By Peter L Duren; Richard Askey; Uta C Merzbach; Harold M Edwards

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6). We get (11. 8 ) u sing (11. 2), (11. 3 ), (6. 2 ). From (11. 8 ) we get not Free x cr c:~ S. Hen ce we o btain (11, 9 ) from (11. 7). - 34 - We get (11. 10) using (11. 1), (11. 2 ) and (6. 3). D. The generalized rules (it), (s t), (bY), (p ) . X (it) X X X X Identity (generalized rule) E =xt a (s t) X t X Substitution E a --..... _, (b~) x aX Renaming of bound variables (generalized r ule) , if not F ree y a (p ) X Partic ularization (generalized rule) - Exa a a-x J ustific a tion of (it) by t h e fo llowing derivation (in a similar way we may X t justi fy (s )) : X E a (assumption) =aa t X (11.

T f- =xt =s s-. X Let be v f x, and not Free v s and not Free v t. Then by (11. 8) not t Free v s-;z. We have the following de riv ation: ( 12. 10) f- =uv =vs =vs =v s =xt t =vs- =ss =xt t =ss- =xt t =ss- u v =s-sx X' (10. 1) (it) X X (ss), (11. 7) v X (10 . 3) X if U f X, V f X . T his i s shown by the following de rivation : u =xu =s s- (1 2 . 9 ) =xv v =s s- (1 2. 9 ) =xu =uv X X u v =s -sX X (10. 5), (10. 6), (10. 3) - 39 - (12. 11) u v =uu =uv =s-sx X =uv u v =s-s- x t u t xu ' f- =s-s-X Let be x, v Vf f f- =tlt2 Let be u l and not Free u s.

48: G. de Rham, S. Maumaryet M. A. Kervaire, Torsion etTypeSimple d 'Homotopie. IV, 101 pages. 1967. OM 9,60 I $ 2. 70 Vol. 49: C. Faith, Lectures on 'Injective Modules and Quotient Rings. 1967. 60 Vol. 50: L Zalcman, Analytic Capacity and Rational Approximation . VI, 155 pages. 1968. 70 Vol. 51: Seminaire de Probabilites II. IV, 199 pages. 1968. OM 14,-1 $ 3. 90 Vol. 52: D. J. Simms, Lie Groups and Quantum Mechanics. IV, 90 pages. 1968. 20 Vol. 53 : J. Cerl, Sur les diffeomorphismes de Ia sphere de dimension trois (14~ 0).