let S be non empty set ; :: thesis: for a 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,a = 0. G
let a 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,a = 0. G
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,a = 0. G
let w be Function of [:S,S:],the carrier of G; :: thesis: ( w is_atlas_of S,G implies w . a,a = 0. G )
assume w is_atlas_of S,G ; :: thesis: w . a,a = 0. G
then (w . a,a) + (w . a,a) = w . a,a by Def3;
hence w . a,a = 0. G by RLVECT_1:22; :: thesis: verum