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)*;
reserve SM for monotone OrderSortedSign,
  o,o1,o2 for OperSymbol of SM,
  w1 for Element of (the carrier of SM)*;
reserve SR for regular monotone OrderSortedSign,
  o,o1,o3,o4 for OperSymbol of SR,
  w1 for Element of (the carrier of SR)*;

theorem Th13:
  o3 has_least_args_for o,w1 & o4 has_least_args_for o,w1 implies o3 = o4
proof
  assume that
A1: o3 has_least_args_for o,w1 and
A2: o4 has_least_args_for o,w1;
A3: o ~= o3 by A1;
A4: o ~= o4 by A2;
  then
A5: o3 ~= o4 by A3,Th2;
  w1 <= the_arity_of o3 by A1;
  then
A6: the_arity_of o4 <= the_arity_of o3 by A2,A3;
  then
A7: the_result_sort_of o4 <= the_result_sort_of o3 by A5,Def7;
  w1 <= the_arity_of o4 by A2;
  then
A8: the_arity_of o3 <= the_arity_of o4 by A1,A4;
  then
A9: the_arity_of o3 = the_arity_of o4 by A6,Th6;
  the_result_sort_of o3 <= the_result_sort_of o4 by A5,A8,Def7;
  then the_result_sort_of o3 = the_result_sort_of o4 by A7,ORDERS_2:2;
  hence thesis by A5,A9,Def3;
end;
