reserve T for non empty RelStr,
  a for Element of T;
reserve a for set;
reserve P for non empty POSet_set,
  A,B for Element of P;

theorem
  for o1,o2 be Object of POSAltCat P, A,B be Element of P st o1 = A & o2
  = B holds <^ o1, o2 ^> c= Funcs (the carrier of A, the carrier of B)
proof
  let o1,o2 be Object of POSAltCat P, A,B be Element of P;
  assume
A1: o1 = A & o2 = B;
  let x be object;
  assume x in <^ o1, o2 ^>;
  then x in MonFuncs (A,B) by A1,Def9;
  then ex f be Function of A, B st f = x & f in Funcs (the carrier of A,the
  carrier of B) & f is monotone by Def6;
  hence thesis;
end;
