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;

theorem Th2:
  for S being non empty non void OverloadedMSSign, o,o1,o2 being
  OperSymbol of S holds o ~= o1 & o1 ~= o2 implies o ~= o2
proof
  let S be non empty non void OverloadedMSSign;
  let o,o1,o2 be OperSymbol of S;
  field the Overloading of S = the carrier' of S by ORDERS_1:12;
  then
A1: the Overloading of S is_transitive_in the carrier' of S by RELAT_2:def 16;
  assume o ~= o1 & o1 ~= o2;
  then [o,o1] in the Overloading of S & [o1,o2] in the Overloading of S;
  then [o,o2] in the Overloading of S by A1,RELAT_2:def 8;
  hence thesis;
end;
