let L be with_infima Poset; :: thesis: for X being Subset of L holds

( X c= uparrow (fininfs X) & ( for F being Filter of L st X c= F holds

uparrow (fininfs X) c= F ) )

let X be Subset of L; :: thesis: ( X c= uparrow (fininfs X) & ( for F being Filter of L st X c= F holds

uparrow (fininfs X) c= F ) )

A1: X c= fininfs X by Th50;

fininfs X c= uparrow (fininfs X) by Th16;

hence X c= uparrow (fininfs X) by A1; :: thesis: for F being Filter of L st X c= F holds

uparrow (fininfs X) c= F

let I be Filter of L; :: thesis: ( X c= I implies uparrow (fininfs X) c= I )

assume A2: X c= I ; :: thesis: uparrow (fininfs X) c= I

let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in uparrow (fininfs X) or x in I )

assume A3: x in uparrow (fininfs X) ; :: thesis: x in I

then reconsider x = x as Element of L ;

consider y being Element of L such that

A4: x >= y and

A5: y in fininfs X by A3, Def16;

consider Y being finite Subset of X such that

A6: y = "/\" (Y,L) and

A7: ex_inf_of Y,L by A5;

set i = the Element of I;

reconsider i = the Element of I as Element of L ;

A8: ex_inf_of {i},L by YELLOW_0:38;

A9: inf {i} = i by YELLOW_0:39;

then y in I by A6, A7, A10, Th43;

hence x in I by A4, Def20; :: thesis: verum

( X c= uparrow (fininfs X) & ( for F being Filter of L st X c= F holds

uparrow (fininfs X) c= F ) )

let X be Subset of L; :: thesis: ( X c= uparrow (fininfs X) & ( for F being Filter of L st X c= F holds

uparrow (fininfs X) c= F ) )

A1: X c= fininfs X by Th50;

fininfs X c= uparrow (fininfs X) by Th16;

hence X c= uparrow (fininfs X) by A1; :: thesis: for F being Filter of L st X c= F holds

uparrow (fininfs X) c= F

let I be Filter of L; :: thesis: ( X c= I implies uparrow (fininfs X) c= I )

assume A2: X c= I ; :: thesis: uparrow (fininfs X) c= I

let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in uparrow (fininfs X) or x in I )

assume A3: x in uparrow (fininfs X) ; :: thesis: x in I

then reconsider x = x as Element of L ;

consider y being Element of L such that

A4: x >= y and

A5: y in fininfs X by A3, Def16;

consider Y being finite Subset of X such that

A6: y = "/\" (Y,L) and

A7: ex_inf_of Y,L by A5;

set i = the Element of I;

reconsider i = the Element of I as Element of L ;

A8: ex_inf_of {i},L by YELLOW_0:38;

A9: inf {i} = i by YELLOW_0:39;

A10: now :: thesis: ( ex_inf_of {} ,L implies "/\" ({},L) in I )

Y c= I
by A2;assume
ex_inf_of {} ,L
; :: thesis: "/\" ({},L) in I

then "/\" ({},L) >= inf {i} by A8, XBOOLE_1:2, YELLOW_0:35;

hence "/\" ({},L) in I by A9, Def20; :: thesis: verum

end;then "/\" ({},L) >= inf {i} by A8, XBOOLE_1:2, YELLOW_0:35;

hence "/\" ({},L) in I by A9, Def20; :: thesis: verum

then y in I by A6, A7, A10, Th43;

hence x in I by A4, Def20; :: thesis: verum