reserve i,j for Nat;
reserve A,B for Ring;

theorem Th15:
  for n be Nat, x be Element of A, p be Polynomial of A st A is Subring of B
  holds In(p.(n-'1),B)*(power B).(In(x,B),n-'1)
  = In(p.(n-'1) * (power A).(x,n-'1),B)
  proof
    let n be Nat,x be Element of A, p be Polynomial of A;
    assume
A0: A is Subring of B; then
    In(p.(n-'1) * (power A).(x,n-'1),B)
    = In(p.(n-'1),B)*In((power A).(x,n-'1),B) by Th13
    .= In(p.(n-'1),B)*(power B).(In(x,B),n -'1) by A0,Th11;
    hence thesis;
  end;
