theorem Th26:
  M is MidSp iff ex G st ex w st w is_atlas_of the carrier of M,G
  & w is associating
proof
  hereby
    assume
A1: M is MidSp;
    thus ex G st ex w st w is_atlas_of the carrier of M,G &
    w is associating
    proof
      reconsider M as MidSp by A1;
      set G = vectgroup(M);
      take G;
      ex 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
      proof
        take vect(M);
        thus thesis by Th21;
      end;
      hence thesis;
    end;
  end;
  given G being midpoint_operator add-associative right_zeroed
right_complementable Abelian non empty addLoopStr, w being Function of [:the
  carrier of M,the carrier of M:],the carrier of G such that
A2: w is_atlas_of the carrier of M,G & w is associating;
  thus thesis by A2,Th20;
end;
