reserve R for non empty Poset,
  S1 for OrderSortedSign;

theorem Th12:
  for U1,U2 being non-empty OSAlgebra of S1 for F being
ManySortedFunction of U1,U2 st F is order-sorted for o1,o2 being OperSymbol of
S1 st o1 <= o2 for x being Element of Args(o1,U1), x1 be Element of Args(o2,U1)
  st x = x1 holds F # x = F # x1
proof
  let U1,U2 be non-empty OSAlgebra of S1;
  let F be ManySortedFunction of U1,U2 such that
A1: F is order-sorted;
  let o1,o2 be OperSymbol of S1 such that
A2: o1 <= o2;
  let x be Element of Args(o1,U1), x1 be Element of Args(o2,U1) such that
A3: x = x1;
A4: dom x = dom (the_arity_of o1) by MSUALG_3:6;
A5: for n being object st n in dom x holds (F # x).n = (F # x1).n
  proof
    let n1 be object such that
A6: n1 in dom x;
    reconsider n2 = n1 as Nat by A6,ORDINAL1:def 12;
    reconsider pi1 = (the_arity_of o1)/.n2, pi2 = (the_arity_of o2)/.n2 as
    Element of S1;
A7: (the_arity_of o1)/.n2 = (the_arity_of o1).n2 by A4,A6,PARTFUN1:def 6;
A8: the_arity_of o1 <= the_arity_of o2 by A2;
    then len (the_arity_of o1) = len (the_arity_of o2);
    then dom (the_arity_of o1) = dom (the_arity_of o2) by FINSEQ_3:29;
    then (the_arity_of o2)/.n2 = (the_arity_of o2).n2 by A4,A6,PARTFUN1:def 6;
    then
A9: pi1 <= pi2 by A4,A6,A8,A7;
    rng (the_arity_of o1) c= the carrier of S1;
    then rng (the_arity_of o1) c= dom (the Sorts of U1) by PARTFUN1:def 2;
    then
A10: n2 in dom ((the Sorts of U1) * (the_arity_of o1)) by A4,A6,RELAT_1:27;
    dom (F.pi1) = (the Sorts of U1).pi1 by FUNCT_2:def 1
      .= (the Sorts of U1).((the_arity_of o1).n2) by A4,A6,PARTFUN1:def 6
      .= ((the Sorts of U1) * (the_arity_of o1)).n2 by A4,A6,FUNCT_1:13;
    then
A11: x1.n2 in dom (F.pi1) by A3,A10,MSUALG_3:6;
    (F # x).n2 = (F.((the_arity_of o1)/.n2)).(x1.n2) by A3,A6,MSUALG_3:def 6
      .= (F.pi2).(x1.n2) by A1,A11,A9
      .= (F # x1).n2 by A3,A6,MSUALG_3:def 6;
    hence thesis;
  end;
  dom x1 = dom (the_arity_of o2) by MSUALG_3:6;
  then
A12: dom (F # x1) = dom x1 by MSUALG_3:6;
  dom (F # x) = dom x by A4,MSUALG_3:6;
  hence thesis by A3,A12,A5,FUNCT_1:2;
end;
