:: LATTICE3 semantic presentation

K118() is set

bool K118() is non empty set

{} is 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

f is strict LattStr

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:]

R is strict LattStr

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

H

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

H

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

(D) is non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr

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

(D) is non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr

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

(D) is non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr

R is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like lower-bounded upper-bounded V81() LattStr

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

(D) is non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like distributive modular lower-bounded upper-bounded V81() complemented Boolean LattStr

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

D is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr

LattRel D is Relation-like 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

{ [b

[:H

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 [:H

f is Relation-like H

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 H

bool H

field f is set

dom f is set

rng f is set

(dom f) \/ (rng f) is set

D is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr

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 join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr

(D) is 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 join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr

(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 join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr

(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

[:H

[:[:H

bool [:[:H

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

H

R is Relation-like [:H

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

H

R is Relation-like [:H

R . (A,A) is Element of the carrier of D

D is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr

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 join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr

(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)

D is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr

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

f is Relation-like D -defined D -valued V14(D) reflexive antisymmetric transitive Element of bool [:D,D:]

f ~ is Relation-like reflexive antisymmetric transitive set

dom f is Element of bool D

bool D is non empty set

f ~ is Relation-like D -defined D -valued reflexive antisymmetric transitive Element of bool [:D,D:]

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 V58() reflexive transitive antisymmetric 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 join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr

(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

R 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

{ b

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

the non empty strict V58() reflexive transitive antisymmetric () RelStr is non empty strict V58() reflexive transitive antisymmetric () RelStr

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 join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr

(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

{ [b

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 join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr

(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 join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr

(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

D is non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr

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 non empty strict join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing Lattice-like LattStr

((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

{ [b

{ [b

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:]<