
theorem Th4:
  for S, T being lower-bounded antisymmetric non empty RelStr holds
  Bottom [:S,T:] = [Bottom S,Bottom T]
proof
  let S, T be lower-bounded antisymmetric non empty RelStr;
A1: for a being Element of [:S,T:] st {} is_<=_than a holds [Bottom S,
  Bottom T] <= a
  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;
    Bottom S <= s & Bottom T <= t by YELLOW_0:44;
    hence thesis by A4,YELLOW_3:11;
  end;
  ex_sup_of {},[:S,T:] & {} is_<=_than [Bottom S,Bottom T] by YELLOW_0:42;
  hence thesis by A1,YELLOW_0:def 9;
end;
