take U1 = U0; :: thesis: ( the Sorts of U1 is MSSubset of U0 & ( for B being MSSubset of U0 st B = the Sorts of U1 holds
( B is opers_closed & the Charact of U1 = Opers (U0,B) ) ) )
thus the Sorts of U1 is MSSubset of U0 by PBOOLE:def 18; :: thesis: for B being MSSubset of U0 st B = the Sorts of U1 holds
( B is opers_closed & the Charact of U1 = Opers (U0,B) )
let B be MSSubset of U0; :: thesis: ( B = the Sorts of U1 implies ( B is opers_closed & the Charact of U1 = Opers (U0,B) ) )
set f1 = the Charact of U1;
set f2 = Opers (U0,B);
assume A1: B = the Sorts of U1 ; :: thesis: ( B is opers_closed & the Charact of U1 = Opers (U0,B) )
hence B is opers_closed by Th3; :: thesis: the Charact of U1 = Opers (U0,B)
for x being object st x in the carrier' of S holds
the Charact of U1 . x = (Opers (U0,B)) . x

proof

let x be object ; :: thesis: ( x in the carrier' of S implies the Charact of U1 . x = (Opers (U0,B)) . x )
assume x in the carrier' of S ; :: thesis: the Charact of U1 . x = (Opers (U0,B)) . x
then reconsider o1 = x as OperSymbol of S ;
( the Charact of U1 . o1 = Den (o1,U1) & (Opers (U0,B)) . o1 = o1 /. B ) by Def8, MSUALG_1:def 6;
hence the Charact of U1 . x = (Opers (U0,B)) . x by A1, Th3; :: thesis: verum

end;

hence the Charact of U1 = Opers (U0,B) ; :: thesis: verum