
theorem
  for L be LATTICE st L is distributive for a,b,c be Element of L holds
  (a "\/" b) \ c = (a \ c) "\/" (b \ c)
proof
  let L be with_suprema with_infima reflexive transitive antisymmetric non
  empty RelStr such that
A1: L is distributive;
  let a,b,c be Element of L;
  thus (a "\/" b) \ c = (a "\/" b) "/\" 'not' c
    .= ('not' c "/\" a) "\/" ('not' c"/\" b) by A1,WAYBEL_1:def 3
    .= (a \ c) "\/" (b \ c);
end;
