reserve X,Y,Z for non trivial RealBanachSpace;

theorem LMTh2:
  for u,v be Point of R_NormSpace_of_BoundedLinearOperators(X,Y)
   st u is invertible
    & ||.v-u.|| < 1 / ||.Inv u .||
  holds
    v is invertible
  & ||.Inv v.|| <= 1 / ( 1 / ||.Inv (u).|| - ||.v-u.|| )
  & ex w be Point of R_Normed_Algebra_of_BoundedLinearOperators X,
     s,I be Point of R_NormSpace_of_BoundedLinearOperators(X,X)
    st w = (Inv u) * (v-u)
     & s = w & I = id X
     & ||.s.|| < 1
     & (-w) GeoSeq is norm_summable
     & I+s is invertible
     & ||.Inv (I+s).|| <= 1 / ( 1 - ||.s.|| )
     & Inv(I+s) = Sum ( (-w) GeoSeq )
     & Inv v = Inv(I+s) * (Inv u)
  proof
    let u,v be Point of R_NormSpace_of_BoundedLinearOperators(X,Y);
    assume
    A1: u is invertible & ||.v-u.|| < 1 / ||.Inv u .||;
    set vu = v-u;
    v = u + vu by RLVECT_4:1;
    hence thesis by A1,Th2;
  end;
