
theorem
  for S being with_infima with_suprema reflexive antisymmetric RelStr, T
being with_infima with_suprema antisymmetric RelStr st [:S,T:] is distributive
  holds T is distributive
proof
  let S be with_infima with_suprema reflexive antisymmetric RelStr, T be
  with_infima with_suprema antisymmetric RelStr such that
A1: for x, y, z being Element of [:S,T:] holds x "/\" (y "\/" z) = (x
  "/\" y) "\/" (x "/\" z);
  set s = the Element of S;
  let x, y, z be Element of T;
A2: s "/\" s = s by YELLOW_0:25;
  s <= s;
  then
A3: s "\/" s = s by YELLOW_0:24;
  thus x "/\" (y "\/" z) = [s,x "/\" (y "\/" z)]`2
    .= ([s,x] "/\" [s,y "\/" z])`2 by A2,Th15
    .= ([s,x] "/\" ([s,y] "\/" [s,z]))`2 by A3,Th16
    .= (([s,x] "/\" [s,y]) "\/" ([s,x] "/\" [s,z]))`2 by A1
    .= (([s,x "/\" y]) "\/" ([s,x] "/\" [s,z]))`2 by A2,Th15
    .= ([s,x "/\" y] "\/" [s,x "/\" z])`2 by A2,Th15
    .= [s,(x "/\" y) "\/" (x "/\" z)]`2 by A3,Th16
    .= (x "/\" y) "\/" (x "/\" z);
end;
