
theorem
  for S being with_infima with_suprema antisymmetric RelStr, T being
  with_infima with_suprema reflexive antisymmetric RelStr st [:S,T:] is
  distributive holds S is distributive
proof
  let S be with_infima with_suprema antisymmetric RelStr, T be with_infima
  with_suprema reflexive antisymmetric RelStr such that
A1: for x, y, z being Element of [:S,T:] holds x "/\" (y "\/" z) = (x
  "/\" y) "\/" (x "/\" z);
  set t = the Element of T;
  let x, y, z be Element of S;
A2: t "/\" t = t by YELLOW_0:25;
  t <= t;
  then
A3: t "\/" t = t by YELLOW_0:24;
  thus x "/\" (y "\/" z) = [x "/\" (y "\/" z),t]`1
    .= ([x,t] "/\" [y "\/" z,t])`1 by A2,Th15
    .= ([x,t] "/\" ([y,t] "\/" [z,t]))`1 by A3,Th16
    .= (([x,t] "/\" [y,t]) "\/" ([x,t] "/\" [z,t]))`1 by A1
    .= (([x "/\" y,t]) "\/" ([x,t] "/\" [z,t]))`1 by A2,Th15
    .= ([x "/\" y,t] "\/" [x "/\" z,t])`1 by A2,Th15
    .= [(x "/\" y) "\/" (x "/\" z),t]`1 by A3,Th16
    .= (x "/\" y) "\/" (x "/\" z);
end;
