
theorem Th42:
  for X be non empty TopSpace
  for x being set st x in CC_0_Functions(X) holds
  x in ComplexBoundedFunctions the carrier of X
proof
  let X be non empty TopSpace;
  let x be set such that
A1:       x in CC_0_Functions(X);
  consider f be Function of the carrier of X,COMPLEX such that
A2:        f=x & f is continuous
           & (ex Y be non empty Subset of X st Y is compact
           & (for A being Subset of X st A=support(f)
                          holds Cl(A) is Subset of Y)) by A1;
  consider Y be non empty Subset of X such that
A3:      Y is compact & (for A being Subset of X st A=support(f)
                       holds Cl(A) is Subset of Y) by A2;
  dom f = the carrier of X by FUNCT_2:def 1;
  then reconsider A= support(f) as Subset of X by PRE_POLY:37;
  |.f.| is Function of the carrier of X,REAL &
        |.f.| is continuous by Th8,A2;
  then consider f1 being Function of the carrier of X,REAL such that
A4:   f1=|.f.| & f1 is continuous;
    f1.:Y is compact by A4,A3,JORDAN_A:1;
  then reconsider Y1 = f1.:Y as non empty real-bounded Subset of REAL
        by RCOMP_1:10;
A5:Y1 c= [. inf Y1,sup Y1 .] by XXREAL_2:69;
  reconsider r1 = inf Y1 as Real;
  reconsider r2 = sup Y1 as Real;
  consider r be Real such that
A6:   r=|.r1.|+|.r2.|+1;
  for p being Element of Y holds r>0 & -r < f1.p & f1.p <r
  proof
    let p be Element of Y;
    f1.p in Y1 by FUNCT_2:35;
    then f1.p in [.r1,r2.] by A5;
    then f1.p in {t where t is Real: r1<=t & t<=r2 } by RCOMP_1:def 1;
    then consider r3 be Real such that
A7:                     r3=f1.p & r1<=r3 & r3<=r2;
    -|.r1.| <= r1 by ABSVALUE:4;
    then -|.r1.| - |.r2.| <= r1 - 0 by XREAL_1:13;
    then -|.r1.| - |.r2.| - 1 < r1 - 0 by XREAL_1:15;
    then
A8:   -r < r1 by A6;
    r2 <= |.r2.| by ABSVALUE:4;
    then r2 + 0 <= |.r2.|+|.r1.| by XREAL_1:7;
    then
A9:  r2 < r by A6,XREAL_1:8;
    -r < f1.p by A7,A8,XXREAL_0:2;
    hence thesis by A7,A9,XXREAL_0:2;
  end;
  then consider r be Real such that
A10: for p being Element of Y holds r>0 & -r< f1.p & f1.p <r;
  for x be Point of X holds -r< f1.x & f1.x <r
  proof
    let x be Point of X;
A11:x in (the carrier of X) \ Y or x in Y by XBOOLE_0:def 5;
    per cases by A11;
    suppose
      x in (the carrier of X) \ Y;
      then
A12:  x in (the carrier of X) & not x in Y by XBOOLE_0:def 5;
      Cl(A) is Subset of Y by A3;
      then
A13:    Cl(A) c= Y;
      support(f) c= Cl(A) by PRE_TOPC:18;
      then
        support(f) c= Y by A13;
      then
A14:    not x in support(f) by A12;
      f.x=0 by A14,PRE_POLY:def 7;
      then |.f.x.| = 0; then
      f1.x = 0 by A4,VALUED_1:18;
      hence -r< f1.x & f1.x <r by A10;
    end;
    suppose x in Y;
      hence thesis by A10;
    end;
  end;
  then
    consider s1 be Real such that
A15:       for x be Point of X holds (-s1< f1.x & f1.x <s1);
  for y being object st y in (the carrier of X) /\ dom f1 holds f1.y<=s1
by A15;
  then
A16:f1|(the carrier of X) is bounded_above by RFUNCT_1:70;
  for y being object st y in (the carrier of X) /\ dom f1
                   holds -s1<=f1.y by A15;
  then
A17:f1|(the carrier of X) is bounded_below by RFUNCT_1:71;
  (the carrier of X ) /\ (the carrier of X ) =(the carrier of X );
  then f1|(the carrier of X ) is bounded by A16,A17,RFUNCT_1:75;
  then |. f|(the carrier of X ) .| is bounded by A4,RFUNCT_1:46;
  then f|(the carrier of X) is bounded by CFUNCT_1:85;
  hence x in ComplexBoundedFunctions the carrier of X by A2;
end;
