
theorem
  for S1, S2 being antisymmetric non empty RelStr for D1 being non empty
  Subset of S1, D2 being non empty Subset of S2 for x being Element of S1, y
being Element of S2 st (for c being Element of S1 st c is_>=_than D1 holds x >=
  c) & for d being Element of S2 st d is_>=_than D2 holds y >= d holds for b
  being Element of [:S1,S2:] st b is_>=_than [:D1,D2:] holds [x,y] >= b
proof
  let S1, S2 be antisymmetric non empty RelStr, D1 be non empty Subset of S1,
D2 be non empty Subset of S2, x be Element of S1, y be Element of S2 such that
A1: for c being Element of S1 st c is_>=_than D1 holds x >= c and
A2: for d being Element of S2 st d is_>=_than D2 holds y >= d;
  let b be Element of [:S1,S2:] such that
A3: b is_>=_than [:D1,D2:];
  the carrier of [:S1,S2:] = [:the carrier of S1, the carrier of S2:] by Def2;
  then
  ex c, d being object st c in the carrier of S1 & d in the carrier of S2 & b
  = [c,d] by ZFMISC_1:def 2;
  then
A4: b = [b`1,b`2];
  then b`2 is_>=_than D2 by A3,Th29;
  then
A5: y >= b`2 by A2;
  b`1 is_>=_than D1 by A3,A4,Th29;
  then x >= b`1 by A1;
  hence thesis by A4,A5,Th11;
end;
