reserve X,Y,Z for non trivial RealBanachSpace;

theorem LM60:
  for u be Point of R_NormSpace_of_BoundedLinearOperators(X,Y)
  st u is invertible
  holds Inv u is invertible & Inv Inv u = u
  proof
    let u be Point of R_NormSpace_of_BoundedLinearOperators(X,Y);
    assume
    A1: u is invertible;
    then A3: Inv u = u" by Def1;
    reconsider Lu = u as Lipschitzian LinearOperator of X,Y by LOPBAN_1:def 9;
    A5: rng Inv u = dom Lu by A1,A3,FUNCT_1:33
    .= the carrier of X by FUNCT_2:def 1;
    A6: (Inv u) " = u by A1,A3,FUNCT_1:43;
    thus Inv u is invertible by A1,A3,A5,FUNCT_1:43;
    hence Inv Inv u = u by A6,Def1;
  end;
