let I be non empty set ; :: thesis: for M being ManySortedSet of I holds Top () = [|M,M|]
let M be ManySortedSet of I; :: thesis: Top () = [|M,M|]
set K = [|M,M|];
[|M,M|] is Equivalence_Relation of M by Th2;
then reconsider K = [|M,M|] as Element of () by MSUALG_5:def 5;
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 M by MSUALG_5:def 5;
A1: now :: thesis: for i being object st i in I holds
(K9 "\/" a9) . i = K9 . i
let i be object ; :: thesis: ( i in I implies (K9 "\/" a9) . i = K9 . i )
A2: ex K1 being ManySortedRelation of M st
( K1 = K9 (\/) a9 & K9 "\/" a9 = EqCl K1 ) by MSUALG_5:def 4;
assume A3: i in I ; :: thesis: (K9 "\/" a9) . i = K9 . i
then reconsider i9 = i as Element of I ;
reconsider K2 = K9 . i9, a2 = a9 . i9 as Equivalence_Relation of (M . i) by MSUALG_4:def 2;
(K9 (\/) a9) . i = (K9 . i) \/ (a9 . i) by
.= [:(M . i),(M . i):] \/ (a9 . i) by
.= (nabla (M . i)) \/ a2 by EQREL_1:def 1
.= nabla (M . i) by EQREL_1:1
.= [:(M . i),(M . i):] by EQREL_1:def 1
.= K9 . i by ;
hence (K9 "\/" a9) . i = EqCl K2 by
.= K9 . i by MSUALG_5:2 ;
:: thesis: verum
end;
thus K "\/" a = the L_join of () . (K,a) by LATTICES:def 1
.= K9 "\/" a9 by MSUALG_5:def 5
.= K by ; :: thesis: a "\/" K = K
hence a "\/" K = K ; :: thesis: verum
end;
hence Top () = [|M,M|] by LATTICES:def 17; :: thesis: verum