reserve r, s, t, g for Real,

          r3, r1, r2, q3, p3 for Real;
reserve T for TopStruct,
  f for RealMap of T;

theorem Th11:
  f is continuous & not 0 in rng f implies Inv f is continuous
proof
  assume that
A1: f is continuous and
A2: not 0 in rng f;
  let X0 be Subset of REAL;
  0 in {0} by TARSKI:def 1;
  then not 0 in X0\{0} by XBOOLE_0:def 5;
  then reconsider X = X0\{0} as without_zero Subset of REAL by MEASURE6:def 2;
  set X9 = Inv X;
A3: X0\{0} c= X0 by XBOOLE_1:36;
  set X9r = X9/\rng f;
  assume
A4: X0 is closed;
  now
    let x be object;
    hereby
A5:   X9r c= Cl X9r by MEASURE6:58;
      assume
A6:   x in X9r;
      then x in rng f by XBOOLE_0:def 4;
      hence x in (Cl X9r) /\ rng f by A6,A5,XBOOLE_0:def 4;
    end;
    assume
A7: x in (Cl X9r) /\ rng f;
    then reconsider s = x as Real;
    x in Cl (X9r) by A7,XBOOLE_0:def 4;
    then consider seq being Real_Sequence such that
A8: rng seq c= X9r and
A9: seq is convergent and
A10: lim seq = s by MEASURE6:64;
    assume
A11: not x in X9r;
A12: x in rng f by A7,XBOOLE_0:def 4;
    now
      rng (seq") c= X
      proof
        let y be object;
        assume y in rng (seq");
        then consider n being object such that
A13:    n in dom (seq") and
A14:    y = (seq").n by FUNCT_1:def 3;
        reconsider n as Element of NAT by A13;
        seq.n in rng seq by FUNCT_2:4;
        then
A15:    1/(1/seq.n) in X9 by A8,XBOOLE_0:def 4;
        (seq").n = (seq.n)" by VALUED_1:10
          .= 1/(seq.n);
        hence thesis by A14,A15,MEASURE6:54;
      end;
      then
A16:  rng (seq") c= X0 by A3;
      assume
A17:  lim seq <> 0;
      now
        let n be Nat;
A18:      n in NAT by ORDINAL1:def 12;
        assume seq.n = 0;
        then 0 in rng seq by FUNCT_2:4,A18;
        hence contradiction by A2,A8,XBOOLE_0:def 4;
      end;
      then
A19:  seq is non-zero by SEQ_1:5;
      then seq" is convergent by A9,A17,SEQ_2:21;
      then
A20:  lim (seq") in X0 by A4,A16;
A21:  lim (seq") = (lim seq)" by A9,A17,A19,SEQ_2:22;
      then lim (seq") <> 0 by A17;
      then not lim (seq") in {0} by TARSKI:def 1;
      then lim (seq") in X by A20,XBOOLE_0:def 5;
      then 1/(lim (seq")) in X9;
      hence contradiction by A12,A10,A11,A21,XBOOLE_0:def 4;
    end;
    hence contradiction by A2,A7,A10,XBOOLE_0:def 4;
  end;
  then
A22: X9r = (Cl X9r) /\ rng f by TARSKI:2;
  f"(Cl X9r) is closed by A1;
  then
A23: f"X9r is closed by A22,RELAT_1:133;
A24: now
    let x be object;
    hereby
      assume
A25:  x in (Inv f)"X0;
      then
A26:  (Inv f).x in X0 by FUNCT_2:38;
      reconsider xx=x as set by TARSKI:1;
      now
        assume not (Inv f).x in X;
        then (Inv f).x in {0} by A26,XBOOLE_0:def 5;
        then (Inv f).x = 0 by TARSKI:def 1;
        then 0 = (f.xx)" by VALUED_1:10;
        hence contradiction by A2,A25,FUNCT_2:4,XCMPLX_1:202;
      end;
      hence x in (Inv f)"X by A25,FUNCT_2:38;
    end;
    (Inv f)"X c= (Inv f)"X0 by RELAT_1:143,XBOOLE_1:36;
    hence x in (Inv f)"X implies x in (Inv f)"X0;
  end;
  f"X9 = f"X9r & (Inv f)"X = f"(Inv X) by MEASURE6:71,RELAT_1:133;
  hence thesis by A23,A24,TARSKI:2;
end;
