I am reading Fulton's book of algebraic curves and on page 37 there is a theorem 3, which says:
The truth is that the proof seems very long and difficult to follow (I am a beginner in algebraic geometry) so I would like to know if there is another proof that is easier to understand and shorter than this. I would appreciate if you could give me references or an answer explaining what it is a better prove, thank you very much!
Not sure whether this will help, but you could give this link a try. You will find your statement as Thm. 4.4 and the proof seems to be essentially the same as the one in Fulton's book just with more details (the author probably used the book by Fulton as a source). Therefore it might be easier for you to follow. The downside is obviously that giving more details does not make the proof shorter though.