
theorem Th33:
  for X being non empty TopSpace
  for f,g being Function of the carrier of X,COMPLEX
                  holds support(f+g) c= support(f) \/ support(g)
proof
  let X be non empty TopSpace;
  let f,g be Function of the carrier of X,COMPLEX;
  set CX= the carrier of X;
  set h=f+g;
A1: rng h c= COMPLEX;
  dom h = dom f /\ dom g by VALUED_1:def 1
     .= (the carrier of X) /\ dom g by PARTFUN1:def 2
     .= (the carrier of X) /\ the carrier of X by PARTFUN1:def 2;
  then reconsider h as Function of the carrier of X,COMPLEX by A1,FUNCT_2:2;
  dom f = CX & dom g = CX & dom h = CX by FUNCT_2:def 1;
  then
A2: support(f) c= CX & support(g) c= CX & support(h) c= CX by PRE_POLY:37;
  now let x be object;
    assume
      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:   x in CX & f.x =0 & (g.x)=0 by PRE_POLY:def 7;
    then
      (f+g).x = ( f.x) + (g.x) by VALUED_1:1
             .= 0 by A3; then
A4:   x in CX & (f+g).x =0 by A3;
    not x in support(f+g) by A4,PRE_POLY:def 7;
    hence x in CX\ support(f+g) by A4,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 A2,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 A2,XBOOLE_1:28;
  hence support(f+g) c= support(f) \/ support(g) by A2,XBOOLE_1:28;
end;
