Normal form for free groups and free product of groups

In mathematics, particularly in combinatorial group theory, a normal form for a free group over a set of generators or for a free product of groups is a representation of an element by a simpler element, the element being either in the free group or free products of group. In case of free group these simpler elements are reduced words and in the case of free product of groups these are reduced sequences. The precise definitions of these are given below. As it turns out, for a free group and for the free product of groups, there exists a unique normal form i.e each element is representable by a simpler element and this representation is unique. This is the Normal Form Theorem for the free groups and for the free product of groups. The proof here of the Normal Form Theorem follows the idea of Artin and van der Waerden.
Normal Form for Free Groups.
Let formula_1 be a free group with generating set formula_2. Each element in formula_1 is represented by a word formula_4 where formula_5 
Definition. A word is called "reduced" if it contains no string of the form formula_6
Definition. A "normal form" for a free group formula_1 with generating set formula_2 is a choice of a reduced word in formula_2 for each element of formula_1.
Proof. An elementary transformation of a word formula_12 consists of inserting or deleting a part of the form formula_13 with formula_14. Two words formula_15 and formula_16 are equivalent, formula_17, if there is a chain of elementary transformations leading from formula_15 to formula_16. This is obviously an equivalence relation on formula_1. Let formula_21 be the set of reduced words. We shall show that each equivalence class of words contains exactly one reduced word. It is clear that each equivalence class contains a reduced word, since successive deletion of parts formula_13 from any word formula_23 must lead to a reduced word. It will suffice then to show that distinct reduced words formula_24 and formula_25 are not equivalent. For each formula_26 define a permutation formula_27 of formula_21 by setting formula_29 if formula_30 is reduced and formula_31 if formula_32. Let formula_33 be the group of permutations of formula_21 generated by the formula_35. Let formula_36 be the multiplicative extension of formula_37 to a map formula_38. If formula_39 then formula_40; moreover formula_41 is reduced with formula_42 It follows that if formula_43 with formula_44 reduced, then formula_45.
Normal Form for Free Products.
Let formula_46 be the free product of groups formula_47 and formula_48. Every element formula_12 is represented by formula_50 where formula_51 for formula_52.
Definition. A "reduced sequence" is a sequence formula_53 such that for formula_52 we have formula_55 and formula_56 are not in the same factor formula_47 or formula_48. The identity element is represented by the empty set.
Definition. A "normal form" for a free product of groups is a representation or choice of a reduced sequence for each element in the free product.
Proof.
Equivalence.
The fact that the second statement implies the first is easy. Now suppose the first statement holds and let: 
This implies 
Hence by first statement left hand side cannot be reduced. This can happen only if formula_71 i.e. formula_72 Proceeding inductively we have formula_73 and formula_74 for all formula_75 This shows both statements are equivalent.
Proof of (2).
Let be the set of all reduced sequences in and be its group of permutations. Define as follows: 
Similarly we define .
It is easy to check that and are homomorphisms. Therefore by universal property of free product we will get a unique map such that 
Now suppose formula_77 where formula_78 is a reduced sequence, then formula_79 Therefore in which contradicts .