theorem
  f is_left_differentiable_in x0 & f.x0 <> 0 implies f^
  is_left_differentiable_in x0 & Ldiff(f^,x0) = - Ldiff(f,x0)/(f.x0)^2
proof
  assume that
A1: f is_left_differentiable_in x0 and
A2: f.x0 <> 0;
  consider r1 such that
A3: r1 > 0 and
  [.x0-r1,x0.] c= dom f and
A4: for g st g in [.x0-r1,x0.] holds f.g <> 0 by A1,A2,Th5,Th6;
  now
    take r1;
    thus r1 > 0 by A3;
    let g;
    assume that
    g in dom f and
A5: g in [.x0 - r1,x0.];
    thus f.g <> 0 by A4,A5;
  end;
  hence thesis by A1,Lm2;
end;
