reserve a,b,c,d for Real;

theorem Th23:
  for X, Y being non empty SubSpace of R^1, f being continuous Function of X,Y
  holds
  (ex a,b being Real st a <= b & [.a,b.] c= the carrier of X &
      [.a,b.] c= the carrier of Y & f.:[.a,b.] c= [.a,b.]) implies
  ex x being Point of X st f.x = x
proof
  let X, Y be non empty SubSpace of R^1, f be continuous Function of X,Y;
  given a,b being Real such that
A1: a <= b and
A2: [.a,b.] c= the carrier of X and
A3: [.a,b.] c= the carrier of Y and
A4: f.:[.a,b.] c= [.a,b.];
  reconsider A = [.a,b.] as non empty Subset of X by A1,A2,XXREAL_1:1;
A5: dom(f|A) = (dom f) /\ A by RELAT_1:61;
  A = (the carrier of X) /\ A & dom f = the carrier of X by FUNCT_2:def 1
,XBOOLE_1:28;
  then
A6: dom(f|A) = the carrier of Closed-Interval-TSpace(a,b) by A1,A5,TOPMETR:18;
A7: A = the carrier of Closed-Interval-TSpace(a,b) by A1,TOPMETR:18;
  then reconsider Z = Closed-Interval-TSpace(a,b) as SubSpace of X by TSEP_1:4;
  rng(f|A) c= the carrier of Closed-Interval-TSpace(a,b) by A4,A7,RELAT_1:115;
  then reconsider g = f|A as Function of Closed-Interval-TSpace(a,b),
  Closed-Interval-TSpace(a,b) by A6,FUNCT_2:def 1,RELSET_1:4;
A8: Z is SubSpace of Y by A3,A7,TSEP_1:4;
  for s being Point of Closed-Interval-TSpace(a,b) holds g is_continuous_at s
  proof
    let s be Point of Closed-Interval-TSpace(a,b);
    reconsider w = s as Point of X by A7,TARSKI:def 3;
    for G being Subset of Closed-Interval-TSpace(a,b) st G is open & g.s
    in G ex H being Subset of Z st H is open & s in H & g.:H c= G
    proof
      let G be Subset of Closed-Interval-TSpace(a,b);
A9:   f is_continuous_at w by TMAP_1:44;
      assume G is open;
      then consider G0 being Subset of Y such that
A10:  G0 is open and
A11:  G0 /\ [#] Closed-Interval-TSpace(a,b) = G by A8,TOPS_2:24;
      assume g.s in G;
      then f.w in G by A7,FUNCT_1:49;
      then f.w in G0 by A11,XBOOLE_0:def 4;
      then consider H0 being Subset of X such that
A12:  H0 is open and
A13:  w in H0 and
A14:  f.:H0 c= G0 by A10,A9,TMAP_1:43;
      now
        reconsider H = H0 /\ [#] Closed-Interval-TSpace(a,b) as Subset of Z;
        take H;
        thus H is open by A12,TOPS_2:24;
        thus s in H by A13,XBOOLE_0:def 4;
        thus g.:H c= G
        proof
          let t be object;
          assume t in g.:H;
          then consider r be object such that
          r in dom g and
A15:      r in H and
A16:      t = g.r by FUNCT_1:def 6;
A17:      r in the carrier of Z by A15;
          reconsider r as Point of Closed-Interval-TSpace(a,b) by A15;
          r in dom g by A17,FUNCT_2:def 1;
          then
A18:      t in g.:(the carrier of Z) by A16,FUNCT_1:def 6;
          reconsider p = r as Point of X by A7,TARSKI:def 3;
          p in [#] X;
          then
A19:      p in dom f by FUNCT_2:def 1;
          t=f.p & p in H0 by A7,A15,A16,FUNCT_1:49,XBOOLE_0:def 4;
          then t in f.:H0 by A19,FUNCT_1:def 6;
          hence thesis by A11,A14,A18,XBOOLE_0:def 4;
        end;
      end;
      hence thesis;
    end;
    hence thesis by TMAP_1:43;
  end;
  then reconsider h = g as continuous Function of Closed-Interval-TSpace(a,b),
  Closed-Interval-TSpace(a,b) by TMAP_1:44;
  now
    consider y being Point of Closed-Interval-TSpace(a,b) such that
A20: h.y = y by A1,Th22;
    reconsider x = y as Point of X by A7,TARSKI:def 3;
    take x;
    thus f.x = x by A7,A20,FUNCT_1:49;
  end;
  hence thesis;
end;
