defpred S₁[ set ] means ex msf being ManySortedFunction of U1,U1 st
( $1 = msf & msf is_isomorphism U1,U1 );
set T = the Sorts of U1;
consider X being set such that
A1: for x being set holds
( x in X iff ( x in product (MSFuncs ( the Sorts of U1, the Sorts of U1)) & S₁[x] ) ) from XBOOLE_0:sch 1();
( id the Sorts of U1 in product (MSFuncs ( the Sorts of U1, the Sorts of U1)) & ex F being ManySortedFunction of U1,U1 st
( id the Sorts of U1 = F & F is_isomorphism U1,U1 ) ) by Th22, MSUALG_3:16;
then reconsider X = X as non empty set by A1;
X is MSFunctionSet of the Sorts of U1, the Sorts of U1

proof

thus the Sorts of U1 is_transformable_to the Sorts of U1 ; :: according to AUTALG_1:def 4 :: thesis: for x being set st x in X holds
x is ManySortedFunction of the Sorts of U1, the Sorts of U1
let q be set ; :: thesis: ( q in X implies q is ManySortedFunction of the Sorts of U1, the Sorts of U1 )
assume q in X ; :: thesis: q is ManySortedFunction of the Sorts of U1, the Sorts of U1
then ex msf being ManySortedFunction of U1,U1 st
( q = msf & msf is_isomorphism U1,U1 ) by A1;
hence q is ManySortedFunction of the Sorts of U1, the Sorts of U1 ; :: thesis: verum

end;

then reconsider X = X as MSFunctionSet of the Sorts of U1, the Sorts of U1 ;
take X ; :: thesis: for h being ManySortedFunction of U1,U1 holds
( h in X iff h is_isomorphism U1,U1 )
let h be ManySortedFunction of U1,U1; :: thesis: ( h in X iff h is_isomorphism U1,U1 )

hereby :: thesis: ( h is_isomorphism U1,U1 implies h in X )

assume h in X ; :: thesis: h is_isomorphism U1,U1
then ex msf being ManySortedFunction of U1,U1 st
( h = msf & msf is_isomorphism U1,U1 ) by A1;
hence h is_isomorphism U1,U1 ; :: thesis: verum

end;

assume A2: h is_isomorphism U1,U1 ; :: thesis: h in X
h in product (MSFuncs ( the Sorts of U1, the Sorts of U1)) by Th22;
hence h in X by A1, A2; :: thesis: verum