
theorem Th14:
  for X being set, L being non empty ZeroStr, p being Series of X,
  L holds p is ConstPoly of X,L iff (p = 0_(X,L) or Support p = {EmptyBag X})
proof
  let n be set, L be non empty ZeroStr, p be Series of n,L;
A1: now
    assume
A2: p is ConstPoly of n,L;
A3: for u being object holds u in Support p implies u in {EmptyBag n}
    proof
      let u be object;
      assume
A4:   u in Support p;
      then reconsider u as Element of Bags n;
      reconsider u9 = u as bag of n;
      p.u9 <> 0.L by A4,POLYNOM1:def 4;
      then u9 = EmptyBag n by A2,Def7;
      hence thesis by TARSKI:def 1;
    end;
    thus Support p = {EmptyBag n} or p = 0_(n,L)
    proof
      assume
A5:   not Support p = {EmptyBag n};
A6:   not EmptyBag n in Support p
      proof
        assume EmptyBag n in Support p;
        then
        for u being object holds u in {EmptyBag n} implies u in Support p by
TARSKI:def 1;
        hence thesis by A3,A5,TARSKI:2;
      end;
A7:   Support p = {}
      proof
        set v = the Element of Support p;
        assume Support p <> {};
        then v in Support p & v in {EmptyBag n} by A3;
        hence thesis by A6,TARSKI:def 1;
      end;
A8:   for b being bag of n holds p.b = 0.L
      proof
        let b be bag of n;
A9:     b is Element of Bags n by PRE_POLY:def 12;
        assume p.b <> 0.L;
        hence thesis by A7,A9,POLYNOM1:def 4;
      end;
A10:  for u being object holds u in rng p implies u in {0.L}
      proof
        let u be object;
        assume u in rng p;
        then consider x being object such that
A11:    x in dom p and
A12:    p.x = u by FUNCT_1:def 3;
        x is bag of n by A11;
        then u = 0.L by A8,A12;
        hence thesis by TARSKI:def 1;
      end;
A13:  dom p = Bags n by FUNCT_2:def 1;
      for u being object holds u in {0.L} implies u in rng p
      proof
        set b = the bag of n;
        let u be object;
        assume u in {0.L};
        then u = 0.L by TARSKI:def 1;
        then
A14:    p.b = u by A8;
        b in dom p by A13,PRE_POLY:def 12;
        hence thesis by A14,FUNCT_1:def 3;
      end;
      then rng p = {0.L} by A10,TARSKI:2;
      then p = (Bags n) --> 0.L by A13,FUNCOP_1:9;
      hence thesis by POLYNOM1:def 8;
    end;
  end;
  now
    assume
A15: p = 0_(n,L) or Support p = {EmptyBag n};
    per cases by A15;
    suppose
      p = 0_(n,L);
      then for b being bag of n st b <> EmptyBag n holds p.b = 0.L by
POLYNOM1:22;
      hence p is ConstPoly of n,L by Def7;
    end;
    suppose
A16:  Support p = {EmptyBag n};
      for b being bag of n st b <> EmptyBag n holds p.b = 0.L
      proof
        let b be bag of n;
        assume
A17:    b <> EmptyBag n;
        reconsider b as Element of Bags n by PRE_POLY:def 12;
        not b in Support p by A16,A17,TARSKI:def 1;
        hence thesis by POLYNOM1:def 4;
      end;
      hence p is ConstPoly of n,L by Def7;
    end;
  end;
  hence thesis by A1;
end;
