reserve X,Y,Z for non trivial RealBanachSpace;

theorem LM400:
  for X,Y be RealNormSpace,
        u be Point of R_NormSpace_of_BoundedLinearOperators(X,Y)
  st u is invertible
  holds (Inv u) * u = id X
      & u * (Inv u) = id Y
  proof
    let X,Y be RealNormSpace,
          v be Point of R_NormSpace_of_BoundedLinearOperators(X,Y);
    assume
    A1: v is invertible;
    v is Lipschitzian LinearOperator of X,Y by LOPBAN_1:def 9;
    then A7: dom v = the carrier of X & rng v = the carrier of Y
        by A1,FUNCT_2:def 1;
    Inv v = v" by A1,Def1;
    then A9: modetrans((Inv v),Y,X) = v" by LOPBAN_1:def 11;
    then A11: (Inv v) * v = v" * v by LOPBAN_1:def 11;
    v * (Inv v) = v * v" by A9,LOPBAN_1:def 11;
    hence thesis by A1,A7,A11,FUNCT_1:39;
  end;
