Ahlfors says that once the existence of the quotient ab has been proven, its value can be found by calculating ab⋅ˉbˉb. Why doesn't this manipulation show the existence of the quotient?
ab=ab⋅ˉbˉb=aˉb⋅b−1⋅ˉb−1=aˉb⋅(bˉb)−1, the last term clearly exists since the thing being inversed is real.
Answer
There's a small logical hitch. If you don't know that ab exists, then you can't begin algebraically manipulating it as you've down to arrive at your last expression, because you don't know you won't arrive at nonsense.
Logically, this boils down to the fact that
X implies Y
is a true statement if both X and Y are false. Here you're starting with a statement X: "ab exists" and concluding that Y: "ab=aˉb.(bˉb)−1
So there's still the job of showing ab does exist.
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