reserve X,Y,Z for non trivial RealBanachSpace;

theorem FUNCT229:
  for X,Y be RealNormSpace,
      f be Point of R_NormSpace_of_BoundedLinearOperators(X,Y)
  st f is one-to-one & rng f = the carrier of Y
  holds f" * f = id X
      & f * f" = id Y
  proof
    let X,Y be RealNormSpace,
          f be Point of R_NormSpace_of_BoundedLinearOperators(X,Y);
    assume
    A1: f is one-to-one & rng f = the carrier of Y;
    A2: f is Lipschitzian LinearOperator of X,Y by LOPBAN_1:def 9;
    hence f" * f = id X by A1,FUNCT_2:29;
    thus f * f" = id Y by A1,A2,FUNCT_2:29;
  end;
