let S be non empty set ; :: thesis: for a, b, c 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 . (b,a) = w . (c,a) holds

b = c

let a, b, c 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 . (b,a) = w . (c,a) holds

b = c

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 . (b,a) = w . (c,a) holds

b = c

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

assume that

A1: w is_atlas_of S,G and

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

w . (a,b) = w . (a,c) by A1, A2, Th5;

hence b = c 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 . (b,a) = w . (c,a) holds

b = c

let a, b, c 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 . (b,a) = w . (c,a) holds

b = c

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 . (b,a) = w . (c,a) holds

b = c

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

assume that

A1: w is_atlas_of S,G and

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

w . (a,b) = w . (a,c) by A1, A2, Th5;

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