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 Th39:
  for U1,U2 being Universal_Algebra
  for S1 being Subset of U1, S2 being Subset of U2 st S1 = S2
  for o1 being operation of U1, o2 being operation of U2 st o1 = o2
  holds S1 is_closed_on o1 implies S2 is_closed_on o2
  proof
    let U1,U2 be Universal_Algebra;
    let S1 be Subset of U1;
    let S2 be Subset of U2;
    assume A1: S1 = S2;
    let o1 be operation of U1;
    let o2 be operation of U2;
    assume A2: o1 = o2;
    assume
A3: for s being FinSequence of S1 st len s = arity o1 holds o1.s in S1;
    let s be FinSequence of S2;
    reconsider s1 = s as FinSequence of S1 by A1;
    assume len s = arity o2;
    hence thesis by A1,A2,A3;
  end;
