theorem Th45:
  for O for x,y being Element of O holds x > y iff not x <= y
     proof
       let O; let x,y be Element of O;
A1:    x <= y & y <= x implies x = y by YELLOW_0:def 3;
       (x <= y or x >= y) & x <= x by WAYBEL_0:def 29;
       hence x > y iff not x <= y by A1,ORDERS_2:def 6;
     end;
