
theorem Th12:
  for m be non zero Element of NAT,
      f be Point of
        R_NormSpace_of_BoundedLinearOperators(REAL-NS m, REAL-NS m),
      M be Matrix of m, F_Real
    st f = Mx2Tran(M)
  holds
    f is invertible
      iff
    M is invertible
proof
  let m be non zero Element of NAT,
      f be Point of
        R_NormSpace_of_BoundedLinearOperators(REAL-NS m, REAL-NS m),
      M be Matrix of m, F_Real;

  assume
  A1: f = Mx2Tran(M);
  A2: the carrier of TOP-REAL m = REAL m by EUCLID:22;
  A3: the carrier of REAL-NS m = REAL m by REAL_NS1:def 4;

  hereby
    assume f is invertible;
    then Mx2Tran(M) is one-to-one
        & rng(Mx2Tran(M)) = the carrier of REAL-NS m by A1,LOPBAN13:def 1;
    then Mx2Tran(M) is bijective by A2,A3,FUNCT_2:def 3;
    hence M is invertible by Th10;
  end;

  assume M is invertible;
  then Mx2Tran(M) is bijective by Th10;
  then f is one-to-one & rng f = the carrier of REAL-NS m
    by A1,A2,A3,FUNCT_2:def 3;
  hence f is invertible by Th11;
end;
