
theorem
  for H being non empty lower-bounded RelStr st H is Heyting for a,b
  being Element of H st a <= b holds 'not' b <= 'not' a
proof
  let H be non empty lower-bounded RelStr such that
A1: H is Heyting;
  let a,b be Element of H;
A2: 'not' b >= 'not' b by A1,ORDERS_2:1;
  assume a <= b;
  then a "/\" 'not' b = (a"/\"b)"/\"'not' b by A1,YELLOW_0:25
    .= a"/\"(b"/\"'not' b) by A1,LATTICE3:16
    .= a"/\"Bottom H by A1,A2,Th82
    .= Bottom H"/\"a by A1,LATTICE3:15
    .= Bottom H by A1,Th3;
  hence thesis by A1,Th82;
end;
