theorem Th19:
  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 for a,b,c,d being Point of M holds (a@b)@(c@
  d) = (a@c)@(b@d)
proof
  let M be non empty MidStr, w be Function of [:the carrier of M,the carrier
  of M:],the carrier of G;
  assume that
A1: w is_atlas_of the carrier of M,G and
A2: w is associating;
  let a,b,c,d be Point of M;
A3: w.(b,a@b) = w.(b,b@a) by A1,A2,Th8
    .= w.(b@a,a) by A2
    .= w.(a@b,a) by A1,A2,Th8;
  set p = (a@b)@(c@d);
A4: w.(c,c@d) = w.(c@d,d) by A2;
A5: w.(b,b@d) = w.(b@d,d) by A2;
  w.(c,a@c) = w.(c,c@a) by A1,A2,Th8
    .= w.(c@a,a) by A2
    .= w.(a@c,a) by A1,A2,Th8;
  then Double w.(a@c,c@d) = w.(a,d) by A1,A4,Th14
    .= -w.(d,a) by A1,Th4
    .= -Double w.(b@d,a@b) by A1,A5,A3,Th14
    .= Double -w.(b@d,a@b) by Th12
    .= Double w.(a@b,b@d) by A1,Th4;
  then w.(a@c,c@d) = w.(a@b,b@d) by Th17;
  then
A6: w.(a@c,p) + w.(p,c@d) = w.(a@b,b@d) by A1
    .= w.(p,b@d) + w.(a@b,p) by A1;
  w.(a@b,p) = w.(p,c@d) by A2;
  then w.(a@c,p) = w.(p,b@d) by A6,RLVECT_1:8;
  hence thesis by A2;
end;
