theorem Th13:
  for 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 iff
  w.(a,d) = w.(c,b))
proof
  let 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;
  thus a@b = c@d implies w.(a,d) = w.(c,b)
  proof
    set p = a@b;
    assume a@b = c@d;
    then
A3: w.(c,p) = w.(p,d) by A2;
    w.(a,p) = w.(p,b) by A2;
    hence w.(a,d) = w.(c,p) + w.(p,b) by A1,A3
      .= w.(c,b) by A1;
  end;
  thus w.(a,d) = w.(c,b) implies a@b = c@d
  proof
    set p = a@b;
    assume
A4: w.(a,d) = w.(c,b);
    w.(p,b) + w.(p,d) = w.(a,p) + w.(p,d) by A2
      .= w.(a,d) by A1
      .= w.(p,b) + w.(c,p) by A1,A4;
    then w.(p,d) = w.(c,p) by RLVECT_1:8;
    hence thesis by A2;
  end;
end;
