reserve R for non empty Poset,
  S1 for OrderSortedSign;

theorem Th2:
  for A be OrderSortedSet of R, B be non-empty OrderSortedSet of R,
  F be ManySortedFunction of A,B holds F is order-sorted iff for s1,s2 being
Element of R st s1 <= s2 holds for a1 being set st a1 in A.s1 holds (F.s1).a1 =
  (F.s2).a1
proof
  let A be OrderSortedSet of R, B be non-empty OrderSortedSet of R, F be
  ManySortedFunction of A,B;
  hereby
    assume
A1: F is order-sorted;
    let s1,s2 be Element of R such that
A2: s1 <= s2;
    let a1 be set;
    assume a1 in A.s1;
    then a1 in dom (F.s1) by FUNCT_2:def 1;
    hence (F.s1).a1 = (F.s2).a1 by A1,A2;
  end;
  assume
A3: for s1,s2 being Element of R st s1 <= s2 holds for a1 being set st
  a1 in A.s1 holds (F.s1).a1 = (F.s2).a1;
  let s1,s2 be Element of R such that
A4: s1 <= s2;
A5: dom (F.s1) = A.s1 & dom (F.s2) = A.s2 by FUNCT_2:def 1;
  let a1 be set such that
A6: a1 in dom (F.s1);
  A.s1 c= A.s2 by A4,OSALG_1:def 16;
  hence a1 in dom (F.s2) by A6,A5;
  thus (F.s1).a1 = (F.s2).a1 by A3,A4,A6;
end;
