
theorem
  for X,Y be non empty LinearTopSpace,
      T be LinearOperator of X,Y,
      S be Function of Y, X
    st T is bijective open & S = T" holds
  S is LinearOperator of Y,X & S is onto continuous
  proof
    let X,Y be non empty LinearTopSpace,
     T be LinearOperator of X,Y,
     S be Function of Y,X;
   assume A1: T is bijective open & S = T"; then
    A2: T" is LinearOperator of Y,X & T" is one-to-one
      & rng(T") = the carrier of X by Th1;
  A3: [#] Y<>{} & [#] X<>{};
    S is continuous
    proof
      now let A be Subset of X;
        assume A4:A is open;
        S"A = S".:A by A1,FUNCT_1:85
           .=T.:A by A1,FUNCT_1:43;
        hence S"A is open by A4, A1,T_0TOPSP:def 2;
      end;
      hence thesis by A3,TOPS_2:43;
    end;
    hence thesis by A2,A1;
  end;
