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 ) & ex_sup_of X,L holds

sup F = sup X

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 ) & ex_sup_of X,L implies sup F = sup X )

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: ( not ex_sup_of X,L or sup F = sup X )

for x being Element of L holds

( x is_>=_than X iff x is_>=_than F ) by A1, A2, A3, Th52;

hence ( not ex_sup_of X,L or sup F = sup X ) by YELLOW_0:47; :: 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 ) & ex_sup_of X,L holds

sup F = sup X

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 ) & ex_sup_of X,L implies sup F = sup X )

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: ( not ex_sup_of X,L or sup F = sup X )

for x being Element of L holds

( x is_>=_than X iff x is_>=_than F ) by A1, A2, A3, Th52;

hence ( not ex_sup_of X,L or sup F = sup X ) by YELLOW_0:47; :: thesis: verum