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

theorem Th20:
  for S1,S2,E1,E2 being Signature
  st the ManySortedSign of S1 = the ManySortedSign of S2 &
  the ManySortedSign of E1 = the ManySortedSign of E2 &
  E1 is Extension of S1
  holds E2 is Extension of S2
  proof
    let S1,S2,E1,E2 be Signature;
    assume A1: the ManySortedSign of S1 = the ManySortedSign of S2;
    assume A2: the ManySortedSign of E1 = the ManySortedSign of E2;
    set f = id the carrier of S1, g = id the carrier' of S1;
    assume
A3: dom f = the carrier of S1 & dom g = the carrier' of S1 &
    rng f c= the carrier of E1 & rng g c= the carrier' of E1 &
    f*the ResultSort of S1 = (the ResultSort of E1)*g &
    for o being set, p being Function
    st o in the carrier' of S1 & p = (the Arity of S1).o
    holds f*p = (the Arity of E1).(g.o);
    thus dom id the carrier of S2 = the carrier of S2;
    thus thesis by A1,A2,A3;
  end;
