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O. Assume F - I' 3 " * o. ",+ Hom (x, c-(r-1* and therefore o. e. =g-a-(r-1*t indecomposabre. 8), t'E" s instead This shows finally t considering c*(r+n-1't" 'r" = o. Now neighbours = O for all of s in 1< s1n. b) Similar. 12 Corollary: and J= ( f, b) P= c) statements 1

2 Proposition: indeconposable, Then there -Fk---€. \-L and assume dim Z = dim X + dim y. - I-<1. Let y be We may assume pos(I) O and si ... u .. e, because s. y ^1 - > O. Thus Z ' = s-;r . . s-]1 z + o . 1) yields an exact sesuence -54- O+4,- sl- ...

E. {,n) a is an exact sequence O - Il dim Y = for such that dim Z = dim y + dim X1 O r Now Y a c- - {. . hence, is by assumption, Y the + I + t + e - Z a X] +O propositl,on a sequence c such that Y, = 7 lr/\-r is Assume we have proved the Therefore there is di-n y = I, there a root. Less of generalety is a root. ,d}. ,Id} if an and a permuta- 1 < t < d; all = is root \"rritten " afrla I S to S d such that is Z-% that Z such that An elementary property If of submodules of ! a sequence o = kt . k2 .

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