let S be non empty non void ManySortedSign ; :: thesis:
let A be non-empty MSAlgebra over S; :: thesis:
set J = id the Sorts of A;
for i being set st i in the carrier of S holds
(id the Sorts of A) . i is Relation of ( the Sorts of A . i)

proof

let i be set ; :: thesis:
assume i in the carrier of S ; :: thesis:
then (id the Sorts of A) . i = id ( the Sorts of A . i) by MSUALG_3:def 1;
hence (id the Sorts of A) . i is Relation of ( the Sorts of A . i) ; :: thesis:

end;

then reconsider J = id the Sorts of A as ManySortedRelation of the Sorts of A by MSUALG_4:def 1;
for i being set
for R being Relation of ( the Sorts of A . i) st i in the carrier of S & J . i = R holds
R is Equivalence_Relation of ( the Sorts of A . i)

proof

let i be set ; :: thesis:
let R be Relation of ( the Sorts of A . i); :: thesis:
assume that
A1: i in the carrier of S and
A2: J . i = R ; :: thesis:
J . i = id ( the Sorts of A . i) by A1, MSUALG_3:def 1;
hence R is Equivalence_Relation of ( the Sorts of A . i) by A2; :: thesis:

end;

then reconsider J = J as MSEquivalence_Relation-like ManySortedRelation of the Sorts of A by MSUALG_4:def 2;
reconsider J = J as ManySortedRelation of A ;
reconsider J = J as MSEquivalence-like ManySortedRelation of A by MSUALG_4:def 3;
for o being OperSymbol of S
for x, y being Element of Args (o,A) st ( for n being Nat st n in dom x holds
[(x . n),(y . n)] in J . ((the_arity_of o) /. n) ) holds
[((Den (o,A)) . x),((Den (o,A)) . y)] in J . (the_result_sort_of o)

proof

let o be OperSymbol of S; :: thesis:
let x, y be Element of Args (o,A); :: thesis:
assume A3: for n being Nat st n in dom x holds
[(x . n),(y . n)] in J . ((the_arity_of o) /. n) ; :: thesis:
A4: dom y = dom (the_arity_of o) by MSUALG_3:6
.= Seg (len (the_arity_of o)) by FINSEQ_1:def 3 ;
A5: dom x = dom (the_arity_of o) by MSUALG_3:6
.= Seg (len (the_arity_of o)) by FINSEQ_1:def 3 ;
then reconsider x9 = x, y9 = y as FinSequence by A4, FINSEQ_1:def 2;

now :: thesis:

let n be Nat; :: thesis:
assume A6: n in dom x ; :: thesis:
J . ((the_arity_of o) /. n) = id ( the Sorts of A . ((the_arity_of o) /. n)) by MSUALG_3:def 1;
then [(x . n),(y . n)] in id ( the Sorts of A . ((the_arity_of o) /. n)) by A3, A6;
hence x9 . n = y9 . n by RELAT_1:def 10; :: thesis:

end;

then A7: x = y by A5, A4, FINSEQ_1:13;
(Den (o,A)) . x in Result (o,A) ;
then A8: (Den (o,A)) . x in ( the Sorts of A * the ResultSort of S) . o by MSUALG_1:def 5;
o in the carrier' of S ;
then o in dom the ResultSort of S by FUNCT_2:def 1;
then (Den (o,A)) . x in the Sorts of A . ( the ResultSort of S . o) by A8, FUNCT_1:13;
then A9: (Den (o,A)) . x in the Sorts of A . (the_result_sort_of o) by MSUALG_1:def 2;
J . (the_result_sort_of o) = id ( the Sorts of A . (the_result_sort_of o)) by MSUALG_3:def 1;
hence [((Den (o,A)) . x),((Den (o,A)) . y)] in J . (the_result_sort_of o) by A7, A9, RELAT_1:def 10; :: thesis:

end;

hence id the Sorts of A is MSCongruence of A by MSUALG_4:def 4; :: thesis: