
theorem
  for R being unital associative commutative non empty multMagma,
  a,b being Element of R, n being Nat holds
  (a * b)|^n = (a|^n) * (b|^n)
proof
  let R be unital associative commutative non empty multMagma, a,b be
  Element of R, n be Nat;
  defpred P[Nat] means power(R).(a * b,$1) = a|^$1 * b|^$1;
A1: now
    let m be Nat;
     reconsider mm=m as Element of NAT by ORDINAL1:def 12;
    assume P[m];
    then
    power(R).(a * b,m+1) = (power(R).(a,mm) * power(R).(b,mm)) * (a * b) by
GROUP_1:def 7
      .= ((power(R).(a,mm) * power(R).(b,mm)) * a) * b by GROUP_1:def 3
      .= ((power(R).(a,mm) * a) * power(R).(b,mm)) * b by GROUP_1:def 3
      .= (power(R).(a,mm) * a) * (power(R).(b,mm) * b) by GROUP_1:def 3
      .= (a|^(m+1)) * (power(R).(b,mm) * b) by GROUP_1:def 7
      .= (a|^(m+1)) * (b|^(m+1)) by GROUP_1:def 7;
    hence P[m+1];
  end;
  power(R).(a * b,0) = 1_R by GROUP_1:def 7
    .= 1_R * 1_R by GROUP_1:def 4
    .= power(R).(a,0) * 1_R by GROUP_1:def 7
    .= power(R).(a,0) * power(R).(b,0) by GROUP_1:def 7;
  then
A2: P[0];
  for m being Nat holds P[m] from NAT_1:sch 2(A2,A1);
  hence thesis;
end;
