let T be non empty TopSpace; :: thesis: for X being Subset-Family of T ex Y being compl-closed all-open-containing closed_for_countable_unions Subset-Family of T st
( X c= Y & ( for Z being compl-closed all-open-containing closed_for_countable_unions Subset-Family of T st X c= Z holds
Y c= Z ) )
let X be Subset-Family of T; :: thesis: ex Y being compl-closed all-open-containing closed_for_countable_unions Subset-Family of T st
( X c= Y & ( for Z being compl-closed all-open-containing closed_for_countable_unions Subset-Family of T st X c= Z holds
Y c= Z ) )
set V = { S where S is Subset-Family of T : ( X c= S & S is compl-closed all-open-containing closed_for_countable_unions Subset-Family of T ) } ;
set Y = meet { S where S is Subset-Family of T : ( X c= S & S is compl-closed all-open-containing closed_for_countable_unions Subset-Family of T ) } ;

A1: now :: thesis: for Z being set st Z in { S where S is Subset-Family of T : ( X c= S & S is compl-closed all-open-containing closed_for_countable_unions Subset-Family of T ) } holds
X c= Z

let Z be set ; :: thesis: ( Z in { S where S is Subset-Family of T : ( X c= S & S is compl-closed all-open-containing closed_for_countable_unions Subset-Family of T ) } implies X c= Z )
assume Z in { S where S is Subset-Family of T : ( X c= S & S is compl-closed all-open-containing closed_for_countable_unions Subset-Family of T ) } ; :: thesis: X c= Z
then ex S being Subset-Family of T st
( Z = S & X c= S & S is compl-closed all-open-containing closed_for_countable_unions Subset-Family of T ) ;
hence X c= Z ; :: thesis: verum

end;

A2: bool the carrier of T in { S where S is Subset-Family of T : ( X c= S & S is compl-closed all-open-containing closed_for_countable_unions Subset-Family of T ) }

proof

set X1 = TotFam T;
TotFam T in { S where S is Subset-Family of T : ( X c= S & S is compl-closed all-open-containing closed_for_countable_unions Subset-Family of T ) } ;
hence bool the carrier of T in { S where S is Subset-Family of T : ( X c= S & S is compl-closed all-open-containing closed_for_countable_unions Subset-Family of T ) } ; :: thesis: verum

end;

A3: for E being Subset of T st E in meet { S where S is Subset-Family of T : ( X c= S & S is compl-closed all-open-containing closed_for_countable_unions Subset-Family of T ) } holds
E ` in meet { S where S is Subset-Family of T : ( X c= S & S is compl-closed all-open-containing closed_for_countable_unions Subset-Family of T ) }

proof

let E be Subset of T; :: thesis: ( E in meet { S where S is Subset-Family of T : ( X c= S & S is compl-closed all-open-containing closed_for_countable_unions Subset-Family of T ) } implies E ` in meet { S where S is Subset-Family of T : ( X c= S & S is compl-closed all-open-containing closed_for_countable_unions Subset-Family of T ) } )
assume A4: E in meet { S where S is Subset-Family of T : ( X c= S & S is compl-closed all-open-containing closed_for_countable_unions Subset-Family of T ) } ; :: thesis: E ` in meet { S where S is Subset-Family of T : ( X c= S & S is compl-closed all-open-containing closed_for_countable_unions Subset-Family of T ) }

now :: thesis: for Z being set st Z in { S where S is Subset-Family of T : ( X c= S & S is compl-closed all-open-containing closed_for_countable_unions Subset-Family of T ) } holds
E ` in Z

let Z be set ; :: thesis: ( Z in { S where S is Subset-Family of T : ( X c= S & S is compl-closed all-open-containing closed_for_countable_unions Subset-Family of T ) } implies E ` in Z )
assume Z in { S where S is Subset-Family of T : ( X c= S & S is compl-closed all-open-containing closed_for_countable_unions Subset-Family of T ) } ; :: thesis: E ` in Z
then ( E in Z & ex S being Subset-Family of T st
( Z = S & X c= S & S is compl-closed all-open-containing closed_for_countable_unions Subset-Family of T ) ) by A4, SETFAM_1:def 1;
hence E ` in Z by PROB_1:def 1; :: thesis: verum

end;

hence E ` in meet { S where S is Subset-Family of T : ( X c= S & S is compl-closed all-open-containing closed_for_countable_unions Subset-Family of T ) } by A2, SETFAM_1:def 1; :: thesis: verum

end;

A5: for BSeq being SetSequence of the carrier of T st rng BSeq c= meet { S where S is Subset-Family of T : ( X c= S & S is compl-closed all-open-containing closed_for_countable_unions Subset-Family of T ) } holds
Intersection BSeq in meet { S where S is Subset-Family of T : ( X c= S & S is compl-closed all-open-containing closed_for_countable_unions Subset-Family of T ) }

proof

let BSeq be SetSequence of the carrier of T; :: thesis: ( rng BSeq c= meet { S where S is Subset-Family of T : ( X c= S & S is compl-closed all-open-containing closed_for_countable_unions Subset-Family of T ) } implies Intersection BSeq in meet { S where S is Subset-Family of T : ( X c= S & S is compl-closed all-open-containing closed_for_countable_unions Subset-Family of T ) } )
assume A6: rng BSeq c= meet { S where S is Subset-Family of T : ( X c= S & S is compl-closed all-open-containing closed_for_countable_unions Subset-Family of T ) } ; :: thesis: Intersection BSeq in meet { S where S is Subset-Family of T : ( X c= S & S is compl-closed all-open-containing closed_for_countable_unions Subset-Family of T ) }

now :: thesis: for Z being set st Z in { S where S is Subset-Family of T : ( X c= S & S is compl-closed all-open-containing closed_for_countable_unions Subset-Family of T ) } holds
Intersection BSeq in Z

let Z be set ; :: thesis: ( Z in { S where S is Subset-Family of T : ( X c= S & S is compl-closed all-open-containing closed_for_countable_unions Subset-Family of T ) } implies Intersection BSeq in Z )
assume A7: Z in { S where S is Subset-Family of T : ( X c= S & S is compl-closed all-open-containing closed_for_countable_unions Subset-Family of T ) } ; :: thesis: Intersection BSeq in Z
then consider S being Subset-Family of T such that
A8: Z = S and
X c= S and
A9: S is compl-closed all-open-containing closed_for_countable_unions Subset-Family of T ;

now :: thesis: for n being Nat holds BSeq . n in Z

let n be Nat; :: thesis: BSeq . n in Z
BSeq . n in rng BSeq by NAT_1:51;
hence BSeq . n in Z by A6, A7, SETFAM_1:def 1; :: thesis: verum

end;

then A10: rng BSeq c= Z by NAT_1:52;
S is SigmaField of T by A9, Th13;
hence Intersection BSeq in Z by A8, A10, PROB_1:def 6; :: thesis: verum

end;

hence Intersection BSeq in meet { S where S is Subset-Family of T : ( X c= S & S is compl-closed all-open-containing closed_for_countable_unions Subset-Family of T ) } by A2, SETFAM_1:def 1; :: thesis: verum

end;

now :: thesis: for Z being set st Z in { S where S is Subset-Family of T : ( X c= S & S is compl-closed all-open-containing closed_for_countable_unions Subset-Family of T ) } holds
{} in Z

let Z be set ; :: thesis: ( Z in { S where S is Subset-Family of T : ( X c= S & S is compl-closed all-open-containing closed_for_countable_unions Subset-Family of T ) } implies {} in Z )
assume Z in { S where S is Subset-Family of T : ( X c= S & S is compl-closed all-open-containing closed_for_countable_unions Subset-Family of T ) } ; :: thesis: {} in Z
then ex S being Subset-Family of T st
( Z = S & X c= S & S is compl-closed all-open-containing closed_for_countable_unions Subset-Family of T ) ;
then Z is Field_Subset of the carrier of T by Th13;
hence {} in Z by PROB_1:4; :: thesis: verum

end;

then {} in meet { S where S is Subset-Family of T : ( X c= S & S is compl-closed all-open-containing closed_for_countable_unions Subset-Family of T ) } by A2, SETFAM_1:def 1;
then reconsider Y = meet { S where S is Subset-Family of T : ( X c= S & S is compl-closed all-open-containing closed_for_countable_unions Subset-Family of T ) } as SigmaField of T by A2, A5, A3, PROB_1:15, SETFAM_1:3;
for A being Subset of T st A is open holds
A in Y

proof

let A be Subset of T; :: thesis: ( A is open implies A in Y )
assume A11: A is open ; :: thesis: A in Y
for Y being set st Y in { S where S is Subset-Family of T : ( X c= S & S is compl-closed all-open-containing closed_for_countable_unions Subset-Family of T ) } holds
A in Y

proof

let Y be set ; :: thesis: ( Y in { S where S is Subset-Family of T : ( X c= S & S is compl-closed all-open-containing closed_for_countable_unions Subset-Family of T ) } implies A in Y )
assume Y in { S where S is Subset-Family of T : ( X c= S & S is compl-closed all-open-containing closed_for_countable_unions Subset-Family of T ) } ; :: thesis: A in Y
then ex S being Subset-Family of the carrier of T st
( Y = S & X c= S & S is compl-closed all-open-containing closed_for_countable_unions Subset-Family of T ) ;
hence A in Y by A11, Def2; :: thesis: verum

end;

hence A in Y by A2, SETFAM_1:def 1; :: thesis: verum

end;

then reconsider Y = Y as compl-closed all-open-containing closed_for_countable_unions Subset-Family of T by Def2;
take Y ; :: thesis: ( X c= Y & ( for Z being compl-closed all-open-containing closed_for_countable_unions Subset-Family of T st X c= Z holds
Y c= Z ) )
for Z being set st X c= Z & Z is compl-closed all-open-containing closed_for_countable_unions Subset-Family of T holds
Y c= Z

proof

let Z be set ; :: thesis: ( X c= Z & Z is compl-closed all-open-containing closed_for_countable_unions Subset-Family of T implies Y c= Z )
assume that
A12: X c= Z and
A13: Z is compl-closed all-open-containing closed_for_countable_unions Subset-Family of T ; :: thesis: Y c= Z
reconsider Z = Z as Subset-Family of T by A13;
Z in { S where S is Subset-Family of T : ( X c= S & S is compl-closed all-open-containing closed_for_countable_unions Subset-Family of T ) } by A12, A13;
hence Y c= Z by SETFAM_1:3; :: thesis: verum

end;

hence ( X c= Y & ( for Z being compl-closed all-open-containing closed_for_countable_unions Subset-Family of T st X c= Z holds
Y c= Z ) ) by A2, A1, SETFAM_1:5; :: thesis: verum