
theorem
  for X being non empty TopSpace
  for f,g be 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;
  reconsider h=f(#)g as Function of the carrier of X,COMPLEX;
  dom f = CX & dom g = CX & dom h = CX by FUNCT_2:def 1;
  then
A1: 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
A2:   x in CX& f.x =0 & g.x=0 by PRE_POLY:def 7;
    (f(#)g).x = 0 * 0 by A2,VALUED_1:5;
    then
A3:   x in CX & (f(#)g).x =0 by A2;
    not x in support(f(#)g) by A3,PRE_POLY:def 7;
    hence x in CX\ support(f(#)g) by A3,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 A1,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 A1,XBOOLE_1:28;
  hence support(f(#)g) c= support(f) \/ support(g) by A1,XBOOLE_1:28;
end;
