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 & w . (a,b) = 0. G holds

a = b

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 & w . (a,b) = 0. G holds

a = b

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 & w . (a,b) = 0. G holds

a = b

let w be Function of [:S,S:], the carrier of G; :: thesis: ( w is_atlas_of S,G & w . (a,b) = 0. G implies a = b )

assume that

A1: w is_atlas_of S,G and

A2: w . (a,b) = 0. G ; :: thesis: a = b

w . (a,b) = w . (a,a) by A1, A2, Th2;

hence a = b by A1; :: thesis: verum

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 & w . (a,b) = 0. G holds

a = b

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 & w . (a,b) = 0. G holds

a = b

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 & w . (a,b) = 0. G holds

a = b

let w be Function of [:S,S:], the carrier of G; :: thesis: ( w is_atlas_of S,G & w . (a,b) = 0. G implies a = b )

assume that

A1: w is_atlas_of S,G and

A2: w . (a,b) = 0. G ; :: thesis: a = b

w . (a,b) = w . (a,a) by A1, A2, Th2;

hence a = b by A1; :: thesis: verum