theorem Th21:
  vect(M) is_atlas_of the carrier of M, vectgroup(M)
proof
  set w = vect(M);
A1: ex q st w.(p,q) = x
  proof
    reconsider xx = x as Vector of M by Th15;
    consider q such that
A2: xx = vect(p,q) by MIDSP_1:35;
    take q;
    thus thesis by A2,Def8;
  end;
A3: w.(p,q) = w.(p,r) implies q = r
  proof
    assume w.(p,q) = w.(p,r);
    then vect(p,q) = w.(p,r) by Def8
      .= vect(p,r) by Def8;
    hence thesis by MIDSP_1:39;
  end;
  w.(p,q) + w.(q,r) = w.(p,r)
  proof
    w.(p,q) = vect(p,q) & w.(q,r) = vect(q,r) by Def8;
    hence w.(p,q) + w.(q,r) = vect(p,q) + vect(q,r) by Th15
      .= vect(p,r) by MIDSP_1:40
      .= w.(p,r) by Def8;
  end;
  hence thesis by A1,A3;
end;
