 reserve
  S for non empty TopSpace,
  T for LinearTopSpace,
  X for non empty Subset of the carrier of S;
 reserve
    S,T for RealNormSpace,
    X for non empty Subset of the carrier of S;

theorem Th57:
  for X be non empty TopSpace,T be NormedLinearTopSpace
  for a be Real,u be Element of RealVectSpace(the carrier of X,T)
     st u in C_0_Functions(X,T) holds a * u in C_0_Functions(X,T)
  proof
    let X be non empty TopSpace,T be NormedLinearTopSpace;
    set W = C_0_Functions(X,T);
    set V = RealVectSpace(the carrier of X,T);
    let a be Real;
    let u be Element of V;
    assume u in W; then
    consider u1 be Function of the carrier of X, the carrier of T such that
A2: u1=u & u1 is continuous
        & ex Y1 be non empty Subset of X st Y1 is compact
        & Cl(support(u1)) c= Y1;
    consider Y1 be non empty Subset of X such that
A3: Y1 is compact & Cl(support(u1)) c=Y1 by A2;
A4: u in ContinuousFunctions(X,T) by A2;
    ContinuousFunctions(X,T) is linearly-closed by Th5; then
    a*u in ContinuousFunctions(X,T) by A4; then
    consider fau be Function of the carrier of X, the carrier of T such that
A5: a*u = fau & fau is continuous;
A6: dom fau = the carrier of X by FUNCT_2:def 1;
A7: dom u1 = the carrier of X by FUNCT_2:def 1;
    Cl(support(a(#)u1)) c= Cl(support(u1)) by Th55,PRE_TOPC:19; then
A8: Cl(support(a(#)u1)) c= Y1 by A3;
    for x be Element of X st x in dom fau
      holds fau/.x = a*u1/.x by LOPBAN_1:12,A2,A5;
    then fau = a(#)u1 by VFUNCT_1:def 4,A7,A6;
    hence thesis by A3,A8,A5;
  end;
