reserve Z for open Subset of REAL;

theorem Th1:
  for x be Real, n be Nat holds |.x |^ n.| = |.x.| |^ n
proof
  let x be Real, n be Nat;
  defpred P[Nat] means |.x |^ $1.| = |.x.| |^ $1;
A1: for k be Nat st P[k] holds P[k+1]
  proof
    let k be Nat such that
A2: P[k];
    |.x |^ (k+1).| = |.x * x |^ k.| by NEWTON:6
      .= |.x.| * |.x |^ k.| by COMPLEX1:65
      .= |.x.| |^ (k+1) by A2,NEWTON:6;
    hence thesis;
  end;
  |.x |^ 0.| = |.1.| by NEWTON:4
    .= 1 by ABSVALUE:def 1;
  then
A3: P[0] by NEWTON:4;
  for n be Nat holds P[n] from NAT_1:sch 2(A3,A1);
  hence thesis;
end;
