Article WAYBEL_7, MML version 4.99.1005
:: WAYBEL_7:th 3
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima complete RelStr
for b2, b3 being set
st b2 c= b3
holds "\/"(b2,b1) <= "\/"(b3,b1) & "/\"(b3,b1) <= "/\"(b2,b1);
:: WAYBEL_7:th 4
theorem
for b1 being set holds
the carrier of BoolePoset b1 = bool b1;
:: WAYBEL_7:th 5
theorem
for b1 being non empty antisymmetric bounded RelStr holds
b1 is trivial
iff
Top b1 = Bottom b1;
:: WAYBEL_7:funcreg 1
registration
let a1 be set;
cluster BoolePoset a1 -> strict Boolean;
end;
:: WAYBEL_7:funcreg 2
registration
let a1 be non empty set;
cluster BoolePoset a1 -> non trivial strict;
end;
:: WAYBEL_7:th 8
theorem
for b1 being non empty reflexive transitive antisymmetric lower-bounded RelStr
for b2 being non empty filtered upper Element of bool the carrier of b1 holds
b2 is proper(bool the carrier of b1)
iff
not Bottom b1 in b2;
:: WAYBEL_7:exreg 1
registration
cluster non empty non trivial strict reflexive transitive antisymmetric Boolean non void with_suprema with_infima RelStr;
end;
:: WAYBEL_7:exreg 2
registration
let a1 be set;
cluster non empty finite Element of bool bool a1;
end;
:: WAYBEL_7:exreg 3
registration
let a1 be 1-sorted;
cluster non empty finite Element of bool bool the carrier of a1;
end;
:: WAYBEL_7:exreg 4
registration
let a1 be non empty non trivial reflexive transitive antisymmetric upper-bounded RelStr;
cluster non empty proper filtered upper Element of bool the carrier of a1;
end;
:: WAYBEL_7:th 9
theorem
for b1 being set
for b2 being Element of the carrier of BoolePoset b1 holds
'not' b2 = b1 \ b2;
:: WAYBEL_7:th 10
theorem
for b1 being set
for b2 being Element of bool the carrier of BoolePoset b1 holds
b2 is lower(BoolePoset b1)
iff
for b3, b4 being set
st b3 c= b4 & b4 in b2
holds b3 in b2;
:: WAYBEL_7:th 11
theorem
for b1 being set
for b2 being Element of bool the carrier of BoolePoset b1 holds
b2 is upper(BoolePoset b1)
iff
for b3, b4 being set
st b3 c= b4 & b4 c= b1 & b3 in b2
holds b4 in b2;
:: WAYBEL_7:th 12
theorem
for b1 being set
for b2 being lower Element of bool the carrier of BoolePoset b1 holds
b2 is directed(BoolePoset b1)
iff
for b3, b4 being set
st b3 in b2 & b4 in b2
holds b3 \/ b4 in b2;
:: WAYBEL_7:th 13
theorem
for b1 being set
for b2 being upper Element of bool the carrier of BoolePoset b1 holds
b2 is filtered(BoolePoset b1)
iff
for b3, b4 being set
st b3 in b2 & b4 in b2
holds b3 /\ b4 in b2;
:: WAYBEL_7:th 14
theorem
for b1 being set
for b2 being non empty lower Element of bool the carrier of BoolePoset b1 holds
b2 is directed(BoolePoset b1)
iff
for b3 being finite Element of bool bool b1
st b3 c= b2
holds union b3 in b2;
:: WAYBEL_7:th 15
theorem
for b1 being set
for b2 being non empty upper Element of bool the carrier of BoolePoset b1 holds
b2 is filtered(BoolePoset b1)
iff
for b3 being finite Element of bool bool b1
st b3 c= b2
holds Intersect b3 in b2;
:: WAYBEL_7:attrnot 1 => WAYBEL_7:attr 1
definition
let a1 be reflexive transitive antisymmetric with_infima RelStr;
let a2 be non empty directed lower Element of bool the carrier of a1;
attr a2 is prime means
for b1, b2 being Element of the carrier of a1
st b1 "/\" b2 in a2 & not b1 in a2
holds b2 in a2;
end;
:: WAYBEL_7:dfs 1
definiens
let a1 be reflexive transitive antisymmetric with_infima RelStr;
let a2 be non empty directed lower Element of bool the carrier of a1;
To prove
a2 is prime
it is sufficient to prove
thus for b1, b2 being Element of the carrier of a1
st b1 "/\" b2 in a2 & not b1 in a2
holds b2 in a2;
:: WAYBEL_7:def 1
theorem
for b1 being reflexive transitive antisymmetric with_infima RelStr
for b2 being non empty directed lower Element of bool the carrier of b1 holds
b2 is prime(b1)
iff
for b3, b4 being Element of the carrier of b1
st b3 "/\" b4 in b2 & not b3 in b2
holds b4 in b2;
:: WAYBEL_7:th 16
theorem
for b1 being reflexive transitive antisymmetric with_infima RelStr
for b2 being non empty directed lower Element of bool the carrier of b1 holds
b2 is prime(b1)
iff
for b3 being non empty finite Element of bool the carrier of b1
st "/\"(b3,b1) in b2
holds ex b4 being Element of the carrier of b1 st
b4 in b3 & b4 in b2;
:: WAYBEL_7:exreg 5
registration
let a1 be reflexive transitive antisymmetric with_suprema with_infima RelStr;
cluster non empty directed lower prime Element of bool the carrier of a1;
end;
:: WAYBEL_7:th 17
theorem
for b1, b2 being reflexive transitive antisymmetric with_suprema with_infima RelStr
st RelStr(#the carrier of b1,the InternalRel of b1#) = RelStr(#the carrier of b2,the InternalRel of b2#)
for b3 being set
st b3 is non empty directed lower prime Element of bool the carrier of b1
holds b3 is non empty directed lower prime Element of bool the carrier of b2;
:: WAYBEL_7:attrnot 2 => WAYBEL_7:attr 2
definition
let a1 be reflexive transitive antisymmetric with_suprema RelStr;
let a2 be non empty filtered upper Element of bool the carrier of a1;
attr a2 is prime means
for b1, b2 being Element of the carrier of a1
st b1 "\/" b2 in a2 & not b1 in a2
holds b2 in a2;
end;
:: WAYBEL_7:dfs 2
definiens
let a1 be reflexive transitive antisymmetric with_suprema RelStr;
let a2 be non empty filtered upper Element of bool the carrier of a1;
To prove
a2 is prime
it is sufficient to prove
thus for b1, b2 being Element of the carrier of a1
st b1 "\/" b2 in a2 & not b1 in a2
holds b2 in a2;
:: WAYBEL_7:def 2
theorem
for b1 being reflexive transitive antisymmetric with_suprema RelStr
for b2 being non empty filtered upper Element of bool the carrier of b1 holds
b2 is prime(b1)
iff
for b3, b4 being Element of the carrier of b1
st b3 "\/" b4 in b2 & not b3 in b2
holds b4 in b2;
:: WAYBEL_7:th 18
theorem
for b1 being reflexive transitive antisymmetric with_suprema RelStr
for b2 being non empty filtered upper Element of bool the carrier of b1 holds
b2 is prime(b1)
iff
for b3 being non empty finite Element of bool the carrier of b1
st "\/"(b3,b1) in b2
holds ex b4 being Element of the carrier of b1 st
b4 in b3 & b4 in b2;
:: WAYBEL_7:exreg 6
registration
let a1 be reflexive transitive antisymmetric with_suprema with_infima RelStr;
cluster non empty filtered upper prime Element of bool the carrier of a1;
end;
:: WAYBEL_7:th 19
theorem
for b1, b2 being reflexive transitive antisymmetric with_suprema with_infima RelStr
st RelStr(#the carrier of b1,the InternalRel of b1#) = RelStr(#the carrier of b2,the InternalRel of b2#)
for b3 being set
st b3 is non empty filtered upper prime Element of bool the carrier of b1
holds b3 is non empty filtered upper prime Element of bool the carrier of b2;
:: WAYBEL_7:th 20
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima RelStr
for b2 being set holds
b2 is non empty directed lower prime Element of bool the carrier of b1
iff
b2 is non empty filtered upper prime Element of bool the carrier of b1 ~;
:: WAYBEL_7:th 21
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima RelStr
for b2 being set holds
b2 is non empty filtered upper prime Element of bool the carrier of b1
iff
b2 is non empty directed lower prime Element of bool the carrier of b1 ~;
:: WAYBEL_7:th 22
theorem
for b1 being reflexive transitive antisymmetric with_infima RelStr
for b2 being non empty directed lower Element of bool the carrier of b1 holds
b2 is prime(b1)
iff
(b2 ` is non empty filtered upper Element of bool the carrier of b1 or b2 ` = {});
:: WAYBEL_7:th 23
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima RelStr
for b2 being non empty directed lower Element of bool the carrier of b1 holds
b2 is prime(b1)
iff
b2 in PRIME InclPoset Ids b1;
:: WAYBEL_7:th 24
theorem
for b1 being reflexive transitive antisymmetric Boolean with_suprema with_infima RelStr
for b2 being non empty filtered upper Element of bool the carrier of b1 holds
b2 is prime(b1)
iff
for b3 being Element of the carrier of b1
st not b3 in b2
holds 'not' b3 in b2;
:: WAYBEL_7:th 25
theorem
for b1 being set
for b2 being non empty filtered upper Element of bool the carrier of BoolePoset b1 holds
b2 is prime(BoolePoset b1)
iff
for b3 being Element of bool b1
st not b3 in b2
holds b1 \ b3 in b2;
:: WAYBEL_7:attrnot 3 => WAYBEL_7:attr 3
definition
let a1 be non empty reflexive transitive antisymmetric RelStr;
let a2 be non empty filtered upper Element of bool the carrier of a1;
attr a2 is ultra means
a2 is proper(bool the carrier of a1) &
(for b1 being non empty filtered upper Element of bool the carrier of a1
st a2 c= b1 & a2 <> b1
holds b1 = the carrier of a1);
end;
:: WAYBEL_7:dfs 3
definiens
let a1 be non empty reflexive transitive antisymmetric RelStr;
let a2 be non empty filtered upper Element of bool the carrier of a1;
To prove
a2 is ultra
it is sufficient to prove
thus a2 is proper(bool the carrier of a1) &
(for b1 being non empty filtered upper Element of bool the carrier of a1
st a2 c= b1 & a2 <> b1
holds b1 = the carrier of a1);
:: WAYBEL_7:def 3
theorem
for b1 being non empty reflexive transitive antisymmetric RelStr
for b2 being non empty filtered upper Element of bool the carrier of b1 holds
b2 is ultra(b1)
iff
b2 is proper(bool the carrier of b1) &
(for b3 being non empty filtered upper Element of bool the carrier of b1
st b2 c= b3 & b2 <> b3
holds b3 = the carrier of b1);
:: WAYBEL_7:condreg 1
registration
let a1 be non empty reflexive transitive antisymmetric RelStr;
cluster non empty filtered upper ultra -> proper (Element of bool the carrier of a1);
end;
:: WAYBEL_7:th 26
theorem
for b1 being reflexive transitive antisymmetric Boolean with_suprema with_infima RelStr
for b2 being non empty filtered upper Element of bool the carrier of b1 holds
b2 is proper(bool the carrier of b1) & b2 is prime(b1)
iff
b2 is ultra(b1);
:: WAYBEL_7:th 27
theorem
for b1 being reflexive transitive antisymmetric distributive with_suprema with_infima RelStr
for b2 being non empty directed lower Element of bool the carrier of b1
for b3 being non empty filtered upper Element of bool the carrier of b1
st b2 misses b3
holds ex b4 being non empty directed lower Element of bool the carrier of b1 st
b4 is prime(b1) & b2 c= b4 & b4 misses b3;
:: WAYBEL_7:th 28
theorem
for b1 being reflexive transitive antisymmetric distributive with_suprema with_infima RelStr
for b2 being non empty directed lower Element of bool the carrier of b1
for b3 being Element of the carrier of b1
st not b3 in b2
holds ex b4 being non empty directed lower Element of bool the carrier of b1 st
b4 is prime(b1) & b2 c= b4 & not b3 in b4;
:: WAYBEL_7:th 29
theorem
for b1 being reflexive transitive antisymmetric distributive with_suprema with_infima RelStr
for b2 being non empty directed lower Element of bool the carrier of b1
for b3 being non empty filtered upper Element of bool the carrier of b1
st b2 misses b3
holds ex b4 being non empty filtered upper Element of bool the carrier of b1 st
b4 is prime(b1) & b3 c= b4 & b2 misses b4;
:: WAYBEL_7:th 30
theorem
for b1 being non trivial reflexive transitive antisymmetric Boolean with_suprema with_infima RelStr
for b2 being non empty proper filtered upper Element of bool the carrier of b1 holds
ex b3 being non empty filtered upper Element of bool the carrier of b1 st
b2 c= b3 & b3 is ultra(b1);
:: WAYBEL_7:prednot 1 => WAYBEL_7:pred 1
definition
let a1 be TopSpace-like TopStruct;
let a2, a3 be set;
pred A3 is_a_cluster_point_of A2,A1 means
for b1 being Element of bool the carrier of a1
st b1 is open(a1) & a3 in b1
for b2 being set
st b2 in a2
holds b1 meets b2;
end;
:: WAYBEL_7:dfs 4
definiens
let a1 be TopSpace-like TopStruct;
let a2, a3 be set;
To prove
a3 is_a_cluster_point_of a2,a1
it is sufficient to prove
thus for b1 being Element of bool the carrier of a1
st b1 is open(a1) & a3 in b1
for b2 being set
st b2 in a2
holds b1 meets b2;
:: WAYBEL_7:def 4
theorem
for b1 being TopSpace-like TopStruct
for b2, b3 being set holds
b3 is_a_cluster_point_of b2,b1
iff
for b4 being Element of bool the carrier of b1
st b4 is open(b1) & b3 in b4
for b5 being set
st b5 in b2
holds b4 meets b5;
:: WAYBEL_7:prednot 2 => WAYBEL_7:pred 2
definition
let a1 be TopSpace-like TopStruct;
let a2, a3 be set;
pred A3 is_a_convergence_point_of A2,A1 means
for b1 being Element of bool the carrier of a1
st b1 is open(a1) & a3 in b1
holds b1 in a2;
end;
:: WAYBEL_7:dfs 5
definiens
let a1 be TopSpace-like TopStruct;
let a2, a3 be set;
To prove
a3 is_a_convergence_point_of a2,a1
it is sufficient to prove
thus for b1 being Element of bool the carrier of a1
st b1 is open(a1) & a3 in b1
holds b1 in a2;
:: WAYBEL_7:def 5
theorem
for b1 being TopSpace-like TopStruct
for b2, b3 being set holds
b3 is_a_convergence_point_of b2,b1
iff
for b4 being Element of bool the carrier of b1
st b4 is open(b1) & b3 in b4
holds b4 in b2;
:: WAYBEL_7:exreg 7
registration
let a1 be non empty set;
cluster non empty filtered upper ultra Element of bool the carrier of BoolePoset a1;
end;
:: WAYBEL_7:th 31
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty filtered upper ultra Element of bool the carrier of BoolePoset the carrier of b1
for b3 being set holds
b3 is_a_cluster_point_of b2,b1
iff
b3 is_a_convergence_point_of b2,b1;
:: WAYBEL_7:th 32
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of the carrier of InclPoset the topology of b1
st b2 is_way_below b3
for b4 being non empty proper filtered upper Element of bool the carrier of BoolePoset the carrier of b1
st b2 in b4
holds ex b5 being Element of the carrier of b1 st
b5 in b3 & b5 is_a_cluster_point_of b4,b1;
:: WAYBEL_7:th 33
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of the carrier of InclPoset the topology of b1
st b2 is_way_below b3
for b4 being non empty filtered upper ultra Element of bool the carrier of BoolePoset the carrier of b1
st b2 in b4
holds ex b5 being Element of the carrier of b1 st
b5 in b3 & b5 is_a_convergence_point_of b4,b1;
:: WAYBEL_7:th 34
theorem
for b1 being non empty TopSpace-like TopStruct
for b2, b3 being Element of the carrier of InclPoset the topology of b1
st b2 c= b3 &
(for b4 being non empty filtered upper ultra Element of bool the carrier of BoolePoset the carrier of b1
st b2 in b4
holds ex b5 being Element of the carrier of b1 st
b5 in b3 & b5 is_a_convergence_point_of b4,b1)
holds b2 is_way_below b3;
:: WAYBEL_7:th 35
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being prebasis of b1
for b3, b4 being Element of the carrier of InclPoset the topology of b1
st b3 c= b4
holds b3 is_way_below b4
iff
for b5 being Element of bool b2
st b4 c= union b5
holds ex b6 being finite Element of bool b5 st
b3 c= union b6;
:: WAYBEL_7:th 36
theorem
for b1 being reflexive transitive antisymmetric distributive with_suprema with_infima complete RelStr
for b2, b3 being Element of the carrier of b1 holds
b2 is_way_below b3
iff
for b4 being non empty directed lower prime Element of bool the carrier of b1
st b3 <= "\/"(b4,b1)
holds b2 in b4;
:: WAYBEL_7:th 37
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima RelStr
for b2 being Element of the carrier of b1
st b2 is prime(b1)
holds downarrow b2 is prime(b1);
:: WAYBEL_7:attrnot 4 => WAYBEL_7:attr 4
definition
let a1 be reflexive transitive antisymmetric with_suprema with_infima RelStr;
let a2 be Element of the carrier of a1;
attr a2 is pseudoprime means
ex b1 being non empty directed lower prime Element of bool the carrier of a1 st
a2 = "\/"(b1,a1);
end;
:: WAYBEL_7:dfs 6
definiens
let a1 be reflexive transitive antisymmetric with_suprema with_infima RelStr;
let a2 be Element of the carrier of a1;
To prove
a2 is pseudoprime
it is sufficient to prove
thus ex b1 being non empty directed lower prime Element of bool the carrier of a1 st
a2 = "\/"(b1,a1);
:: WAYBEL_7:def 6
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima RelStr
for b2 being Element of the carrier of b1 holds
b2 is pseudoprime(b1)
iff
ex b3 being non empty directed lower prime Element of bool the carrier of b1 st
b2 = "\/"(b3,b1);
:: WAYBEL_7:th 38
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima RelStr
for b2 being Element of the carrier of b1
st b2 is prime(b1)
holds b2 is pseudoprime(b1);
:: WAYBEL_7:th 39
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima continuous RelStr
for b2 being Element of the carrier of b1
st b2 is pseudoprime(b1)
for b3 being non empty finite Element of bool the carrier of b1
st "/\"(b3,b1) is_way_below b2
holds ex b4 being Element of the carrier of b1 st
b4 in b3 & b4 <= b2;
:: WAYBEL_7:th 40
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima continuous RelStr
for b2 being Element of the carrier of b1
st (b2 = Top b1 implies Top b1 is not compact(b1)) &
(for b3 being non empty finite Element of bool the carrier of b1
st "/\"(b3,b1) is_way_below b2
holds ex b4 being Element of the carrier of b1 st
b4 in b3 & b4 <= b2)
holds uparrow fininfs ((downarrow b2) `) misses waybelow b2;
:: WAYBEL_7:th 41
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima continuous RelStr
st Top b1 is compact(b1)
holds (for b2 being non empty finite Element of bool the carrier of b1
st "/\"(b2,b1) is_way_below Top b1
holds ex b3 being Element of the carrier of b1 st
b3 in b2 & b3 <= Top b1) &
uparrow fininfs ((downarrow Top b1) `) meets waybelow Top b1;
:: WAYBEL_7:th 42
theorem
for b1 being reflexive transitive antisymmetric with_suprema with_infima continuous RelStr
for b2 being Element of the carrier of b1
st uparrow fininfs ((downarrow b2) `) misses waybelow b2
for b3 being non empty finite Element of bool the carrier of b1
st "/\"(b3,b1) is_way_below b2
holds ex b4 being Element of the carrier of b1 st
b4 in b3 & b4 <= b2;
:: WAYBEL_7:th 43
theorem
for b1 being reflexive transitive antisymmetric distributive with_suprema with_infima continuous RelStr
for b2 being Element of the carrier of b1
st uparrow fininfs ((downarrow b2) `) misses waybelow b2
holds b2 is pseudoprime(b1);
:: WAYBEL_7:attrnot 5 => WAYBEL_7:attr 5
definition
let a1 be non empty RelStr;
let a2 be Relation of the carrier of a1,the carrier of a1;
attr a2 is multiplicative means
for b1, b2, b3 being Element of the carrier of a1
st [b1,b2] in a2 & [b1,b3] in a2
holds [b1,b2 "/\" b3] in a2;
end;
:: WAYBEL_7:dfs 7
definiens
let a1 be non empty RelStr;
let a2 be Relation of the carrier of a1,the carrier of a1;
To prove
a2 is multiplicative
it is sufficient to prove
thus for b1, b2, b3 being Element of the carrier of a1
st [b1,b2] in a2 & [b1,b3] in a2
holds [b1,b2 "/\" b3] in a2;
:: WAYBEL_7:def 7
theorem
for b1 being non empty RelStr
for b2 being Relation of the carrier of b1,the carrier of b1 holds
b2 is multiplicative(b1)
iff
for b3, b4, b5 being Element of the carrier of b1
st [b3,b4] in b2 & [b3,b5] in b2
holds [b3,b4 "/\" b5] in b2;
:: WAYBEL_7:funcreg 3
registration
let a1 be reflexive transitive antisymmetric lower-bounded with_suprema RelStr;
let a2 be auxiliary Relation of the carrier of a1,the carrier of a1;
let a3 be Element of the carrier of a1;
cluster a2 -above a3 -> upper;
end;
:: WAYBEL_7:th 44
theorem
for b1 being reflexive transitive antisymmetric lower-bounded with_suprema with_infima RelStr
for b2 being auxiliary Relation of the carrier of b1,the carrier of b1 holds
b2 is multiplicative(b1)
iff
for b3 being Element of the carrier of b1 holds
b2 -above b3 is filtered(b1);
:: WAYBEL_7:th 45
theorem
for b1 being reflexive transitive antisymmetric lower-bounded with_suprema with_infima RelStr
for b2 being auxiliary Relation of the carrier of b1,the carrier of b1 holds
b2 is multiplicative(b1)
iff
for b3, b4, b5, b6 being Element of the carrier of b1
st [b3,b5] in b2 & [b4,b6] in b2
holds [b3 "/\" b4,b5 "/\" b6] in b2;
:: WAYBEL_7:th 46
theorem
for b1 being reflexive transitive antisymmetric lower-bounded with_suprema with_infima RelStr
for b2 being auxiliary Relation of the carrier of b1,the carrier of b1 holds
b2 is multiplicative(b1)
iff
for b3 being full SubRelStr of [:b1,b1:]
st the carrier of b3 = b2
holds b3 is meet-inheriting([:b1,b1:]);
:: WAYBEL_7:th 47
theorem
for b1 being reflexive transitive antisymmetric lower-bounded with_suprema with_infima RelStr
for b2 being auxiliary Relation of the carrier of b1,the carrier of b1 holds
b2 is multiplicative(b1)
iff
b2 -below is meet-preserving(b1, InclPoset Ids b1);
:: WAYBEL_7:th 48
theorem
for b1 being reflexive transitive antisymmetric lower-bounded with_suprema with_infima continuous RelStr
st b1 -waybelow is multiplicative(b1)
for b2 being Element of the carrier of b1 holds
b2 is pseudoprime(b1)
iff
for b3, b4 being Element of the carrier of b1
st b3 "/\" b4 is_way_below b2 & not b3 <= b2
holds b4 <= b2;
:: WAYBEL_7:th 49
theorem
for b1 being reflexive transitive antisymmetric lower-bounded with_suprema with_infima continuous RelStr
st b1 -waybelow is multiplicative(b1)
for b2 being Element of the carrier of b1
st b2 is pseudoprime(b1)
holds b2 is prime(b1);
:: WAYBEL_7:th 50
theorem
for b1 being reflexive transitive antisymmetric lower-bounded distributive with_suprema with_infima continuous RelStr
st for b2 being Element of the carrier of b1
st b2 is pseudoprime(b1)
holds b2 is prime(b1)
holds b1 -waybelow is multiplicative(b1);