Order 8 of the following sentences so that they form a logical proof by contradiction of the statement:

If the sum of two integers is even then they have the same parity.

Choose from these sentences:
1. $\exists k, j \in \mathbb{Z}$ such that $x = 2k+1$ and $y = 2j$
2. Assume $x + y$ even implies $x$ and $y$ have the same parity.
3. Without loss of generality, assume $x$ is odd and $y$ is even
4. Presume the provided statement is false.
5. Let $x$ and $y$ be integers with the same parity but with an odd sum.
6. Parity is not knowable without a paring knife
7. $x + y$ is odd
8. Let $x$ and $y$ be opposite parity integers with even sum.
9. $x$ is even and $y$ is odd or $x$ is odd and $y$ is even
10. Hence $x$ and $y$ have the same parity
11. $x + y = 2k+1 + 2j = 2(k+j) + 1$
Your Proof:
Choose from these sentences:
1. $\exists k, j \in \mathbb{Z}$ such that $x = 2k+1$ and $y = 2j$
2. Assume $x + y$ even implies $x$ and $y$ have the same parity.
3. Without loss of generality, assume $x$ is odd and $y$ is even
4. Presume the provided statement is false.
5. Let $x$ and $y$ be integers with the same parity but with an odd sum.
6. Parity is not knowable without a paring knife
7. $x + y$ is odd
8. Let $x$ and $y$ be opposite parity integers with even sum.
9. $x$ is even and $y$ is odd or $x$ is odd and $y$ is even
10. Hence $x$ and $y$ have the same parity
11. $x + y = 2k+1 + 2j = 2(k+j) + 1$
Your Proof: