let L be non empty transitive RelStr ; :: thesis: for k being Function of L,L st k is directed-sups-preserving holds
corestr k is directed-sups-preserving
let k be Function of L,L; :: thesis: ( k is directed-sups-preserving implies corestr k is directed-sups-preserving )
assume A1: k is directed-sups-preserving ; :: thesis: corestr k is directed-sups-preserving
let X be Subset of L; :: according to WAYBEL_0:def 37 :: thesis: ( X is empty or not X is directed or corestr k preserves_sup_of X )
assume ( not X is empty & X is directed ) ; :: thesis: corestr k preserves_sup_of X
then A2: k preserves_sup_of X by A1, WAYBEL_0:def 37;
set f = corestr k;
A3: k = corestr k by WAYBEL_1:32;
assume A4: ex_sup_of X,L ; :: according to WAYBEL_0:def 31 :: thesis: ( ex_sup_of (corestr k) .: X, Image k & "\/" ((corestr k) .: X),(Image k) = (corestr k) . ("\/" X,L) )
then A5: ex_sup_of k .: X,L by A2, WAYBEL_0:def 31;
reconsider fX = (corestr k) .: X as Subset of (Image k) ;
dom k = the carrier of L by FUNCT_2:def 1;
then ( rng k = the carrier of (Image k) & k . (sup X) in rng k ) by FUNCT_1:def 5, YELLOW_0:def 15;
then "\/" fX,L is Element of (Image k) by A2, A3, A4, WAYBEL_0:def 31;
hence ex_sup_of (corestr k) .: X, Image k by A3, A5, YELLOW_0:65; :: thesis: "\/" ((corestr k) .: X),(Image k) = (corestr k) . ("\/" X,L)
sup (k .: X) = k . (sup X) by A2, A4, WAYBEL_0:def 31;
hence "\/" ((corestr k) .: X),(Image k) = (corestr k) . ("\/" X,L) by A3, A5, YELLOW_0:65; :: thesis: verum