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

p @ q = q @ p

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

p @ q = q @ p

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

p @ q = q @ p

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 p @ q = q @ p )

assume that

A1: w is_atlas_of the carrier of M,G and

A2: w is associating ; :: thesis: p @ q = q @ p

set r = p @ q;

w . (p,(p @ q)) = w . ((p @ q),q) by A2;

then w . ((p @ q),p) = w . (q,(p @ q)) by A1, Th5;

hence p @ q = q @ p by A2; :: thesis: verum

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

p @ q = q @ p

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

p @ q = q @ p

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

p @ q = q @ p

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 p @ q = q @ p )

assume that

A1: w is_atlas_of the carrier of M,G and

A2: w is associating ; :: thesis: p @ q = q @ p

set r = p @ q;

w . (p,(p @ q)) = w . ((p @ q),q) by A2;

then w . ((p @ q),p) = w . (q,(p @ q)) by A1, Th5;

hence p @ q = q @ p by A2; :: thesis: verum