
theorem Th9:
  for L being complemented' join-commutative meet-commutative
  distributive upper-bounded' lower-bounded' distributive' non empty LattStr
  holds L is join-idempotent
proof
  let L be complemented' join-commutative meet-commutative distributive
  upper-bounded' lower-bounded' distributive' non empty LattStr;
  let x be Element of L;
  thus x "\/" x = (x "\/" x) "/\" Top' L by Def2
    .= (x "\/" x) "/\" (x "\/" x`# ) by Th2
    .= x "\/" (x "/\" x`# ) by Def5
    .= x "\/" Bot' L by Th3
    .= x by Def4;
end;
