
theorem Th9:
  for X being set, L being non empty ZeroStr, a being Element of L,
  b being bag of X holds coefficient(Monom(a,b)) = a
proof
  let n be set, L be non empty ZeroStr, a be Element of L, b be bag of n;
  set m = 0_(n,L)+*(b,a);
  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: b in dom(b .--> a) by TARSKI:def 1;
  dom(0_(n,L)) = dom((Bags n) --> 0.L) by POLYNOM1:def 8
    .= Bags n;
  then
A3: m.b = (0_(n,L)+*(b .--> a)).b by A1,FUNCT_7:def 3
    .= (b .--> a).b by A2,FUNCT_4:13
    .= a by FUNCOP_1:72;
  per cases;
  suppose
    m.b <> 0.L;
    hence thesis by A3,Def5;
  end;
  suppose
    m.b = 0.L;
    hence thesis by A3,Th8;
  end;
end;
