reserve A for non empty set;
reserve a,b,c,x,y,z for Element of A;
reserve o,o9 for Element of LinPreorders A;
reserve o99 for Element of LinOrders A;

theorem Th13:
  for o being Element of LinOrders A, o9 being Element of LinPreorders A holds
  (for a,b st a <_o, b holds a <_o9, b) iff
  for a,b holds a <_o, b iff a <_o9, b
proof
  let o be Element of LinOrders A, o9 be Element of LinPreorders A;
  hereby
    assume
A1: for a,b st a <_o, b holds a <_o9, b;
    let a,b;
    per cases by Th6;
    suppose
   a <_o, b;
      hence a <_o, b iff a <_o9, b by A1;
    end;
    suppose
   a = b;
      hence a <_o, b iff a <_o9, b by Th3;
    end;
    suppose
A2:   b <_o, a;
then    b <_o9, a by A1;
      hence a <_o, b iff a <_o9, b by A2,Th4;
    end;
  end;
  thus thesis;
end;
