theorem Th18:
  the L_join of L is having_a_unity implies Bottom L =
  the_unity_wrt the L_join of L
proof
  set J = the L_join of L;
  given u such that
A1: u is_a_unity_wrt J;
  (the L_join of L).(u,v) = v by A1,BINOP_1:4;
  then u = Bottom L by Th15;
  hence thesis by A1,BINOP_1:def 8;
end;
