In Computability and Unsolvability (Martin Davis), we have theorem 1.9 on page 87. It states, for every normal system T, we can construct a normal system T', whose alphabet consists of two letters, such that $S_T=S_{T'}$. This is the link to the book and page: https://books.google.com/books?id=TMLgEIOGhBMC&pg=PA87&lpg=PA87&dq=hilary+putnam+combinatorial+normal&source=bl&ots=d7tBQUnshV&sig=epWIg7_7NO5Z3q31jkG9QGbG4Ts&hl=en&sa=X&ved=0ahUKEwi-h9iU-JbUAhWE2SYKHfq4BFsQ6AEIIzAA#v=onepage&q=hilary%20putnam%20combinatorial%20normal&f=false
I understand how we get $\vdash_{T^*}(1b1)^{n+1} \to \vdash_{T'} (b11)^{n+1} \to \vdash_ {T'}(1)^{n+1}$. I don't understand, however, how we get $\vdash_ {T'}(1)^{n+1} \to \vdash_{T'} (b11)^{n+1} \to \vdash_{T^*}(1b1)^{n+1}$. Can someone explain to me where this comes from? Thanks.