Article LATTICE4, MML version 4.99.1005
:: LATTICE4:th 3
theorem
for b1 being set
st b1 <> {} &
(for b2 being set
st b2 <> {} & b2 c= b1 & b2 is c=-linear
holds ex b3 being set st
b3 in b1 &
(for b4 being set
st b4 in b2
holds b4 c= b3))
holds ex b2 being set st
b2 in b1 &
(for b3 being set
st b3 in b1 & b3 <> b2
holds not b2 c= b3);
:: LATTICE4:th 4
theorem
for b1 being non empty Lattice-like LattStr holds
<.b1.) is prime(b1);
:: LATTICE4:th 5
theorem
for b1 being non empty Lattice-like LattStr
for b2, b3 being Filter of b1 holds
b2 c= <.b2 \/ b3.) & b3 c= <.b2 \/ b3.);
:: LATTICE4:th 6
theorem
for b1 being non empty Lattice-like LattStr
for b2 being Filter of b1
for b3, b4 being Element of the carrier of b1
st b3 in <.<.b4.) \/ b2.)
holds ex b5 being Element of the carrier of b1 st
b5 in b2 & b4 "/\" b5 [= b3;
:: LATTICE4:modenot 1 => LATTICE4:mode 1
definition
let a1, a2 be non empty Lattice-like LattStr;
mode Homomorphism of A1,A2 -> Function-like quasi_total Relation of the carrier of a1,the carrier of a2 means
for b1, b2 being Element of the carrier of a1 holds
it . (b1 "\/" b2) = (it . b1) "\/" (it . b2) &
it . (b1 "/\" b2) = (it . b1) "/\" (it . b2);
end;
:: LATTICE4:dfs 1
definiens
let a1, a2 be non empty Lattice-like LattStr;
let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
To prove
a3 is Homomorphism of a1,a2
it is sufficient to prove
thus for b1, b2 being Element of the carrier of a1 holds
a3 . (b1 "\/" b2) = (a3 . b1) "\/" (a3 . b2) &
a3 . (b1 "/\" b2) = (a3 . b1) "/\" (a3 . b2);
:: LATTICE4:def 1
theorem
for b1, b2 being non empty Lattice-like LattStr
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 holds
b3 is Homomorphism of b1,b2
iff
for b4, b5 being Element of the carrier of b1 holds
b3 . (b4 "\/" b5) = (b3 . b4) "\/" (b3 . b5) &
b3 . (b4 "/\" b5) = (b3 . b4) "/\" (b3 . b5);
:: LATTICE4:th 7
theorem
for b1, b2 being non empty Lattice-like LattStr
for b3, b4 being Element of the carrier of b1
for b5 being Homomorphism of b1,b2
st b3 [= b4
holds b5 . b3 [= b5 . b4;
:: LATTICE4:attrnot 1 => LATTICE4:attr 1
definition
let a1, a2 be non empty Lattice-like LattStr;
let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
attr a3 is monomorphism means
a3 is one-to-one;
end;
:: LATTICE4:dfs 2
definiens
let a1, a2 be non empty Lattice-like LattStr;
let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
To prove
a3 is monomorphism
it is sufficient to prove
thus a3 is one-to-one;
:: LATTICE4:def 2
theorem
for b1, b2 being non empty Lattice-like LattStr
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 holds
b3 is monomorphism(b1, b2)
iff
b3 is one-to-one;
:: LATTICE4:attrnot 2 => LATTICE4:attr 2
definition
let a1, a2 be non empty Lattice-like LattStr;
let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
attr a3 is epimorphism means
proj2 a3 = the carrier of a2;
end;
:: LATTICE4:dfs 3
definiens
let a1, a2 be non empty Lattice-like LattStr;
let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
To prove
a3 is epimorphism
it is sufficient to prove
thus proj2 a3 = the carrier of a2;
:: LATTICE4:def 3
theorem
for b1, b2 being non empty Lattice-like LattStr
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 holds
b3 is epimorphism(b1, b2)
iff
proj2 b3 = the carrier of b2;
:: LATTICE4:th 8
theorem
for b1, b2 being non empty Lattice-like LattStr
for b3, b4 being Element of the carrier of b1
for b5 being Homomorphism of b1,b2
st b5 is monomorphism(b1, b2)
holds b3 [= b4
iff
b5 . b3 [= b5 . b4;
:: LATTICE4:th 9
theorem
for b1, b2 being non empty Lattice-like LattStr
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
st b3 is epimorphism(b1, b2)
for b4 being Element of the carrier of b2 holds
ex b5 being Element of the carrier of b1 st
b4 = b3 . b5;
:: LATTICE4:attrnot 3 => LATTICE4:attr 3
definition
let a1, a2 be non empty Lattice-like LattStr;
let a3 be Homomorphism of a1,a2;
attr a3 is isomorphism means
a3 is monomorphism(a1, a2) & a3 is epimorphism(a1, a2);
end;
:: LATTICE4:dfs 4
definiens
let a1, a2 be non empty Lattice-like LattStr;
let a3 be Homomorphism of a1,a2;
To prove
a3 is isomorphism
it is sufficient to prove
thus a3 is monomorphism(a1, a2) & a3 is epimorphism(a1, a2);
:: LATTICE4:def 4
theorem
for b1, b2 being non empty Lattice-like LattStr
for b3 being Homomorphism of b1,b2 holds
b3 is isomorphism(b1, b2)
iff
b3 is monomorphism(b1, b2) & b3 is epimorphism(b1, b2);
:: LATTICE4:prednot 1 => FILTER_1:pred 1
definition
let a1, a2 be non empty Lattice-like LattStr;
pred A1,A2 are_isomorphic means
ex b1 being Homomorphism of a1,a2 st
b1 is isomorphism(a1, a2);
symmetry;
:: for a1, a2 being non empty Lattice-like LattStr
:: st a1,a2 are_isomorphic
:: holds a2,a1 are_isomorphic;
reflexivity;
:: for a1 being non empty Lattice-like LattStr holds
:: a1,a1 are_isomorphic;
end;
:: LATTICE4:dfs 5
definiens
let a1, a2 be non empty Lattice-like LattStr;
To prove
a1,a2 are_isomorphic
it is sufficient to prove
thus ex b1 being Homomorphism of a1,a2 st
b1 is isomorphism(a1, a2);
:: LATTICE4:def 5
theorem
for b1, b2 being non empty Lattice-like LattStr holds
b1,b2 are_isomorphic
iff
ex b3 being Homomorphism of b1,b2 st
b3 is isomorphism(b1, b2);
:: LATTICE4:prednot 2 => LATTICE4:pred 1
definition
let a1, a2 be non empty Lattice-like LattStr;
let a3 be Homomorphism of a1,a2;
pred A3 preserves_implication means
for b1, b2 being Element of the carrier of a1 holds
a3 . (b1 => b2) = (a3 . b1) => (a3 . b2);
end;
:: LATTICE4:dfs 6
definiens
let a1, a2 be non empty Lattice-like LattStr;
let a3 be Homomorphism of a1,a2;
To prove
a3 preserves_implication
it is sufficient to prove
thus for b1, b2 being Element of the carrier of a1 holds
a3 . (b1 => b2) = (a3 . b1) => (a3 . b2);
:: LATTICE4:def 6
theorem
for b1, b2 being non empty Lattice-like LattStr
for b3 being Homomorphism of b1,b2 holds
b3 preserves_implication
iff
for b4, b5 being Element of the carrier of b1 holds
b3 . (b4 => b5) = (b3 . b4) => (b3 . b5);
:: LATTICE4:prednot 3 => LATTICE4:pred 2
definition
let a1, a2 be non empty Lattice-like LattStr;
let a3 be Homomorphism of a1,a2;
pred A3 preserves_top means
a3 . Top a1 = Top a2;
end;
:: LATTICE4:dfs 7
definiens
let a1, a2 be non empty Lattice-like LattStr;
let a3 be Homomorphism of a1,a2;
To prove
a3 preserves_top
it is sufficient to prove
thus a3 . Top a1 = Top a2;
:: LATTICE4:def 7
theorem
for b1, b2 being non empty Lattice-like LattStr
for b3 being Homomorphism of b1,b2 holds
b3 preserves_top
iff
b3 . Top b1 = Top b2;
:: LATTICE4:prednot 4 => LATTICE4:pred 3
definition
let a1, a2 be non empty Lattice-like LattStr;
let a3 be Homomorphism of a1,a2;
pred A3 preserves_bottom means
a3 . Bottom a1 = Bottom a2;
end;
:: LATTICE4:dfs 8
definiens
let a1, a2 be non empty Lattice-like LattStr;
let a3 be Homomorphism of a1,a2;
To prove
a3 preserves_bottom
it is sufficient to prove
thus a3 . Bottom a1 = Bottom a2;
:: LATTICE4:def 8
theorem
for b1, b2 being non empty Lattice-like LattStr
for b3 being Homomorphism of b1,b2 holds
b3 preserves_bottom
iff
b3 . Bottom b1 = Bottom b2;
:: LATTICE4:prednot 5 => LATTICE4:pred 4
definition
let a1, a2 be non empty Lattice-like LattStr;
let a3 be Homomorphism of a1,a2;
pred A3 preserves_complement means
for b1 being Element of the carrier of a1 holds
a3 . (b1 `) = (a3 . b1) `;
end;
:: LATTICE4:dfs 9
definiens
let a1, a2 be non empty Lattice-like LattStr;
let a3 be Homomorphism of a1,a2;
To prove
a3 preserves_complement
it is sufficient to prove
thus for b1 being Element of the carrier of a1 holds
a3 . (b1 `) = (a3 . b1) `;
:: LATTICE4:def 9
theorem
for b1, b2 being non empty Lattice-like LattStr
for b3 being Homomorphism of b1,b2 holds
b3 preserves_complement
iff
for b4 being Element of the carrier of b1 holds
b3 . (b4 `) = (b3 . b4) `;
:: LATTICE4:modenot 2 => LATTICE4:mode 2
definition
let a1 be non empty Lattice-like LattStr;
mode ClosedSubset of A1 -> Element of bool the carrier of a1 means
for b1, b2 being Element of the carrier of a1
st b1 in it & b2 in it
holds b1 "/\" b2 in it & b1 "\/" b2 in it;
end;
:: LATTICE4:dfs 10
definiens
let a1 be non empty Lattice-like LattStr;
let a2 be Element of bool the carrier of a1;
To prove
a2 is ClosedSubset of a1
it is sufficient to prove
thus for b1, b2 being Element of the carrier of a1
st b1 in a2 & b2 in a2
holds b1 "/\" b2 in a2 & b1 "\/" b2 in a2;
:: LATTICE4:def 10
theorem
for b1 being non empty Lattice-like LattStr
for b2 being Element of bool the carrier of b1 holds
b2 is ClosedSubset of b1
iff
for b3, b4 being Element of the carrier of b1
st b3 in b2 & b4 in b2
holds b3 "/\" b4 in b2 & b3 "\/" b4 in b2;
:: LATTICE4:th 10
theorem
for b1 being non empty Lattice-like LattStr holds
the carrier of b1 is ClosedSubset of b1;
:: LATTICE4:exreg 1
registration
let a1 be non empty Lattice-like LattStr;
cluster non empty ClosedSubset of a1;
end;
:: LATTICE4:th 11
theorem
for b1 being non empty Lattice-like LattStr
for b2 being Filter of b1 holds
b2 is ClosedSubset of b1;
:: LATTICE4:funcnot 1 => LATTICE4:func 1
definition
let a1 be non empty Lattice-like LattStr;
let a2 be Element of Fin the carrier of a1;
func FinJoin A2 -> Element of the carrier of a1 equals
FinJoin(a2,id a1);
end;
:: LATTICE4:def 12
theorem
for b1 being non empty Lattice-like LattStr
for b2 being Element of Fin the carrier of b1 holds
FinJoin b2 = FinJoin(b2,id b1);
:: LATTICE4:funcnot 2 => LATTICE4:func 2
definition
let a1 be non empty Lattice-like LattStr;
let a2 be Element of Fin the carrier of a1;
func FinMeet A2 -> Element of the carrier of a1 equals
FinMeet(a2,id a1);
end;
:: LATTICE4:def 13
theorem
for b1 being non empty Lattice-like LattStr
for b2 being Element of Fin the carrier of b1 holds
FinMeet b2 = FinMeet(b2,id b1);
:: LATTICE4:th 16
theorem
for b1 being non empty Lattice-like LattStr
for b2 being Element of the carrier of b1 holds
FinJoin {.b2.} = b2;
:: LATTICE4:th 17
theorem
for b1 being non empty Lattice-like LattStr
for b2 being Element of the carrier of b1 holds
FinMeet {.b2.} = b2;
:: LATTICE4:th 18
theorem
for b1 being non empty Lattice-like LattStr
for b2 being non empty Lattice-like distributive LattStr
for b3 being Homomorphism of b2,b1
st b3 is epimorphism(b2, b1)
holds b1 is distributive;
:: LATTICE4:th 19
theorem
for b1 being non empty Lattice-like LattStr
for b2 being non empty Lattice-like lower-bounded LattStr
for b3 being Homomorphism of b2,b1
st b3 is epimorphism(b2, b1)
holds b1 is lower-bounded & b3 preserves_bottom;
:: LATTICE4:th 20
theorem
for b1 being non empty Lattice-like lower-bounded LattStr
for b2 being Element of Fin the carrier of b1
for b3 being Element of the carrier of b1
for b4 being Function-like quasi_total Relation of the carrier of b1,the carrier of b1 holds
FinJoin(b2 \/ {.b3.},b4) = (FinJoin(b2,b4)) "\/" (b4 . b3);
:: LATTICE4:th 21
theorem
for b1 being non empty Lattice-like lower-bounded LattStr
for b2 being Element of Fin the carrier of b1
for b3 being Element of the carrier of b1 holds
FinJoin (b2 \/ {.b3.}) = (FinJoin b2) "\/" b3;
:: LATTICE4:th 22
theorem
for b1 being non empty Lattice-like lower-bounded LattStr
for b2, b3 being Element of Fin the carrier of b1 holds
(FinJoin b2) "\/" FinJoin b3 = FinJoin (b2 \/ b3);
:: LATTICE4:th 23
theorem
for b1 being non empty Lattice-like lower-bounded LattStr holds
FinJoin {}. the carrier of b1 = Bottom b1;
:: LATTICE4:th 24
theorem
for b1 being non empty Lattice-like lower-bounded LattStr
for b2 being ClosedSubset of b1
st Bottom b1 in b2
for b3 being Element of Fin the carrier of b1
st b3 c= b2
holds FinJoin b3 in b2;
:: LATTICE4:th 25
theorem
for b1 being non empty Lattice-like LattStr
for b2 being non empty Lattice-like upper-bounded LattStr
for b3 being Homomorphism of b2,b1
st b3 is epimorphism(b2, b1)
holds b1 is upper-bounded & b3 preserves_top;
:: LATTICE4:th 26
theorem
for b1 being non empty Lattice-like upper-bounded LattStr holds
FinMeet {}. the carrier of b1 = Top b1;
:: LATTICE4:th 27
theorem
for b1 being non empty Lattice-like upper-bounded LattStr
for b2 being Element of Fin the carrier of b1
for b3 being Element of the carrier of b1
for b4 being Function-like quasi_total Relation of the carrier of b1,the carrier of b1 holds
FinMeet(b2 \/ {.b3.},b4) = (FinMeet(b2,b4)) "/\" (b4 . b3);
:: LATTICE4:th 28
theorem
for b1 being non empty Lattice-like upper-bounded LattStr
for b2 being Element of Fin the carrier of b1
for b3 being Element of the carrier of b1 holds
FinMeet (b2 \/ {.b3.}) = (FinMeet b2) "/\" b3;
:: LATTICE4:th 29
theorem
for b1 being non empty Lattice-like upper-bounded LattStr
for b2 being Element of Fin the carrier of b1
for b3, b4 being Function-like quasi_total Relation of the carrier of b1,the carrier of b1 holds
FinMeet(b3 .: b2,b4) = FinMeet(b2,b4 * b3);
:: LATTICE4:th 30
theorem
for b1 being non empty Lattice-like upper-bounded LattStr
for b2, b3 being Element of Fin the carrier of b1 holds
(FinMeet b2) "/\" FinMeet b3 = FinMeet (b2 \/ b3);
:: LATTICE4:th 31
theorem
for b1 being non empty Lattice-like upper-bounded LattStr
for b2 being ClosedSubset of b1
st Top b1 in b2
for b3 being Element of Fin the carrier of b1
st b3 c= b2
holds FinMeet b3 in b2;
:: LATTICE4:th 32
theorem
for b1 being non empty Lattice-like distributive upper-bounded LattStr
for b2 being Element of Fin the carrier of b1
for b3 being Element of the carrier of b1 holds
(FinMeet b2) "\/" b3 = FinMeet (((the L_join of b1) [:](id b1,b3)) .: b2);
:: LATTICE4:th 33
theorem
for b1 being non empty Lattice-like bounded complemented LattStr
for b2 being non empty Lattice-like implicative LattStr
for b3 being Homomorphism of b2,b1
for b4, b5 being Element of the carrier of b2 holds
(b3 . b4) "/\" (b3 . (b4 => b5)) [= b3 . b5;
:: LATTICE4:th 34
theorem
for b1 being non empty Lattice-like bounded complemented LattStr
for b2 being non empty Lattice-like implicative LattStr
for b3 being Homomorphism of b2,b1
for b4, b5, b6 being Element of the carrier of b2
st b3 is monomorphism(b2, b1) &
(b3 . b4) "/\" (b3 . b5) [= b3 . b6
holds b3 . b5 [= b3 . (b4 => b6);
:: LATTICE4:th 35
theorem
for b1 being non empty Lattice-like bounded complemented LattStr
for b2 being non empty Lattice-like implicative LattStr
for b3 being Homomorphism of b2,b1
st b3 is isomorphism(b2, b1)
holds b1 is implicative & b3 preserves_implication;
:: LATTICE4:th 36
theorem
for b1 being non empty Lattice-like Boolean LattStr holds
(Top b1) ` = Bottom b1;
:: LATTICE4:th 37
theorem
for b1 being non empty Lattice-like Boolean LattStr holds
(Bottom b1) ` = Top b1;
:: LATTICE4:th 38
theorem
for b1 being non empty Lattice-like bounded complemented LattStr
for b2 being non empty Lattice-like Boolean LattStr
for b3 being Homomorphism of b2,b1
st b3 is epimorphism(b2, b1)
holds b1 is Boolean & b3 preserves_complement;
:: LATTICE4:modenot 3 => LATTICE4:mode 3
definition
let a1 be non empty Lattice-like Boolean LattStr;
mode Field of A1 -> non empty Element of bool the carrier of a1 means
for b1, b2 being Element of the carrier of a1
st b1 in it & b2 in it
holds b1 "/\" b2 in it & b1 ` in it;
end;
:: LATTICE4:dfs 13
definiens
let a1 be non empty Lattice-like Boolean LattStr;
let a2 be non empty Element of bool the carrier of a1;
To prove
a2 is Field of a1
it is sufficient to prove
thus for b1, b2 being Element of the carrier of a1
st b1 in a2 & b2 in a2
holds b1 "/\" b2 in a2 & b1 ` in a2;
:: LATTICE4:def 14
theorem
for b1 being non empty Lattice-like Boolean LattStr
for b2 being non empty Element of bool the carrier of b1 holds
b2 is Field of b1
iff
for b3, b4 being Element of the carrier of b1
st b3 in b2 & b4 in b2
holds b3 "/\" b4 in b2 & b3 ` in b2;
:: LATTICE4:th 39
theorem
for b1 being non empty Lattice-like Boolean LattStr
for b2, b3 being Element of the carrier of b1
for b4 being Field of b1
st b2 in b4 & b3 in b4
holds b2 "\/" b3 in b4;
:: LATTICE4:th 40
theorem
for b1 being non empty Lattice-like Boolean LattStr
for b2, b3 being Element of the carrier of b1
for b4 being Field of b1
st b2 in b4 & b3 in b4
holds b2 => b3 in b4;
:: LATTICE4:th 41
theorem
for b1 being non empty Lattice-like Boolean LattStr holds
the carrier of b1 is Field of b1;
:: LATTICE4:th 42
theorem
for b1 being non empty Lattice-like Boolean LattStr
for b2 being Field of b1 holds
b2 is ClosedSubset of b1;
:: LATTICE4:funcnot 3 => LATTICE4:func 3
definition
let a1 be non empty Lattice-like Boolean LattStr;
let a2 be non empty Element of bool the carrier of a1;
func field_by A2 -> Field of a1 means
a2 c= it &
(for b1 being Field of a1
st a2 c= b1
holds it c= b1);
end;
:: LATTICE4:def 15
theorem
for b1 being non empty Lattice-like Boolean LattStr
for b2 being non empty Element of bool the carrier of b1
for b3 being Field of b1 holds
b3 = field_by b2
iff
b2 c= b3 &
(for b4 being Field of b1
st b2 c= b4
holds b3 c= b4);
:: LATTICE4:funcnot 4 => LATTICE4:func 4
definition
let a1 be non empty Lattice-like Boolean LattStr;
let a2 be non empty Element of bool the carrier of a1;
func SetImp A2 -> Element of bool the carrier of a1 equals
{b1 => b2 where b1 is Element of the carrier of a1, b2 is Element of the carrier of a1: b1 in a2 & b2 in a2};
end;
:: LATTICE4:def 16
theorem
for b1 being non empty Lattice-like Boolean LattStr
for b2 being non empty Element of bool the carrier of b1 holds
SetImp b2 = {b3 => b4 where b3 is Element of the carrier of b1, b4 is Element of the carrier of b1: b3 in b2 & b4 in b2};
:: LATTICE4:funcreg 1
registration
let a1 be non empty Lattice-like Boolean LattStr;
let a2 be non empty Element of bool the carrier of a1;
cluster SetImp a2 -> non empty;
end;
:: LATTICE4:th 43
theorem
for b1 being set
for b2 being non empty Lattice-like Boolean LattStr
for b3 being non empty Element of bool the carrier of b2 holds
b1 in SetImp b3
iff
ex b4, b5 being Element of the carrier of b2 st
b1 = b4 => b5 & b4 in b3 & b5 in b3;
:: LATTICE4:th 44
theorem
for b1 being non empty Lattice-like Boolean LattStr
for b2 being non empty Element of bool the carrier of b1
for b3 being Element of the carrier of b1 holds
b3 in SetImp b2
iff
ex b4, b5 being Element of the carrier of b1 st
b3 = b4 ` "\/" b5 & b4 in b2 & b5 in b2;
:: LATTICE4:funcnot 5 => LATTICE4:func 5
definition
let a1 be non empty Lattice-like Boolean LattStr;
func comp A1 -> Function-like quasi_total Relation of the carrier of a1,the carrier of a1 means
for b1 being Element of the carrier of a1 holds
it . b1 = b1 `;
end;
:: LATTICE4:def 17
theorem
for b1 being non empty Lattice-like Boolean LattStr
for b2 being Function-like quasi_total Relation of the carrier of b1,the carrier of b1 holds
b2 = comp b1
iff
for b3 being Element of the carrier of b1 holds
b2 . b3 = b3 `;
:: LATTICE4:th 45
theorem
for b1 being non empty Lattice-like Boolean LattStr
for b2 being Element of the carrier of b1
for b3 being Element of Fin the carrier of b1 holds
FinJoin(b3 \/ {.b2.},comp b1) = (FinJoin(b3,comp b1)) "\/" (b2 `);
:: LATTICE4:th 46
theorem
for b1 being non empty Lattice-like Boolean LattStr
for b2 being Element of Fin the carrier of b1 holds
(FinJoin b2) ` = FinMeet(b2,comp b1);
:: LATTICE4:th 47
theorem
for b1 being non empty Lattice-like Boolean LattStr
for b2 being Element of the carrier of b1
for b3 being Element of Fin the carrier of b1 holds
FinMeet(b3 \/ {.b2.},comp b1) = (FinMeet(b3,comp b1)) "/\" (b2 `);
:: LATTICE4:th 48
theorem
for b1 being non empty Lattice-like Boolean LattStr
for b2 being Element of Fin the carrier of b1 holds
(FinMeet b2) ` = FinJoin(b2,comp b1);
:: LATTICE4:th 49
theorem
for b1 being non empty Lattice-like Boolean LattStr
for b2 being non empty ClosedSubset of b1
st Bottom b1 in b2 & Top b1 in b2
for b3 being Element of Fin the carrier of b1
st b3 c= SetImp b2
holds ex b4 being Element of Fin the carrier of b1 st
b4 c= SetImp b2 & FinJoin(b3,comp b1) = FinMeet b4;
:: LATTICE4:th 50
theorem
for b1 being non empty Lattice-like Boolean LattStr
for b2 being non empty ClosedSubset of b1
st Bottom b1 in b2 & Top b1 in b2
holds {FinMeet b3 where b3 is Element of Fin the carrier of b1: b3 c= SetImp b2} = field_by b2;