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)*;

theorem Th12:
  for SM being monotone OrderSortedSign holds SM is op-discrete
  implies SM is regular
proof
  let SM be monotone OrderSortedSign;
  assume
A1: SM is op-discrete;
  let om be OperSymbol of SM;
  thus om is monotone;
  let wm1 be Element of (the carrier of SM)* such that
A2: wm1 <= the_arity_of om;
  om has_least_args_for om,wm1
  by A2,A1,Th3;
  hence thesis;
end;
