let L be up-complete Semilattice; :: thesis: for D being non empty directed Subset of [:L,L:] holds "\/" ( { (sup X) where X is non empty directed Subset of L : ex x being Element of L st

( X = {x} "/\" (proj2 D) & x in proj1 D ) } ,L) <= "\/" ((union { X where X is non empty directed Subset of L : ex x being Element of L st

( X = {x} "/\" (proj2 D) & x in proj1 D ) } ),L)

let D be non empty directed Subset of [:L,L:]; :: thesis: "\/" ( { (sup X) where X is non empty directed Subset of L : ex x being Element of L st

( X = {x} "/\" (proj2 D) & x in proj1 D ) } ,L) <= "\/" ((union { X where X is non empty directed Subset of L : ex x being Element of L st

( X = {x} "/\" (proj2 D) & x in proj1 D ) } ),L)

defpred S_{1}[ set ] means ex x being Element of L st

( $1 = {x} "/\" (proj2 D) & x in proj1 D );

A1: "\/" ((union { X where X is non empty directed Subset of L : S_{1}[X] } ),L) is_>=_than { (sup X) where X is non empty directed Subset of L : S_{1}[X] }
_{1}[X] } ,L
by Th8;

hence "\/" ( { (sup X) where X is non empty directed Subset of L : ex x being Element of L st

( X = {x} "/\" (proj2 D) & x in proj1 D ) } ,L) <= "\/" ((union { X where X is non empty directed Subset of L : ex x being Element of L st

( X = {x} "/\" (proj2 D) & x in proj1 D ) } ),L) by A1, YELLOW_0:def 9; :: thesis: verum

( X = {x} "/\" (proj2 D) & x in proj1 D ) } ,L) <= "\/" ((union { X where X is non empty directed Subset of L : ex x being Element of L st

( X = {x} "/\" (proj2 D) & x in proj1 D ) } ),L)

let D be non empty directed Subset of [:L,L:]; :: thesis: "\/" ( { (sup X) where X is non empty directed Subset of L : ex x being Element of L st

( X = {x} "/\" (proj2 D) & x in proj1 D ) } ,L) <= "\/" ((union { X where X is non empty directed Subset of L : ex x being Element of L st

( X = {x} "/\" (proj2 D) & x in proj1 D ) } ),L)

defpred S

( $1 = {x} "/\" (proj2 D) & x in proj1 D );

A1: "\/" ((union { X where X is non empty directed Subset of L : S

proof

ex_sup_of { (sup X) where X is non empty directed Subset of L : S
let a be Element of L; :: according to LATTICE3:def 9 :: thesis: ( not a in { (sup X) where X is non empty directed Subset of L : S_{1}[X] } or a <= "\/" ((union { X where X is non empty directed Subset of L : S_{1}[X] } ),L) )

assume a in { (sup X) where X is non empty directed Subset of L : S_{1}[X] }
; :: thesis: a <= "\/" ((union { X where X is non empty directed Subset of L : S_{1}[X] } ),L)

then consider q being non empty directed Subset of L such that

A2: a = sup q and

A3: S_{1}[q]
;

A4: q in { X where X is non empty directed Subset of L : S_{1}[X] }
by A3;

( ex_sup_of q,L & ex_sup_of union { X where X is non empty directed Subset of L : S_{1}[X] } ,L )
by Th7, WAYBEL_0:75;

hence a <= "\/" ((union { X where X is non empty directed Subset of L : S_{1}[X] } ),L)
by A2, A4, YELLOW_0:34, ZFMISC_1:74; :: thesis: verum

end;assume a in { (sup X) where X is non empty directed Subset of L : S

then consider q being non empty directed Subset of L such that

A2: a = sup q and

A3: S

A4: q in { X where X is non empty directed Subset of L : S

( ex_sup_of q,L & ex_sup_of union { X where X is non empty directed Subset of L : S

hence a <= "\/" ((union { X where X is non empty directed Subset of L : S

hence "\/" ( { (sup X) where X is non empty directed Subset of L : ex x being Element of L st

( X = {x} "/\" (proj2 D) & x in proj1 D ) } ,L) <= "\/" ((union { X where X is non empty directed Subset of L : ex x being Element of L st

( X = {x} "/\" (proj2 D) & x in proj1 D ) } ),L) by A1, YELLOW_0:def 9; :: thesis: verum