let S be non empty non void ManySortedSign ; :: thesis: for A being non-empty MSAlgebra over S holds Bottom () = id the Sorts of A
let A be non-empty MSAlgebra over S; :: thesis: Bottom () = id the Sorts of A
set K = id the Sorts of A;
id the Sorts of A is MSCongruence of A by Th17;
then reconsider K = id the Sorts of A as Element of () by Def6;
A1: the L_meet of () = the L_meet of (EqRelLatt the Sorts of A) || the carrier of () by NAT_LAT:def 12;
now :: thesis: for a being Element of () holds
( K "/\" a = K & a "/\" K = K )
let a be Element of (); :: thesis: ( K "/\" a = K & a "/\" K = K )
reconsider K9 = K, a9 = a as Equivalence_Relation of the Sorts of A by Def6;
A2: [K,a] in [: the carrier of (), the carrier of ():] by ZFMISC_1:87;
A3: now :: thesis: for i being object st i in the carrier of S holds
(K9 (/\) a9) . i = K9 . i
let i be object ; :: thesis: ( i in the carrier of S implies (K9 (/\) a9) . i = K9 . i )
assume A4: i in the carrier of S ; :: thesis: (K9 (/\) a9) . i = K9 . i
then reconsider i9 = i as Element of S ;
reconsider a2 = a9 . i9 as Equivalence_Relation of ( the Sorts of A . i) by MSUALG_4:def 2;
thus (K9 (/\) a9) . i = (K9 . i) /\ (a9 . i) by
.= (id ( the Sorts of A . i)) /\ a2 by MSUALG_3:def 1
.= id ( the Sorts of A . i) by EQREL_1:10
.= K9 . i by ; :: thesis: verum
end;
thus K "/\" a = the L_meet of () . (K,a) by LATTICES:def 2
.= the L_meet of (EqRelLatt the Sorts of A) . (K,a) by
.= K9 (/\) a9 by Def5
.= K by ; :: thesis: a "/\" K = K
hence a "/\" K = K ; :: thesis: verum
end;
hence Bottom () = id the Sorts of A by LATTICES:def 16; :: thesis: verum