reserve X,Y,Z for non trivial RealBanachSpace;

theorem Th3:
  for S be Subset of R_NormSpace_of_BoundedLinearOperators(X,Y)
  st S = {v where v is Point of R_NormSpace_of_BoundedLinearOperators(X,Y)
          : v is invertible}
  holds S is open
  proof
    let S be Subset of R_NormSpace_of_BoundedLinearOperators(X,Y);
    assume
    A1: S = {v where v is Point of R_NormSpace_of_BoundedLinearOperators(X,Y)
             : v is invertible};
    set P = R_NormSpace_of_BoundedLinearOperators(X,Y);
    for u be Point of P st u in S holds
    ex r be Real st r > 0 & Ball(u,r) c= S
    proof
      let u be Point of P;
      assume u in S; then
      A2: ex v be Point of P
          st u = v & v is invertible by A1; then
      A3: 0 < ||.Inv u .|| by LM50;
      set r = 1 / ||.Inv u .||;
      take r;
      thus 0 < r by A3,XREAL_1:139;
      now
        let x be object;
        assume x in Ball(u,r); then
        x in {y where y is Point of P: ||.y - u.|| < r} by NDIFF_8:17; then
        consider v be Point of P such that
        A4: x = v & ||.v - u.|| < r;
        v = u + (v-u) by RLVECT_4:1; then
        v is invertible by A2,A4,Th2;
        hence x in S by A1,A4;
      end;
      hence Ball(u,r) c= S by TARSKI:def 3;
    end;
    hence S is open by NDIFF_8:20;
  end;
