
theorem
  for H being non empty RelStr st H is Heyting & H is lower-bounded for
a being Element of H holds 'not' a is_maximum_of {x where x is Element of H: a
  "/\"x = Bottom H}
proof
  let H be non empty RelStr such that
A1: H is Heyting and
A2: H is lower-bounded;
  let a be Element of H;
  set X = {x where x is Element of H: a"/\"x = Bottom H}, Y = {x where x is
  Element of H: a"/\"x <= Bottom H};
A3: X = Y
  proof
    hereby
      let y be object;
      assume y in X;
      then consider x being Element of H such that
A4:   y = x and
A5:   a"/\"x = Bottom H;
      a"/\"x <= Bottom H by A1,A5,ORDERS_2:1;
      hence y in Y by A4;
    end;
    let y be object;
    assume y in Y;
    then consider x being Element of H such that
A6: y = x and
A7: a"/\"x <= Bottom H;
    Bottom H <= a"/\"x by A1,A2,YELLOW_0:44;
    then Bottom H = a"/\"x by A1,A7,ORDERS_2:2;
    hence thesis by A6;
  end;
A8: now
    a => (Bottom H) <= a => (Bottom H) by A1,ORDERS_2:1;
    then a"/\"'not' a <= Bottom H by A1,Th67;
    then
A9: 'not' a in X by A3;
    let b be Element of H;
    assume b is_>=_than X;
    hence 'not' a <= b by A9;
  end;
A10: ex_max_of X,H by A1,A3,Th62;
  hence ex_sup_of X,H;
  'not' a is_>=_than X
  proof
    let b be Element of H;
    assume b in X;
    then ex x being Element of H st x = b & a"/\"x <= Bottom H by A3;
    hence thesis by A1,Th67;
  end;
  hence 'not' a = "\/"(X,H) by A1,A8,YELLOW_0:30;
  thus thesis by A10;
end;
