reserve a,b,c,d for Real;

theorem Th22:
  a <= b implies for f being continuous Function of
  Closed-Interval-TSpace(a,b), Closed-Interval-TSpace(a,b)
  ex x being Point of Closed-Interval-TSpace(a,b) st f.x = x
proof
  assume
A1: a <= b;
  let f be continuous Function of Closed-Interval-TSpace(a,b),
  Closed-Interval-TSpace(a,b);
  now
    per cases by A1,XXREAL_0:1;
    suppose
A2:   a < b;
      set L = L[01]((#)(a,b),(a,b)(#)), P = P[01](a,b,(#)(0,1),(0,1)(#));
A3:   P is continuous Function of Closed-Interval-TSpace(a,b),
      Closed-Interval-TSpace(0,1) by A2,Th12;
      set g = (P * f) * L;
A4:   id Closed-Interval-TSpace(a,b) = L * P by A2,Th15;
      then
A5:   f = (L * P) * f by FUNCT_2:17
        .= L * (P * f) by RELAT_1:36
        .= L * ((P * f) * (L * P)) by A4,FUNCT_2:17
        .= L * (g * P) by RELAT_1:36
        .= (L * g) * P by RELAT_1:36;
      L is continuous Function of Closed-Interval-TSpace(0,1),
      Closed-Interval-TSpace(a,b) by A1,Th8;
      then consider y be Point of Closed-Interval-TSpace(0,1) such that
A6:   g.y = y by A3,Th21,TOPMETR:20;
A7:   id Closed-Interval-TSpace(0,1) = P * L by A2,Th15;
      now
        take x = L.y;
        thus f.x = (((L * g) * P) * L).y by A5,FUNCT_2:15
          .= ((L * g) *(id Closed-Interval-TSpace(0,1))).y by A7,RELAT_1:36
          .= (L * g).y by FUNCT_2:17
          .= x by A6,FUNCT_2:15;
      end;
      hence thesis;
    end;
    suppose
A8:   a = b;
      then [.a,b.] = {a} & a = (#)(a,b) by Def1,XXREAL_1:17;
      then
A9:   the carrier of Closed-Interval-TSpace(a,b) = {(#)(a,b)} by A8,TOPMETR:18;
      now
        take x = (#)(a,b);
        thus f.x = x by A9,TARSKI:def 1;
      end;
      hence thesis;
    end;
  end;
  hence thesis;
end;
