let L be non empty reflexive transitive RelStr ; :: thesis: for X, F being Subset of L st ( for Y being finite Subset of X st Y <> {} holds
ex_sup_of Y,L ) & ( for x being Element of L st x in F holds
ex Y being finite Subset of X st
( ex_sup_of Y,L & x = "\/" Y,L ) ) & ( for Y being finite Subset of X st Y <> {} holds
"\/" Y,L in F ) holds
for x being Element of L holds
( x is_>=_than X iff x is_>=_than F )
let X, F be Subset of L; :: thesis: ( ( for Y being finite Subset of X st Y <> {} holds
ex_sup_of Y,L ) & ( for x being Element of L st x in F holds
ex Y being finite Subset of X st
( ex_sup_of Y,L & x = "\/" Y,L ) ) & ( for Y being finite Subset of X st Y <> {} holds
"\/" Y,L in F ) implies for x being Element of L holds
( x is_>=_than X iff x is_>=_than F ) )
assume that
A1: for Y being finite Subset of X st Y <> {} holds
ex_sup_of Y,L and
A2: for x being Element of L st x in F holds
ex Y being finite Subset of X st
( ex_sup_of Y,L & x = "\/" Y,L ) and
A3: for Y being finite Subset of X st Y <> {} holds
"\/" Y,L in F ; :: thesis: for x being Element of L holds
( x is_>=_than X iff x is_>=_than F )
let x be Element of L; :: thesis: ( x is_>=_than X iff x is_>=_than F )
thus ( x is_>=_than X implies x is_>=_than F ) :: thesis: ( x is_>=_than F implies x is_>=_than X )

proof

assume A4: x is_>=_than X ; :: thesis: x is_>=_than F
let y be Element of L; :: according to LATTICE3:def 9 :: thesis: ( not y in F or y <= x )
assume y in F ; :: thesis: y <= x
then consider Y being finite Subset of X such that
A5: ( ex_sup_of Y,L & y = "\/" Y,L ) by A2;
x is_>=_than Y by A4, YELLOW_0:9;
hence y <= x by A5, YELLOW_0:def 9; :: thesis: verum

end;

assume A6: x is_>=_than F ; :: thesis: x is_>=_than X
let y be Element of L; :: according to LATTICE3:def 9 :: thesis: ( not y in X or y <= x )
assume y in X ; :: thesis: y <= x
then {y} c= X by ZFMISC_1:37;
then ( sup {y} in F & ex_sup_of {y},L ) by A1, A3;
then A7: ( {y} is_<=_than sup {y} & sup {y} <= x ) by A6, LATTICE3:def 9, YELLOW_0:def 9;
then y <= sup {y} by YELLOW_0:7;
hence y <= x by A7, ORDERS_2:26; :: thesis: verum