reserve i for Nat, x,y for set;
reserve S for non empty non void ManySortedSign;
reserve X for non-empty ManySortedSet of S;

theorem Th40:
  for U1,U2 being Universal_Algebra st the UAStr of U1 = the UAStr of U2
  for S1 being Subset of U1, S2 being Subset of U2 st S1 = S2
  holds S1 is opers_closed implies S2 is opers_closed
  proof
    let U1,U2 be Universal_Algebra;
    assume A1: the UAStr of U1 = the UAStr of U2;
    let S1 be Subset of U1;
    let S2 be Subset of U2;
    assume A2: S1 = S2;
    assume
A3: for o be operation of U1 holds S1 is_closed_on o;
    let o be operation of U2;
    reconsider o1 = o as operation of U1 by A1;
    thus thesis by A1,A2,A3,Th39;
  end;
