sentences:  From these, it follows that
  (Indeed, suppose α("k","h"1)=α("k","h"2).Then "h"1=α("k"-1"k","h"1) = α("k"-1, α("k","h"1))=α("k"-1, α("k","h"2))="h"2.This establishes injectivity, and for surjectivity, use "h"=α("k",α("k"-1,"h")).)