
theorem Th36:
  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_inf_of D1,S1 & ex_inf_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_inf_of D1,S1 and
A2: ex_inf_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: inf D2 is_<=_than D2 by A2,YELLOW_0:31;
    let b be Element of S1;
    assume b is_<=_than D1;
    then [x,y] >= [b,inf D2] by A3,A4,Th33;
    hence thesis by Th11;
  end;
A5: inf D1 is_<=_than D1 by A1,YELLOW_0:31;
  let b be Element of S2;
  assume b is_<=_than D2;
  then [x,y] >= [inf D1,b] by A3,A5,Th33;
  hence thesis by Th11;
end;
