
theorem
  for X being set, L being non empty ZeroStr, p being Series of X,L
  holds p is ConstPoly of X,L iff ex a being Element of L st p = a |(X,L)
proof
  let X be set, L be non empty ZeroStr, p be Series of X,L;
  now
    assume
A1: p is ConstPoly of X,L;
    now
      per cases by A1,Th14;
      case
        p = 0_(X,L);
        then p = 0.L |(X,L) by Lm3;
        hence ex a being Element of L st p = a |(X,L);
      end;
      case
A2:     Support p = {EmptyBag X};
        set q = 0_(X,L)+*(EmptyBag X,p.(EmptyBag X));
A3:     now
          let x be object;
          assume x in Bags X;
          then reconsider x9 = x as bag of X;
A4:       dom(0_(X,L)) = dom((Bags X) --> 0.L) by POLYNOM1:def 8
            .= Bags X;
          then
A5:       q = 0_(X,L)+*(EmptyBag X .--> p.(EmptyBag X)) by FUNCT_7:def 3;
A6:       EmptyBag X in dom(EmptyBag X .--> p.(EmptyBag X)) by TARSKI:def 1;
A7:       q.(EmptyBag X) = (0_(X,L)+*(EmptyBag X .--> p.(EmptyBag X))).(
          EmptyBag X) by A4,FUNCT_7:def 3
            .= (EmptyBag X .--> p.(EmptyBag X)).(EmptyBag X) by A6,FUNCT_4:13
            .= p.(EmptyBag X) by FUNCOP_1:72;
          now
            per cases;
            case
              x9 = EmptyBag X;
              hence p.x = q.x by A7;
            end;
            case
A8:           x9 <> EmptyBag X;
A9:          x9 is Element of Bags X by PRE_POLY:def 12;
              not x9 in Support p by A2,A8,TARSKI:def 1;
              then
A10:          p.x9 = 0.L by A9,POLYNOM1:def 4;
              not x9 in dom(EmptyBag X .--> p.(EmptyBag X)) by A8,TARSKI:def 1;
              then q.x9 = (0_(X,L)).x9 by A5,FUNCT_4:11;
              hence p.x = q.x by A10,POLYNOM1:22;
            end;
          end;
          hence p.x = q.x;
        end;
A11:    Bags X = dom q by FUNCT_2:def 1;
        q = p.(EmptyBag X) |(X,L) & Bags X = dom p by FUNCT_2:def 1;
        hence ex a being Element of L st p = a |(X,L) by A11,A3,FUNCT_1:2;
      end;
    end;
    hence ex a being Element of L st p = a |(X,L);
  end;
  hence thesis;
end;
