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 Th3:
  for S being non empty non void OverloadedMSSign holds S is
  op-discrete iff for x,y be OperSymbol of S st x ~= y holds x = y
proof
  let S be non empty non void OverloadedMSSign;
  set d = id the carrier' of S;
  set opss = the carrier' of S;
  set ol = the Overloading of S;
  thus S is op-discrete implies for x,y be OperSymbol of S st x ~= y holds x =
  y
  by RELAT_1:def 10;
  assume
A1: for x,y be OperSymbol of S st x ~= y holds x = y;
  now
    let x,y be object;
    thus [x,y] in ol implies x in opss & x = y
    proof
      assume
A2:   [x,y] in ol;
      then ex x1,y1 being object
st [x,y] = [x1,y1] & x1 in opss & y1 in opss by
RELSET_1:2;
      then reconsider x2 = x, y2 = y as OperSymbol of S by XTUPLE_0:1;
      x2 ~= y2 by A2;
      hence thesis by A1;
    end;
    assume x in opss & x = y;
    hence [x,y] in ol by Def2;
  end;
  hence the Overloading of S = d by RELAT_1:def 10;
end;
