reserve A,O for non empty set,
  R for Order of A,
  Ol for Equivalence_Relation of O,
  f for Function of O,A*,
  g for Function of O,A;
reserve S for OverloadedRSSign;
reserve S0 for non empty non void ManySortedSign;
reserve S for non empty Poset;
reserve s1,s2 for Element of S;
reserve w1,w2 for Element of (the carrier of S)*;
reserve S for OrderSortedSign;
reserve o,o1,o2 for OperSymbol of S;
reserve w1 for Element of (the carrier of S)*;

theorem Th8:
  S is discrete & o1 ~= o2 & (the_arity_of o1) <= (the_arity_of o2)
  & the_result_sort_of o1 <= the_result_sort_of o2 implies o1 = o2
proof
  assume
A1: S is discrete;
  then reconsider S1 = S as discrete OrderSortedSign;
  reconsider s1 = the_result_sort_of o1, s2 = the_result_sort_of o2 as
  SortSymbol of S1;
  assume that
A2: o1 ~= o2 and
A3: (the_arity_of o1) <= (the_arity_of o2) and
A4: the_result_sort_of o1 <= the_result_sort_of o2;
A5: s1 = s2 by A4,ORDERS_3:1;
  (the_arity_of o1) = (the_arity_of o2) by A1,A3,Th7;
  hence thesis by A2,A5,Def3;
end;
