reserve h,h1,h2 for 0-convergent non-zero Real_Sequence,
  c,c1 for constant Real_Sequence,
  f,f1,f2 for PartFunc of REAL,REAL,
  x0,r,r0,r1,r2,g,g1,g2 for Real,
  n0,k,n,m for Element of NAT,
  a,b,d for Real_Sequence,
  x for set;

theorem
  f is_right_differentiable_in x0 & f.x0 <> 0 implies f^
  is_right_differentiable_in x0 & Rdiff(f^,x0) = - Rdiff(f,x0)/(f.x0)^2
proof
  assume that
A1: f is_right_differentiable_in x0 and
A2: f.x0 <> 0;
  consider r1 such that
A3: r1 > 0 and
  [.x0,x0 + r1.] c= dom f and
A4: for g st g in [.x0,x0 + r1.] holds f.g <> 0 by A1,A2,Th7,Th8;
  now
    take r1;
    thus r1 > 0 by A3;
    let g;
    assume that
    g in dom f and
A5: g in [.x0,x0+r1.];
    thus f.g <> 0 by A4,A5;
  end;
  hence thesis by A1,Lm4;
end;
