请帮忙分析下面这句英文的语法，谢谢！

Question

I cherish a belief that rewards, although many people do good deeds not to gain any reward, will gradually and eventually increase the number of people who expect to help others.
请问这句是状语从句吗？中间的小句子是句完整的话么？里面not to gain any reward作什么成分呢？

Answer 1

答：插入语，也是状语从句。去掉它：
I cherish a belief that rewards will gradually and eventually increase the number of people who expect to help others.
则句子结构就清楚多了：I cherish a belief +that-同位语从句。我抱有这么一种信念，那就是。。。。
分析一下这个同位语从句：
主 语：rewards
谓语：will increase
宾语：the number of people
定语从句：who expect to help others修饰people
状语：gradually and eventually 修饰谓语动词。追问

中间的小句子是句完整的话么？里面not to gain any reward作什么成分呢？

追答

那就是：奖赏将会让那些想帮助他人的人越来越多。
当作者讲到奖赏时，突然想到：尽管很多人并不是为了奖赏而去做好事的。
这个是让步状语从句，不叫小句子。though是连词，引导从句：
many people do good deeds not to gain any reward
主语：many people
谓语：do
宾语：good deeds
状语：not to gain any reward （目的，否定，不是为了。。。）

Answer 2

although many people （who）do good deeds not to gain any reward,
这里是不是应该把not to gain any reward理解为动词不定式作主语补足语
我也不是很确定，期待标准答案中O(∩_∩)O~

Answer 3

这句话首先有一个that 引导的同位语从句（做belief的同位语），在同位语从句里面有一个让步状语从句“rewards will gradually and eventually increase the number of people who expect to help other, although many people do good deeds not to gain any reward”，注意这个从句的前半句还有一个定语从句“who expect to help others”。“not to gain any reward”是一个表目的的短语