Question

$prove space that space space increment space equals space open vertical bar table row cell straight a plus straight b space straight x end cell cell straight c plus straight d space straight x end cell cell space space space straight p plus straight q space straight x end cell row cell straight a space straight x plus straight b end cell cell straight c space straight x plus straight d end cell cell space space straight p space straight x plus straight q end cell row straight upsilon straight nu straight omega end table close vertical bar space equals space left parenthesis 1 minus straight x squared right parenthesis space space open vertical bar table row straight a cell space straight c end cell cell space space straight p end cell row straight b cell space space space straight d space space end cell cell space straight q end cell row straight u cell space straight v end cell cell space space straight w end cell end table close vertical bar$

Solution

space increment space equals space open vertical bar table row cell straight a plus straight b space straight x end cell cell straight c plus straight d space straight x end cell cell space space space straight p plus straight q space straight x end cell row cell straight a space straight x plus straight b end cell cell straight c space straight x plus straight d end cell cell space space straight p space straight x plus straight q end cell row straight upsilon straight nu straight omega end table close vertical bar space space space space space equals space open vertical bar table row cell straight a left parenthesis 1 minus straight x squared right parenthesis end cell cell space space straight c left parenthesis 1 minus straight x squared right parenthesis end cell cell space space space straight p left parenthesis 1 minus straight x squared right parenthesis end cell row cell straight a space straight x plus straight b end cell cell straight c space straight x space plus straight d end cell cell straight p space straight x plus straight q end cell row straight u straight v straight w end table close vertical bar space space by space straight R subscript 1 rightwards arrow straight R subscript 2 minus space straight x space straight R subscript 2 space space space space equals space left parenthesis 1 minus straight x squared right parenthesis space space open vertical bar table row cell straight a plus straight b space straight x end cell cell straight c plus straight d space straight x end cell cell space space space straight p plus straight q space straight w end cell row cell straight a space straight x plus straight b end cell cell straight c space straight x plus straight d end cell cell space space straight p space straight x plus straight q end cell row straight upsilon straight nu straight omega end table close vertical bar space space space space space equals space left parenthesis 1 minus straight x squared right parenthesis space space open vertical bar table row straight a cell space space straight c end cell cell space space space straight p end cell row straight b cell space space straight d end cell cell space space straight q end cell row straight upsilon cell space space straight nu end cell cell space straight omega end cell end table close vertical bar space comma space by space straight R subscript 2 rightwards arrow space straight R subscript 2 space minus space straight x space straight R subscript 1

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