process

112 F. Muro and A. TonksNow we are ready to prove Theorem 7.1.Proof of Theorem 7.1. We define the homomorphism by89 8p ˆ= ˆ; ˆ: A _ A1A _ A A;AHere the second equality follows from (S1) and (S2) applied to the following pairs ofweak cofiber sequences9>=:>; A 11 A A;1A 1 ; A _ A 1 1 A _ A A _ A;1A _ A A;Ap 1 A 1 A i 2 A _ A A;i 1 p 2A 1A 1 1 A A;1A 1p 1 A 1 A i 2 A _ A A:i 2 p 1A 1Given a cofiber sequence A j B r C the equation ŒCCŒA D ŒB followsfrom (S1) and (S2) applied to the following pairs of weak cofiber sequencesp 1 A 1 A i 2 A _ A A;i 1 p 2A 1r C 1 A j B j C;rC 1A _ A p 1 B 1 B i 2 B _ B B;i 1 j _j j _jp 2B 1r_r C _ C 1 B _ B C _ C;r_rC _ C 1p 1 C 1 C i 2 C _ C C;i 1 p 2C 1r C 1 A j B j C:rC 1Given a weak equivalence w W A ! A 0 the equation ŒA D ŒA 0 follows from(S1) and (S2) applied to the following pairs of weak cofiber sequences ; A 0 w 1 A 0 A;1A 0 w A 0A 0 1 i p 1 2 A i 0 _ A 0 A10 ; p 2A 0 1A 0 _ A 01 w_w A 0 _ A 0 A _ A;1A 0 _ A 0 w_wp 1 A 1 A i 2 A _ A A;i 1 p 2A 1A 0 w 1 A 0 A:1A 0 w