
theorem Th8:
  for m be non zero Element of NAT,
      f be LinearOperator of REAL-NS m,REAL-NS m
    st f is bijective
  holds
    ex g be Point of R_NormSpace_of_BoundedLinearOperators(REAL-NS m,REAL-NS m)
    st g = f & g is invertible
proof
  let m be non zero Element of NAT,
      f be LinearOperator of REAL-NS m,REAL-NS m;
  assume
  A1: f is bijective;
  then consider h be Lipschitzian LinearOperator of REAL-NS m,REAL-NS m
  such that
  A2: h = f" & h is one-to-one onto by Th7;

  REAL-NS m is finite-dimensional
  & dim(REAL-NS m) = m by REAL_NS2:62;
  then f is Lipschitzian by LOPBAN15:2;
  then reconsider g = f as Point of
    R_NormSpace_of_BoundedLinearOperators(REAL-NS m,REAL-NS m)
    by LOPBAN_1:def 9;

  take g;
  thus g = f;

  A3: rng g = the carrier of REAL-NS m by A1,FUNCT_2:def 3;
  h in BoundedLinearOperators(REAL-NS m,REAL-NS m) by LOPBAN_1:def 9;
  hence thesis by A1,A2,A3,LOPBAN13:def 1;
end;
