sentences:  The assumption implies that there exist surjective partial isometries from "H" to "G" and from "G" to "H".Fix two such partial isometries for the argument.One has
  
  In turn,
  
  By induction,
  
  and
  
  Now each additional summand in the direct sum expression is obtained using one of the two fixed partial isometries, so
  
  This proves the theorem.