Pythagorean Theorem is one of the most proven concepts in mathematics, and for the calculus to reason to it, proves the Calculus works. So also, the calculus is a proof in itself, being so founded in empirical knowledge, that the Pythagorean Theorem works. It's proving itself empirically. Just like the kindergarten proof we use with the apples, for addition and subtraction.

Calculus is an empirical method. Just like you can prove Quadratic Equations, by simply using it to do rudimentary cuts. So also Geometry. It's a different form of proof... more useful in many ways. But that's how Euclid proved his Theorems in Elements. It wasn't with algebra, but with a compass and ruler and piece of string.

As a basis, we know the Calculus and Algebra work. But, the fact that this method turns into the Pythagorean Theorem, is in itself a proof of Pythagorean Theorem, because Calculus is an empirical science, and is simply explaining the axiom of a Right Triangle. As it should. If it didn't, we'd have to question whether the method worked.

 In logic it's called a biconditional. And they would prove themselves by both being tautologies.

It's also premised on the notion of Equalities in Algebra and Calculus. Really the fundamental aspect of all mathematics. And since the Logic of Calculus explains the axiom of Pythagorean Theorem, we know the Calculus works, too.

At the base of the issue, it's proven through observation. Both. Calculus is one of the most proven things in history, and so is Pythagorean Theorem. It only makes sense that they'd both compliment each other.

Rooted in the empirical observations of both, and how each is so well proven, they both contiguously prove the other.

If their proof were purely deductive, it would be circular, but it's empirical. It's a posteriori and not a priori, therefore, it cannot be circular. The empirical fact is rooted in the Calculus observing the axiom of Pythagorean Theorem, which is proven over three hundred times. So it is itself trustworthy. So also, the axiom of Pythagorean Theorem is proven, by the fact that Calculus is an empirically proven method, used in all forms of mathematics, without which, no engineering could flourish.

 It should be standard at every college math program in the STEM field. It's an important concept that I'm trying to tease out of it. It is philosophy, so to speak, but one needs philosophy to truly understand mathematics. 

Just the sheer saliency of the proof, in that it must work. That's something the world sorely needs in modern day. One can apply the same reasoning to literature, and get true ideas, too. As we kind of are hitting a brick wall, because nobody understands math empirically. They understand it deductively, and that's why so many people are lost in the weeds, and are trying to reinvent algebra or number systems.

It uses calculus to define the parameterisations. It is an elegant proof of all mathematical concepts, at once. At least, I'll make sure it's understood that way.

Two black maidens set their minds to proof...
They do their math, and prove.
Calculus, they use, to prove Pythagorean Theorem,
Yet Calculus is proven so much, so
That Pythagorean Theorem is proven too!

It is a biconditional,
Two Tautologies that necessitate.
The very crux of Equalities,
And the very crux of all mathematics.
It is how, oh my souls,
That science knows.

Ride, Abraham! and pursue those Elamite foes.
Make haste to avenge Ur, to impose that Amorite King
Melchizedek, king of Salem, to instill Babylonian rule.

Read once.
Read twice.
Then labor over every line.
Look at every comma, 
And learn every word in time.
Find every allusion,
Find every hidden word---
Read over a lifetime,
And you shall know that verse.

Am I just an obsessed hobbyist?
Spending hours a day,
Surfing for inspiration...
Is it only money that makes a career?

Music,
Art,
Science,
Literature,
Poetry,
History,
Psychology,
Sociology,
Maths,

For what?

Would I have been better off
Learning how to play a video game real well?
To know nothing about our world?
Twiddle my thumbs,
And I could have made a fortune.
I could have been a good poker player,
Or a good chess player,
Or a good COD player,
Or I  could play pinochle,
Or I could play Fortnite,
Or I could have made a YouTube channel
Where I got really good at Mario or Rome Total War,
And made a fortune?
I could have beat myself,
Made a fool out of myself,
Like a Jackass
And give my exorbitant riches to the poor.

No, instead I chose to grow up.
And am punished for it,
By not being allowed to.

Do you not know? The robot does not dream?
Do you not know, that it plagiarizes you, and yes me?
The swirling subconsciousness of a hundred million artists
Get copied and pasted over a sentence or line of text.
The author of that text believes it is the magic of mind---
He believes it is intellect, and creativity of the machine.
No... it is the intellect and creativity of millions of artists
Which the AI picks and chooses, and warps
Around a few platitudes of thought.
And the author of such thought says, "I can not do better."
Yet, I, being an artist, see only the wish fulfillment
Of having one's immediate, and simplistic fancy
Shown as a flashy, surrealist portrait.

No, my loves, the true artist is you who wrote the prompt.
For that subconscious thought then gets woven
By the machine, to create what is pretty,
But not sublime.

I look at all the philosophers handed down through history,
And I wonder, "How simple it would be, if their teacher were Christ alone?"
Then, I suppose, there would not be any confusion about the most basic truths.

My vision is near complete.
The enemies, the hordes,
They break through...
Soon, I shall awaken from this dream.
And soon, I shall be at peace.

Brother and Sister of the True vine,
That Olive and Fig---
The Olive with his fatness,
The Fig with her sweetness;
Gideon and Deborah,
Elijah and Mary Magdalene;
What distinguishes thou?
When offered a crown, ye cast it;
Ye forsake the world, and worldly authority.
Oh, the Vine, when pierced---
You True Vine, you who merry the hearts of God and Man,
You too had been lifted up,
And would not take Your Kingdom with the Twelve Legions of Angels;
Not before you were lifted up.
Yet, the Bramble with his Shadow
Says, "Give me authority,
"And I shall guide ye well!"
And he is a fire which burns like hell's black flame.

I will never mystify you with how I create.
Call it an inner voice, call it a conscience,
Call it the voice of God...
But, I don't seek to persuade you that it's anymore than
Genius, like such which causes the grapes to grow
Or the performance to be good,
Or the wine to accent meat with its berry.
I will never use mystic words
Or try to dupe you into believing this comes from me.
It comes from practice---
Not some bold power of self will.
Just practice, and a little help from God.