Доказать, что [latex] \sqrt{11...1 - 22...2}=33...3 [/latex]. 11....1-100 раз, 22...2-50 раз и 33...3-50 раз

[latex] \sqrt{ \underbrace{11...1}\limits_{100} - \underbrace{22...2}\limits_{50}} = \underbrace{33...3}\limits_{50}, \\ \sqrt{\underbrace{11...1}\limits_{50}\cdot(1\underbrace{00...0}\limits_{49}1 - 2)} = \sqrt{\underbrace{11...1}\limits_{50}\cdot\underbrace{99...9}\limits_{50}} = \sqrt{\underbrace{11...1}\limits_{50}\cdot\underbrace{11...1}\limits_{50}\cdot9} = \\ = \sqrt{(\underbrace{11...1}\limits_{50})^2\cdot9} = \underbrace{11...1}\limits_{50}\cdot3 = \underbrace{33...3}\limits_{50}[/latex]