reserve A for set,
  C for non empty set,
  B for Subset of A,
  x for Element of A,
  f,g for Function of A,C;
reserve B for Element of Fin A;
reserve L for non empty LattStr,
  a,b,c for Element of L;
reserve L for Lattice;
reserve a,b,c,u,v for Element of L;

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;
