
theorem
  for S, T being bounded with_infima with_suprema antisymmetric RelStr
  st [:S,T:] is complemented holds S is complemented & T is complemented
proof
  let S, T be bounded with_infima with_suprema antisymmetric RelStr;
  set s = the Element of S;
  assume
A1: for x being Element of [:S,T:] ex y being Element of [:S,T:] st y
  is_a_complement_of x;
  thus S is complemented
  proof
    set t = the Element of T;
    let s be Element of S;
    consider y being Element of [:S,T:] such that
A2: y is_a_complement_of [s,t] by A1;
    take y`1;
    [s,t]`1 = s;
    hence y`1 is_a_complement_of s by A2,Th17;
  end;
  let t be Element of T;
  consider y being Element of [:S,T:] such that
A3: y is_a_complement_of [s,t] by A1;
  take y`2;
  [s,t]`2 = t;
  hence y`2 is_a_complement_of t by A3,Th17;
end;
