theorem
  f1 is_left_differentiable_in x0 & f2 is_left_differentiable_in x0 & f2
.x0 <> 0 implies f1/f2 is_left_differentiable_in x0 & Ldiff(f1/f2,x0) = (Ldiff(
  f1,x0)*f2.x0 - Ldiff(f2,x0)*f1.x0)/(f2.x0)^2
proof
  assume that
A1: f1 is_left_differentiable_in x0 and
A2: f2 is_left_differentiable_in x0 and
A3: f2.x0 <> 0;
  consider r1 such that
A4: r1 > 0 and
  [.x0-r1,x0.] c= dom f2 and
A5: for g st g in [.x0-r1,x0.] holds f2.g <> 0 by A2,A3,Th5,Th6;
  now
    take r1;
    thus r1 > 0 by A4;
    let g;
    assume that
    g in dom f2 and
A6: g in [.x0 - r1,x0.];
    thus f2.g <> 0 by A5,A6;
  end;
  hence thesis by A1,A2,Lm1;
end;
