reserve i,j,n,n1,n2,m,k,l,u for Nat,
        i1,i2,i3,i4,i5,i6 for Element of n,
        p,q for n-element XFinSequence of NAT,
        a,b,c,d,e,f for Integer;

theorem Th3:
  for n being set, p being Series of n, F_Real holds
    Support p = Support |. p .|
proof
  let n be set, p be Series of n, F_Real;
  A1: dom p = Bags n = dom |. p .| by FUNCT_2:def 1;
  thus Support p c= Support |. p .|
  proof
    let x be object;
    assume x in Support p;
    then x in dom p & p.x <> 0.F_Real &
    |. p.x .| = |.p.| .x by Def1,POLYNOM1:def 3;
    hence thesis by A1,POLYNOM1:def 3;
  end;
  let x be object;
  assume x in Support |.p.|;
  then x in dom |.p.| & |.p.| .x <> 0.F_Real &
  |. p.x .| = |.p.| . x by Def1,POLYNOM1:def 3;
  then x in dom p & p.x <> 0.F_Real by A1;
  hence thesis by POLYNOM1:def 3;
end;
