reserve G for non empty addLoopStr;
reserve x for Element of G;
reserve M for non empty MidStr;
reserve p,q,r for Point of M;
reserve w for Function of [:the carrier of M,the carrier of M:], the carrier
  of G;
reserve S for non empty set;
reserve a,b,b9,c,c9,d for Element of S;
reserve w for Function of [:S,S:],the carrier of G;
reserve G for add-associative right_zeroed right_complementable non empty
  addLoopStr;
reserve x for Element of G;
reserve w for Function of [:S,S:],the carrier of G;
reserve G for add-associative right_zeroed right_complementable Abelian non
  empty addLoopStr;
reserve x for Element of G;

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;
