
theorem Th11:
  for m be non zero Element of NAT,
      f be Point of R_NormSpace_of_BoundedLinearOperators(REAL-NS m,REAL-NS m)
    st f is one-to-one & rng f = the carrier of REAL-NS m
  holds f is invertible
proof
  let m be non zero Element of NAT;
  let f be Point of
    R_NormSpace_of_BoundedLinearOperators(REAL-NS m,REAL-NS m);

  assume
  A1: f is one-to-one & rng f = the carrier of REAL-NS m;

  reconsider g = f as Lipschitzian LinearOperator of REAL-NS m,REAL-NS m
    by LOPBAN_1:def 9;

  g is bijective by A1,FUNCT_2:def 3;
  then
  ex f be Point of R_NormSpace_of_BoundedLinearOperators(REAL-NS m,REAL-NS m)
  st f = g & f is invertible by Th8;
  hence thesis;
end;
