process

98 F. Muro and A. TonksTherefore we obtain the corresponding relationsŒA X D 0 C Œl A Œw A ŒA0 D ŒA 0 X A 000 _ D 0 C ŒA 000 A 000 _ D 0 D 0 ŒA0 ;ŒB 0 X A 000 C Œl 0 Œw 0 ŒA0 D ŒA 0 X A 000 _ D 0 C ŒD 0 A 000 _ D 0 A 000 ŒA0 ;ŒX B D 000 C ŒA X D 0 D Œl Œw ŒA C ŒD 0 D D 000 ŒAD Œl C .Œl C Œw C ŒC 0 Œw 000 ŒC 0 Œw 0 / ŒA ;ŒX B D 000 C ŒB 0 X A 000 D Œl B Œw B ŒB0 C ŒA 000 B 000 D 000 ŒB0 D Œl B C .Œl 00 Œw 00 ŒA00 Œw D ŒA00 Œw A / ŒB0 :(c)(d)(e)(f)Here we have used equations (a) and (b). Now .c/ C .d/ and .e/ C .f/ yieldŒw A ŒA0 Œl A ŒA X D 0 C ŒB 0 X A 000 C Œl 0 Œw 0 ŒA0 D ŒA 000 A 000 _ D 0 D 0 ŒA0 C ŒD 0 A 000 _ D 0 A 000 ŒA0 DhŒD 0 ; ŒA 000 i;(g)ŒA X D 0 C ŒB 0 X A 000 D Œw 0 ŒA C a ŒC 0 CŒAŒl C ŒA Œl C Œl B C Œl 00 ŒB0 a ŒA00 CŒB 0 (h)Œw A ŒB0 ;where we use (R9) and Remark 1.2, and we write a D Œw 000 C Œw C D Œw D C Œw 00 ,by (R6). In fact it is easy to check by using Remark 1.2 and the laws of stable quadraticmodules that the terms a ŒC 0 CŒA and a ŒA00 CŒB 0 cancel in this expression, since@ Œl C ŒA Œl C Œl B C Œl 00 ŒB0 D .ŒC 0 C ŒA/ C .ŒA 00 C ŒB 0 /:Substituting (h) into the left hand side of (g) we obtainŒw A ŒA0 Œl A C Œw 0 ŒA Œl C ŒA Œl C Œl B C Œl 00 ŒB0 DhŒD 0 ; ŒA 000 i;Œw A ŒB0 C Œl 0 Œw 0 ŒA0 which, using again Definition 1.1 and Remark 1.2, may be rewritten asŒl A Œl C ŒA Œl C Œl B C Œl 00 ŒB0 C Œl 0 D Œw 0 ŒA00 CŒA 0 Œw A ŒA0 hŒA 000 ; ŒD 0 iCŒw 0 ŒA0 C Œw A ŒC 0 CŒA 0 :The result then follows from the identityh@Œw A ; @Œw 0 ihŒA 00 ; @Œw 0 iChŒC 0 ; @Œw A iDhŒC 0 ; ŒA 00 iChŒA 000 ; ŒD 0 i: