:: PCOMPS_2 semantic presentation

theorem Th1: :: PCOMPS_2:1
canceled;

theorem Th2: :: PCOMPS_2:2
canceled;

theorem Th3: :: PCOMPS_2:3
for r, u being real number st r > 0 & u > 0 holds
ex k being Element of NAT st u / (2 |^ k) <= r by PREPOWER:106;

theorem Th4: :: PCOMPS_2:4
for r being real number
for k, n being Element of NAT st k >= n & r >= 1 holds
r |^ k >= r |^ n by PREPOWER:107;

theorem Th5: :: PCOMPS_2:5
for R being Relation
for A being set st R well_orders A holds
( R |_2 A well_orders A & A = field (R |_2 A) )
proof end;

scheme :: PCOMPS_2:sch 58
s58{ F1() -> set , F2() -> Relation, P1[ set ] } :
ex X being set st
( X in F1() & P1[X] & ( for Y being set st Y in F1() & P1[Y] holds
[X,Y] in F2() ) )
provided
E22: F2() well_orders F1() and
E23: ex X being set st
( X in F1() & P1[X] )
proof end;

definition
let FX be set ;
let R be Relation;
let B be Element of FX;
func PartUnion c3,c2 -> set equals :: PCOMPS_2:def 1
union (R -Seg B);
coherence
union (R -Seg B) is set
;
end;

:: deftheorem Def1 defines PartUnion PCOMPS_2:def 1 :
for FX being set
for R being Relation
for B being Element of FX holds PartUnion B,R = union (R -Seg B);

definition
let FX be set ;
let R be Relation;
func DisjointFam c1,c2 -> set means :: PCOMPS_2:def 2
for A being set holds
( A in it iff ex B being Element of FX st
( B in FX & A = B \ (PartUnion B,R) ) );
existence
ex b1 being set st
for A being set holds
( A in b1 iff ex B being Element of FX st
( B in FX & A = B \ (PartUnion B,R) ) )
proof end;
uniqueness
for b1, b2 being set st ( for A being set holds
( A in b1 iff ex B being Element of FX st
( B in FX & A = B \ (PartUnion B,R) ) ) ) & ( for A being set holds
( A in b2 iff ex B being Element of FX st
( B in FX & A = B \ (PartUnion B,R) ) ) ) holds
b1 = b2
proof end;
end;

:: deftheorem Def2 defines DisjointFam PCOMPS_2:def 2 :
for FX being set
for R being Relation
for b3 being set holds
( b3 = DisjointFam FX,R iff for A being set holds
( A in b3 iff ex B being Element of FX st
( B in FX & A = B \ (PartUnion B,R) ) ) );

definition
let X be set ;
let n be Element of NAT ;
let f be Function of NAT , bool X;
func PartUnionNat c2,c3 -> set equals :: PCOMPS_2:def 3
union (f .: ((Seg n) \ {n}));
coherence
union (f .: ((Seg n) \ {n})) is set
;
end;

:: deftheorem Def3 defines PartUnionNat PCOMPS_2:def 3 :
for X being set
for n being Element of NAT
for f being Function of NAT , bool X holds PartUnionNat n,f = union (f .: ((Seg n) \ {n}));

theorem Th6: :: PCOMPS_2:6
for PT being non empty TopSpace st PT is_T3 holds
for FX being Subset-Family of PT st FX is_a_cover_of PT & FX is open holds
ex HX being Subset-Family of PT st
( HX is open & HX is_a_cover_of PT & ( for V being Subset of PT st V in HX holds
ex W being Subset of PT st
( W in FX & Cl V c= W ) ) )
proof end;

theorem Th7: :: PCOMPS_2:7
for PT being non empty TopSpace
for FX being Subset-Family of PT st PT is_T2 & PT is paracompact & FX is_a_cover_of PT & FX is open holds
ex GX being Subset-Family of PT st
( GX is open & GX is_a_cover_of PT & clf GX is_finer_than FX & GX is locally_finite )
proof end;

theorem Th8: :: PCOMPS_2:8
for PT being non empty TopSpace
for PM being MetrSpace
for f being Function of [:the carrier of PT,the carrier of PT:], REAL st f is_metric_of the carrier of PT & PM = SpaceMetr the carrier of PT,f holds
the carrier of PM = the carrier of PT
proof end;

theorem Th9: :: PCOMPS_2:9
canceled;

theorem Th10: :: PCOMPS_2:10
canceled;

theorem Th11: :: PCOMPS_2:11
for PT being non empty TopSpace
for PM being MetrSpace
for FX being Subset-Family of PT
for f being Function of [:the carrier of PT,the carrier of PT:], REAL st f is_metric_of the carrier of PT & PM = SpaceMetr the carrier of PT,f holds
( FX is Subset-Family of PT iff FX is Subset-Family of PM ) by ;

definition
let PM be non empty set ;
let g be Function of NAT ,(bool (bool PM)) * ;
let n be Element of NAT ;
redefine func . as c2 . c3 -> FinSequence of bool (bool a1);
coherence
g . n is FinSequence of bool (bool PM)
proof end;
end;

scheme :: PCOMPS_2:sch 83
s83{ F1() -> non empty set , F2() -> Subset-Family of F1(), F3( set , set ) -> Subset of F1(), P1[ set ], P2[ set , set , set , set ] } :
ex f being Function of NAT , bool (bool F1()) st
( f . 0 = F2() & ( for n being Element of NAT holds f . (n + 1) = { (union { F3(c,n) where c is Element of F1() : for fq being Subset-Family of F1()
for q being Element of NAT st q <= n & fq = f . q holds
P2[c,V,n,fq]
}
)
where V is Subset of F1() : P1[V]
}
) )
proof end;

scheme :: PCOMPS_2:sch 102
s102{ F1() -> non empty set , F2() -> Subset-Family of F1(), F3( set , set ) -> Subset of F1(), P1[ set ], P2[ set , set , set ] } :
ex f being Function of NAT , bool (bool F1()) st
( f . 0 = F2() & ( for n being Element of NAT holds f . (n + 1) = { (union { F3(c,n) where c is Element of F1() : ( P2[c,V,n] & not c in union { (union (f . q)) where q is Element of NAT : q <= n } ) } ) where V is Subset of F1() : P1[V] } ) )
proof end;

theorem Th12: :: PCOMPS_2:12
for PT being non empty TopSpace st PT is metrizable holds
for FX being Subset-Family of PT st FX is_a_cover_of PT & FX is open holds
ex GX being Subset-Family of PT st
( GX is open & GX is_a_cover_of PT & GX is_finer_than FX & GX is locally_finite )
proof end;

theorem Th13: :: PCOMPS_2:13
for PT being non empty TopSpace st PT is metrizable holds
PT is paracompact
proof end;