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 ) & ex_inf_of X,L holds
inf F = inf X
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 ) & ex_inf_of X,L implies inf F = inf X )
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: ( not ex_inf_of X,L or inf F = inf X )
for x being Element of L holds
( x is_<=_than X iff x is_<=_than F ) by A1, A2, A3, Th57;
hence ( not ex_inf_of X,L or inf F = inf X ) by YELLOW_0:49; :: thesis: verum