reserve a for set,
  i for Nat;

theorem Th10:
  for U1,U2 being Universal_Algebra st U1 is SubAlgebra of U2 for
  B being MSSubset of MSAlg U2 st B = the Sorts of MSAlg U1 holds B is
  opers_closed
proof
  let U1,U2 be Universal_Algebra such that
A1: U1 is SubAlgebra of U2;
  let B be MSSubset of MSAlg U2 such that
A2: B = the Sorts of MSAlg U1;
  let o be OperSymbol of MSSign U2;
  reconsider a = o as Element of the carrier' of MSSign U1 by A1,Th7;
  set S = (B * the ResultSort of MSSign U2).a;
  S = ((the Sorts of MSAlg U1) * the ResultSort of MSSign U1).a by A1,A2,Th7;
  then
A3: S = Result(a,MSAlg U1) by MSUALG_1:def 5;
  set Z = rng ((Den(o,MSAlg U2))|((B# * the Arity of MSSign U2).a));
  MSSign U1 = MSSign U2 by A1,Th7;
  then
  rng Den(a,MSAlg U1) c= Result(a,MSAlg U1) & Z = rng ((Den(o,MSAlg U2))|(
  Args (a,MSAlg U1))) by A2,MSUALG_1:def 4,RELAT_1:def 19;
  then Z c= Result (a,MSAlg U1) by A1,A2,Th8;
  hence thesis by A3;
end;
