
theorem Th13:
  for a, b, c, d, x1 being Real, f being Function of
  Closed-Interval-TSpace(a,b), Closed-Interval-TSpace(c,d), x being Point of
Closed-Interval-TSpace(a,b), g being PartFunc of REAL, REAL st a < b & c < d &
  f is_continuous_at x & f.a = c & f.b = d & f is one-to-one & f = g & x = x1
  holds g| [.a,b.] is_continuous_in x1
proof
  let a, b, c, d, x1 be Real;
  let f be Function of Closed-Interval-TSpace(a,b), Closed-Interval-TSpace(c,d
  ), x be Point of Closed-Interval-TSpace(a,b);
  let g be PartFunc of REAL, REAL;
  assume that
A1: a < b and
A2: c < d and
A3: f is_continuous_at x and
A4: f.a = c and
A5: f.b = d and
A6: f is one-to-one and
A7: f = g and
A8: x = x1;
A9: for c be Element of REAL st c in dom g holds g/.c = g/.c;
  dom g = the carrier of Closed-Interval-TSpace(a,b) by A7,FUNCT_2:def 1;
  then dom g = [.a,b.] by A1,TOPMETR:18;
  then dom g = dom g /\ [.a,b.];
  then
A10: g = g| [.a,b.] by A9,PARTFUN2:15;
  per cases;
  suppose
A11: x1 = a;
    for N1 being Neighbourhood of g.x1 ex N being Neighbourhood of x1 st
    g.:N c= N1
    proof
      reconsider f0 = f.a as Point of Closed-Interval-TSpace(c,d) by A2,A4,Lm6;
      let N1 be Neighbourhood of g.x1;
      reconsider N2 = N1 as Subset of RealSpace by METRIC_1:def 13;
      set NN1 = N1 /\ [.c,d.];
      N2 in Family_open_set RealSpace by Lm3;
      then
A12:  N2 in the topology of TopSpaceMetr RealSpace by TOPMETR:12;
      NN1 = N1 /\ the carrier of Closed-Interval-TSpace(c,d) by A2,TOPMETR:18;
      then reconsider NN = NN1 as Subset of Closed-Interval-TSpace(c,d) by
XBOOLE_1:17;
      NN1 = N1 /\ [#] Closed-Interval-TSpace(c,d) by A2,TOPMETR:18;
      then NN in the topology of Closed-Interval-TSpace(c,d) by A12,
PRE_TOPC:def 4,TOPMETR:def 6;
      then
A13:  NN is open;
      f.a in the carrier of Closed-Interval-TSpace(c,d) by A2,A4,Lm6;
      then g.x1 in N1 & g.x1 in [.c,d.] by A2,A7,A11,RCOMP_1:16,TOPMETR:18;
      then g.x1 in NN1 by XBOOLE_0:def 4;
      then reconsider N19 = NN as a_neighborhood of f0 by A7,A11,A13,CONNSP_2:3
;
      consider H being a_neighborhood of x such that
A14:  f.:H c= N19 by A3,A8,A11,TMAP_1:def 2;
      consider H1 being Subset of Closed-Interval-TSpace(a,b) such that
A15:  H1 is open and
A16:  H1 c= H and
A17:  x in H1 by CONNSP_2:6;
      H1 in the topology of Closed-Interval-TSpace(a,b) by A15;
      then consider Q being Subset of R^1 such that
A18:  Q in the topology of R^1 and
A19:  H1 = Q /\ [#] Closed-Interval-TSpace(a,b) by PRE_TOPC:def 4;
      reconsider Q9 = Q as Subset of RealSpace by TOPMETR:12,def 6;
      reconsider Q1 = Q9 as Subset of REAL by METRIC_1:def 13;
      Q9 in Family_open_set RealSpace by A18,TOPMETR:12,def 6;
      then
A20:  Q1 is open by Lm4;
      x1 in Q1 by A8,A17,A19,XBOOLE_0:def 4;
      then consider N being Neighbourhood of x1 such that
A21:  N c= Q1 by A20,RCOMP_1:18;
      take N;
      g.:N c= N1
      proof
        let aa be object;
        assume
A22:    aa in g.:N;
        then reconsider a9 = aa as Element of REAL;
        consider cc be Element of REAL such that
A23:    cc in dom g and
A24:    cc in N and
A25:    a9 = g.cc by A22,PARTFUN2:59;
        cc in the carrier of Closed-Interval-TSpace(a,b) by A7,A23,
FUNCT_2:def 1;
        then cc in H1 by A19,A21,A24,XBOOLE_0:def 4;
        then g.cc in f.:H by A7,A16,FUNCT_2:35;
        hence thesis by A14,A25,XBOOLE_0:def 4;
      end;
      hence thesis;
    end;
    hence thesis by A10,FCONT_1:5;
  end;
  suppose
A26: x1 = b;
    for N1 being Neighbourhood of g.x1 ex N being Neighbourhood of x1 st
    g.:N c= N1
    proof
      reconsider f0 = f.b as Point of Closed-Interval-TSpace(c,d) by A2,A5,Lm6;
      let N1 be Neighbourhood of g.x1;
      reconsider N2 = N1 as Subset of RealSpace by METRIC_1:def 13;
      set NN1 = N1 /\ [#]Closed-Interval-TSpace(c,d);
      reconsider NN = NN1 as Subset of Closed-Interval-TSpace(c,d);
      N2 in Family_open_set RealSpace by Lm3;
      then N2 in the topology of TopSpaceMetr RealSpace by TOPMETR:12;
      then NN in the topology of Closed-Interval-TSpace(c,d) by PRE_TOPC:def 4
,TOPMETR:def 6;
      then
A27:  NN is open;
      g.x1 in N1 & g.x1 in [#] Closed-Interval-TSpace(c,d) by A2,A5,A7,A26,Lm6,
RCOMP_1:16;
      then g.x1 in NN1 by XBOOLE_0:def 4;
      then reconsider N19 = NN as a_neighborhood of f0 by A7,A26,A27,CONNSP_2:3
;
      consider H being a_neighborhood of x such that
A28:  f.:H c= N19 by A3,A8,A26,TMAP_1:def 2;
      consider H1 being Subset of Closed-Interval-TSpace(a,b) such that
A29:  H1 is open and
A30:  H1 c= H and
A31:  x in H1 by CONNSP_2:6;
      H1 in the topology of Closed-Interval-TSpace(a,b) by A29;
      then consider Q being Subset of R^1 such that
A32:  Q in the topology of R^1 and
A33:  H1 = Q /\ [#](Closed-Interval-TSpace(a,b)) by PRE_TOPC:def 4;
      reconsider Q9 = Q as Subset of RealSpace by TOPMETR:12,def 6;
      reconsider Q1 = Q9 as Subset of REAL by METRIC_1:def 13;
      Q9 in Family_open_set RealSpace by A32,TOPMETR:12,def 6;
      then
A34:  Q1 is open by Lm4;
      x1 in Q1 by A8,A31,A33,XBOOLE_0:def 4;
      then consider N being Neighbourhood of x1 such that
A35:  N c= Q1 by A34,RCOMP_1:18;
      take N;
      g.:N c= N1
      proof
        let aa be object;
        assume
A36:    aa in g.:N;
        then reconsider a9 = aa as Element of REAL;
        consider cc be Element of REAL such that
A37:    cc in dom g and
A38:    cc in N and
A39:    a9 = g.cc by A36,PARTFUN2:59;
        cc in the carrier of Closed-Interval-TSpace(a,b) by A7,A37,
FUNCT_2:def 1;
        then cc in H1 by A33,A35,A38,XBOOLE_0:def 4;
        then g.cc in f.:H by A7,A30,FUNCT_2:35;
        hence thesis by A28,A39,XBOOLE_0:def 4;
      end;
      hence thesis;
    end;
    hence thesis by A10,FCONT_1:5;
  end;
  suppose
    x1 <> a & x1 <> b;
    hence thesis by A1,A2,A3,A4,A5,A6,A7,A8,A10,Lm7;
  end;
end;
