
theorem Th11:
  for L being Boolean LATTICE, x,y being Element of L holds y
  is_a_complement_of x iff y = 'not' x
proof
  let L be Boolean LATTICE, x,y be Element of L;
A1: for x being Element of L holds 'not' 'not' x = x by WAYBEL_1:87;
  then
A2: 'not' x is_a_complement_of x by WAYBEL_1:86;
  then
A3: x "/\" 'not' x = Bottom L by WAYBEL_1:def 23;
A4: x "\/" 'not' x = Top L by A2,WAYBEL_1:def 23;
  hereby
    assume
A5: y is_a_complement_of x;
    then
A6: x "/\" y = Bottom L by WAYBEL_1:def 23;
A7: Top L >= 'not' x by YELLOW_0:45;
A8: Bottom L <= y"/\"'not' x by YELLOW_0:44;
    Top L >= y by YELLOW_0:45;
    then
A9: y = (x"\/"'not' x)"/\"y by A4,YELLOW_0:25
      .= (x"/\"y)"\/"(y"/\"'not' x) by WAYBEL_1:def 3
      .= y"/\"'not' x by A6,A8,YELLOW_0:24;
    x "\/" y = Top L by A5,WAYBEL_1:def 23;
    then 'not' x = (x"\/"y)"/\"'not' x by A7,YELLOW_0:25
      .= (x"/\"'not' x)"\/"(y"/\"'not' x) by WAYBEL_1:def 3
      .= y"/\"'not' x by A3,A8,YELLOW_0:24;
    hence y = 'not' x by A9;
  end;
  thus thesis by A1,WAYBEL_1:86;
end;
