reserve a,a1,a2,b,c,d for Ordinal,
  n,m,k for Nat,
  x,y,z,t,X,Y,Z for set;
reserve f,g for Function;
reserve A,B,C for array;
reserve O for connected non empty Poset;
reserve R,Q for array of O;

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;
