
theorem
  for X being set, R being non empty ZeroStr holds R is non trivial iff
  ex s being Series of X,R st Support(s) <> {}
proof
  let X be set, R be non empty ZeroStr;
A1: now
    set x = EmptyBag X;
    assume R is non trivial;
    then consider a being Element of R such that
A2: a <> 0.R;
    take s = (Bags X) --> a;
    s.x = a;
    then EmptyBag X in Support s by A2,POLYNOM1:def 4;
    hence ex s being Series of X,R st Support(s) <> {};
  end;
  now
    assume ex s being Series of X,R st Support(s) <> {};
    then consider s being Series of X,R such that
A3: Support(s) <> {};
    set b = the Element of Support s;
    b in Support s by A3;
    then reconsider b as Element of Bags X;
    now
      given x being object such that
A4:   the carrier of R = {x};
      0.R = x by A4,TARSKI:def 1
        .= s.b by A4,TARSKI:def 1;
      hence contradiction by A3,POLYNOM1:def 4;
    end;
    hence R is non trivial by ZFMISC_1:131;
  end;
  hence thesis by A1;
end;
