
theorem Th21:
  for S, T being up-complete non empty Poset, x, y being Element
  of [:S,T:] holds x << y iff x`1 << y`1 & x`2 << y`2
proof
  let S, T be up-complete non empty Poset, x, y be Element of [:S,T:];
  the carrier of [:S,T:] = [:the carrier of S,the carrier of T:] by
YELLOW_3:def 2;
  then x = [x`1,x`2] & y = [y`1,y`2] by MCART_1:21;
  hence thesis by Th19;
end;
