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)*;
reserve R for non empty Poset;
reserve z for non empty set;
reserve s1,s2 for SortSymbol of S,
  o,o1,o2,o3 for OperSymbol of S,
  w1,w2 for Element of (the carrier of S)*;
reserve CH for ManySortedFunction of ConstOSSet(S,z)# * the Arity of S,
  ConstOSSet(S,z) * the ResultSort of S;
reserve A for OSAlgebra of S;
reserve M for MSAlgebra over S0;
reserve A for OSAlgebra of S;
reserve op1,op2 for OperSymbol of S;

theorem
  for S being regular monotone OrderSortedSign, o being OperSymbol of
S, w1 being Element of (the carrier of S)* st w1 <= the_arity_of o holds LBound
  (o,w1) <= o
proof
  let S being regular monotone OrderSortedSign, o being OperSymbol of S, w1
  being Element of (the carrier of S)* such that
A1: w1 <= the_arity_of o;
  set lo = LBound(o,w1);
A2: lo has_least_rank_for o,w1 by A1,Th14;
  then lo has_least_sort_for o,w1;
  then
A3: the_result_sort_of lo <= the_result_sort_of o by A1;
A4: lo has_least_args_for o,w1 by A2;
  then
A5: o ~= lo;
  the_arity_of lo <= the_arity_of o by A1,A4;
  hence thesis by A5,A3;
end;
