reserve X,Y,Z for non trivial RealBanachSpace;

theorem Th1:
  for v,I be Point of R_NormSpace_of_BoundedLinearOperators(X,X)
  st I = id X
   & ||.v.|| < 1
  holds I+v is invertible
     & ||.Inv (I+v).|| <= 1 / ( 1 - ||.v.|| )
     & ex w be Point of R_Normed_Algebra_of_BoundedLinearOperators X
       st w = v
        & (-w) GeoSeq is norm_summable
        & Inv(I+v) = Sum( (-w) GeoSeq )
  proof
    let v,I be Point of R_NormSpace_of_BoundedLinearOperators(X,X);
    assume that
    A1: I = id X and
    A2: ||.v.|| < 1;
    set S = R_Normed_Algebra_of_BoundedLinearOperators X;
    reconsider w = v as Point of S;
    set x = 1.S + w;
    ||.w.|| < 1 by A2; then
    A3: x is invertible
      & (-w) GeoSeq is norm_summable
      & x" = Sum ( (-w) GeoSeq )
      & ||.x".|| <= 1 / ( 1 - ||.w.|| ) by LM2;
    hence
    A6: I+v is invertible by A1,LM4;
    A7: (I+v)" = x" by A1,A3,LM4;
    hence ||.Inv (I+v).|| <= 1 / ( 1 - ||.v.|| ) by A3,A6,Def1;
    thus thesis by A3,A6,A7,Def1;
  end;
