
theorem
  for S, T being non empty RelStr st [:S,T:] is upper-bounded holds S is
  upper-bounded & T is upper-bounded
proof
  let S, T be non empty RelStr;
  given x being Element of [:S,T:] such that
A1: x is_>=_than the carrier of [:S,T:];
  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: x = [s,t] by ZFMISC_1:def 2;
  reconsider t as Element of T by A3;
  reconsider s as Element of S by A2;
A5: the carrier of S c= the carrier of S & the carrier of T c= the carrier
  of T;
A6: [s,t] is_>=_than [:the carrier of S,the carrier of T:] by A1,A4;
  thus S is upper-bounded
  proof
    take s;
    thus thesis by A5,A6,YELLOW_3:29;
  end;
  take t;
  thus thesis by A5,A6,YELLOW_3:29;
end;
