
theorem
  for a, b, c, d, e, f, g, h being Real
  for F being Function of
Closed-Interval-TSpace (a,b), Closed-Interval-TSpace (c,d) st a < b & c < d & e
< f & a <= e & f <= b & F is being_homeomorphism & F.a = c & F.b = d & g = F.e
  & h = F.f holds F.:[.e, f.] = [.g, h.]
proof
  let a, b, c, d, e, f, g, h be Real;
  let F be Function of Closed-Interval-TSpace (a,b), Closed-Interval-TSpace (c
  ,d);
  assume that
A1: a < b and
A2: c < d and
A3: e < f and
A4: a <= e and
A5: f <= b and
A6: F is being_homeomorphism and
A7: F.a = c & F.b = d and
A8: g = F.e and
A9: h = F.f;
  a <= f by A3,A4,XXREAL_0:2;
  then f in { l1 where l1 is Real: a <= l1 & l1 <= b } by A5;
  then
A10: f in [.a,b.] by RCOMP_1:def 1;
  then f in the carrier of Closed-Interval-TSpace (a,b) by A1,TOPMETR:18;
  then h in the carrier of Closed-Interval-TSpace (c,d) by A9,FUNCT_2:5;
  then
A11: h in [.c,d.] by A2,TOPMETR:18;
  e <= b by A3,A5,XXREAL_0:2;
  then e in { l1 where l1 is Real: a <= l1 & l1 <= b } by A4;
  then
A12: e in [.a,b.] by RCOMP_1:def 1;
  then e in the carrier of Closed-Interval-TSpace (a,b) by A1,TOPMETR:18;
  then g in the carrier of Closed-Interval-TSpace (c,d) by A8,FUNCT_2:5;
  then g in [.c,d.] by A2,TOPMETR:18;
  then [.g,h.] c= [.c,d.] by A11,XXREAL_2:def 12;
  then
A13: [.g,h.] c= the carrier of Closed-Interval-TSpace (c,d) by A2,TOPMETR:18;
A14: F is continuous one-to-one by A6,TOPS_2:def 5;
A15: [.g, h.] c= F.:[.e, f.]
  proof
    let aa be object;
A16: F is one-to-one by A6,TOPS_2:def 5;
    assume aa in [.g,h.];
    then aa in { l1 where l1 is Real: g <= l1 & l1 <= h }
         by RCOMP_1:def 1;
    then consider l be Real such that
A17: aa = l and
A18: g <= l and
A19: l <= h;
A20: rng F = [#]Closed-Interval-TSpace (c,d) by A6,TOPS_2:def 5;
    l in { l1 where l1 is Real: g <= l1 & l1 <= h } by A18,A19;
    then
A21: l in [.g,h.] by RCOMP_1:def 1;
    reconsider x = F".l as Real;
    F is onto by A20,FUNCT_2:def 3;
    then
A22: x = (F qua Function)".l by A16,TOPS_2:def 4;
    then
A23: x in dom F by A13,A16,A20,A21,FUNCT_1:32;
    dom F = [#]Closed-Interval-TSpace (a,b) by A6,TOPS_2:def 5;
    then reconsider
    e9 = e, f9 = f, x9 = x as Point of Closed-Interval-TSpace (a,b)
    by A1,A12,A10,A13,A16,A20,A21,A22,FUNCT_1:32,TOPMETR:18;
    reconsider g9 = F.e9, h9 = F.f9, l9 = F.x9 as Point of
    Closed-Interval-TSpace (c,d);
    reconsider gg = g9, hh = h9, ll = l9 as Real;
A24: x <= f
    proof
      assume x > f;
      then ll > hh by A1,A2,A7,A14,Th15;
      hence thesis by A9,A13,A19,A16,A20,A21,A22,FUNCT_1:32;
    end;
    e <= x
    proof
      assume e > x;
      then gg > ll by A1,A2,A7,A14,Th15;
      hence thesis by A8,A13,A18,A16,A20,A21,A22,FUNCT_1:32;
    end;
    then x in { l1 where l1 is Real: e <= l1 & l1 <= f } by A24;
    then
A25: x in [.e, f.] by RCOMP_1:def 1;
    aa = F.x by A13,A17,A16,A20,A21,A22,FUNCT_1:32;
    hence thesis by A23,A25,FUNCT_1:def 6;
  end;
  F.:[.e, f.] c= [.g, h.]
  proof
    let aa be object;
    assume aa in F.:[.e, f.];
    then consider x be object such that
A26: x in dom F and
A27: x in [.e, f.] and
A28: aa = F.x by FUNCT_1:def 6;
    x in { l where l is Real: e <= l & l <= f } by A27,RCOMP_1:def 1;
    then consider x9 be Real such that
A29: x9 = x and
A30: e <= x9 and
A31: x9 <= f;
    reconsider Fx = F.x9 as Real;
    reconsider e1 = e, f1 = f, x1 = x9 as Point of Closed-Interval-TSpace (a,
    b) by A1,A12,A10,A26,A29,TOPMETR:18;
    reconsider gg = F.e1, hh = F.f1, FFx = F.x1 as Real;
A32: Fx <= h
    proof
      per cases;
      suppose
        f = x;
        hence thesis by A9,A29;
      end;
      suppose
        f <> x;
        then f > x9 by A29,A31,XXREAL_0:1;
        then hh > FFx by A1,A2,A7,A14,Th15;
        hence thesis by A9;
      end;
    end;
    g <= Fx
    proof
      per cases;
      suppose
        e = x;
        hence thesis by A8,A29;
      end;
      suppose
        e <> x;
        then e < x9 by A29,A30,XXREAL_0:1;
        then gg < FFx by A1,A2,A7,A14,Th15;
        hence thesis by A8;
      end;
    end;
    then Fx in { l1 where l1 is Real: g <= l1 & l1 <= h } by A32;
    hence thesis by A28,A29,RCOMP_1:def 1;
  end;
  hence thesis by A15;
end;
