theorem Th17:
  for G being Fanoian add-associative right_zeroed
  right_complementable Abelian non empty addLoopStr, x,y being Element of G
  holds Double x = Double y implies x = y
proof
  let G be Fanoian add-associative right_zeroed right_complementable Abelian
  non empty addLoopStr, x,y be Element of G;
  assume Double x = Double y;
  then 0.G = (x+x)+-(y+y) by RLVECT_1:def 10
    .= x+x+(-y+-y) by RLVECT_1:31
    .= x+(x+(-y+-y)) by RLVECT_1:def 3
    .= x+(x+-y+-y) by RLVECT_1:def 3
    .= (x+-y)+(x+-y) by RLVECT_1:def 3;
  then -y+x = 0.G by VECTSP_1:def 18;
  hence x = -(-y) by RLVECT_1:6
    .= y by RLVECT_1:17;
end;
