
theorem Th1:
  for X being set, R being non empty ZeroStr, s being Series of X,R
  holds s = 0_(X,R) iff Support s = {}
proof
  let X be set, R be non empty ZeroStr, s be Series of X,R;
A1: now
    assume
A2: Support s = {};
    now
      let x be object;
      assume x in Bags X;
      then reconsider x9 = x as Element of Bags X;
      s.x9 = 0.R by A2,POLYNOM1:def 4;
      hence s.x = (0_(X,R)).x by POLYNOM1:22;
    end;
    hence s = 0_(X,R);
  end;
  now
    assume
A3: s = 0_(X,R);
    now
      set x = the Element of Support s;
      assume Support s <> {};
      then
A4:   x in Support s;
      then reconsider x as bag of X;
      s.x <> 0.R by A4,POLYNOM1:def 4;
      hence contradiction by A3,POLYNOM1:22;
    end;
    hence Support s = {};
  end;
  hence thesis by A1;
end;
