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)*;

theorem Th9:
  S is op-discrete implies S is monotone
proof
  set ol = the Overloading of S;
  assume S is op-discrete;
  then
A1: ol = id the carrier' of S;
  let o be OperSymbol of S;
  let o2 be OperSymbol of S;
  assume o ~= o2;
  then [o,o2] in ol;
  hence thesis by A1,RELAT_1:def 10;
end;
