
theorem Th6:
  for X being set, L being non empty ZeroStr, p being Series of X,L
  holds p is Monomial of X,L iff (Support p = {} or ex b being bag of X st
  Support p = {b})
proof
  let n be set, L be non empty ZeroStr, p be Series of n,L;
A1: now
    assume ex b being bag of n st Support p = {b};
    then consider b being bag of n such that
A2: Support p = {b};
    for b9 being bag of n st b9 <> b holds p.b9 = 0.L
    proof
      let b9 be bag of n;
      assume
A3:   b9 <> b;
      assume
A4:   p.b9 <> 0.L;
      reconsider b9 as Element of Bags n by PRE_POLY:def 12;
      b9 in Support p by A4,POLYNOM1:def 4;
      hence thesis by A2,A3,TARSKI:def 1;
    end;
    hence p is Monomial of n,L by Def3;
  end;
A5: now
    assume p is Monomial of n,L;
    then consider b being bag of n such that
A6: for b9 being bag of n st b9 <> b holds p.b9 = 0.L by Def3;
    now
      per cases;
      case
A7:     p.b <> 0.L;
A8:     for u being object holds u in {b} implies u in Support p
        proof
          let u be object;
          assume
A9:       u in {b};
          then u = b by TARSKI:def 1;
          then reconsider u9 = u as Element of Bags n by PRE_POLY:def 12;
          p.u9 <> 0.L by A7,A9,TARSKI:def 1;
          hence thesis by POLYNOM1:def 4;
        end;
        for u being object holds u in Support p implies u in {b}
        proof
          let u be object;
          assume
A10:      u in Support p;
          then reconsider u9 = u as bag of n;
          p.u <> 0.L by A10,POLYNOM1:def 4;
          then u9 = b by A6;
          hence thesis by TARSKI:def 1;
        end;
        then Support p = {b} by A8,TARSKI:2;
        hence ex b being bag of n st Support p = {b};
      end;
      case
A11:    p.b = 0.L;
        thus Support p = {}
        proof
          assume Support p <> {};
          then reconsider sp = Support p as non empty Subset of Bags n;
          set c = the Element of sp;
          p.c <> 0.L by POLYNOM1:def 4;
          hence thesis by A6,A11;
        end;
      end;
    end;
    hence Support p = {} or ex b being bag of n st Support p = {b};
  end;
  now
    set b = the bag of n;
    assume
A12: Support p = {};
    for b9 being bag of n st b9 <> b holds p.b9 = 0.L
    proof
      let b9 be bag of n;
      assume b9 <> b;
      reconsider c = b9 as Element of Bags n by PRE_POLY:def 12;
      assume p.b9 <> 0.L;
      then c in Support p by POLYNOM1:def 4;
      hence thesis by A12;
    end;
    hence p is Monomial of n,L by Def3;
  end;
  hence thesis by A1,A5;
end;
