let G be non empty right_complementable Abelian add-associative right_zeroed midpoint_operator addLoopStr ; :: thesis: for M being non empty MidStr
for w being Function of [: the carrier of M, the carrier of M:], the carrier of G
for a, c, b being Point of M st w is_atlas_of the carrier of M,G & M,G are_associated_wrp w holds
( a @ c = b iff w . (a,c) = Double (w . (a,b)) )
let M be non empty MidStr ; :: thesis: for w being Function of [: the carrier of M, the carrier of M:], the carrier of G
for a, c, b being Point of M st w is_atlas_of the carrier of M,G & M,G are_associated_wrp w holds
( a @ c = b iff w . (a,c) = Double (w . (a,b)) )
let w be Function of [: the carrier of M, the carrier of M:], the carrier of G; :: thesis: for a, c, b being Point of M st w is_atlas_of the carrier of M,G & M,G are_associated_wrp w holds
( a @ c = b iff w . (a,c) = Double (w . (a,b)) )
let a, c, b be Point of M; :: thesis: ( w is_atlas_of the carrier of M,G & M,G are_associated_wrp w implies ( a @ c = b iff w . (a,c) = Double (w . (a,b)) ) )
assume A1: ( w is_atlas_of the carrier of M,G & M,G are_associated_wrp w ) ; :: thesis: ( a @ c = b iff w . (a,c) = Double (w . (a,b)) )
then reconsider MM = M as MidSp by Th23;
reconsider bb = b as Point of MM ;
bb @ bb = bb by MIDSP_1:def 3;
hence ( a @ c = b iff w . (a,c) = Double (w . (a,b)) ) by A1, Th34; :: thesis: verum