theorem Th20:
  for M being non empty MidStr, w being Function of [:the carrier
of M,the carrier of M:],the carrier of G holds w is_atlas_of the carrier of M,G
  & w is associating implies M is MidSp
proof
  let M be non empty MidStr, w be Function of [:the carrier of M,the carrier
  of M:],the carrier of G;
  assume w is_atlas_of the carrier of M,G & w is associating;
  then
  for a,b,c,d being Point of M holds a@a = a & a@b = b@a & (a@b)@(c@d) = (
  a@c)@(b@d) & ex x being Point of M st x@a = b by Th1,Th8,Th9,Th19;
  hence thesis by MIDSP_1:def 3;
end;
