process

On K 1 of a Waldhausen category 97such that the induced map j [ A 0 j B W A [ A 0 B 0 B is a cofibration. Denote thehorizontal weak cofiber sequences by l A , l B , l D and l C , and the vertical ones by l 0 , l,l 000 and l 00 . Then the following equation holds in D C.Œl A Œl C ŒA Œl C Œl B C Œl 00 ŒB0 C Œl 0 DhŒC 0 ; ŒA 00 i:Proof. Applying (R7) to the two obvious equivalences of cofiber sequences givesŒD 0 D D 000 D Œw C Œl C Œw C ŒC 0 Œw 000 ŒC 0 Œw 0 ;(a)ŒA 000 B 000 D 000 D Œw B C Œl 00 Œw 00 ŒA00 Œw D ŒA00 Œw A ;(b)using the notation of (1.1).Now consider the following commutative diagrams:A 0 j AA A 0 j AA j 0 B 0 push X jj 0 B 0 push X j BB,i 2 r 0i 1 r A A 000 _ D 0 .We now have four commutative diagrams of cofiber sequences as in (R8).c) D0d) A000A 000 A 000 _ D0A 0 A Xe) D 000D 0 DA X BD0 A 000 _ D0A 0 B0 Xf) D000A 000 B000B 0 X B