let S be non empty non void ManySortedSign ; :: thesis: for A being non-empty MSAlgebra over S holds [| the Sorts of A, the Sorts of A|] is MSCongruence of A
let A be non-empty MSAlgebra over S; :: thesis: [| the Sorts of A, the Sorts of A|] is MSCongruence of A
set J = [| the Sorts of A, the Sorts of A|];
for i being set st i in the carrier of S holds
[| the Sorts of A, the Sorts of A|] . i is Relation of ( the Sorts of A . i)
proof
let i be set ; :: thesis: ( i in the carrier of S implies [| the Sorts of A, the Sorts of A|] . i is Relation of ( the Sorts of A . i) )
assume i in the carrier of S ; :: thesis: [| the Sorts of A, the Sorts of A|] . i is Relation of ( the Sorts of A . i)
then [| the Sorts of A, the Sorts of A|] . i c= [:( the Sorts of A . i),( the Sorts of A . i):] by PBOOLE:def 16;
hence [| the Sorts of A, the Sorts of A|] . i is Relation of ( the Sorts of A . i) ; :: thesis: verum
end;
then reconsider J = [| the Sorts of A, 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: 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)

let R be Relation of ( the Sorts of A . i); :: thesis: ( i in the carrier of S & J . i = R implies R is Equivalence_Relation of ( the Sorts of A . i) )
assume that
A1: i in the carrier of S and
A2: J . i = R ; :: thesis: R is Equivalence_Relation of ( the Sorts of A . i)
J . i = nabla ( the Sorts of A . i) by ;
hence R is Equivalence_Relation of ( the Sorts of A . i) by A2; :: thesis: verum
end;
then reconsider J = J as MSEquivalence_Relation-like ManySortedRelation of the Sorts of A by MSUALG_4:def 2;
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 . (() /. n) ) holds
[((Den (o,A)) . x),((Den (o,A)) . y)] in J .
proof
let o be OperSymbol of S; :: thesis: 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 . (() /. n) ) holds
[((Den (o,A)) . x),((Den (o,A)) . y)] in J .

let x, y be Element of Args (o,A); :: thesis: ( ( for n being Nat st n in dom x holds
[(x . n),(y . n)] in J . (() /. n) ) implies [((Den (o,A)) . x),((Den (o,A)) . y)] in J . )

assume for n being Nat st n in dom x holds
[(x . n),(y . n)] in J . (() /. n) ; :: thesis: [((Den (o,A)) . x),((Den (o,A)) . y)] in J .
o in the carrier' of S ;
then A3: o in dom the ResultSort of S by FUNCT_2:def 1;
(Den (o,A)) . y in Result (o,A) ;
then (Den (o,A)) . y in ( the Sorts of A * the ResultSort of S) . o by MSUALG_1:def 5;
then (Den (o,A)) . y in the Sorts of A . ( the ResultSort of S . o) by ;
then A4: (Den (o,A)) . y in the Sorts of A . by MSUALG_1:def 2;
(Den (o,A)) . x in Result (o,A) ;
then (Den (o,A)) . x in ( the Sorts of A * the ResultSort of S) . o by MSUALG_1:def 5;
then (Den (o,A)) . x in the Sorts of A . ( the ResultSort of S . o) by ;
then A5: (Den (o,A)) . x in the Sorts of A . by MSUALG_1:def 2;
J . = [:( the Sorts of A . ),( the Sorts of A . ):] by PBOOLE:def 16;
hence [((Den (o,A)) . x),((Den (o,A)) . y)] in J . by ; :: thesis: verum
end;
hence [| the Sorts of A, the Sorts of A|] is MSCongruence of A by MSUALG_4:def 4; :: thesis: verum