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 Th15:
  (for v holds (the L_join of L).(u,v) = v) implies u = Bottom L
proof
  assume for v holds (the L_join of L).(u,v) = v;
  then for v holds u "\/" v = v;
  hence thesis by Th14;
end;
