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

theorem Th17:
  for n be Ordinal,
      L be right_zeroed add-associative right_complementable
              well-unital distributive Abelian non trivial commutative
              associative non empty doubleLoopStr
  for x being Function of n, L
    for b be bag of n, i be object,j be Nat st i in n holds
  eval (b+*(i,j),x) * (power L).(x/.i, b.i) = eval (b,x) * (power L).(x/.i, j)
  proof
    let n be Ordinal,
      L be right_zeroed add-associative right_complementable
              well-unital distributive Abelian non trivial commutative
              associative non empty doubleLoopStr;
    let x be Function of n, L;
    set E=EmptyBag n;
    let b be bag of n,i be object ,j be Nat such that
A1: i in n;
A2: i is set by TARSKI:1;
    set d = b+*(i,j);
    d = d +*(i,0) + E+*(i,d.i) by Th15;
    then d = b +*(i,0) + E+*(i,d.i) by A2,FUNCT_7:34;
    then
A3: eval (b+*(i,j),x) * (power L).(x/.i, b.i) =
    eval(b +*(i,0),x) * eval(E+*(i,d.i),x) * (power L).(x/.i, b.i)
    by POLYNOM2:16;
    dom b = n =dom x by PARTFUN1:def 2;
    then
A5: d.i = j & x/.i = x.i by A1,FUNCT_7:31,PARTFUN1:def 6;
A10: b = (b +*(i,0)) + E+*(i,b.i) by Th15;
     (power L).(x/.i, b.i) = eval(E +* (i,b.i),x) by A5,Th14,A1;
     then
A14: eval(b +*(i,0),x) * (power L).(x/.i, b.i) =
     eval(b,x) by POLYNOM2:16,A10;
     thus eval (b+*(i,j),x) * (power L).(x/.i, b.i) =
     eval(b +*(i,0),x) * (power L).(x/.i, b.i) * eval(E+*(i,j),x)
     by A5,A3,GROUP_1:def 3
     .= eval(b,x) * (power L).(x/.i, j) by A14,A5,A1,Th14;
   end;
