reserve x,y for object;
reserve S for non void non empty ManySortedSign,
  o for OperSymbol of S,
  U0,U1, U2 for MSAlgebra over S;

theorem Th4:
  for B being MSSubset of U0 st B=the Sorts of U0 holds Opers(U0,B)
  = the Charact of U0
proof
  let B be MSSubset of U0;
  set f1 = the Charact of U0, f2 = Opers(U0,B);
  assume
A1: B=the Sorts of U0;
  for x being object st x in (the carrier' of S) holds f1.x = f2.x
  proof
    let x be object;
    assume x in (the carrier' of S);
    then reconsider o1 = x as OperSymbol of S;
    f1.o1 = Den(o1,U0) & f2.o1 = o1/.B by Def8,MSUALG_1:def 6;
    hence thesis by A1,Th3;
  end;
  hence thesis;
end;
