Give a Hilbert proof to show the following: (Please show each step and what method was used for each step in the proof.)

(pq)(q(pq))

Your Hilbert proof cannot use any shortcuts; i.e., it must be a Hilbert proof by Definition 1.4.5 in the book which is Theorem calculations or proofs..

(Hint: It might help to first write out a proof in a looser style and then fill in the shortcuts used later. Be careful not to skip steps; the shortest proof I found uses 12 lines).

1.4.5 Definition. (Theorem-Calculationsor Proofs) Let I be an arbitrary, given set of formulae. A theorem-calculation (or proof) from I is any finite (ordered) sequence of for- mulae that we may write respecting the following two requirements: In any stage we may write down Pr1 Any member of A or Pr2 Any formula that appears in the denominator of an instance of a rule Inf1-Inf2 as long as all the formulae in the numerator of the same instance of the (same) rule have already been written down at an earlier stage We may call a proof from I by the alternative name T-proof. 1.4.5 Definition. (Theorem-Calculationsor Proofs) Let I be an arbitrary, given set of formulae. A theorem-calculation (or proof) from I is any finite (ordered) sequence of for- mulae that we may write respecting the following two requirements: In any stage we may write down Pr1 Any member of A or Pr2 Any formula that appears in the denominator of an instance of a rule Inf1-Inf2 as long as all the formulae in the numerator of the same instance of the (same) rule have already been written down at an earlier stage We may call a proof from I by the alternative name T-proof