"Lamb's book of life" is only mentioned once n the Book of Revelation, then "book of life of the Lamb" is also mentioned once.
"Book of life", on the other hand, is mentioned six times.
A believer's name will not be removed from the Book of life . Revelation 3:5 Believers' names....
Will never be removed....
Rev 3:5
He that overcometh, Who is an overcomer? One who has faith....a believer. 1 John 5:4
the same shall be clothed in white raiment;
and I will not blot out his name out of the book of life, Whose name will He NOT blot out? 1 John 5:4....a believer.
but I will confess his name before my Father, and before his angels. Faith...overcomes the world....an "overcomer" cannot be lost because he is born of God (1 John 5:4) Unbelievers' names
However, the above verse, by it's very mention does convey the possiblilty of names being blotted out..... unbelievers' names.
Therefore, my opinion, is that everyones' name is originally written in the "book of life" and unbeliever's names get blotted out. Here's some more verses for you to read.
Exodus 32:32-33.
32Yet now, if thou wilt forgive their sin--; and if not, blot me, I pray thee, out of thy book which thou hast written.
33And the LORD said unto Moses, Whosoever hath sinned against me, him will I blot out of my book. Your name has to already be there, for it to be blotted out.
Psalm 69:28,
Let them be blotted out of the book of the living, and not be written with the righteous. Same here
Rev 22:19
And if any man shall take away from the words of the book of this prophecy, God shall take away his part out of the book of life, and out of the holy city, and [from] the things which are written in this book. You must already have "a part" in the book of life, for it to be taken away.