
theorem
  for S, T being non empty antisymmetric RelStr, x, y being Element of
[:S,T:] holds ex_sup_of {x,y},[:S,T:] iff ex_sup_of {x`1,y`1}, S & ex_sup_of {x
  `2,y`2}, T
proof
  let S, T be non empty antisymmetric RelStr, 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;
  then proj1 {x,y} = {x`1,y`1} & proj2 {x,y} = {x`2,y`2} by FUNCT_5:13;
  hence thesis by YELLOW_3:41;
end;
