let V be non empty right_complementable add-associative right_zeroed addLoopStr ; :: thesis: for v being Element of V holds
( Sum <*v,(- v)*> = 0. V & Sum <*(- v),v*> = 0. V )
let v be Element of V; :: thesis: ( Sum <*v,(- v)*> = 0. V & Sum <*(- v),v*> = 0. V )
A1: v + (- v) = 0. V by Def10;
hence Sum <*v,(- v)*> = 0. V by Th45; :: thesis: Sum <*(- v),v*> = 0. V
(- v) + v = 0. V by A1, Lm1;
hence Sum <*(- v),v*> = 0. V by Th45; :: thesis: verum