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Prove 0 = 1
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![\underline{Substracting \: \: \: b {}^{2} \: \: \: from \: \: both \: \: sides \: \: of \: \:eq. (2)} \\ \\ We \: \: get ,\\ \\ a {}^{2} - b {}^{2} = ab - b {}^{2} \\ \\ = > (a + b)( a- b) = b(a - b) \\ \\ = > a + b = b \\ \\ = > a + a = a \: \: \: \: [as \: \: \: a = b \: \: from \: (1)] \\ \\ = > 2 a= a \: \: \: \\ \\ = > 2 = 1 \\ \\ = > 2 - 1 = 1 - 1 \\ \\ = > 1 = 0 \\ \\ = > 0 = 1 \underline{Substracting \: \: \: b {}^{2} \: \: \: from \: \: both \: \: sides \: \: of \: \:eq. (2)} \\ \\ We \: \: get ,\\ \\ a {}^{2} - b {}^{2} = ab - b {}^{2} \\ \\ = > (a + b)( a- b) = b(a - b) \\ \\ = > a + b = b \\ \\ = > a + a = a \: \: \: \: [as \: \: \: a = b \: \: from \: (1)] \\ \\ = > 2 a= a \: \: \: \\ \\ = > 2 = 1 \\ \\ = > 2 - 1 = 1 - 1 \\ \\ = > 1 = 0 \\ \\ = > 0 = 1](https://tex.z-dn.net/?f=%5Cunderline%7BSubstracting+%5C%3A+%5C%3A+%5C%3A+b+%7B%7D%5E%7B2%7D+%5C%3A+%5C%3A+%5C%3A+from+%5C%3A+%5C%3A+both+%5C%3A+%5C%3A+sides+%5C%3A+%5C%3A+of+%5C%3A+%5C%3Aeq.+%282%29%7D+%5C%5C+%5C%5C+We+%5C%3A+%5C%3A+get+%2C%5C%5C+%5C%5C+a+%7B%7D%5E%7B2%7D+-+b+%7B%7D%5E%7B2%7D+%3D+ab+-+b+%7B%7D%5E%7B2%7D+%5C%5C+%5C%5C+%3D+%26gt%3B+%28a+%2B+b%29%28+a-+b%29+%3D+b%28a+-+b%29+%5C%5C+%5C%5C+%3D+%26gt%3B+a+%2B+b+%3D+b+%5C%5C+%5C%5C+%3D+%26gt%3B+a+%2B+a+%3D+a+%5C%3A+%5C%3A+%5C%3A+%5C%3A+%5Bas+%5C%3A+%5C%3A+%5C%3A+a+%3D+b+%5C%3A+%5C%3A+from+%5C%3A+%281%29%5D+%5C%5C+%5C%5C+%3D+%26gt%3B+2+a%3D+a+%5C%3A+%5C%3A+%5C%3A+%5C%5C+%5C%5C+%3D+%26gt%3B+2+%3D+1+%5C%5C+%5C%5C+%3D+%26gt%3B+2+-+1+%3D+1+-+1+%5C%5C+%5C%5C+%3D+%26gt%3B+1+%3D+0+%5C%5C+%5C%5C+%3D+%26gt%3B+0+%3D+1)

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