
theorem Th29:
  for R being Skew-Field, x being Element of MultGroup R, y being
Element of R st y = x for k being Nat holds (power (MultGroup R)).(x
  ,k) = (power R).(y,k)
proof
  let R be Skew-Field, x be Element of MultGroup R, y be Element of R such
  that
A1: y = x;
  defpred P[Nat] means (power (MultGroup R)).(x,$1) = (power R).(y,$1);
A2: for k be Nat st P[k] holds P[k+1]
  proof
    let k be Nat such that
A3: P[k];
     reconsider kk=k as Element of NAT by ORDINAL1:def 12;
    thus (power (MultGroup R)).(x,k+1) = (power (MultGroup R)).(x,kk) * x
                  by GROUP_1:def 7
      .= (power R).(y,kk) * y by A1,A3,Th16
      .= (power R).(y,k+1) by GROUP_1:def 7;
  end;
  (power (MultGroup R)).(x,0) = 1_MultGroup R & (power R).(y,0) = 1_R by
GROUP_1:def 7;
  then
A4: P[0] by Th17;
  thus for k be Nat holds P[k] from NAT_1:sch 2(A4,A2);
end;
