reserve P for non empty Poset,
  i, j, k for Element of P;
reserve S for non void non empty ManySortedSign;
reserve OAF for OrderedAlgFam of P, S;
reserve B for Binding of OAF;

theorem
  for DP be discrete non empty Poset, S for OAF be OrderedAlgFam of DP,S
  for B be normalized Binding of OAF holds InvLim B = product OAF
proof
  let DP be discrete non empty Poset, S;
  let OAF be OrderedAlgFam of DP,S;
  let B be normalized Binding of OAF;
A1: for s be object st s in the carrier of S holds (the Sorts of InvLim B).s =
  (the Sorts of product OAF).s
  proof
    let a be object;
    assume a in the carrier of S;
    then reconsider s = a as SortSymbol of S;
    thus (the Sorts of InvLim B).a c= (the Sorts of product OAF).a
    proof
      let e be object;
      (the Sorts of InvLim B) is MSSubset of product OAF by MSUALG_2:def 9;
      then (the Sorts of InvLim B) c= the Sorts of product OAF by PBOOLE:def 18
;
      then
A2:   (the Sorts of InvLim B).s c= (the Sorts of product OAF).s by PBOOLE:def 2
;
      assume e in (the Sorts of InvLim B).a;
      hence thesis by A2;
    end;
    let e be object;
    assume e in (the Sorts of product OAF).a;
    then reconsider f = e as Element of (SORTS OAF).s by PRALG_2:12;
    for i,j be Element of DP st i >= j holds (bind (B,i,j).s).(f.i) = f.j
    proof
      let i,j be Element of DP;
      assume i >= j;
      then
A3:   i = j by ORDERS_3:1;
      f in (SORTS OAF).s;
      then dom (Carrier (OAF,s)) = the carrier of DP & f in product Carrier (
      OAF,s) by PARTFUN1:def 2,PRALG_2:def 10;
      then
A4:   f.i in (Carrier (OAF,s)).i by CARD_3:9;
      bind (B,i,i) = B.(i,i) by Def3,ORDERS_2:1
        .= id (the Sorts of OAF.i) by Def4; then
A5:   (bind (B,i,i).s) = id ((the Sorts of OAF.i).s) by MSUALG_3:def 1;
      ex U0 being MSAlgebra over S st U0 = OAF.i & (Carrier ( OAF,s)).i =
      ((the Sorts of U0).s) by PRALG_2:def 9;
      hence thesis by A3,A5,A4,FUNCT_1:18;
    end;
    hence thesis by Def6;
  end;
  product OAF is MSSubAlgebra of product OAF by MSUALG_2:5;
  hence thesis by A1,MSUALG_2:9,PBOOLE:3;
end;
