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 Th23:
  a>0 & n>=1 implies n -Root (1/a) = 1/(n -Root a)
proof
  assume that
A1: a>0 and
A2: n>=1 and
A3: n -Root (1/a) <> 1/(n -Root a);
A4: n -Root a > 0 by A1,A2,Def2;
A5: (1/(n -Root a)) |^ n = 1 / (n -Root a) |^ n by Th7
    .= 1/a by A1,A2,Lm2;
A6: n -Root (1/a) > 0 by A1,A2,Def2;
  per cases by A3,XXREAL_0:1;
  suppose
    n -Root (1/a) > 1/(n -Root a);
    hence contradiction by A2,A4,A5,Lm2;
  end;
  suppose
    n -Root (1/a) < 1/(n -Root a);
    hence contradiction by A2,A5,A6,Lm2;
  end;
end;
