
theorem Th35:
  for S1, S2 being antisymmetric non empty RelStr for D1 being
Subset of S1, D2 being Subset of S2 for x being Element of S1, y being Element
  of S2 st ex_sup_of D1,S1 & ex_sup_of D2,S2 & for b being Element of [:S1,S2:]
st b is_>=_than [:D1,D2:] holds [x,y] <= b holds (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
proof
  let S1, S2 be antisymmetric non empty RelStr, D1 be Subset of S1, D2 be
  Subset of S2, x be Element of S1, y be Element of S2 such that
A1: ex_sup_of D1,S1 and
A2: ex_sup_of D2,S2 and
A3: for b being Element of [:S1,S2:] st b is_>=_than [:D1,D2:] holds [x,
  y] <= b;
  thus for c being Element of S1 st c is_>=_than D1 holds x <= c
  proof
A4: sup D2 is_>=_than D2 by A2,YELLOW_0:30;
    let b be Element of S1;
    assume b is_>=_than D1;
    then [x,y] <= [b,sup D2] by A3,A4,Th30;
    hence thesis by Th11;
  end;
A5: sup D1 is_>=_than D1 by A1,YELLOW_0:30;
  let b be Element of S2;
  assume b is_>=_than D2;
  then [x,y] <= [sup D1,b] by A3,A5,Th30;
  hence thesis by Th11;
end;
