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application of step apply the collecting arbitrary assume by induction basic commutators basic products basic sequence Basis Theorem collected product collecting process combination of monomials commutators of weight congruence following CORNELL UNIVERSITY cu,cv define definition element f factor of 77 finite number finite product form a basis form an additive four steps free Abelian group free group group with free Hall is invalid Hence application integers integral linear combination investigation of step least one representation Lemma Mod Fn+1 Mod P5 Mod Pn+i monomials of degree non-zero element number of basic number of steps Pn+1 positive word product always yields products of weight Proof ring commutator rn monomials s.s.b. product sequence of basic simply ordered suppose terms ck thesis trouble can occur uncollected factor unique member unique representation v"1 Mod weight greater weight less weight n form whence y,vt yields a s.s.b.