reserve X,Y for set, x,y,z for object, i,j,n for natural number;

theorem Th33:
  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;
