deffunc H₁( Element of S) -> Element of K6(K7((the Sorts of U1 . S),(the Sorts of U1 . S))) = id (the Sorts of U1 . S);
consider f being Function such that
A1: ( dom f = the carrier of S & ( for d being Element of S holds f . d = H₁(d) ) ) from FUNCT_1:sch 4();
reconsider f = f as ManySortedSet of by A1, PARTFUN1:def 4, RELAT_1:def 18;
for x being set st x in dom f holds
f . x is Relation

let x be set ; :: thesis:
assume x in dom f ; :: thesis:
then f . x = id (the Sorts of U1 . x) by A1;
hence f . x is Relation ; :: thesis:

end;

then reconsider f = f as ManySortedRelation of by Def1;
for i being set st i in the carrier of S holds
f . i is Relation of (the Sorts of U1 . i),(the Sorts of U1 . i)

let i be set ; :: thesis:
assume i in the carrier of S ; :: thesis:
then f . i = id (the Sorts of U1 . i) by A1;
hence f . i is Relation of (the Sorts of U1 . i),(the Sorts of U1 . i) ; :: thesis:

end;

then reconsider f = f as ManySortedRelation of the Sorts of U1,the Sorts of U1 by Def2;
reconsider f = f as ManySortedRelation of U1 ;
for i being set
for R being Relation of (the Sorts of U1 . i) st i in the carrier of S & f . i = R holds
R is Equivalence_Relation of (the Sorts of U1 . i)

let i be set ; :: thesis:
let R be Relation of (the Sorts of U1 . i); :: thesis:
assume ( i in the carrier of S & f . i = R ) ; :: thesis:
then R = id (the Sorts of U1 . i) by A1;
hence R is Equivalence_Relation of (the Sorts of U1 . i) ; :: thesis:

end;

then f is MSEquivalence_Relation-like by Def3;
then reconsider f = f as MSEquivalence-like ManySortedRelation of U1 by Def5;
take f ; :: thesis:
for o being OperSymbol of S
for x, y being Element of Args o,U1 st ( for n being Nat st n in dom x holds
[(x . n),(y . n)] in f . ((the_arity_of o) /. n) ) holds
[((Den o,U1) . x),((Den o,U1) . y)] in f . (the_result_sort_of o)

let o be OperSymbol of S; :: thesis:
let x, y be Element of Args o,U1; :: thesis:
assume A2: for n being Nat st n in dom x holds
[(x . n),(y . n)] in f . ((the_arity_of o) /. n) ; :: thesis:
A3: ( dom x = dom (the_arity_of o) & dom y = dom (the_arity_of o) ) by MSUALG_3:6;
for a being set st a in dom (the_arity_of o) holds
x . a = y . a

let a be set ; :: thesis:
assume A4: a in dom (the_arity_of o) ; :: thesis:
then reconsider n = a as Nat by ORDINAL1:def 13;
A5: (the_arity_of o) /. n = (the_arity_of o) . n by A4, PARTFUN1:def 8;
set ao = the_arity_of o;
(the_arity_of o) . n in rng (the_arity_of o) by A4, FUNCT_1:def 5;
then A6: f . ((the_arity_of o) . n) = id (the Sorts of U1 . ((the_arity_of o) . n)) by A1;
[(x . n),(y . n)] in f . ((the_arity_of o) . n) by A2, A3, A4, A5;
hence x . a = y . a by A6, RELAT_1:def 10; :: thesis:

end;

then A7: (Den o,U1) . x = (Den o,U1) . y by A3, FUNCT_1:9;
set r = the_result_sort_of o;
A8: f . (the_result_sort_of o) = id (the Sorts of U1 . (the_result_sort_of o)) by A1;
A10: ( dom the ResultSort of S = the carrier' of S & rng the ResultSort of S c= the carrier of S ) by FUNCT_2:def 1;
then A11: dom (the Sorts of U1 * the ResultSort of S) = dom the ResultSort of S by PARTFUN1:def 4;
Result o,U1 = (the Sorts of U1 * the ResultSort of S) . o by MSUALG_1:def 10
.= the Sorts of U1 . (the ResultSort of S . o) by A10, A11, FUNCT_1:22
.= the Sorts of U1 . (the_result_sort_of o) by MSUALG_1:def 7 ;
hence [((Den o,U1) . x),((Den o,U1) . y)] in f . (the_result_sort_of o) by A7, A8, RELAT_1:def 10; :: thesis:

end;

hence f is MSCongruence-like by Def6; :: thesis: