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,S99 be non empty ManySortedSign st S <= S9 & S9 <= S99 holds
  S <= S99
proof
  let S,S9,S99 be non empty ManySortedSign;
  assume that
A1: S <= S9 and
A2: S9 <= S99;
  the carrier of S c= the carrier of S9 & the carrier of S9 c= the carrier
  of S99 by A1,A2;
  hence the carrier of S c= the carrier of S99;
A3: the carrier' of S c= the carrier' of S9 by A1;
  the carrier' of S9 c= the carrier' of S99 by A2;
  hence the carrier' of S c= the carrier' of S99 by A3;
  thus (the Arity of S99)|the carrier' of S = (the Arity of S99)|((the
  carrier' of S9)/\(the carrier' of S)) by A3,XBOOLE_1:28
    .= the Arity of S by A1,A2,RELAT_1:71;
  thus (the ResultSort of S99)|the carrier' of S = (the ResultSort of S99)|((
  the carrier' of S9) /\ (the carrier' of S)) by A3,XBOOLE_1:28
    .= the ResultSort of S by A1,A2,RELAT_1:71;
end;
