let L1, L2, T1, T2 be non empty antisymmetric RelStr ; for f being Function of L1,T1
for g being Function of L2,T2 st f is sups-preserving & g is sups-preserving holds
[:f,g:] is sups-preserving
let f be Function of L1,T1; for g being Function of L2,T2 st f is sups-preserving & g is sups-preserving holds
[:f,g:] is sups-preserving
let g be Function of L2,T2; ( f is sups-preserving & g is sups-preserving implies [:f,g:] is sups-preserving )
assume that
A1:
f is sups-preserving
and
A2:
g is sups-preserving
; [:f,g:] is sups-preserving
let X be Subset of [:L1,L2:]; WAYBEL_0:def 33 [:f,g:] preserves_sup_of X
A3:
f preserves_sup_of proj1 X
by A1;
A4:
g preserves_sup_of proj2 X
by A2;
set iX = [:f,g:] .: X;
A5:
( dom f = the carrier of L1 & dom g = the carrier of L2 )
by FUNCT_2:def 1;
assume A6:
ex_sup_of X,[:L1,L2:]
; WAYBEL_0:def 31 ( ex_sup_of [:f,g:] .: X,[:T1,T2:] & "\/" (([:f,g:] .: X),[:T1,T2:]) = [:f,g:] . ("\/" (X,[:L1,L2:])) )
then A7:
ex_sup_of proj1 X,L1
by YELLOW_3:41;
A8:
ex_sup_of proj2 X,L2
by A6, YELLOW_3:41;
X c= the carrier of [:L1,L2:]
;
then A9:
X c= [: the carrier of L1, the carrier of L2:]
by YELLOW_3:def 2;
then A10:
proj2 ([:f,g:] .: X) = g .: (proj2 X)
by A5, Th4;
then A11:
ex_sup_of proj2 ([:f,g:] .: X),T2
by A4, A8;
A12:
proj1 ([:f,g:] .: X) = f .: (proj1 X)
by A5, A9, Th4;
then
ex_sup_of proj1 ([:f,g:] .: X),T1
by A3, A7;
hence
ex_sup_of [:f,g:] .: X,[:T1,T2:]
by A11, YELLOW_3:41; "\/" (([:f,g:] .: X),[:T1,T2:]) = [:f,g:] . ("\/" (X,[:L1,L2:]))
hence sup ([:f,g:] .: X) =
[(sup (f .: (proj1 X))),(sup (g .: (proj2 X)))]
by A12, A10, Th8
.=
[(f . (sup (proj1 X))),(sup (g .: (proj2 X)))]
by A3, A7
.=
[(f . (sup (proj1 X))),(g . (sup (proj2 X)))]
by A4, A8
.=
[:f,g:] . ((sup (proj1 X)),(sup (proj2 X)))
by A5, FUNCT_3:def 8
.=
[:f,g:] . (sup X)
by A6, Th8
;
verum