reserve i,j,k,n,m for Nat,
        X for set,
        b,s for bag of X,
        x for object;

theorem Th27:
  for n be Nat, p be object st p in n
  for i be integer Element of F_Real
   for x be Function of n,F_Real holds
     eval( Monom(i,(EmptyBag n) +*(p,1)),x) = i*(x/.p)
proof
  let n be Nat, p be object such that
A1: p in n;
  let i be integer Element of F_Real;
  let x be Function of n,F_Real;
  set Ep=(EmptyBag n) +*(p,1);
  n =dom x by PARTFUN1:def 2;
  then
A2: x.p = x/.p by A1,PARTFUN1:def 6;
  eval(Ep,x) = power(F_Real).(x.p,1) by A1,Th14
  .= x/.p by A2,GROUP_1:50;
  hence thesis by POLYNOM7:13;
end;
