reserve X,Y,Z for non trivial RealBanachSpace;

theorem LM50:
  for u be Point of R_NormSpace_of_BoundedLinearOperators(X,Y)
  st u is invertible
  holds 0 < ||.u.|| & 0 < ||.Inv u.||
  proof
    let u be Point of R_NormSpace_of_BoundedLinearOperators(X,Y);
    assume
    A1: u is invertible;
    set S = R_Normed_Algebra_of_BoundedLinearOperators X;
    reconsider Lu = u as Lipschitzian LinearOperator of X,Y by LOPBAN_1:def 9;
    reconsider LInvu = (Inv u) as Lipschitzian LinearOperator of Y,X
      by LOPBAN_1:def 9;
    A8: BoundedLinearOperatorsNorm(X,X).(LInvu*Lu)
      <= (BoundedLinearOperatorsNorm(Y,X).LInvu)
       * (BoundedLinearOperatorsNorm(X,Y).Lu) by LOPBAN_2:2;
    LInvu = u" by A1,Def1; then
    BoundedLinearOperatorsNorm(X,X).(LInvu*Lu)
     = ||.1.S.|| by A1,FUNCT_2:29
    .= 1 by LOPBAN_2:def 10;
    then ||.Inv u.|| <> 0 & ||.u.|| <> 0 by A8;
    hence 0 < ||.u.|| & 0 < ||.Inv u.||;
  end;
