
theorem Th80:
  for H being non empty RelStr st H is Heyting & H is
  lower-bounded holds 'not' Bottom H = Top H & 'not' Top H = Bottom H
proof
  let H be non empty RelStr such that
A1: H is Heyting and
A2: H is lower-bounded;
  (Top H) => (Bottom H) <= (Top H) => (Bottom H) by A1,ORDERS_2:1;
  then
A3: Bottom H >= Top H "/\" 'not' Top H by A1,Th67;
  Bottom H >= Bottom H "/\" Top H by A1,YELLOW_0:23;
  then
A4: Top H <= (Bottom H) => (Bottom H) by A1,Th67;
  Bottom H <= Top H "/\" 'not' Top H by A1,A2,YELLOW_0:44;
  then
A5: Bottom H = Top H "/\" 'not' Top H by A1,A3,ORDERS_2:2;
  'not' Bottom H <= Top H by A1,YELLOW_0:45;
  hence Top H = 'not' Bottom H by A1,A4,ORDERS_2:2;
  'not' Top H <= Top H by A1,YELLOW_0:45;
  hence 'not' Top H = 'not' Top H"/\"Top H by A1,YELLOW_0:25
    .= Bottom H by A1,A5,LATTICE3:15;
end;
