Theorem (can be generalized to arbitrary number of attributes):
  Suppose r(A,B,C) is a relation.
  Let r_AB be the projection of r to AB, and 
      r_AC be the projection of r to AC.
  Then 
    (1) r is a subset of r_AB join r_AC
    (2) if the dependency A -> B holds,
        then r_AB join r_AC is a subset of r
            (and the join thus lossless)
Proof:
  First (1): 
   let t = (a,b,c) belong to r.
   Then (a,b) belongs to r_AB, and (a,c) belongs to r_AC,
    so t = (a,b,c) belongs to the join of r_AB and r_AC, as desired.  
  Next (2):
   let t = (a,b,c) belong to the join of r_AB and r_AC,
    implying that 
         (a,b) belongs to r_AB and 
         (a,c) belongs to r_AC.
    We infer that there exists c' and b' such that
         (a,b,c') belongs to r and
         (a,b',c) belongs to r
    Since A -> B holds, we now infer that b = b'.
      But then t = (a,b,c) belongs to r, as desired.