let S be non empty set ; :: thesis: for a, b being Element of S
for G being non empty right_complementable add-associative right_zeroed addLoopStr
for w being Function of [:S,S:], the carrier of G st w is_atlas_of S,G holds
w . (a,b) = - (w . (b,a))
let a, b be Element of S; :: thesis: for G being non empty right_complementable add-associative right_zeroed addLoopStr
for w being Function of [:S,S:], the carrier of G st w is_atlas_of S,G holds
w . (a,b) = - (w . (b,a))
let G be non empty right_complementable add-associative right_zeroed addLoopStr ; :: thesis: for w being Function of [:S,S:], the carrier of G st w is_atlas_of S,G holds
w . (a,b) = - (w . (b,a))
let w be Function of [:S,S:], the carrier of G; :: thesis: ( w is_atlas_of S,G implies w . (a,b) = - (w . (b,a)) )
assume A1: w is_atlas_of S,G ; :: thesis: w . (a,b) = - (w . (b,a))
then (w . (b,a)) + (w . (a,b)) = w . (b,b)
.= 0. G by A1, Th2 ;
then - (w . (b,a)) = - (- (w . (a,b))) by RLVECT_1:6;
hence w . (a,b) = - (w . (b,a)) by RLVECT_1:17; :: thesis: verum