Sentence:  If ( remember this is an assumption ) the minimal polynomial for T decomposes * * f where * * f are distinct <WSD>elements</WSD> of F , then we shall show that the space V is the direct sum of the null spaces of * * f . 
Meaning of the word marked with <WSD> tags:  an artifact that is one of the individual parts of which a composite entity is made up; especially a part that can be separated from or attached to a system