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

theorem Th14:
  for n be Ordinal,
      L be well-unital non trivial doubleLoopStr,
      y be Function of n, L st x in n holds
  eval((EmptyBag n) +*(x,i),y) = power(L).(y.x,i)
proof
  let n be Ordinal, L be well-unital non trivial doubleLoopStr,
  y be Function of n, L such that
A1: x in n;
A2: n=dom y by PARTFUN1:def 2;
  reconsider x as set by TARSKI:1;
  set E = EmptyBag n,Ex = E+*(x,i);
  per cases;
  suppose A3:i<>0;
A4: dom E = n by PARTFUN1:def 2;
   support(Ex)={x} by Th13,A1,A3;
   then eval(Ex,y) = power(L).(y.x,Ex.x) by POLYNOM2:15;
    hence thesis by A4,A1,FUNCT_7:31;
  end;
  suppose
A5: i=0;
    then E = E +*(x,E.x) & E.x = i by FUNCT_7:35;
    then eval(E +*(x,i),y) = 1_L by POLYNOM2:14
    .= (power L).(y/.x, i) by A5,GROUP_1:def 7;
    hence thesis by A2,A1,PARTFUN1:def 6;
  end;
end;
