
theorem Th28:
  for L be distributive LATTICE for p,q,u be Element of L st p < q
& (for s be Element of L st p < s holds q <= s) & not u <= p holds not u "/\" q
  <= p
proof
  let L be distributive LATTICE;
  let p,q,u be Element of L;
  assume that
A1: p < q and
A2: ( for s be Element of L st p < s holds q <= s)& not u <= p and
A3: u "/\" q <= p;
A4: p <= q by A1,ORDERS_2:def 6;
  p = p "\/" (u "/\" q) by A3,YELLOW_0:24
    .= (p "\/" u) "/\" (q "\/" p) by WAYBEL_1:5
    .= (p "\/" u) "/\" q by A4,YELLOW_0:24
    .= q "/\" (q "\/" u) by A1,A2,Th27
    .= q by LATTICE3:18;
  hence contradiction by A1;
end;
