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 )

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 A8: {y} c= X by ZFMISC_1:31;

then A9: sup {y} in F by A3;

ex_sup_of {y},L by A1, A8;

then A10: {y} is_<=_than sup {y} by YELLOW_0:def 9;

A11: sup {y} <= x by A7, A9;

y <= sup {y} by A10, YELLOW_0:7;

hence y <= x by A11, ORDERS_2:3; :: thesis: verum

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 A7:
x is_>=_than F
; :: thesis: x is_>=_than X
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 and

A6: y = "\/" (Y,L) by A2;

x is_>=_than Y by A4;

hence y <= x by A5, A6, YELLOW_0:def 9; :: thesis: verum

end;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 and

A6: y = "\/" (Y,L) by A2;

x is_>=_than Y by A4;

hence y <= x by A5, A6, YELLOW_0:def 9; :: thesis: verum

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 A8: {y} c= X by ZFMISC_1:31;

then A9: sup {y} in F by A3;

ex_sup_of {y},L by A1, A8;

then A10: {y} is_<=_than sup {y} by YELLOW_0:def 9;

A11: sup {y} <= x by A7, A9;

y <= sup {y} by A10, YELLOW_0:7;

hence y <= x by A11, ORDERS_2:3; :: thesis: verum