Question

Show that:
$open vertical bar table row cell straight b plus straight c end cell cell straight c plus straight a end cell cell straight a plus straight b end cell row cell straight b apostrophe plus straight c apostrophe end cell cell straight c apostrophe plus straight a apostrophe end cell cell straight a apostrophe plus straight b apostrophe end cell row cell straight b " plus straight c " end cell cell straight c " plus straight a " end cell cell straight a " plus straight b " end cell end table close vertical bar space equals space 2 open vertical bar table row straight a straight b straight c row cell straight a apostrophe end cell cell straight b apostrophe end cell cell straight c apostrophe end cell row cell straight a " end cell cell straight b " end cell cell straight c " end cell end table close vertical bar$

Solution

Let $increment space equals space open vertical bar table row cell straight b plus straight c end cell cell straight c plus straight a end cell cell straight a plus straight b end cell row cell straight b apostrophe plus straight c apostrophe end cell cell straight c apostrophe plus straight a apostrophe end cell cell straight a apostrophe plus straight b apostrophe end cell row cell straight b " plus straight c " end cell cell straight c " plus straight a " end cell cell straight a " plus straight b " end cell end table close vertical bar$
$equals space open vertical bar table row straight b cell straight c plus straight a end cell cell straight a plus straight b end cell row cell straight b apostrophe end cell cell straight c apostrophe plus straight a apostrophe end cell cell straight a apostrophe plus straight b apostrophe end cell row cell straight b " end cell cell straight c " plus straight a " end cell cell straight a " plus straight b " end cell end table close vertical bar space plus space open vertical bar table row straight c cell straight c plus straight a end cell cell straight a plus straight b end cell row cell straight c apostrophe end cell cell straight c apostrophe plus straight a apostrophe end cell cell straight a apostrophe plus straight b apostrophe end cell row cell straight c " end cell cell straight c " plus straight a " end cell cell straight a " plus straight b " end cell end table close vertical bar equals space open vertical bar table row straight b straight c cell straight a plus straight b end cell row cell straight b apostrophe end cell cell straight c apostrophe end cell cell straight a apostrophe plus straight b apostrophe end cell row cell straight b " end cell cell straight c " end cell cell straight a " plus straight b " end cell end table close vertical bar space plus space open vertical bar table row straight b straight a cell straight a plus straight b end cell row cell straight b apostrophe end cell cell straight a apostrophe end cell cell straight a apostrophe plus straight b apostrophe end cell row cell straight b " end cell cell straight a " end cell cell straight a " plus straight b " end cell end table close vertical bar plus space open vertical bar table row straight c straight c cell straight a plus straight b end cell row cell straight c apostrophe end cell cell straight c apostrophe end cell cell apostrophe straight a plus straight b apostrophe end cell row cell straight c " end cell cell straight c " end cell cell straight a " plus straight b " end cell end table close vertical bar plus open vertical bar table row straight c straight a cell straight a plus straight b end cell row cell straight c apostrophe end cell cell straight a apostrophe end cell cell straight a apostrophe plus straight b apostrophe end cell row cell straight c " end cell cell straight a " end cell cell straight a " plus straight b " end cell end table close vertical bar equals space open vertical bar table row straight b straight c straight a row cell straight b apostrophe end cell cell straight c apostrophe end cell cell straight a apostrophe end cell row cell straight b " end cell cell straight c " end cell cell straight a " end cell end table close vertical bar plus 0 plus 0 plus 0 plus 0 plus 0 plus open vertical bar table row straight c straight a straight b row cell straight c apostrophe end cell cell straight a apostrophe end cell cell straight b apostrophe end cell row cell straight c " end cell cell straight a " end cell cell straight b " end cell end table close vertical bar equals space open vertical bar table row straight a straight b straight c row cell straight a apostrophe end cell cell straight b apostrophe end cell cell straight c apostrophe end cell row cell straight a " end cell cell straight b " end cell cell straight c " end cell end table close vertical bar plus space open vertical bar table row straight a straight b straight c row cell straight a apostrophe end cell cell straight b apostrophe end cell cell straight c apostrophe end cell row cell straight a " end cell cell straight b " end cell cell straight c " end cell end table close vertical bar equals space 2 space open vertical bar table row straight a straight b straight c row cell straight a apostrophe end cell cell straight b apostrophe end cell cell straight c apostrophe end cell row cell straight a " end cell cell straight b " end cell cell straight c " end cell end table close vertical bar$

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