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PDF-2CLAYSHONKWILERTherefore,theabovecontradictionimpliesthatthereexist ; PDF document

xj 0det yi xj 0LetcbeapathinVfromq12U Vtoq22U VsuchthatcdoesnotpassthroughpThentheU covercandciscompactsoa nitesubcollectionU1UkcoverscwhereU1U andUkU Sincedetyi xj10anddety

@xj !0det @yi @xj !0:LetcbeapathinVfromq12U \Vtoq22U \Vsuchthatcdoesnotpassthroughp.ThentheU covercandciscompact soa nitesubcollectionU1;:::;Ukcoversc whereU1=U andUk=U .Sincedet@yi @xj10anddet@y

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2CLAYSHONKWILERTherefore,theabovecontradictionimpliesthatthereexist ;: Transcript

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