:: YELLOW14 semantic presentation

theorem Th1: :: YELLOW14:1
bool 1 = {0,1} by CARD_5:1, ZFMISC_1:30;

theorem Th2: :: YELLOW14:2
for b1 being set
for b2 being Subset of b1 holds rng ((id b1) | b2) = b2
proof end;

theorem Th3: :: YELLOW14:3
for b1 being Function
for b2, b3 being set holds (b1 +* (b2 .--> b3)) . b2 = b3 by FUNCT_7:96;

registration
cluster empty strict RelStr ;
existence
ex b1 being RelStr st
( b1 is strict & b1 is empty )
proof end;
end;

theorem Th4: :: YELLOW14:4
for b1 being empty 1-sorted
for b2 being 1-sorted
for b3 being Function of b1,b2 holds
( rng b3 = [#] b2 implies b2 is empty )
proof end;

theorem Th5: :: YELLOW14:5
for b1 being 1-sorted
for b2 being empty 1-sorted
for b3 being Function of b1,b2 holds
( dom b3 = [#] b1 implies b1 is empty )
proof end;

theorem Th6: :: YELLOW14:6
for b1 being non empty 1-sorted
for b2 being 1-sorted
for b3 being Function of b1,b2 holds
not ( dom b3 = [#] b1 & b2 is empty )
proof end;

theorem Th7: :: YELLOW14:7
for b1 being 1-sorted
for b2 being non empty 1-sorted
for b3 being Function of b1,b2 holds
not ( rng b3 = [#] b2 & b1 is empty )
proof end;

definition
let c1 be non empty reflexive RelStr ;
let c2 be non empty RelStr ;
let c3 be Function of c1,c2;
redefine attr a3 is directed-sups-preserving means :: YELLOW14:def 1
for b1 being non empty directed Subset of a1 holds a3 preserves_sup_of b1;
compatibility
( c3 is directed-sups-preserving iff for b1 being non empty directed Subset of c1 holds c3 preserves_sup_of b1 )
proof end;
end;

:: deftheorem Def1 defines directed-sups-preserving YELLOW14:def 1 :
for b1 being non empty reflexive RelStr
for b2 being non empty RelStr
for b3 being Function of b1,b2 holds
( b3 is directed-sups-preserving iff for b4 being non empty directed Subset of b1 holds b3 preserves_sup_of b4 );

definition
let c1 be 1-sorted ;
let c2 be NetStr of c1;
attr a2 is Function-yielding means :Def2: :: YELLOW14:def 2
the mapping of a2 is Function-yielding;
end;

:: deftheorem Def2 defines Function-yielding YELLOW14:def 2 :
for b1 being 1-sorted
for b2 being NetStr of b1 holds
( b2 is Function-yielding iff the mapping of b2 is Function-yielding );

registration
cluster strict non empty constituted-Functions 1-sorted ;
existence
ex b1 being 1-sorted st
( b1 is strict & not b1 is empty & b1 is constituted-Functions )
proof end;
end;

registration
cluster non empty strict constituted-Functions RelStr ;
existence
ex b1 being RelStr st
( b1 is strict & not b1 is empty & b1 is constituted-Functions )
proof end;
end;

registration
let c1 be constituted-Functions 1-sorted ;
cluster -> Function-yielding NetStr of a1;
coherence
for b1 being NetStr of c1 holds b1 is Function-yielding
proof end;
end;

registration
let c1 be constituted-Functions 1-sorted ;
cluster strict Function-yielding NetStr of a1;
existence
ex b1 being NetStr of c1 st
( b1 is strict & b1 is Function-yielding )
proof end;
end;

registration
let c1 be non empty constituted-Functions 1-sorted ;
cluster non empty strict Function-yielding NetStr of a1;
existence
ex b1 being NetStr of c1 st
( b1 is strict & not b1 is empty & b1 is Function-yielding )
proof end;
end;

registration
let c1 be constituted-Functions 1-sorted ;
let c2 be Function-yielding NetStr of c1;
cluster the mapping of a2 -> Function-yielding ;
coherence
the mapping of c2 is Function-yielding
by Def2;
end;

registration
let c1 be non empty constituted-Functions 1-sorted ;
cluster strict Function-yielding NetStr of a1;
existence
ex b1 being net of c1 st
( b1 is strict & b1 is Function-yielding )
proof end;
end;

registration
let c1 be non empty 1-sorted ;
let c2 be non empty NetStr of c1;
cluster rng the mapping of a2 -> non empty ;
coherence
not rng the mapping of c2 is empty
;
end;

registration
let c1 be non empty 1-sorted ;
let c2 be non empty NetStr of c1;
cluster rng (netmap a2,a1) -> non empty ;
coherence
not rng (netmap c2,c1) is empty
;
end;

theorem Th8: :: YELLOW14:8
for b1, b2, b3 being non empty RelStr
for b4 being Function of b2,b3
for b5, b6 being Function of b1,b2 holds
( b5 <= b6 & b4 is monotone implies b4 * b5 <= b4 * b6 )
proof end;

theorem Th9: :: YELLOW14:9
for b1 being non empty TopSpace
for b2 being non empty TopSpace-like TopRelStr
for b3, b4 being Function of b1,b2
for b5, b6 being Element of (ContMaps b1,b2) holds
( b5 = b3 & b6 = b4 implies ( b5 <= b6 iff b3 <= b4 ) )
proof end;

definition
let c1 be non empty set ;
let c2 be non empty RelStr ;
let c3 be Element of (c2 |^ c1);
let c4 be Element of c1;
redefine func . as c3 . c4 -> Element of a2;
coherence
c3 . c4 is Element of c2
proof end;
end;

theorem Th10: :: YELLOW14:10
for b1, b2 being RelStr
for b3 being Function of b1,b2 holds
( b3 is isomorphic implies b3 is onto )
proof end;

registration
let c1, c2 be RelStr ;
cluster isomorphic -> onto Relation of the carrier of a1,the carrier of a2;
coherence
for b1 being Function of c1,c2 holds
( b1 is isomorphic implies b1 is onto )
by Th10;
end;

theorem Th11: :: YELLOW14:11
for b1, b2 being non empty RelStr
for b3 being Function of b1,b2 holds
( b3 is isomorphic implies b3 /" is isomorphic )
proof end;

theorem Th12: :: YELLOW14:12
for b1, b2 being non empty RelStr holds
( b1,b2 are_isomorphic & b1 is with_infima implies b2 is with_infima )
proof end;

theorem Th13: :: YELLOW14:13
for b1, b2 being non empty RelStr holds
( b1,b2 are_isomorphic & b1 is with_suprema implies b2 is with_suprema )
proof end;

theorem Th14: :: YELLOW14:14
for b1 being RelStr holds
( b1 is empty implies b1 is bounded )
proof end;

registration
cluster empty -> bounded RelStr ;
coherence
for b1 being RelStr holds
( b1 is empty implies b1 is bounded )
by Th14;
end;

theorem Th15: :: YELLOW14:15
for b1, b2 being RelStr holds
( b1,b2 are_isomorphic & b1 is lower-bounded implies b2 is lower-bounded )
proof end;

theorem Th16: :: YELLOW14:16
for b1, b2 being RelStr holds
( b1,b2 are_isomorphic & b1 is upper-bounded implies b2 is upper-bounded )
proof end;

theorem Th17: :: YELLOW14:17
for b1, b2 being non empty RelStr
for b3 being Subset of b1
for b4 being Function of b1,b2 holds
( b4 is isomorphic & ex_sup_of b3,b1 implies ex_sup_of b4 .: b3,b2 )
proof end;

theorem Th18: :: YELLOW14:18
for b1, b2 being non empty RelStr
for b3 being Subset of b1
for b4 being Function of b1,b2 holds
( b4 is isomorphic & ex_inf_of b3,b1 implies ex_inf_of b4 .: b3,b2 )
proof end;

theorem Th19: :: YELLOW14:19
for b1, b2 being TopStruct holds
( not ( not b1,b2 are_homeomorphic & ( for b3 being Function of b1,b2 holds
not ( dom b3 = [#] b1 & rng b3 = [#] b2 ) ) ) implies ( b1 is empty iff b2 is empty ) )
proof end;

theorem Th20: :: YELLOW14:20
for b1 being non empty TopSpace holds b1, TopStruct(# the carrier of b1,the topology of b1 #) are_homeomorphic
proof end;

registration
let c1 be non empty reflexive Scott TopRelStr ;
cluster open -> upper inaccessible Element of bool the carrier of a1;
coherence
for b1 being Subset of c1 holds
( b1 is open implies ( b1 is inaccessible_by_directed_joins & b1 is upper ) )
by WAYBEL11:def 4;
cluster upper inaccessible -> open Element of bool the carrier of a1;
coherence
for b1 being Subset of c1 holds
( b1 is inaccessible_by_directed_joins & b1 is upper implies b1 is open )
by WAYBEL11:def 4;
end;

theorem Th21: :: YELLOW14:21
for b1 being TopStruct
for b2, b3 being Point of b1
for b4, b5 being Subset of b1 holds
( b4 = {b2} & Cl b4 c= Cl b5 implies b2 in Cl b5 )
proof end;

theorem Th22: :: YELLOW14:22
for b1 being TopStruct
for b2, b3 being Point of b1
for b4, b5 being Subset of b1 holds
( b4 = {b3} & b2 in Cl b4 & b5 is open & b2 in b5 implies b3 in b5 )
proof end;

theorem Th23: :: YELLOW14:23
for b1 being TopStruct
for b2, b3 being Point of b1
for b4, b5 being Subset of b1 holds
( b4 = {b2} & b5 = {b3} & ( for b6 being Subset of b1 holds
( b6 is open & b2 in b6 implies b3 in b6 ) ) implies Cl b4 c= Cl b5 )
proof end;

theorem Th24: :: YELLOW14:24
for b1, b2 being non empty TopSpace
for b3 being irreducible Subset of b1
for b4 being Subset of b2 holds
( b3 = b4 & TopStruct(# the carrier of b1,the topology of b1 #) = TopStruct(# the carrier of b2,the topology of b2 #) implies b4 is irreducible )
proof end;

theorem Th25: :: YELLOW14:25
for b1, b2 being non empty TopSpace
for b3 being Point of b1
for b4 being Point of b2
for b5 being Subset of b1
for b6 being Subset of b2 holds
( b3 = b4 & b5 = b6 & TopStruct(# the carrier of b1,the topology of b1 #) = TopStruct(# the carrier of b2,the topology of b2 #) & b3 is_dense_point_of b5 implies b4 is_dense_point_of b6 )
proof end;

theorem Th26: :: YELLOW14:26
for b1, b2 being TopStruct
for b3 being Subset of b1
for b4 being Subset of b2 holds
( b3 = b4 & TopStruct(# the carrier of b1,the topology of b1 #) = TopStruct(# the carrier of b2,the topology of b2 #) & b3 is compact implies b4 is compact )
proof end;

theorem Th27: :: YELLOW14:27
for b1, b2 being non empty TopSpace holds
( TopStruct(# the carrier of b1,the topology of b1 #) = TopStruct(# the carrier of b2,the topology of b2 #) & b1 is sober implies b2 is sober )
proof end;

theorem Th28: :: YELLOW14:28
for b1, b2 being non empty TopSpace holds
( TopStruct(# the carrier of b1,the topology of b1 #) = TopStruct(# the carrier of b2,the topology of b2 #) & b1 is locally-compact implies b2 is locally-compact )
proof end;

theorem Th29: :: YELLOW14:29
for b1, b2 being TopStruct holds
( TopStruct(# the carrier of b1,the topology of b1 #) = TopStruct(# the carrier of b2,the topology of b2 #) & b1 is compact implies b2 is compact )
proof end;

definition
let c1 be non empty set ;
let c2 be non empty TopSpace;
let c3 be Point of (product (c1 --> c2));
let c4 be Element of c1;
redefine func . as c3 . c4 -> Element of a2;
coherence
c3 . c4 is Element of c2
proof end;
end;

theorem Th30: :: YELLOW14:30
for b1 being non empty set
for b2 being non-Empty TopSpace-yielding ManySortedSet of b1
for b3, b4 being Point of (product b2) holds
( b3 in Cl {b4} iff for b5 being Element of b1 holds b3 . b5 in Cl {(b4 . b5)} )
proof end;

theorem Th31: :: YELLOW14:31
for b1 being non empty set
for b2 being non empty TopSpace
for b3, b4 being Point of (product (b1 --> b2)) holds
( b3 in Cl {b4} iff for b5 being Element of b1 holds b3 . b5 in Cl {(b4 . b5)} )
proof end;

theorem Th32: :: YELLOW14:32
for b1 being non empty set
for b2 being Element of b1
for b3 being non-Empty TopSpace-yielding ManySortedSet of b1
for b4 being Point of (product b3) holds pi (Cl {b4}),b2 = Cl {(b4 . b2)}
proof end;

theorem Th33: :: YELLOW14:33
for b1 being non empty set
for b2 being Element of b1
for b3 being non empty TopSpace
for b4 being Point of (product (b1 --> b3)) holds pi (Cl {b4}),b2 = Cl {(b4 . b2)}
proof end;

theorem Th34: :: YELLOW14:34
for b1, b2 being non empty TopStruct
for b3 being Function of b1,b2
for b4 being Function of b2,b1 holds
( b3 = id b1 & b4 = id b1 & b3 is continuous & b4 is continuous implies TopStruct(# the carrier of b1,the topology of b1 #) = TopStruct(# the carrier of b2,the topology of b2 #) )
proof end;

theorem Th35: :: YELLOW14:35
for b1, b2 being non empty TopSpace
for b3 being Function of b1,b2 holds
( corestr b3 is continuous implies b3 is continuous )
proof end;

registration
let c1 be non empty TopSpace;
let c2 be non empty SubSpace of c1;
cluster incl a2 -> continuous ;
coherence
incl c2 is continuous
by TMAP_1:98;
end;

theorem Th36: :: YELLOW14:36
for b1 being non empty TopSpace
for b2 being Function of b1,b1 holds
( b2 * b2 = b2 implies (corestr b2) * (incl (Image b2)) = id (Image b2) )
proof end;

theorem Th37: :: YELLOW14:37
for b1 being non empty TopSpace
for b2 being non empty SubSpace of b1 holds corestr (incl b2) is_homeomorphism
proof end;

theorem Th38: :: YELLOW14:38
for b1 being non empty set
for b2 being non-Empty TopSpace-yielding ManySortedSet of b1 holds
( ( for b3 being Element of b1 holds
b2 . b3 is T_0 TopSpace ) implies product b2 is T_0 )
proof end;

registration
let c1 be non empty set ;
let c2 be non empty T_0 TopSpace;
cluster product (a1 --> a2) -> T_0 ;
coherence
product (c1 --> c2) is T_0
proof end;
end;

theorem Th39: :: YELLOW14:39
for b1 being non empty set
for b2 being non-Empty TopSpace-yielding ManySortedSet of b1 holds
( ( for b3 being Element of b1 holds
( b2 . b3 is being_T1 & b2 . b3 is TopSpace-like ) ) implies product b2 is_T1 )
proof end;

registration
let c1 be non empty set ;
let c2 be non empty being_T1 TopSpace;
cluster product (a1 --> a2) -> being_T1 ;
coherence
product (c1 --> c2) is being_T1
proof end;
end;