let L1, L2, T1, T2 be non empty reflexive antisymmetric RelStr ; :: thesis: for f being Function of L1,T1
for g being Function of L2,T2 st f is directed-sups-preserving & g is directed-sups-preserving holds
[:f,g:] is directed-sups-preserving
let f be Function of L1,T1; :: thesis: for g being Function of L2,T2 st f is directed-sups-preserving & g is directed-sups-preserving holds
[:f,g:] is directed-sups-preserving
let g be Function of L2,T2; :: thesis: ( f is directed-sups-preserving & g is directed-sups-preserving implies [:f,g:] is directed-sups-preserving )
assume A1: ( f is directed-sups-preserving & g is directed-sups-preserving ) ; :: thesis: [:f,g:] is directed-sups-preserving
let X be Subset of [:L1,L2:]; :: according to WAYBEL_0:def 37 :: thesis: ( X is empty or not X is directed or [:f,g:] preserves_sup_of X )
assume ( not X is empty & X is directed ) ; :: thesis: [:f,g:] preserves_sup_of X
then ( not proj1 X is empty & proj1 X is directed & not proj2 X is empty & proj2 X is directed ) by YELLOW_3:21, YELLOW_3:22;
then A2: ( f preserves_sup_of proj1 X & g preserves_sup_of proj2 X ) by A1, WAYBEL_0:def 37;
assume A3: ex_sup_of X,[:L1,L2:] ; :: according to WAYBEL_0:def 31 :: thesis: ( ex_sup_of [:f,g:] .: X,[:T1,T2:] & "\/" ([:f,g:] .: X),[:T1,T2:] = [:f,g:] . ("\/" X,[:L1,L2:]) )
then A4: ( ex_sup_of proj1 X,L1 & ex_sup_of proj2 X,L2 ) by YELLOW_3:41;
set iX = [:f,g:] .: X;
A5: ( dom f = the carrier of L1 & dom g = the carrier of L2 ) by FUNCT_2:def 1;
X c= the carrier of [:L1,L2:] ;
then X c= [:the carrier of L1,the carrier of L2:] by YELLOW_3:def 2;
then A6: ( proj1 ([:f,g:] .: X) = f .: (proj1 X) & proj2 ([:f,g:] .: X) = g .: (proj2 X) ) by A5, Th4;
then ( ex_sup_of proj1 ([:f,g:] .: X),T1 & ex_sup_of proj2 ([:f,g:] .: X),T2 ) by A2, A4, WAYBEL_0:def 31;
hence ex_sup_of [:f,g:] .: X,[:T1,T2:] by YELLOW_3:41; :: thesis: "\/" ([:f,g:] .: X),[:T1,T2:] = [:f,g:] . ("\/" X,[:L1,L2:])
hence sup ([:f,g:] .: X) = [(sup (f .: (proj1 X))),(sup (g .: (proj2 X)))] by A6, Th8
.= [(f . (sup (proj1 X))),(sup (g .: (proj2 X)))] by A2, A4, WAYBEL_0:def 31
.= [(f . (sup (proj1 X))),(g . (sup (proj2 X)))] by A2, A4, WAYBEL_0:def 31
.= [:f,g:] . (sup (proj1 X)),(sup (proj2 X)) by A5, FUNCT_3:def 9
.= [:f,g:] . (sup X) by A3, Th8 ;
:: thesis: verum