let L1, L2, L3 be non empty reflexive antisymmetric RelStr ; :: thesis: for f being Function of L1,L2
for g being Function of L2,L3 st f is filtered-infs-preserving & g is filtered-infs-preserving holds
g * f is filtered-infs-preserving
let f be Function of L1,L2; :: thesis: for g being Function of L2,L3 st f is filtered-infs-preserving & g is filtered-infs-preserving holds
g * f is filtered-infs-preserving
let g be Function of L2,L3; :: thesis: ( f is filtered-infs-preserving & g is filtered-infs-preserving implies g * f is filtered-infs-preserving )
assume that
A1: f is filtered-infs-preserving and
A2: g is filtered-infs-preserving ; :: thesis: g * f is filtered-infs-preserving
set gf = g * f;
let X be Subset of L1; :: according to WAYBEL_0:def 36 :: thesis: ( X is empty or not X is filtered or g * f preserves_inf_of X )
assume that
A3: ( not X is empty & X is filtered ) and
A4: ex_inf_of X,L1 ; :: according to WAYBEL_0:def 30 :: thesis: ( ex_inf_of (g * f) .: X,L3 & "/\" ((g * f) .: X),L3 = (g * f) . ("/\" X,L1) )
A5: dom f = the carrier of L1 by FUNCT_2:def 1;
set gfX = (g * f) .: X;
set fX = f .: X;
A6: f preserves_inf_of X by A1, A3, WAYBEL_0:def 36;
consider xx being Element of X;
( xx in X & X c= the carrier of L1 ) by A3;
then f . xx in f .: X by FUNCT_2:43;
then ( not f .: X is empty & f .: X is filtered ) by A1, A3, Th24, Th25;
then A7: g preserves_inf_of f .: X by A2, WAYBEL_0:def 36;
A8: (g * f) .: X = g .: (f .: X) by RELAT_1:159;
A9: ex_inf_of f .: X,L2 by A4, A6, WAYBEL_0:def 30;
hence ex_inf_of (g * f) .: X,L3 by A7, A8, WAYBEL_0:def 30; :: thesis: "/\" ((g * f) .: X),L3 = (g * f) . ("/\" X,L1)
thus inf ((g * f) .: X) = g . (inf (f .: X)) by A7, A8, A9, WAYBEL_0:def 30
.= g . (f . (inf X)) by A4, A6, WAYBEL_0:def 30
.= (g * f) . (inf X) by A5, FUNCT_1:23 ; :: thesis: verum