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 NB be normalized Binding of OAF for i, j st i >= j holds (
  Normalized NB).(j,i) = NB.(j,i)
proof
  let NB be normalized Binding of OAF;
  let i, j;
  assume
A1: i >= j;
  per cases;
  suppose
    i <> j;
    hence thesis by A1,Th3;
  end;
  suppose
A2: i = j;
    (Normalized NB).(j,i) = IFEQ (j, i, id (the Sorts of OAF.i), bind (NB,
    i,j) ** id (the Sorts of OAF.i) ) by A1,Def5;
    then (Normalized NB).(j,i) = id (the Sorts of OAF.i) by A2,FUNCOP_1:def 8;
    hence thesis by A2,Def4;
  end;
end;
