defpred S_{1}[ set ] means ex A1 being Subset of X st

( A1 = $1 & A1 is open & A1 is closed & x in $1 );

consider F being Subset-Family of X such that

A1: for A being Subset of X holds

( A in F iff S_{1}[A] )
from SUBSET_1:sch 3();

reconsider S = meet F as Subset of X ;

take S ; :: thesis: ex F being Subset-Family of X st

( ( for A being Subset of X holds

( A in F iff ( A is open & A is closed & x in A ) ) ) & meet F = S )

take F ; :: thesis: ( ( for A being Subset of X holds

( A in F iff ( A is open & A is closed & x in A ) ) ) & meet F = S )

thus for A being Subset of X holds

( A in F iff ( A is open & A is closed & x in A ) ) :: thesis: meet F = S

( A1 = $1 & A1 is open & A1 is closed & x in $1 );

consider F being Subset-Family of X such that

A1: for A being Subset of X holds

( A in F iff S

reconsider S = meet F as Subset of X ;

take S ; :: thesis: ex F being Subset-Family of X st

( ( for A being Subset of X holds

( A in F iff ( A is open & A is closed & x in A ) ) ) & meet F = S )

take F ; :: thesis: ( ( for A being Subset of X holds

( A in F iff ( A is open & A is closed & x in A ) ) ) & meet F = S )

thus for A being Subset of X holds

( A in F iff ( A is open & A is closed & x in A ) ) :: thesis: meet F = S

proof

thus
meet F = S
; :: thesis: verum
let A be Subset of X; :: thesis: ( A in F iff ( A is open & A is closed & x in A ) )

thus ( A in F implies ( A is open & A is closed & x in A ) ) :: thesis: ( A is open & A is closed & x in A implies A in F )

end;thus ( A in F implies ( A is open & A is closed & x in A ) ) :: thesis: ( A is open & A is closed & x in A implies A in F )

proof

thus
( A is open & A is closed & x in A implies A in F )
by A1; :: thesis: verum
assume
A in F
; :: thesis: ( A is open & A is closed & x in A )

then ex A1 being Subset of X st

( A1 = A & A1 is open & A1 is closed & x in A ) by A1;

hence ( A is open & A is closed & x in A ) ; :: thesis: verum

end;then ex A1 being Subset of X st

( A1 = A & A1 is open & A1 is closed & x in A ) by A1;

hence ( A is open & A is closed & x in A ) ; :: thesis: verum