How to make oia appeal

I was expelled from the University of Sheffield due to the failure on one resit exam after have repeated once. I have devoted to much on my study in mathematics, and I also have made appeal inside school twice, but they all turned me down, I wonder why I should be treated cruelly, I had always been working hard to complete my study but no one helped me during the school, if they think I am unqualified, why they did not refuse me to repeat at first? Why they gave me hope and then tore it into pieces?

They DID give u a chance to repeat and I think that's fair.
Check the school policy about this issue (academic resit), more likely than not it should be clearly written there.
Unless you have any extenuating circumstances that is particularly compelling (you have to provide proof for that), I think all you can do is to reflect on why you could not satisfy the requirements having been given another chance.
No matter what, I know this is harsh, but this is the reality of life.

This is entirely standard at basically every institution in Britain.

I have heard a lots of students facing the similar or even more terrible situation as me, also try to have another chance to complete their study, so I do not care whether it is acceptable to the British or not, one thing is certain: I will be doomed without postgraduate degree in China, my mother said she want to kill me almost everyday.

Sure, but more opportunities are only permitted at undergraduate, and even then with caveats.
Postgraduate degrees have a higher pass threshold and generally allow only two attempts at any given exam - a first sit, and one resit.
Unless you are not telling us something, it is not the fault of the university that you failed twice. I appreciate that it is a tough pill to swallow but it is true. I am 99% certain that any appeal will fall flat.

Yes, it is my fault not telling you all. My appeal is that I want repeat, not resit, I just want to finish my study, at least the school should allowed me to have second repeat, which my tutor have told me I might have last year, before my first repeat ended.

See above.

Why they cheat me? And twice?

See above.

Oh, I see, they just thought I was too stupid to believe their words, and actually I did, I bought that I could have second repeat, and I bought that at least I can have diploma. And no one think they did wrong to a student with over-zealous passion for study, it is alright to get rid of her, because the dump girl is so annoying!
That should be my most valuable EDUCATION in UK.

I'm not sure what distinction you are making. At postgrad only two attempts are allowed so it sounds like procedure has been followed correctly

I mean, they have told me that I can have another repeat, I should not ask for resit in the summer, and after I failed on one exam again, they told me I can have diploma, I should not try to make academic appeal.

I'm not in a position to judge any of this. Sometimes there are possibilities at one stage which are ruled out as time goes on, not everything is a stitch up.
I ask.again, what distinction are you.making between a resit and a repeat? What is the difference?

Not a trap? So I can kill someone and said "there is probability that he might alive" to get away from death penalty? If the school believe us should know every rules, the tutors should understand the rules better than us.
The difference between resit and repeat is that latter costs me a lot money and time.

You are being too obtuse for me to offer anything of substance. Calm down for a minute and actually reply to what I am asking you instead of being a smart arse. There is no point being angry at me. Do you mean a full year repeat, and am I to undsrstand that you have already done this once?
Tutors rarely know the regulations - they have administrators for that, and that is a sad fact. I see this every day. Because my specialism.is student support, I am, by far, the best versed in my departmental and school-level regulations out of 60-odd academics in my dept. The rest are cclueless. This is effrctively the same everywhere.