ex c being Element of (EqRelLatt M) st
for a being Element of (EqRelLatt M) holds
( c "\/" a = c & a "\/" c = c )

proof

reconsider c9 = [|M,M|] as Equivalence_Relation of M by Th2;
reconsider c = c9 as Element of (EqRelLatt M) by MSUALG_5:def 5;
take c ; :: thesis: for a being Element of (EqRelLatt M) holds
( c "\/" a = c & a "\/" c = c )
let a be Element of (EqRelLatt M); :: thesis: ( c "\/" a = c & a "\/" c = c )
reconsider a9 = a as Equivalence_Relation of M by MSUALG_5:def 5;

A1: now :: thesis: for i being object st i in I holds
(c9 "\/" a9) . i = c9 . i

let i be object ; :: thesis: ( i in I implies (c9 "\/" a9) . i = c9 . i )
A2: ex K1 being ManySortedRelation of M st
( K1 = c9 (\/) a9 & c9 "\/" a9 = EqCl K1 ) by MSUALG_5:def 4;
assume A3: i in I ; :: thesis: (c9 "\/" a9) . i = c9 . i
then reconsider i9 = i as Element of I ;
reconsider K2 = c9 . i9, a2 = a9 . i9 as Equivalence_Relation of (M . i) by MSUALG_4:def 2;
(c9 (\/) a9) . i = (c9 . i) \/ (a9 . i) by A3, PBOOLE:def 4
.= [:(M . i),(M . i):] \/ (a9 . i) by A3, PBOOLE:def 16
.= (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
.= c9 . i by A3, PBOOLE:def 16 ;
hence (c9 "\/" a9) . i = EqCl K2 by A2, MSUALG_5:def 3
.= c9 . i by MSUALG_5:2 ;
:: thesis: verum

end;

thus c "\/" a = the L_join of (EqRelLatt M) . (c,a) by LATTICES:def 1
.= c9 "\/" a9 by MSUALG_5:def 5
.= c by A1, PBOOLE:3 ; :: thesis: a "\/" c = c
hence a "\/" c = c ; :: thesis: verum

end;

then A4: EqRelLatt M is upper-bounded by LATTICES:def 14;
ex c being Element of (EqRelLatt M) st
for a being Element of (EqRelLatt M) holds
( c "/\" a = c & a "/\" c = c )

proof

reconsider c9 = id M as Equivalence_Relation of M by Th1;
reconsider c = c9 as Element of (EqRelLatt M) by MSUALG_5:def 5;
take c ; :: thesis: for a being Element of (EqRelLatt M) holds
( c "/\" a = c & a "/\" c = c )
let a be Element of (EqRelLatt M); :: thesis: ( c "/\" a = c & a "/\" c = c )
reconsider a9 = a as Equivalence_Relation of M by MSUALG_5:def 5;

A5: now :: thesis: for i being object st i in I holds
(c9 (/\) a9) . i = c9 . i

let i be object ; :: thesis: ( i in I implies (c9 (/\) a9) . i = c9 . i )
assume A6: i in I ; :: thesis: (c9 (/\) a9) . i = c9 . i
then reconsider i9 = i as Element of I ;
reconsider a2 = a9 . i9 as Equivalence_Relation of (M . i) by MSUALG_4:def 2;
thus (c9 (/\) a9) . i = (c9 . i) /\ (a9 . i) by A6, PBOOLE:def 5
.= (id (M . i)) /\ a2 by MSUALG_3:def 1
.= id (M . i) by EQREL_1:10
.= c9 . i by A6, MSUALG_3:def 1 ; :: thesis: verum

end;

thus c "/\" a = the L_meet of (EqRelLatt M) . (c,a) by LATTICES:def 2
.= c9 (/\) a9 by MSUALG_5:def 5
.= c by A5, PBOOLE:3 ; :: thesis: a "/\" c = c
hence a "/\" c = c ; :: thesis: verum

end;

then EqRelLatt M is lower-bounded by LATTICES:def 13;
hence EqRelLatt M is bounded by A4; :: thesis: verum