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 Th11:
  SM is regular iff for o,w1 st w1 <= the_arity_of o ex o1 st o1
  has_least_rank_for o,w1
proof
  hereby
    assume
A1: SM is regular;
    let o,w1;
    assume
A2: w1 <= the_arity_of o;
    o is regular by A1;
    then consider o1 such that
A3: o1 has_least_args_for o,w1 by A2;
    take o1;
    o1 has_least_sort_for o,w1
    proof
      thus
A4:   o ~= o1 & w1 <= the_arity_of o1 by A3;
      let o2;
      assume that
A5:   o ~= o2 and
A6:   w1 <= the_arity_of o2;
A7:   o1 ~= o2 by A4,A5,Th2;
      the_arity_of o1 <= the_arity_of o2 by A3,A5,A6;
      hence thesis by A7,Def7;
    end;
    hence o1 has_least_rank_for o,w1 by A3;
  end;
  assume
A8: for o,w1 st w1 <= the_arity_of o ex o1 st o1 has_least_rank_for o, w1;
  let o;
  thus o is monotone;
  let w1;
  assume w1 <= the_arity_of o;
  then consider o1 such that
A9: o1 has_least_rank_for o,w1 by A8;
  take o1;
  thus thesis by A9;
end;
