 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 Th54:
  for X be non empty TopSpace, T be RealLinearSpace
  for f,g be Function of X,T holds
    support(f+g) c= support(f) \/ support(g)
proof
  let X be non empty TopSpace,T be RealLinearSpace;
  let f,g be Function of X,T;
  set CX = the carrier of X;
  reconsider h=f+g as Function of X,T;
A1:dom f = CX & dom g = CX & dom h = CX by FUNCT_2:def 1;
  now let x be object;
    assume A2:x in (CX\ support(f)) /\ (CX\ support(g)); then
    x in (CX\ support(f)) & x in (CX\ support(g)) by XBOOLE_0:def 4; then
    x in CX & not x in support(f) & x in CX & not x in support(g)
      by XBOOLE_0:def 5; then
A3: (not x in dom f or f/.x = 0.T) & (not x in dom g or g/.x=0.T) by Def10;
A4: (f+g)/.x= 0.T+ 0.T by A3,A2,A1,VFUNCT_1:def 1;
    not x in support(f+g) by A4,Def10;
    hence x in CX\ support(f+g) by A2,XBOOLE_0:def 5;
  end; then
  (CX\ support(f)) /\ (CX \ support(g)) c= CX\ support(f+g); then
  CX\ ( support(f) \/ support(g) ) c= CX\ support(f+g) by XBOOLE_1:53; then
  CX\ (CX\ support(f+g)) c=CX\ (CX\ (support(f) \/ support(g)))
                                               by XBOOLE_1:34; then
  CX/\ support(f+g) c=CX\ (CX\ (support(f) \/ support(g))) by XBOOLE_1:48;
  then
  CX/\ support(f+g) c= CX/\ (support(f) \/ support(g)) by XBOOLE_1:48;
  then support(f+g) c= CX/\ (support(f) \/ support(g)) by XBOOLE_1:28; then
  support(f+g) c= (CX/\ support(f)) \/ (CX/\ support(g)) by XBOOLE_1:23;
  then support(f+g) c= support(f) \/ (CX/\ support(g)) by XBOOLE_1:28;
  hence thesis by XBOOLE_1:28;
end;
