reserve x,y for object, X for set;

theorem Th28:
  for p be Prime, z be Element of Z/Z*(p), y be Element of
INT.Ring(p) st z=y holds for n be Element of NAT holds (power Z/Z*(p)).(z,n) =
  (power INT.Ring(p)).(y,n)
proof
  let p be Prime, z be Element of Z/Z*(p), y be Element of INT.Ring(p);
  defpred P[Nat] means (power Z/Z*(p)).(z,$1) =(power INT.Ring(p)).(y,$1);
  assume
A1: z=y;
A2: now
    let k be Nat;
    assume
A3: P[k];
    (power Z/Z*(p)).(z,k+1) = (power Z/Z*(p)).(z,k)*z by GROUP_1:def 7
      .=(power INT.Ring(p)).(y,k)*y by A1,A3,Lm12
      .=(power INT.Ring(p)).(y,k+1) by GROUP_1:def 7;
    hence P[k+1];
  end;
  (power Z/Z*(p)).(z,0)=1_(Z/Z*(p)) by GROUP_1:def 7
    .= 1_(INT.Ring(p)) by Th21
    .= (power INT.Ring(p)).(y,0) by GROUP_1:def 7;
  then
A4: P[ 0 ];
  for k be Nat holds P[k] from NAT_1:sch 2(A4,A2);
  hence thesis;
end;
