
theorem Th1:
  for f, g being Function holds support (f +* g) c= support f \/ support g
proof
  let f, g be Function;
  let a be object;
  assume a in support (f +* g);
  then
A1: (f +* g).a <> 0 by PRE_POLY:def 7;
  assume
A2: not a in support f \/ support g;
  then not a in support f by XBOOLE_0:def 3;
  then
A3: f.a = 0 by PRE_POLY:def 7;
  not a in support g by A2,XBOOLE_0:def 3;
  then
A4: g.a = 0 by PRE_POLY:def 7;
  a in dom g or not a in dom g;
  hence contradiction by A1,A3,A4,FUNCT_4:11,13;
end;
