let L be non empty 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
F is directed
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 F is directed )
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: F is directed
let x, y be Element of L; :: according to WAYBEL_0:def 1 :: thesis: ( x in F & y in F implies ex z being Element of L st
( z in F & x <= z & y <= z ) )
assume A4: x in F ; :: thesis: ( not y in F or ex z being Element of L st
( z in F & x <= z & y <= z ) )
then consider Y1 being finite Subset of X such that
A5: ex_sup_of Y1,L and
A6: x = "\/" Y1,L by A2;
assume y in F ; :: thesis: ex z being Element of L st
( z in F & x <= z & y <= z )
then consider Y2 being finite Subset of X such that
A7: ex_sup_of Y2,L and
A8: y = "\/" Y2,L by A2;
take z = "\/" (Y1 \/ Y2),L; :: thesis: ( z in F & x <= z & y <= z )
A9: ( ( Y1 = {} & Y2 = {} & {} \/ {} = {} ) or Y1 \/ Y2 <> {} ) ;
hence z in F by A3, A4, A6; :: thesis: ( x <= z & y <= z )
ex_sup_of Y1 \/ Y2,L by A1, A5, A9;
hence ( x <= z & y <= z ) by A5, A6, A7, A8, XBOOLE_1:7, YELLOW_0:34; :: thesis: verum