Sentence:  If T is a linear operator on an arbitrary vector space and if there is a monic polynomial p such that * * f , then parts ( a ) and ( b ) of Theorem 12 <WSD>are</WSD> valid for T with the proof which we gave . 
Meaning of the word marked with <WSD> tags:  have the quality of being; (copula, used with an adjective or a predicate noun)