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 filtered-infs-preserving & g is filtered-infs-preserving holds
[:f,g:] is filtered-infs-preserving
let f be Function of L1,T1; :: thesis: for g being Function of L2,T2 st f is filtered-infs-preserving & g is filtered-infs-preserving holds
[:f,g:] is filtered-infs-preserving
let g be Function of L2,T2; :: thesis: ( f is filtered-infs-preserving & g is filtered-infs-preserving implies [:f,g:] is filtered-infs-preserving )
assume that
A1: f is filtered-infs-preserving and
A2: g is filtered-infs-preserving ; :: thesis: [:f,g:] is filtered-infs-preserving
let X be Subset of [:L1,L2:]; :: according to WAYBEL_0:def 36 :: thesis: ( X is empty or not X is filtered or [:f,g:] preserves_inf_of X )
assume A3: ( not X is empty & X is filtered ) ; :: thesis: [:f,g:] preserves_inf_of X
then ( not proj1 X is empty & proj1 X is filtered ) by YELLOW_3:21, YELLOW_3:24;
then A4: f preserves_inf_of proj1 X by A1;
( not proj2 X is empty & proj2 X is filtered ) by A3, YELLOW_3:21, YELLOW_3:24;
then A5: g preserves_inf_of proj2 X by A2;
set iX = [:f,g:] .: X;
A6: ( dom f = the carrier of L1 & dom g = the carrier of L2 ) by FUNCT_2:def 1;
assume A7: ex_inf_of X,[:L1,L2:] ; :: according to WAYBEL_0:def 30 :: thesis: ( ex_inf_of [:f,g:] .: X,[:T1,T2:] & "/\" (([:f,g:] .: X),[:T1,T2:]) = [:f,g:] . ("/\" (X,[:L1,L2:])) )
then A8: ex_inf_of proj1 X,L1 by YELLOW_3:42;
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 A6, Th4;
A11: ex_inf_of proj2 X,L2 by A7, YELLOW_3:42;
then A12: ex_inf_of proj2 ([:f,g:] .: X),T2 by A5, A10;
A13: proj1 ([:f,g:] .: X) = f .: (proj1 X) by A6, A9, Th4;
then ex_inf_of proj1 ([:f,g:] .: X),T1 by A4, A8;
hence ex_inf_of [:f,g:] .: X,[:T1,T2:] by A12, YELLOW_3:42; :: thesis: "/\" (([:f,g:] .: X),[:T1,T2:]) = [:f,g:] . ("/\" (X,[:L1,L2:]))
hence inf ([:f,g:] .: X) = [(inf (f .: (proj1 X))),(inf (g .: (proj2 X)))] by A13, A10, Th7
.= [(f . (inf (proj1 X))),(inf (g .: (proj2 X)))] by A4, A8
.= [(f . (inf (proj1 X))),(g . (inf (proj2 X)))] by A5, A11
.= [:f,g:] . ((inf (proj1 X)),(inf (proj2 X))) by A6, FUNCT_3:def 8
.= [:f,g:] . (inf X) by A7, Th7 ;
:: thesis: verum