theorem Th11:
  inverse_op G is one-to-one
proof
  set f = inverse_op G;
  let x1, x2 be object;
  assume that
A1: x1 in dom f & x2 in dom f and
A2: f.x1 = f.x2;
  reconsider a = x1, b = x2 as Element of G by A1;
  f.a = a" & f.b = b" by GROUP_1:def 6;
  hence thesis by A2,GROUP_1:9;
end;
