A CHARACTERISTIC PROPERTY OF SURFACES OF NEGATIVE CURVATUREf

From the definition of the "terms" of a phrase, it is clear that if a phrase is free of parentheses all its terms are letters. Hence, by II , any phrase-equation E whose members are free of parentheses is interderivable with any phrase-equation resulting from permutation of non-initial letters within members of E. Therefore any phrase-equation whose members are free of parentheses and have like initial letters is interderivable with an equation of canonical form. It then follows, by III and IV, that every phrase-equation is reducible to canonical form. In view of §§3-4, this concludes the proof that upon elimination of abbreviations all homogeneous linear identities with rational coefficients are generable by (R) and (R') from (A) and (B).