let V be non empty right_complementable Abelian 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 )
thus Sum <*v,(- v)*> = v + (- v) by Th45
.= 0. V by Th5 ; :: thesis: Sum <*(- v),v*> = 0. V
hence Sum <*(- v),v*> = 0. V by Th54; :: thesis: verum