Question

Prove that:
$open vertical bar table row straight a cell space space space straight b minus straight c end cell cell space space space straight c minus straight b end cell row cell straight a minus straight c end cell straight b cell space space space space straight c minus straight a end cell row cell straight a minus straight b end cell cell space space space straight b minus straight a end cell cell space space space straight c end cell end table close vertical bar space equals space left parenthesis straight a plus straight b minus straight c right parenthesis thin space left parenthesis straight b plus straight c minus straight a right parenthesis thin space left parenthesis straight c plus straight a minus straight b right parenthesis$

Solution

Let $increment space equals space open vertical bar table row straight a cell space space space straight b minus straight c end cell cell space space space straight c minus straight b end cell row cell straight a minus straight c end cell straight b cell space space space space straight c minus straight a end cell row cell straight a minus straight b end cell cell space space space straight b minus straight a end cell cell space space space straight c end cell end table close vertical bar space$
$equals space open vertical bar table row cell straight a plus straight b minus straight c end cell cell space space straight b minus straight c end cell 0 row cell straight a plus straight b minus straight c end cell straight b cell space space space straight b plus straight c minus straight a end cell row 0 cell space straight b minus straight a end cell cell space straight b plus straight c minus straight a end cell end table close vertical bar comma space by space straight C subscript 1 plus straight C subscript 2 plus straight C subscript 3 plus straight C subscript 2 space equals space left parenthesis straight a plus straight b minus straight c right parenthesis thin space left parenthesis straight b plus straight c minus straight a right parenthesis thin space open vertical bar table row 1 cell space space space space space straight b minus straight c end cell cell space space 0 end cell row 1 cell space straight b end cell cell space space 1 end cell row 0 cell space space space straight b minus straight a end cell cell space space 1 end cell end table close vertical bar space equals space left parenthesis straight a plus straight b minus straight c right parenthesis thin space left parenthesis straight b plus straight c minus straight a right parenthesis space open vertical bar table row 1 cell space space space straight b minus straight c end cell cell space 0 end cell row 0 straight c cell space 1 end cell row 0 cell space straight b minus straight a end cell cell space 1 end cell end table close vertical bar comma space space space by space straight R subscript 2 space minus straight R subscript 1 equals space left parenthesis straight a plus straight b minus straight c right parenthesis thin space left parenthesis straight b plus straight c minus straight a right parenthesis thin space open vertical bar table row straight c cell space 1 end cell row cell straight b minus straight a end cell cell space 1 end cell end table close vertical bar equals space left parenthesis straight a plus straight b minus straight c right parenthesis thin space left parenthesis straight b plus straight c minus straight a right parenthesis thin space left parenthesis straight c minus straight b plus straight a right parenthesis space equals space left parenthesis straight a plus straight b minus straight c right parenthesis thin space left parenthesis straight b plus straight c minus straight a right parenthesis thin space left parenthesis straight c plus straight a minus straight b right parenthesis therefore space space space space open vertical bar table row straight a cell space space straight b minus straight c end cell cell straight c minus straight b end cell row cell straight a minus straight c end cell straight b cell straight c minus straight a end cell row cell straight a minus straight b end cell cell straight b minus straight a end cell straight c end table close vertical bar space equals space left parenthesis straight a plus straight b minus straight c right parenthesis thin space left parenthesis straight b plus straight c minus straight a right parenthesis thin space left parenthesis straight c plus straight a minus straight b right parenthesis$

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