set Y = the carrier' of S;
defpred S₁[ set , set ] means for o being OperSymbol of S st $1 = o holds
$2 = DenOp o,X;
A1: for x being set st x in the carrier' of S holds
ex y being set st S₁[x,y]

proof

let x be set ; :: thesis: ( x in the carrier' of S implies ex y being set st S₁[x,y] )
assume x in the carrier' of S ; :: thesis: ex y being set st S₁[x,y]
then reconsider o = x as OperSymbol of S ;
take DenOp o,X ; :: thesis: S₁[x, DenOp o,X]
thus S₁[x, DenOp o,X] ; :: thesis: verum

end;

consider f being Function such that
A2: ( dom f = the carrier' of S & ( for x being set st x in the carrier' of S holds
S₁[x,f . x] ) ) from CLASSES1:sch 1(A1);
reconsider f = f as ManySortedSet of by A2, PARTFUN1:def 4, RELAT_1:def 18;
for x being set st x in dom f holds
f . x is Function

proof

let x be set ; :: thesis: ( x in dom f implies f . x is Function )
assume x in dom f ; :: thesis: f . x is Function
then reconsider o = x as OperSymbol of S by A2;
f . o = DenOp o,X by A2;
hence f . x is Function ; :: thesis: verum

end;

then reconsider f = f as ManySortedFunction of by FUNCOP_1:def 6;
for x being set st x in the carrier' of S holds
f . x is Function of ((((FreeSort X) # ) * the Arity of S) . x),(((FreeSort X) * the ResultSort of S) . x)

proof

let x be set ; :: thesis: ( x in the carrier' of S implies f . x is Function of ((((FreeSort X) # ) * the Arity of S) . x),(((FreeSort X) * the ResultSort of S) . x) )
assume x in the carrier' of S ; :: thesis: f . x is Function of ((((FreeSort X) # ) * the Arity of S) . x),(((FreeSort X) * the ResultSort of S) . x)
then reconsider o = x as OperSymbol of S ;
f . o = DenOp o,X by A2;
hence f . x is Function of ((((FreeSort X) # ) * the Arity of S) . x),(((FreeSort X) * the ResultSort of S) . x) ; :: thesis: verum

end;

then reconsider f = f as ManySortedFunction of ((FreeSort X) # ) * the Arity of S,(FreeSort X) * the ResultSort of S by PBOOLE:def 18;
take f ; :: thesis: for o being OperSymbol of S holds f . o = DenOp o,X
let o be OperSymbol of S; :: thesis: f . o = DenOp o,X
thus f . o = DenOp o,X by A2; :: thesis: verum