let M be non empty MidStr ; :: thesis: for p, q being Point of M
for G being non empty right_complementable add-associative right_zeroed addLoopStr
for w being Function of [: the carrier of M, the carrier of M:], the carrier of G st w is_atlas_of the carrier of M,G & w is associating holds
ex r being Point of M st r @ p = q
let p, q be Point of M; :: thesis: for G being non empty right_complementable add-associative right_zeroed addLoopStr
for w being Function of [: the carrier of M, the carrier of M:], the carrier of G st w is_atlas_of the carrier of M,G & w is associating holds
ex r being Point of M st r @ p = q
let G be non empty right_complementable add-associative right_zeroed addLoopStr ; :: thesis: for w being Function of [: the carrier of M, the carrier of M:], the carrier of G st w is_atlas_of the carrier of M,G & w is associating holds
ex r being Point of M st r @ p = q
let w be Function of [: the carrier of M, the carrier of M:], the carrier of G; :: thesis: ( w is_atlas_of the carrier of M,G & w is associating implies ex r being Point of M st r @ p = q )
assume that
A1: w is_atlas_of the carrier of M,G and
A2: w is associating ; :: thesis: ex r being Point of M st r @ p = q
consider r being Point of M such that
A3: w . (r,q) = w . (q,p) by A1, Th6;
take r ; :: thesis: r @ p = q
thus r @ p = q by A2, A3; :: thesis: verum