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But sometimes, for flavor, they can get a little more smug, such as,

"indeed," - my topology professor. You had to hear him say it.

"clearly," - pretty smug

"it is clear that"

then, we start getting into the more epic:

"Bear witness to the fact that,"

And these can even be combined:

"Indeed, let us bear witness to the fact that"

But perhaps the most epic one of all, was an algebra professor at my school (Arturo Magidin, for the math.SE posters):

"We let x stand sentinel to the fact that"

Wow, straight out of JRR Tolkein.

So, for fun, I want to see what you can come up with. For the sake of participation, I have included a small proof in which you can "fill in the blanks" with the most flamboyant, pompous proof phrases you can think of.

example:Claim: Any multiple of an even integer is even.

Proof:

Let k be an even integer.

______, there is exist an integer n such that k = 2n.

______, let a be an integer. Then consider ak.

______, ak = 2(an).

______, ak is even by definition

Let k be an even integer.

Indubitably, there is exist an integer n such that k = 2n.

To further embark on our quest for truth, let a be an integer. Then consider ak.

It is abundantly clear as the full moon on a pale October night, ak = 2(an)

Rightfully so, it is indeed shown for all to see that ak is even by definition