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Explain the Proof Step-by-step : If n is not a perfect square, then n^(1/2) is irrational.

Recall that a perfect sqaure is a natural number n such that n = (k^2), for some natural number k.
Theorem.
If the natural number n is not a perfect square, then n^(1/2) is irrational.

Proof.
S(1):
Suppose n^(1/2) = r/s for some natural numbers r and s.

S(2):
We may assume that r and s have no prime factors in common, since any
common prime factors may be cancelled.

S(3):
From the first step, we have (s^2)*n = r^2.

S(4):
Suppose that s>1 and p is a prime factor of s.

S(5):
Then p is a prime factor of s^2.

S(6):
Hence p is a prime factor of r^2 = (s^2)*n.

S(7):
It follows that p is a prime factor of r.

S(8):
This contradicts our assumption that r and s have no prime factors in
common, and so s = 1.

S(9):
Therefore, n = (r^2), so n is a perfect square.

Explain all S(1) ~ S(9)

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