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 Th27:
  0<=a & a<=b & n>=1 implies n -Root a <= n -Root b
proof
  assume that
A1: a>=0 and
A2: a<=b and
A3: n>=1 and
A4: n -Root a > n -Root b;
A5: now
    per cases by A1,A2;
    suppose
      b > 0;
      hence n -Root b >= 0 by A3,Def2;
    end;
    suppose
      b=0;
      hence n -Root b >= 0 by A3,Def2;
    end;
  end;
  (n -Root a) |^ n = a by A1,A3,Th19;
  then (n -Root a) |^ n <= (n -Root b) |^ n by A1,A2,A3,Th19;
  hence contradiction by A3,A4,A5,Th10;
end;
