
theorem Th3:
  for S, T being upper-bounded antisymmetric non empty RelStr holds
  Top [:S,T:] = [Top S,Top T]
proof
  let S, T be upper-bounded antisymmetric non empty RelStr;
A1: for a being Element of [:S,T:] st {} is_>=_than a holds a <= [Top S, Top
  T ]
  proof
    let a be Element of [:S,T:];
    assume {} is_>=_than a;
    the carrier of [:S,T:] = [:the carrier of S, the carrier of T:] by
YELLOW_3:def 2;
    then consider s, t being object such that
A2: s in the carrier of S and
A3: t in the carrier of T and
A4: a = [s,t] by ZFMISC_1:def 2;
    reconsider t as Element of T by A3;
    reconsider s as Element of S by A2;
    s <= Top S & t <= Top T by YELLOW_0:45;
    hence thesis by A4,YELLOW_3:11;
  end;
  ex_inf_of {},[:S,T:] & {} is_>=_than [Top S,Top T] by YELLOW_0:43;
  hence thesis by A1,YELLOW_0:def 10;
end;
