
theorem Th87:
  for L being bounded LATTICE holds L is distributive complemented
  iff L is Heyting & for x being Element of L holds 'not' 'not' x = x
proof
  let L be bounded LATTICE;
  hereby
    assume L is distributive complemented;
    then for x being Element of L ex x9 being Element of L st for y being
Element of L holds (y "\/" x9) "/\" x <= y & y <= (y "/\" x) "\/" x9 by Lm6;
    hence L is Heyting & for x being Element of L holds 'not' 'not' x = x by
Lm7;
  end;
  assume that
A1: L is Heyting and
A2: for x being Element of L holds 'not' 'not' x = x;
  thus L is distributive by A1;
  let x be Element of L;
  take 'not' x;
  thus thesis by A1,A2,Th86;
end;
