of D is Relation-like the carrier of D -defined the carrier of D -valued Element of bool [: the carrier of D, the carrier of D:]
[: the carrier of D, the carrier of D:] is set
bool [: the carrier of D, the carrier of D:] is non empty set
the InternalRel of D ~ is Relation-like the carrier of D -defined the carrier of D -valued Element of bool [: the carrier of D, the carrier of D:]
RelStr(# the carrier of D,( the InternalRel of D ~) #) is strict RelStr
((D)) is strict RelStr
the carrier of (D) is set
the InternalRel of (D) is Relation-like the carrier of (D) -defined the carrier of (D) -valued Element of bool [: the carrier of (D), the carrier of (D):]
[: the carrier of (D), the carrier of (D):] is set
bool [: the carrier of (D), the carrier of (D):] is non empty set
the InternalRel of (D) ~ is Relation-like the carrier of (D) -defined the carrier of (D) -valued Element of bool [: the carrier of (D), the carrier of (D):]
RelStr(# the carrier of (D),( the InternalRel of (D) ~) #) is strict RelStr
RelStr(# the carrier of D, the InternalRel of D #) is strict RelStr
D is RelStr
the carrier of D is set
f is Element of the carrier of D
(D) is strict RelStr
the InternalRel of D is Relation-like the carrier of D -defined the carrier of D -valued Element of bool [: the carrier of D, the carrier of D:]
[: the carrier of D, the carrier of D:] is set
bool [: the carrier of D, the carrier of D:] is non empty set
the InternalRel of D ~ is Relation-like the carrier of D -defined the carrier of D -valued Element of bool [: the carrier of D, the carrier of D:]
RelStr(# the carrier of D,( the InternalRel of D ~) #) is strict RelStr
the carrier of (D) is set
D is RelStr
(D) is strict RelStr
the carrier of D is set
the InternalRel of D is Relation-like the carrier of D -defined the carrier of D -valued Element of bool [: the carrier of D, the carrier of D:]
[: the carrier of D, the carrier of D:] is set
bool [: the carrier of D, the carrier of D:] is non empty set
the InternalRel of D ~ is Relation-like the carrier of D -defined the carrier of D -valued Element of bool [: the carrier of D, the carrier of D:]
RelStr(# the carrier of D,( the InternalRel of D ~) #) is strict RelStr
the carrier of (D) is set
f is Element of the carrier of (D)
D is RelStr
the carrier of D is set
(D) is strict RelStr
the InternalRel of D is Relation-like the carrier of D -defined the carrier of D -valued Element of bool [: the carrier of D, the carrier of D:]
[: the carrier of D, the carrier of D:] is set
bool [: the carrier of D, the carrier of D:] is non empty set
the InternalRel of D ~ is Relation-like the carrier of D -defined the carrier of D -valued Element of bool [: the carrier of D, the carrier of D:]
RelStr(# the carrier of D,( the InternalRel of D ~) #) is strict RelStr
f is Element of the carrier of D
(D,f) is Element of the carrier of (D)
the carrier of (D) is set
R is Element of the carrier of D
(D,R) is Element of the carrier of (D)
[f,R] is set
{f,R} is non empty set
{f} is non empty set
{{f,R},{f}} is non empty set
[(D,R),(D,f)] is set
{(D,R),(D,f)} is non empty set
{(D,R)} is non empty set
{{(D,R),(D,f)},{(D,R)}} is non empty set
the InternalRel of (D) is Relation-like the carrier of (D) -defined the carrier of (D) -valued Element of bool [: the carrier of (D), the carrier of (D):]
[: the carrier of (D), the carrier of (D):] is set
bool [: the carrier of (D), the carrier of (D):] is non empty set
D is RelStr
the carrier of D is set
D is RelStr
the carrier of D is set
D is RelStr
the carrier of D is set
the Element of the carrier of D is Element of the carrier of D
R is Element of the carrier of D
[ the Element of the carrier of D,R] is set
{ the Element of the carrier of D,R} is non empty set
{ the Element of the carrier of D} is non empty set
{{ the Element of the carrier of D,R},{ the Element of the carrier of D}} is non empty set
the InternalRel of D is Relation-like the carrier of D -defined the carrier of D -valued Element of bool [: the carrier of D, the carrier of D:]
[: the carrier of D, the carrier of D:] is set
bool [: the carrier of D, the carrier of D:] is non empty set
D is RelStr
the carrier of D is set
the Element of the carrier of D is Element of the carrier of D
R is Element of the carrier of D
[R, the Element of the carrier of D] is set
{R, the Element of the carrier of D} is non empty set
{R} is non empty set
{{R, the Element of the carrier of D},{R}} is non empty set
the InternalRel of D is Relation-like the carrier of D -defined the carrier of D -valued Element of bool [: the carrier of D, the carrier of D:]
[: the carrier of D, the carrier of D:] is set
bool [: the carrier of D, the carrier of D:] is non empty set
D is RelStr
(D) is strict RelStr
the carrier of D is set
the InternalRel of D is Relation-like the carrier of D -defined the carrier of D -valued Element of bool [: the carrier of D, the carrier of D:]
[: the carrier of D, the carrier of D:] is set
bool [: the carrier of D, the carrier of D:] is non empty set
the InternalRel of D ~ is Relation-like the carrier of D -defined the carrier of D -valued Element of bool [: the carrier of D, the carrier of D:]
RelStr(# the carrier of D,( the InternalRel of D ~) #) is strict RelStr
the carrier of (D) is set
f is Element of the carrier of (D)
R is Element of the carrier of (D)
(D,f) is Element of the carrier of D
(D,R) is Element of the carrier of D
R is Element of the carrier of D
(D,R) is Element of the carrier of (D)
A is Element of the carrier of (D)
(D,(D,f)) is Element of the carrier of (D)
(D,(D,R)) is Element of the carrier of (D)
A is Element of the carrier of (D)
(D,A) is Element of the carrier of D
(D,(D,A)) is Element of the carrier of (D)
the carrier of (D) is set
f is Element of the carrier of D
R is Element of the carrier of D
(D,f) is Element of the carrier of (D)
(D,R) is Element of the carrier of (D)
R is Element of the carrier of (D)
(D,R) is Element of the carrier of D
A is Element of the carrier of D
(D,(D,R)) is Element of the carrier of (D)
A is Element of the carrier of D
(D,A) is Element of the carrier of (D)

(D) is non empty strict V58() reflexive transitive antisymmetric RelStr
the carrier of D is non empty set
(D) is Relation-like the carrier of D -defined the carrier of D -valued V14( the carrier of D) reflexive antisymmetric transitive Element of bool [: the carrier of D, the carrier of D:]
[: the carrier of D, the carrier of D:] is non empty set
bool [: the carrier of D, the carrier of D:] is non empty set
RelStr(# the carrier of D,(D) #) is non empty strict V58() reflexive transitive antisymmetric RelStr
the carrier of (D) is non empty set
f is Element of the carrier of (D)
R is Element of the carrier of (D)
(D,f) is Element of the carrier of D
(D,R) is Element of the carrier of D
(D,f) "\/" (D,R) is Element of the carrier of D
the L_join of D is Relation-like [: the carrier of D, the carrier of D:] -defined the carrier of D -valued Function-like V18([: the carrier of D, the carrier of D:], the carrier of D) commutative associative idempotent Element of bool [:[: the carrier of D, the carrier of D:], the carrier of D:]
[:[: the carrier of D, the carrier of D:], the carrier of D:] is non empty set
bool [:[: the carrier of D, the carrier of D:], the carrier of D:] is non empty set
the L_join of D . ((D,f),(D,R)) is Element of the carrier of D
(D,((D,f) "\/" (D,R))) is Element of the carrier of (D)
R is Element of the carrier of (D)
(D,R) "\/" (D,f) is Element of the carrier of D
the L_join of D . ((D,R),(D,f)) is Element of the carrier of D
(D,(D,f)) is Element of the carrier of (D)
(D,(D,R)) is Element of the carrier of (D)
A is Element of the carrier of (D)
(D,A) is Element of the carrier of D
(D,(D,A)) is Element of the carrier of (D)
the carrier of (D) is non empty set
f is Element of the carrier of (D)
R is Element of the carrier of (D)
(D,f) is Element of the carrier of D
(D,R) is Element of the carrier of D
(D,f) "/\" (D,R) is Element of the carrier of D
the L_meet of D is Relation-like [: the carrier of D, the carrier of D:] -defined the carrier of D -valued Function-like V18([: the carrier of D, the carrier of D:], the carrier of D) commutative associative idempotent Element of bool [:[: the carrier of D, the carrier of D:], the carrier of D:]
[:[: the carrier of D, the carrier of D:], the carrier of D:] is non empty set
bool [:[: the carrier of D, the carrier of D:], the carrier of D:] is non empty set
the L_meet of D . ((D,f),(D,R)) is Element of the carrier of D
(D,((D,f) "/\" (D,R))) is Element of the carrier of (D)
R is Element of the carrier of (D)
(D,R) "/\" (D,f) is Element of the carrier of D
the L_meet of D . ((D,R),(D,f)) is Element of the carrier of D
(D,(D,f)) is Element of the carrier of (D)
(D,(D,R)) is Element of the carrier of (D)
A is Element of the carrier of (D)
(D,A) is Element of the carrier of D
(D,(D,A)) is Element of the carrier of (D)
the set is set
{ the set } is non empty set
[:{ the set },{ the set }:] is non empty set
bool [:{ the set },{ the set }:] is non empty set
the Relation-like { the set } -defined { the set } -valued V14({ the set }) reflexive antisymmetric transitive Element of bool [:{ the set },{ the set }:] is Relation-like { the set } -defined { the set } -valued V14({ the set }) reflexive antisymmetric transitive Element of bool [:{ the set },{ the set }:]
RelStr(# { the set }, the Relation-like { the set } -defined { the set } -valued V14({ the set }) reflexive antisymmetric transitive Element of bool [:{ the set },{ the set }:] #) is non empty strict V58() reflexive transitive antisymmetric RelStr

the carrier of R is non empty set
A is Element of the carrier of R
A is set
L is Element of the carrier of R
X is Element of the carrier of R
X is Element of the carrier of R
D is non empty RelStr
the carrier of D is non empty set
f is Element of the carrier of D
R is Element of the carrier of D
{f,R} is non empty Element of bool the carrier of D
bool the carrier of D is non empty set
R is Element of the carrier of D
A is Element of the carrier of D
A is Element of the carrier of D
f is Element of the carrier of D
R is Element of the carrier of D
{ b1 where b1 is Element of the carrier of D : ( b1 <= f & b1 <= R ) } is set
A is Element of the carrier of D
A is Element of the carrier of D
L is Element of the carrier of D
A is Element of the carrier of D
L is Element of the carrier of D
A is Element of the carrier of D

D is RelStr
the carrier of D is set
f is Element of the carrier of D
R is Element of the carrier of D
R is Element of the carrier of D
R is Element of the carrier of D
A is Element of the carrier of D
D is non empty antisymmetric () RelStr
the carrier of D is non empty set
f is Element of the carrier of D
R is Element of the carrier of D
(D,f,R) is Element of the carrier of D
R is Element of the carrier of D
A is Element of the carrier of D
A is Element of the carrier of D
L is Element of the carrier of D
D is RelStr
the carrier of D is set
f is Element of the carrier of D
R is Element of the carrier of D
R is Element of the carrier of D
R is Element of the carrier of D
A is Element of the carrier of D
D is non empty antisymmetric () RelStr
the carrier of D is non empty set
f is Element of the carrier of D
R is Element of the carrier of D
(D,f,R) is Element of the carrier of D
R is Element of the carrier of D
A is Element of the carrier of D
A is Element of the carrier of D
L is Element of the carrier of D
D is non empty antisymmetric () RelStr
the carrier of D is non empty set
f is Element of the carrier of D
R is Element of the carrier of D
(D,f,R) is Element of the carrier of D
(D,R,f) is Element of the carrier of D
R is Element of the carrier of D
D is non empty antisymmetric () RelStr
the carrier of D is non empty set
f is Element of the carrier of D
R is Element of the carrier of D
(D,f,R) is Element of the carrier of D
R is Element of the carrier of D
(D,(D,f,R),R) is Element of the carrier of D
(D,R,R) is Element of the carrier of D
(D,f,(D,R,R)) is Element of the carrier of D
A is Element of the carrier of D
D is non empty antisymmetric () RelStr
the carrier of D is non empty set
f is Element of the carrier of D
R is Element of the carrier of D
(D,f,R) is Element of the carrier of D
(D,R,f) is Element of the carrier of D
R is Element of the carrier of D
D is non empty antisymmetric () RelStr
the carrier of D is non empty set
f is Element of the carrier of D
R is Element of the carrier of D
(D,f,R) is Element of the carrier of D
R is Element of the carrier of D
(D,(D,f,R),R) is Element of the carrier of D
(D,R,R) is Element of the carrier of D
(D,f,(D,R,R)) is Element of the carrier of D
A is Element of the carrier of D
D is non empty antisymmetric () RelStr
the carrier of D is non empty set
R is Element of the carrier of D
A is Element of the carrier of D
(D,R,A) is Element of the carrier of D
(D,A,R) is Element of the carrier of D
D is non empty antisymmetric () RelStr
the carrier of D is non empty set
R is Element of the carrier of D
A is Element of the carrier of D
(D,R,A) is Element of the carrier of D
(D,A,R) is Element of the carrier of D
D is non empty V58() reflexive antisymmetric () () RelStr
the carrier of D is non empty set
f is Element of the carrier of D
R is Element of the carrier of D
(D,f,R) is Element of the carrier of D
(D,(D,f,R),R) is Element of the carrier of D
R is Element of the carrier of D
D is non empty V58() reflexive antisymmetric () () RelStr
the carrier of D is non empty set
f is Element of the carrier of D
R is Element of the carrier of D
(D,f,R) is Element of the carrier of D
(D,f,(D,f,R)) is Element of the carrier of D
R is Element of the carrier of D
D is non empty V58() reflexive transitive antisymmetric () () RelStr
the carrier of D is non empty set
the InternalRel of D is Relation-like the carrier of D -defined the carrier of D -valued V14( the carrier of D) reflexive antisymmetric transitive Element of bool [: the carrier of D, the carrier of D:]
[: the carrier of D, the carrier of D:] is non empty set
bool [: the carrier of D, the carrier of D:] is non empty set
RelStr(# the carrier of D, the InternalRel of D #) is non empty strict V58() reflexive transitive antisymmetric RelStr
f is Element of the carrier of D
R is Element of the carrier of D
R is Element of the carrier of D
A is Element of the carrier of D
(D,R,A) is Element of the carrier of D
A is Element of the carrier of D
L is Element of the carrier of D
(D,A,L) is Element of the carrier of D
[:[: the carrier of D, the carrier of D:], the carrier of D:] is non empty set
bool [:[: the carrier of D, the carrier of D:], the carrier of D:] is non empty set
f is Relation-like [: the carrier of D, the carrier of D:] -defined the carrier of D -valued Function-like V18([: the carrier of D, the carrier of D:], the carrier of D) Element of bool [:[: the carrier of D, the carrier of D:], the carrier of D:]
R is Element of the carrier of D
R is Element of the carrier of D
A is Element of the carrier of D
A is Element of the carrier of D
(D,A,A) is Element of the carrier of D
L is Element of the carrier of D
X is Element of the carrier of D
(D,L,X) is Element of the carrier of D
R is Relation-like [: the carrier of D, the carrier of D:] -defined the carrier of D -valued Function-like V18([: the carrier of D, the carrier of D:], the carrier of D) Element of bool [:[: the carrier of D, the carrier of D:], the carrier of D:]
LattStr(# the carrier of D,f,R #) is non empty strict LattStr
the carrier of LattStr(# the carrier of D,f,R #) is non empty set
A is Element of the carrier of LattStr(# the carrier of D,f,R #)
A is Element of the carrier of LattStr(# the carrier of D,f,R #)
L is Element of the carrier of D
X is Element of the carrier of D
f . (L,X) is Element of the carrier of D
(D,L,X) is Element of the carrier of D
A "\/" A is Element of the carrier of LattStr(# the carrier of D,f,R #)
the L_join of LattStr(# the carrier of D,f,R #) is Relation-like [: the carrier of LattStr(# the carrier of D,f,R #), the carrier of LattStr(# the carrier of D,f,R #):] -defined the carrier of LattStr(# the carrier of D,f,R #) -valued Function-like V18([: the carrier of LattStr(# the carrier of D,f,R #), the carrier of LattStr(# the carrier of D,f,R #):], the carrier of LattStr(# the carrier of D,f,R #)) Element of bool [:[: the carrier of LattStr(# the carrier of D,f,R #), the carrier of LattStr(# the carrier of D,f,R #):], the carrier of LattStr(# the carrier of D,f,R #):]
[: the carrier of LattStr(# the carrier of D,f,R #), the carrier of LattStr(# the carrier of D,f,R #):] is non empty set
[:[: the carrier of LattStr(# the carrier of D,f,R #), the carrier of LattStr(# the carrier of D,f,R #):], the carrier of LattStr(# the carrier of D,f,R #):] is non empty set
bool [:[: the carrier of LattStr(# the carrier of D,f,R #), the carrier of LattStr(# the carrier of D,f,R #):], the carrier of LattStr(# the carrier of D,f,R #):] is non empty set
the L_join of LattStr(# the carrier of D,f,R #) . (A,A) is Element of the carrier of LattStr(# the carrier of D,f,R #)
A "\/" A is Element of the carrier of LattStr(# the carrier of D,f,R #)
the L_join of LattStr(# the carrier of D,f,R #) . (A,A) is Element of the carrier of LattStr(# the carrier of D,f,R #)
A is Element of the carrier of LattStr(# the carrier of D,f,R #)
A is Element of the carrier of LattStr(# the carrier of D,f,R #)
L is Element of the carrier of LattStr(# the carrier of D,f,R #)
A "\/" L is Element of the carrier of LattStr(# the carrier of D,f,R #)
the L_join of LattStr(# the carrier of D,f,R #) . (A,L) is Element of the carrier of LattStr(# the carrier of D,f,R #)
A "\/" (A "\/" L) is Element of the carrier of LattStr(# the carrier of D,f,R #)
the L_join of LattStr(# the carrier of D,f,R #) . (A,(A "\/" L)) is Element of the carrier of LattStr(# the carrier of D,f,R #)
X is Element of the carrier of D
Y is Element of the carrier of D
a is Element of the carrier of D
(D,Y,a) is Element of the carrier of D
f . (X,(D,Y,a)) is Element of the carrier of D
(D,X,(D,Y,a)) is Element of the carrier of D
(D,X,Y) is Element of the carrier of D
(D,(D,X,Y),a) is Element of the carrier of D
f . ((D,X,Y),a) is Element of the carrier of D
A "\/" A is Element of the carrier of LattStr(# the carrier of D,f,R #)
the L_join of LattStr(# the carrier of D,f,R #) . (A,A) is Element of the carrier of LattStr(# the carrier of D,f,R #)
(A "\/" A) "\/" L is Element of the carrier of LattStr(# the carrier of D,f,R #)
the L_join of LattStr(# the carrier of D,f,R #) . ((A "\/" A),L) is Element of the carrier of LattStr(# the carrier of D,f,R #)
A is Element of the carrier of LattStr(# the carrier of D,f,R #)
A is Element of the carrier of LattStr(# the carrier of D,f,R #)
A "/\" A is Element of the carrier of LattStr(# the carrier of D,f,R #)
the L_meet of LattStr(# the carrier of D,f,R #) is Relation-like [: the carrier of LattStr(# the carrier of D,f,R #), the carrier of LattStr(# the carrier of D,f,R #):] -defined the carrier of LattStr(# the carrier of D,f,R #) -valued Function-like V18([: the carrier of LattStr(# the carrier of D,f,R #), the carrier of LattStr(# the carrier of D,f,R #):], the carrier of LattStr(# the carrier of D,f,R #)) Element of bool [:[: the carrier of LattStr(# the carrier of D,f,R #), the carrier of LattStr(# the carrier of D,f,R #):], the carrier of LattStr(# the carrier of D,f,R #):]
the L_meet of LattStr(# the carrier of D,f,R #) . (A,A) is Element of the carrier of LattStr(# the carrier of D,f,R #)
(A "/\" A) "\/" A is Element of the carrier of LattStr(# the carrier of D,f,R #)
the L_join of LattStr(# the carrier of D,f,R #) . ((A "/\" A),A) is Element of the carrier of LattStr(# the carrier of D,f,R #)
L is Element of the carrier of D
X is Element of the carrier of D
(D,L,X) is Element of the carrier of D
f . ((D,L,X),X) is Element of the carrier of D
(D,(D,L,X),X) is Element of the carrier of D
A is Element of the carrier of LattStr(# the carrier of D,f,R #)
A is Element of the carrier of LattStr(# the carrier of D,f,R #)
L is Element of the carrier of D
X is Element of the carrier of D
R . (L,X) is Element of the carrier of D
(D,L,X) is Element of the carrier of D
A "/\" A is Element of the carrier of LattStr(# the carrier of D,f,R #)
the L_meet of LattStr(# the carrier of D,f,R #) . (A,A) is Element of the carrier of LattStr(# the carrier of D,f,R #)
A "/\" A is Element of the carrier of LattStr(# the carrier of D,f,R #)
the L_meet of LattStr(# the carrier of D,f,R #) . (A,A) is Element of the carrier of LattStr(# the carrier of D,f,R #)
A is Element of the carrier of LattStr(# the carrier of D,f,R #)
A is Element of the carrier of LattStr(# the carrier of D,f,R #)
L is Element of the carrier of LattStr(# the carrier of D,f,R #)
A "/\" L is Element of the carrier of LattStr(# the carrier of D,f,R #)
the L_meet of LattStr(# the carrier of D,f,R #) . (A,L) is Element of the carrier of LattStr(# the carrier of D,f,R #)
A "/\" (A "/\" L) is Element of the carrier of LattStr(# the carrier of D,f,R #)
the L_meet of LattStr(# the carrier of D,f,R #) . (A,(A "/\" L)) is Element of the carrier of LattStr(# the carrier of D,f,R #)
X is Element of the carrier of D
Y is Element of the carrier of D
a is Element of the carrier of D
(D,Y,a) is Element of the carrier of D
R . (X,(D,Y,a)) is Element of the carrier of D
(D,X,(D,Y,a)) is Element of the carrier of D
(D,X,Y) is Element of the carrier of D
(D,(D,X,Y),a) is Element of the carrier of D
R . ((D,X,Y),a) is Element of the carrier of D
A "/\" A is Element of the carrier of LattStr(# the carrier of D,f,R #)
the L_meet of LattStr(# the carrier of D,f,R #) . (A,A) is Element of the carrier of LattStr(# the carrier of D,f,R #)
(A "/\" A) "/\" L is Element of the carrier of LattStr(# the carrier of D,f,R #)
the L_meet of LattStr(# the carrier of D,f,R #) . ((A "/\" A),L) is Element of the carrier of LattStr(# the carrier of D,f,R #)
A is Element of the carrier of LattStr(# the carrier of D,f,R #)
A is Element of the carrier of LattStr(# the carrier of D,f,R #)
A "\/" A is Element of the carrier of LattStr(# the carrier of D,f,R #)
the L_join of LattStr(# the carrier of D,f,R #) . (A,A) is Element of the carrier of LattStr(# the carrier of D,f,R #)
A "/\" (A "\/" A) is Element of the carrier of LattStr(# the carrier of D,f,R #)
the L_meet of LattStr(# the carrier of D,f,R #) . (A,(A "\/" A)) is Element of the carrier of LattStr(# the carrier of D,f,R #)
L is Element of the carrier of D
X is Element of the carrier of D
(D,L,X) is Element of the carrier of D
R . (L,(D,L,X)) is Element of the carrier of D
(D,L,(D,L,X)) is Element of the carrier of D

(A) is non empty strict V58() reflexive transitive antisymmetric RelStr
the carrier of A is non empty set
(A) is Relation-like the carrier of A -defined the carrier of A -valued V14( the carrier of A) reflexive antisymmetric transitive Element of bool [: the carrier of A, the carrier of A:]
[: the carrier of A, the carrier of A:] is non empty set
bool [: the carrier of A, the carrier of A:] is non empty set
RelStr(# the carrier of A,(A) #) is non empty strict V58() reflexive transitive antisymmetric RelStr
{ [b1,b2] where b1, b2 is Element of the carrier of A : b1 [= b2 } is set
A is set
L is set
[A,L] is set
{A,L} is non empty set
{A} is non empty set
{{A,L},{A}} is non empty set
X is Element of the carrier of A
Y is Element of the carrier of A
[X,Y] is Element of the carrier of [:A,A:]
[:A,A:] is strict LattStr
the carrier of [:A,A:] is set
{X,Y} is non empty set
{X} is non empty set
{{X,Y},{X}} is non empty set
a is Element of the carrier of D
p is Element of the carrier of D
(D,a,p) is Element of the carrier of D
X "\/" Y is Element of the carrier of A
the L_join of A is Relation-like [: the carrier of A, the carrier of A:] -defined the carrier of A -valued Function-like V18([: the carrier of A, the carrier of A:], the carrier of A) commutative associative idempotent Element of bool [:[: the carrier of A, the carrier of A:], the carrier of A:]
[:[: the carrier of A, the carrier of A:], the carrier of A:] is non empty set
bool [:[: the carrier of A, the carrier of A:], the carrier of A:] is non empty set
the L_join of A . (X,Y) is Element of the carrier of A
X is set
Y is set
[X,Y] is set
{X,Y} is non empty set
{X} is non empty set
{{X,Y},{X}} is non empty set
a is Element of the carrier of D
p is Element of the carrier of D
(D,a,p) is Element of the carrier of D
q is Element of the carrier of A
y is Element of the carrier of A
q "\/" y is Element of the carrier of A
the L_join of A is Relation-like [: the carrier of A, the carrier of A:] -defined the carrier of A -valued Function-like V18([: the carrier of A, the carrier of A:], the carrier of A) commutative associative idempotent Element of bool [:[: the carrier of A, the carrier of A:], the carrier of A:]
[:[: the carrier of A, the carrier of A:], the carrier of A:] is non empty set
bool [:[: the carrier of A, the carrier of A:], the carrier of A:] is non empty set
the L_join of A . (q,y) is Element of the carrier of A
D is RelStr
the carrier of D is set
the InternalRel of D is Relation-like the carrier of D -defined the carrier of D -valued Element of bool [: the carrier of D, the carrier of D:]
[: the carrier of D, the carrier of D:] is set
bool [: the carrier of D, the carrier of D:] is non empty set
RelStr(# the carrier of D, the InternalRel of D #) is strict RelStr

(f) is non empty strict V58() reflexive transitive antisymmetric RelStr
the carrier of f is non empty set
(f) is Relation-like the carrier of f -defined the carrier of f -valued V14( the carrier of f) reflexive antisymmetric transitive Element of bool [: the carrier of f, the carrier of f:]
[: the carrier of f, the carrier of f:] is non empty set
bool [: the carrier of f, the carrier of f:] is non empty set
RelStr(# the carrier of f,(f) #) is non empty strict V58() reflexive transitive antisymmetric RelStr

(R) is non empty strict V58() reflexive transitive antisymmetric RelStr
the carrier of R is non empty set
(R) is Relation-like the carrier of R -defined the carrier of R -valued V14( the carrier of R) reflexive antisymmetric transitive Element of bool [: the carrier of R, the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty set
bool [: the carrier of R, the carrier of R:] is non empty set
RelStr(# the carrier of R,(R) #) is non empty strict V58() reflexive transitive antisymmetric RelStr

the carrier of D is non empty set
(D) is Relation-like the carrier of D -defined the carrier of D -valued V14( the carrier of D) reflexive antisymmetric transitive Element of bool [: the carrier of D, the carrier of D:]
[: the carrier of D, the carrier of D:] is non empty set
bool [: the carrier of D, the carrier of D:] is non empty set
( the carrier of D,(D)) is Relation-like the carrier of D -defined the carrier of D -valued V14( the carrier of D) reflexive antisymmetric transitive Element of bool [: the carrier of D, the carrier of D:]

((D .:)) is Relation-like the carrier of (D .:) -defined the carrier of (D .:) -valued V14( the carrier of (D .:)) reflexive antisymmetric transitive Element of bool [: the carrier of (D .:), the carrier of (D .:):]
the carrier of (D .:) is non empty set
[: the carrier of (D .:), the carrier of (D .:):] is non empty set
bool [: the carrier of (D .:), the carrier of (D .:):] is non empty set
(D) is non empty strict V58() reflexive transitive antisymmetric RelStr
RelStr(# the carrier of D,(D) #) is non empty strict V58() reflexive transitive antisymmetric RelStr
((D)) is non empty strict V58() reflexive transitive antisymmetric RelStr
the carrier of (D) is non empty set
the InternalRel of (D) is Relation-like the carrier of (D) -defined the carrier of (D) -valued V14( the carrier of (D)) reflexive antisymmetric transitive Element of bool [: the carrier of (D), the carrier of (D):]
[: the carrier of (D), the carrier of (D):] is non empty set
bool [: the carrier of (D), the carrier of (D):] is non empty set
the InternalRel of (D) ~ is Relation-like the carrier of (D) -defined the carrier of (D) -valued reflexive antisymmetric transitive Element of bool [: the carrier of (D), the carrier of (D):]
RelStr(# the carrier of (D),( the InternalRel of (D) ~) #) is non empty strict RelStr
((D .:)) is non empty strict V58() reflexive transitive antisymmetric RelStr
RelStr(# the carrier of (D .:),((D .:)) #) is non empty strict V58() reflexive transitive antisymmetric RelStr
{ [b1,b2] where b1, b2 is Element of the carrier of D : b1 [= b2 } is set
{ [b1,b2] where b1, b2 is Element of the carrier of (D .:) : b1 [= b2 } is set
the L_meet of D is Relation-like [: the carrier of D, the carrier of D:] -defined the carrier of D -valued Function-like V18([: the carrier of D, the carrier of D:], the carrier of D) commutative associative idempotent Element of bool [:[: the carrier of D, the carrier of D:], the carrier of D:]
[:[: the carrier of D, the carrier of D:], the carrier of D:] is non empty set
bool [:[: the carrier of D, the carrier of D:], the carrier of D:] is non empty set
the L_join of D is Relation-like [: the carrier of D, the carrier of D:] -defined the carrier of D -valued Function-like V18([: the carrier of D, the carrier of D:], the carrier of D) commutative associative idempotent Element of bool [:[: the carrier of D, the carrier of D:]<