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 Th6:
  for n being natural Number holds 0 < a implies 0 < a |^ n
proof
  let n be natural Number;
A1: n is Nat by TARSKI:1;
  defpred P[Nat] means 0 < a |^ $1;
  assume
A2: 0<a;
A3: for m be Nat st P[m] holds P[m+1]
  proof
    let m be Nat;
    assume a |^ m > 0;
    then (a |^ m)*a > 0*a by A2;
    hence thesis by NEWTON:6;
  end;
A4: P[0] by NEWTON:4;
  for m be Nat holds P[m] from NAT_1:sch 2(A4,A3);
  hence thesis by A1;
end;
