In german what is the use of the word "mal" in sentences like these?

In your examples, the word "mal" cannot be translated. These are simply idiomatic expressions in which "mal" adds a kind of emphasis. This is not unlike "just" in English (see third example). This is a part of language learning that is difficult to teach, you simply have to pick it up by hearing it and considering the context. There are several other German words which are used this way including "wohl", "aber", "ja", "doch". Each has a dictionary meaning, but also has uses which can't be easily translated.

Mal is often short for einmal especially when it's used to modify a period of time. So in the first statement what you're saying is actually "It goes to prove once again." 'Mal' is modifying again to produce the repetition of time. Your second phrase uses mal to modify the period of time again by functioning as a noun "It's just like when..." or "It's just like." The third phrase uses mal to modify the period of time as an adjective. Denk mal an translates literally to "Think about it for some time." The final phrase uses mal to say "for once". It translates to "Be serious for once." or "But, be serious for a moment." Mal is used in these sentences to define the period of time in which the action occurs or is to occur.

Mal means times or time. But it depends on what context or sentence you put it in. The german language is hard to grasp. Sometimes two words mean different meanings.

its used poliety so senrances dont sound so hard. sometimes its just added to the sentance to be less harsh and not sound so demanding. its a weird concept since its an actual work but if you go to germany, its used a lot in phrases that could sound demanding or not casual enough

We need it because without it the meaning would be different. It can't be readily said in english.As you progress in german, you will learn to feel the meaning of this word and several others. Several of them are listed on wikipedia under "Modalpartikel", with some explanation

usually it means time

Mal is often short for einmal especially when it's used to modify a period of time. So in the first statement what you're saying is actually "It goes to prove once again." 'Mal' is modifying again to produce the repetition of time. Your second phrase uses mal to modify the period of time again by functioning as a noun "It's just like when..." or "It's just like." The third phrase uses mal to modify the period of time as an adjective. Denk mal an translates literally to "Think about it for some time." The final phrase uses mal to say "for once". It translates to "Be serious for once." or "But, be serious for a moment." Mal is used in these sentences to define the period of time in which the action occurs or is to occur.

Mal means times or time. But it depends on what context or sentence you put it in. The german language is hard to grasp. Sometimes two words mean different meanings.

We need it because without it the meaning would be different. It can't be readily said in english.As you progress in german, you will learn to feel the meaning of this word and several others. Several of them are listed on wikipedia under "Modalpartikel", with some explanation