 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 Th56:
  for X be non empty TopSpace,T be NormedLinearTopSpace
  for v,u be Element of RealVectSpace(the carrier of X,T)
     st v in C_0_Functions(X,T) & u in C_0_Functions(X,T)
                   holds v + 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 u,v be Element of V such that
A1: u in W & v in W;
    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 by A1;
    consider Y1 be non empty Subset of X such that
A3: Y1 is compact & Cl(support(u1)) c= Y1 by A2;
    consider v1 be Function of the carrier of X, the carrier of T such that
A4: v1=v & v1 is continuous
    & ex Y2 be non empty Subset of X st Y2 is compact
          & Cl(support(v1)) c= Y2 by A1;
    consider Y2 be non empty Subset of X such that
A5: Y2 is compact & Cl(support(v1)) c= Y2 by A4;
A6: ContinuousFunctions(X,T) is linearly-closed by Th5;
A7: u in ContinuousFunctions(X,T) by A2;
    v in ContinuousFunctions(X,T) by A4; then
    v + u in ContinuousFunctions(X,T) by A7,A6; then
    consider fvu be Function of the carrier of X, the carrier of T such that
A9: v+u = fvu & fvu is continuous;
    reconsider Y12 = Y1 \/ Y2 as non empty Subset of X;
A10:Y12 is compact by A3,A5,COMPTS_1:10;
    reconsider A1= support(u1),A2= support(v1) as Subset of X;
A11:dom v1 /\ dom u1 = (the carrier of X) /\ dom u1 by FUNCT_2:def 1
                    .= (the carrier of X) /\ (the carrier of X)
                                                   by FUNCT_2:def 1
                    .= the carrier of X;
    Cl(support(v1+u1)) c= Cl(support(u1) \/ support(v1)) by Th54,PRE_TOPC:19;
      then
A12:Cl(support(v1+u1)) c= Cl(support(u1)) \/ Cl(support(v1)) by PRE_TOPC:20;
    Cl(support(v1)) \/ Cl(support(u1)) c= Y1 \/ Y2 by A3,A5,XBOOLE_1:13; then
A13:Cl(support(v1+u1)) c= Y12 by A12;
A14:dom fvu = the carrier of X by FUNCT_2:def 1;
    for x be Element of X st x in dom fvu
      holds fvu/.x = v1/.x + u1/.x by LOPBAN_1:11,A9,A2,A4; then
    fvu = v1+u1 by VFUNCT_1:def 1,A14,A11;
    hence thesis by A9,A10,A13;
  end;
