
theorem Th8:
  for X being set, L being non empty ZeroStr, b being bag of X
  holds coefficient(Monom(0.L,b)) = 0.L & term(Monom(0.L,b)) = EmptyBag X
proof
  let n be set, L be non empty ZeroStr, b be bag of n;
  set m = 0_(n,L)+*(b,0.L);
  reconsider m as Function of Bags n, the carrier of L;
  reconsider m as Function of Bags n, L;
  reconsider m as Series of n, L;
A1: b in Bags n by PRE_POLY:def 12;
A2: dom(0_(n,L)) = dom((Bags n) --> 0.L) by POLYNOM1:def 8
    .= Bags n;
  then
A3: m = 0_(n,L)+*(b .--> 0.L) by A1,FUNCT_7:def 3;
A4: b in dom(b .--> 0.L) by TARSKI:def 1;
A5: m.b = (0_(n,L)+*(b .--> 0.L)).b by A2,A1,FUNCT_7:def 3
    .= (b .--> 0.L).b by A4,FUNCT_4:13
    .= 0.L by FUNCOP_1:72;
A6: now
    let b9 be bag of n;
    now
      per cases;
      case
        b9 = b;
        hence m.b9 = 0.L by A5;
      end;
      case
        b9 <> b;
        then not b9 in dom(b .--> 0.L) by TARSKI:def 1;
        hence m.b9 = (0_(n,L)).b9 by A3,FUNCT_4:11
          .= 0.L by POLYNOM1:22;
      end;
    end;
    hence m.b9 = 0.L;
  end;
  hence coefficient(Monom(0.L,b)) = 0.L;
  (Monom(0.L,b)).(term(Monom(0.L,b))) = 0.L by A6;
  hence thesis by Def5;
end;
