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_inf_of Y,L ) & ( for x being Element of L st x in F holds
ex Y being finite Subset of X st
( ex_inf_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_inf_of Y,L ) & ( for x being Element of L st x in F holds
ex Y being finite Subset of X st
( ex_inf_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_inf_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_inf_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 8 :: thesis: ( not y in F or x <= y )
assume y in F ; :: thesis: x <= y
then consider Y being finite Subset of X such that
A5: ex_inf_of Y,L and
A6: y = "/\" (Y,L) by A2;
x is_<=_than Y by A4;
hence x <= y by A5, A6, YELLOW_0:def 10; :: thesis: verum
end;
assume A7: x is_<=_than F ; :: thesis: x is_<=_than X
let y be Element of L; :: according to LATTICE3:def 8 :: thesis: ( not y in X or x <= y )
assume y in X ; :: thesis: x <= y
then A8: {y} c= X by ZFMISC_1:31;
then A9: inf {y} in F by A3;
ex_inf_of {y},L by A1, A8;
then A10: {y} is_>=_than inf {y} by YELLOW_0:def 10;
A11: inf {y} >= x by A7, A9;
y >= inf {y} by A10, YELLOW_0:7;
hence x <= y by A11, ORDERS_2:3; :: thesis: verum