reserve k,m,n for Nat;
reserve R for commutative Ring,
        p,q for Polynomial of R,
        z0,z1 for Element of R;

theorem Th8:
  for L being Abelian add-associative right_zeroed right_complementable
    well-unital associative commutative distributive non empty doubleLoopStr
  for x being Element of L
  holds <%x%>`^n = <%x|^n%>
proof
  let L be Abelian add-associative right_zeroed right_complementable
    well-unital associative commutative distributive
    non empty doubleLoopStr;let x be Element of L;
    set X = <%x%>;
    defpred P[Nat] means X`^$1 = <%x |^ $1 %>;
    A1:P[0]
    proof
      A2:<%x |^ 0 %> = <%1_L%> by BINOM:8;
      1_.L = <%1_L%>
      proof
        let n be Element of NAT;
        per cases by NAT_1:14;
        suppose  n = 0;
          then <%1_L%>.n = 1_L & (1_.(L)).n = 1.L by POLYNOM5:32,POLYNOM3:30;
          hence thesis;
        end;
        suppose n >= 1;
          then <%1_L%>.n = 0.L & (1_.(L)).n = 0.L by POLYNOM5:32,POLYNOM3:30;
          hence thesis;
        end;
      end;
      hence thesis by POLYNOM5:15,A2;
    end;
    A3:for n holds P[n] implies P[n+1]
    proof let n;
      set n1=n+1;
      assume A4: P[n];
      A5:X`^n1 = (X`^n) *' X by POLYNOM5:19;
      let k be Element of NAT;
      per cases by NAT_1:14;
      suppose  k = 0;
        then A6:(X`^n).k  = (x|^n) &  <%x |^ n1 %>.k = x |^ n1 &
        x|^1 =x by A4,POLYNOM5:32,BINOM:8;
        then (X`^n1).k = (x|^n)*(x|^1) by A5,Th7;
        hence thesis by A6,BINOM:10;
      end;
      suppose A7:k >= 1;
        then ( X`^n).k = 0.L by A4,POLYNOM5:32;
        then ( X`^n1).k = 0.L * x by A5,Th7;
        hence thesis by A7,POLYNOM5:32;
      end;
    end;
    for n holds P[n] from NAT_1:sch 2(A1,A3);
    hence thesis;
  end;
