reserve U1,U2,U3 for Universal_Algebra,
  m,n for Nat,
  a for set,
  A for non empty set,
  h for Function of U1,U2;

theorem
  for S,S9 be strict non empty ManySortedSign st S <= S9 & S9 <= S holds S = S9
proof
  let S,S9 be strict non empty ManySortedSign;
  assume that
A1: S <= S9 and
A2: S9 <= S;
A3: the carrier' of S9 c= the carrier' of S by A2;
A4: dom (the ResultSort of S9) = the carrier' of S9 by FUNCT_2:def 1;
  (the ResultSort of S9)|the carrier' of S = the ResultSort of S by A1;
  then
A5: the ResultSort of S = the ResultSort of S9 by A3,A4,RELAT_1:68;
A6: dom (the Arity of S9) = the carrier' of S9 by FUNCT_2:def 1;
  (the Arity of S9)|the carrier' of S = the Arity of S by A1;
  then
A7: the Arity of S = the Arity of S9 by A3,A6,RELAT_1:68;
  the carrier' of S c= the carrier' of S9 by A1;
  then
A8: the carrier' of S = the carrier' of S9 by A3,XBOOLE_0:def 10;
  the carrier of S c= the carrier of S9 & the carrier of S9 c= the carrier
  of S by A1,A2;
  hence thesis by A8,A7,A5,XBOOLE_0:def 10;
end;
