reserve R for non empty Poset,
  S1 for OrderSortedSign;

theorem Th1:
  for F being ManySortedFunction of the carrier of R st F is
order-sorted for s1,s2 being Element of R st s1 <= s2 holds dom (F.s1) c= dom (
  F.s2) & F.s1 c= F.s2
proof
  let F be ManySortedFunction of the carrier of R such that
A1: F is order-sorted;
  let s1,s2 be Element of R such that
A2: s1 <= s2;
  thus dom (F.s1) c= dom (F.s2)
  by A1,A2;
  for a,b being object holds [a,b] in F.s1 implies [a,b] in F.s2
  proof
    let y,z be object such that
A3: [y,z] in F.s1;
    y in dom (F.s1) by A3,XTUPLE_0:def 12;
    then
A4: y in dom (F.s2) & (F.s1).y = (F.s2).y by A1,A2;
    (F.s1).y = z by A3,FUNCT_1:1;
    hence thesis by A4,FUNCT_1:1;
  end;
  hence thesis by RELAT_1:def 3;
end;
