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 Th19:
  for n being Nat st a>=0 & n>=1 holds (n -Root a) |^ n = a & n
  -Root (a |^ n) = a
proof
  let n be Nat;
  assume that
A1: a>=0 and
A2: n>=1;
  per cases by A1;
  suppose
    a>0;
    hence thesis by A2,Lm2;
  end;
  suppose
A3: a=0;
    reconsider k=n, k1=1 as Integer;
    reconsider m=k-k1 as Element of NAT by A2,INT_1:5;
A4: 0 |^ n = 0 |^ (m+1) .= 0 |^ m * 0 |^ 1 by NEWTON:8
      .= 0 |^ m * 0 GeoSeq.(0+1) by Def1
      .= 0 |^ m * (0 GeoSeq.0 * 0) by Th3
      .= 0;
    hence (n -Root a) |^ n = a by A2,A3,Def2;
    thus thesis by A2,A3,A4,Def2;
  end;
end;
