
theorem
  for R being antisymmetric with_suprema transitive non empty RelStr, x,
  y being Element of R holds uparrow (x"\/"y) = (uparrow x) /\ uparrow y
proof
  let R be antisymmetric with_suprema transitive non empty RelStr, x,y be
  Element of R;
  now
    let z be object;
    hereby
      assume
A1:   z in uparrow (x"\/"y);
      then reconsider z9 = z as Element of R;
A2:   z9 >= (x"\/"y) by A1,WAYBEL_0:18;
      (x"\/"y) >= y by YELLOW_0:22;
      then z9 >= y by A2,YELLOW_0:def 2;
      then
A3:   z9 in uparrow y by WAYBEL_0:18;
      (x"\/"y) >= x by YELLOW_0:22;
      then z9 >= x by A2,YELLOW_0:def 2;
      then z9 in uparrow x by WAYBEL_0:18;
      hence z in (uparrow x) /\ uparrow y by A3,XBOOLE_0:def 4;
    end;
    assume
A4: z in (uparrow x) /\ uparrow y;
    then reconsider z9 = z as Element of R;
    z in uparrow y by A4,XBOOLE_0:def 4;
    then
A5: z9 >= y by WAYBEL_0:18;
    z in (uparrow x) by A4,XBOOLE_0:def 4;
    then z9 >= x by WAYBEL_0:18;
    then x"\/"y <= z9 by A5,YELLOW_0:22;
    hence z in uparrow (x"\/"y) by WAYBEL_0:18;
  end;
  hence thesis by TARSKI:2;
end;
