
theorem Th22:
  for L be non degenerated commutative
    Abelian add-associative right_zeroed right_complementable
      associative well-unital distributive non empty doubleLoopStr
  for p be Polynomial of L for x be Element of L
  for n be Nat holds eval(p`^n,x) = (power L).(eval(p,x),n)
proof
  let L be non degenerated commutative
    Abelian add-associative right_zeroed right_complementable
      associative well-unital distributive non empty doubleLoopStr;
  let p be Polynomial of L;
  let x be Element of L;
  defpred P[Nat] means eval(p`^$1,x) = (power L).(eval(p,x),$1);
A1: now
    let n be Nat;
    assume
A2: P[n];
    eval(p`^(n+1),x) = eval((p`^n)*'p,x) by Th19
      .= (power L).(eval(p,x),n)*eval(p,x) by A2,POLYNOM4:24
      .= (power L).(eval(p,x),n+1) by GROUP_1:def 7;
    hence P[n+1];
  end;
  eval(p`^0,x) = eval(1_.(L),x) by Th15
    .= 1_(L) by POLYNOM4:18
    .= (power L).(eval(p,x),0) by GROUP_1:def 7;
  then
A3: P[0];
  thus for n be Nat holds P[n] from NAT_1:sch 2(A3,A1);
end;
