
theorem Th1:
  for L being unital associative non empty multMagma, a being Element of L,
    n,m being Element of NAT holds
  power(L).(a,n+m) = power(L).(a,n) * power(L).(a,m)
proof
  let L be unital associative non empty multMagma, a be Element of L, n,m be
  Element of NAT;
  defpred P[Nat] means power(L).(a,n+$1) = power(L).(a,n) * power(L).(a,$1);
A1: now
    let m be Nat;
    assume
A2: P[m];
    power(L).(a,n+(m+1)) = power(L).(a,(n+m)+1)
      .= (power(L).(a,n) * power(L).(a,m)) * a by A2,GROUP_1:def 7
      .= power(L).(a,n) * (power(L).(a,m) * a) by GROUP_1:def 3
      .= power(L).(a,n) * power(L).(a,(m+1)) by GROUP_1:def 7;
    hence P[m+1];
  end;
  power(L).(a,n + 0) = power(L).(a,n) * 1_L by GROUP_1:def 4
    .= power(L).(a,n) * power(L).(a,0) by GROUP_1:def 7;
  then
A3: P[0];
  for m being Nat holds P[m] from NAT_1:sch 2(A3,A1);
  hence thesis;
end;
