deffunc H₁( Element of S) -> Function of ( the Sorts of U1 . $1),(OSClass (R,$1)) = OSNat_Hom (U1,R,$1);
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 the carrier of S by A1, PARTFUN1:def 2, RELAT_1:def 18;
for x being object st x in dom f holds
f . x is Function

proof

let x be object ; :: thesis: ( x in dom f implies f . x is Function )
assume x in dom f ; :: thesis: f . x is Function
then reconsider y = x as Element of S ;
f . y = OSNat_Hom (U1,R,y) by A1;
hence f . x is Function ; :: thesis: verum

end;

then reconsider f = f as ManySortedFunction of the carrier of S by FUNCOP_1:def 6;
for i being object st i in the carrier of S holds
f . i is Function of ( the Sorts of U1 . i),((OSClass R) . i)

proof

let i be object ; :: thesis: ( i in the carrier of S implies f . i is Function of ( the Sorts of U1 . i),((OSClass R) . i) )
assume i in the carrier of S ; :: thesis: f . i is Function of ( the Sorts of U1 . i),((OSClass R) . i)
then reconsider s = i as Element of S ;
( OSClass (R,s) = (OSClass R) . i & f . s = OSNat_Hom (U1,R,s) ) by A1, Def11;
hence f . i is Function of ( the Sorts of U1 . i),((OSClass R) . i) ; :: thesis: verum

end;

then reconsider f = f as ManySortedFunction of the Sorts of U1, OSClass R by PBOOLE:def 15;
reconsider f = f as ManySortedFunction of U1,(QuotOSAlg (U1,R)) ;
take f ; :: thesis: for s being Element of S holds f . s = OSNat_Hom (U1,R,s)
thus for s being Element of S holds f . s = OSNat_Hom (U1,R,s) by A1; :: thesis: verum