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;
