let V be non empty right_complementable add-associative right_zeroed addLoopStr ; :: thesis: for v being Element of V holds
( Sum <*(0. V),(0. V),v*> = v & Sum <*(0. V),v,(0. V)*> = v & Sum <*v,(0. V),(0. V)*> = v )
let v be Element of V; :: thesis: ( Sum <*(0. V),(0. V),v*> = v & Sum <*(0. V),v,(0. V)*> = v & Sum <*v,(0. V),(0. V)*> = v )
thus Sum <*(0. V),(0. V),v*> = ((0. V) + (0. V)) + v by Th63
.= (0. V) + v by Th10
.= v by Th10 ; :: thesis: ( Sum <*(0. V),v,(0. V)*> = v & Sum <*v,(0. V),(0. V)*> = v )
thus Sum <*(0. V),v,(0. V)*> = ((0. V) + v) + (0. V) by Th63
.= (0. V) + v by Th10
.= v by Th10 ; :: thesis: Sum <*v,(0. V),(0. V)*> = v
thus Sum <*v,(0. V),(0. V)*> = (v + (0. V)) + (0. V) by Th63
.= v + (0. V) by Th10
.= v by Th10 ; :: thesis: verum