
theorem
  for H being non empty lower-bounded RelStr st H is Heyting for a,b
  being Element of H holds 'not' (a"/\"b) >= 'not' a"\/"'not' b
proof
  let H be non empty lower-bounded RelStr;
  assume
A1: H is Heyting;
  then
A2: Bottom H<=Bottom H by ORDERS_2:1;
  let a,b be Element of H;
A3: 'not' a <= 'not' a by A1,ORDERS_2:1;
A4: 'not' b <= 'not' b by A1,ORDERS_2:1;
  (a"/\"b)"/\"('not' a"\/"'not' b) = ((a"/\"b)"/\"'not' a)"\/"((a"/\"b)
  "/\" 'not' b) by A1,Def3
    .= ((b"/\"a)"/\"'not' a)"\/"((a"/\"b)"/\" 'not' b) by A1,LATTICE3:15
    .= (b"/\"(a"/\"'not' a))"\/"((a"/\"b)"/\" 'not' b) by A1,LATTICE3:16
    .= (b"/\"(a"/\"'not' a))"\/"(a"/\"(b"/\" 'not' b)) by A1,LATTICE3:16
    .= (b"/\"Bottom H)"\/"(a"/\"(b"/\"'not' b)) by A1,A3,Th82
    .= (b"/\"Bottom H)"\/"(a"/\"Bottom H) by A1,A4,Th82
    .= (Bottom H"/\"b)"\/"(a"/\"Bottom H) by A1,LATTICE3:15
    .= (Bottom H"/\"b)"\/"(Bottom H"/\"a) by A1,LATTICE3:15
    .= Bottom H"\/"(Bottom H"/\"a) by A1,Th3
    .= Bottom H"\/"Bottom H by A1,Th3
    .= Bottom H by A1,A2,YELLOW_0:24;
  hence thesis by A1,Th82;
end;
