let S be non empty set ; :: thesis: for b, a, 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 b, a, 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, Th7;
hence b = c by A1, Def3; :: thesis: verum