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 Th42:
  for U1,U2 be Universal_Algebra st the UAStr of U1 = the UAStr of U2
  holds signature U1 = signature U2
  proof
    let U1,U2 be Universal_Algebra;
    assume A1: the UAStr of U1 = the UAStr of U2;
A2: len signature U2 = len the charact of U1 by A1,UNIALG_1:def 4;
    for i st i in dom signature U2
    for h be homogeneous non empty PartFunc of (the carrier of U1)*,
    the carrier of U1 st h = (the charact of U1).i holds
    (signature U2).i = arity h by A1,UNIALG_1:def 4;
    hence signature U1 = signature U2 by A2,UNIALG_1:def 4;
  end;
