defpred S1[ object ] means ex msf being ManySortedFunction of U1,U1 st
( \$1 = msf & msf is_homomorphism U1,U1 );
consider X being set such that
A1: for x being object holds
( x in X iff ( x in MSFuncs ( the Sorts of U1, the Sorts of U1) & S1[x] ) ) from ( id the Sorts of U1 in 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_homomorphism U1,U1 ) ) by ;
then reconsider X = X as non empty set by A1;
X c= MSFuncs ( the Sorts of U1, the Sorts of U1) by A1;
then reconsider X = X as MSFunctionSet of U1,U1 ;
take X ; :: thesis: ( ( for f being Element of X holds f is ManySortedFunction of U1,U1 ) & ( for h being ManySortedFunction of U1,U1 holds
( h in X iff h is_homomorphism U1,U1 ) ) )

thus for f being Element of X holds f is ManySortedFunction of U1,U1 ; :: thesis: for h being ManySortedFunction of U1,U1 holds
( h in X iff h is_homomorphism U1,U1 )

let h be ManySortedFunction of U1,U1; :: thesis: ( h in X iff h is_homomorphism U1,U1 )
hereby :: thesis: ( h is_homomorphism U1,U1 implies h in X )
assume h in X ; :: thesis: h is_homomorphism U1,U1
then ex msf being ManySortedFunction of U1,U1 st
( h = msf & msf is_homomorphism U1,U1 ) by A1;
hence h is_homomorphism U1,U1 ; :: thesis: verum
end;
h in MSFuncs ( the Sorts of U1, the Sorts of U1) by AUTALG_1:20;
hence ( h is_homomorphism U1,U1 implies h in X ) by A1; :: thesis: verum