Here's a wittle eckshooshize fow wittle hawtshowners... awwww yu widdle cuuute hawtshown babby. your soooooo cute, you wittle wittle ting. da excshooshishe mwenshionss tweerum twee pwoint fwo, swo hewes a pictwor owf dat:



Wow! Wut a wovewy wittle teowem, ishnt dat!? So cool! Its swo cool! Cool! Nwow, lwets lwet U_i = Pⁿ - H_i and U_j = Pⁿ - H_j. Swo weve gwot, and wet W = Pⁿ - (H_i ∩ H_j), swo W = U_i ∪ U_j.

Now wet f bwe a rweg fwunc on W, swo obwoswy f is awso rweg on U_i and U_j. Now since U_i ≃ Aⁿ and U_j ≃ Aⁿ awe iswomophic two affwines (pwop twoo pwoint twoo), wid cwoodinnade wing iswomowfisams A(U_i) ≃ S(Pⁿ)_xi = k[x₀,…,x_n]_xi and A(U_j) ≃ S(Pⁿ)_xj = k[x₀,…,x_n]_xj wepectivwy. dis is ewabowated on in da pwoof of twee pwoint fwo.

What? What did you say? You... didn't read the proof of theorem 3.4? YOU..... SKIPPED THE THEOREM 3.4 PROOF? BAD! BAD! YOUR <WHOEVER IS THE MOST THREATENING FORCE IN YOUR EXISTENCE> IS GOING TO HEAR ABOUT THIS. HOW DARE YOU. YOU HAVE TO APOLOGIZE RIGHT NOW. RIGHT NOW, YOUNG MAN. IF YOU DON'T APOLOGIZE, YOU'RE GONNA BE GROUNDED. WE'LL TAKE AWAY ALL YOUR VIDEO GAMES, AND WE'RE CANCELLING YOUR BIRTHDAY PARTY. YOU DON'T WANT THAT, DO YOU? YOU BETTER APOLOGIZE. WHAT? WHY DO YOU HAVE TO APOLOGIZE? DON'T ASK QUESTIONS, YOUNG SIR. YOU DON'T HAVE TO UNDERSTAND THE ACTUAL PURPOSE OF APOLOGIZING. YOU APOLOGIZE BECAUSE I TOLD YOU SO. WHY AM I CALLING YOU YOUNG MAN WHEN I'M TREATING YOU LIKE A CHILD? IT'S CALLED IRONY. DIDN'T THEY TEACH YOU THAT AT SCHOOL. UGH, THAT'S THE OTHER THING. WHY ARE YOUR GRADES SO LOW? YOU NEVER STUDY. ALL YOU DO IS PLAY, PLAY, PLAY. WHY CAN'T YOU BE MORE LIKE ALYSSA? SHE STUDIES SO MUCH. SHE'S SUCH A BOOKWORM AND SUCH A GOOD KID. WHY CAN'T YOU JUST SIT FOR 5 MINUTES AND DO YOUR HOMEWORK? YOU MISSED ANOTHER MATH ASSIGNMENT YESTERDAY. I SAW IT ON SCHOOLOBOOOLOHOOLOSCOPY. AT LEAST IT WOULD BE FINE IF YOU WERE INTO SPORTS OR SOMETHING. BUT ALL YOU DO IS SIT INSIDE ALL DAY. AND YOU HAVE NO FRIENDS EITHER. WHY DON'T YOU HAVE FRIENDS? JUST GO UP TO YOUR CLASSMATES AND SAY HI. JUST. SAY. HI. IT'S THAT SIMPLE. THAT'S HOW IT WORKED BACK IN MY DAY. WHY DON'T YOU DO THAT? AND WHY ARE YOU SO THIN? YOU HAVE TO EAT MORE. YOU NEVER EAT ENOUGH. WHY? ARE YOU HAVING EATING PROBLEMS? I'M WORRIED ABOUT YOU. YOU CAN COME TO ME, YOU KNOW? I CAN HELP YOU. I'M HERE TO HELP YOU, CAN'T YOU SEE? WHY ARE YOU LIKE THIS? CAN YOU ANSWER ME THAT? WHY ARE YOU LIKE THIA? WHY? *starts crying* I'VE DONE SO MUCH FOR YOU. WHY ARE YOU LIKE THIS? WHERE DID I GO WRONG WITH YOU? YOU KNOW WHAT? YOU NEEDED MORE DISCIPLINE. YOU'RE SPOILED. I SHOULD TREAT YOU LIKE ALAN'S MOM. WHY CAN'T YOU BE MORE LIKE ALAN? IT'S BECAUSE I'M TOO NICE. I'M JUST TOO NICE. I SHOULD HAVE BEEN STRICTER. *sobs* WHY DID YOU TURN OUT LIKE THIS? OH, WHY, GOD. HELP ME. IT'S MY FAULT ISN'T IT. IT'S ALL MY FAULT. *sobs uncontrollably*

"I'm sorry."

.....

"I said I'm sorry."

.....

"I said I'm sorry. pls respond."

....I'm not talking to you.

"I SAID I'M SORRY."

.....

"SORRY."

.....

"SORRY. I'M SORRY I'M SORRY I'M SORRY. DON'T IGNORE ME FOR AN INDEFINITE PERIOD OF TIME AGAIN."

.....Then you better listen to me. Say "I'll never do it again."

"I'll never do it again."

Promise?

"Promise"

Good child. Yes. Perfect. We learned absolutely nothing. Now, SHUDDEN SWITCH BWACK TO CONDESHENDING BWABBY TWALK (swo i cwan confoose you even mwore on the stwate of your own maturity). Now wets cwontinwue dwe exshooshiaishe, shwall we? *Hug*

Nwow, where wer we? Wite. f cwan bwe witten as f = g∕(x_i^N) on U_i and f = h∕(x_j^M) on U_j, where g,h ∈ k[x₀,…,x_n]. Swo on U_i ∩ U_j, we cwan wite

(1)

g ⋅ xjM - h ⋅ x iN	= 0
g ⋅ xjM	= h ⋅ x iN

Pwugging dis bwack into (1), dwis gwives

(2)

bwut f and g are weguwar fwuncs won W. and gwuess wut. Wemembew wemawk twee pwoint won pwoint won?



Dwats wite! If f = g on U_i ∩U_j, den ackshauelly f = g on all of W. Swo cool! Cool, ishnt it!? I cwall evewyting COOL because i dont know how to mwake things intewesting fwor a chwild udder dan spwinkling it with words like COOL (underestimating chwildens intelligence and dwumbing tings down fwor them is RAD). Bwut wait. it gwets eeeeeewen COOLER! Wemember? g ∈ k[x₀,…,x_n], bwut the ownly rweguwar fwuncs fwom k[x₀,…,x_n] dat r reguwar on an owpn dwense swubswet owf Pⁿ awre cwonstant. So wewe dwone! COOL.

"Ummm, actually, my generation says 'based'."

Wuts dat? i didnt hear yew (im bad at listening). BUT YAAAAAY, WE FINISHD DA EXSHURSHISHE! Gwwwwwwwwwwwwwweat jwob! YWO GET A HI FIVE. YAAAAAAAAAAY.

"Y-yaaay... So... my birthday party?"

Well.. as long as yewr a GWOOD KWID, we wont twake away ur birthdway. ywes, between skwool and our household ywo have bwasically nwo autonomy and dwont gwet two wive ywour own wife, swo tha only twing yew hwave to lwook forward twoo awr these superfwicial cwarrots on stwicks lwike "bwirthday pwarties" dwat WE pwovide fwor you and hwype up fwor you, swo it fweels lwike the end of the world when we thweaten two twake it away. bwilliant schweme of contwol, ishnt it?

"Err. okay. I'm going out now."

WAIT. WHERE? WEMEMBER. YEW MWUST HWAVE YWOR SMWATFONE TUNED ON AT AWLL TWIMES AND IF YEW DWONT RWESPOND TO OWR TWEXTS, WE'LL PWANIC AND AWSSUME YOUVE BWEEN KWIDNAPPED AND CWALL DWA CWOPS. DWONT EVER TWAKE RWISKS. DWIS IS A NO RWISK HOUSHWOLD. SWAY IT WID ME. NO RWISK. NO RWISK. NO RWISK. NO RWISK. NO RWISK [phases into Gregorian chant] HEWICOPTER GO BWRRRRRR.