reserve x for set;
reserve a, b, c for Real;
reserve m, n, m1, m2 for Nat;
reserve k, l for Integer;
reserve p, q for Rational;
reserve s1, s2 for Real_Sequence;

theorem Th7:
  for a being Complex, n being natural Number holds (1/a) |^ n = 1 / a |^ n
proof
  let a be Complex;
  let n be natural Number;
A1: n is Nat by TARSKI:1;
  defpred P[Nat] means (1/a) |^ $1 = 1/ a |^ $1;
A2: for m be Nat st P[m] holds P[m+1]
  proof
    let m be Nat;
    assume (1/a) |^ m = 1 / a |^ m;
    hence (1/a) |^ (m+1) = 1 / a |^ m * (1/a) by NEWTON:6
      .= 1*1 / (a |^ m * a) by XCMPLX_1:76
      .= 1 / a |^ (m+1) by NEWTON:6;
  end;
  (1/a) |^ 0 = 1 by NEWTON:4
    .= 1/1
    .= 1/ a |^ 0 by NEWTON:4;
  then
A3: P[0];
  for m be Nat holds P[m] from NAT_1:sch 2(A3,A2);
  hence thesis by A1;
end;
