reserve x for set;
reserve a, b, c, d, e for Real;
reserve m, n, m1, m2 for Nat;
reserve k, l for Integer;
reserve p for Rational;

theorem
  a>-1 & a<=0 & n is odd implies a >= n-root a & n-root a > -1
proof
  assume that
A1: a>-1 and
A2: a<=0;
  assume n is odd;
  then
A3: ex m st n=2*m+1 by ABIAN:9;
then A4: n>=1 & -a<1 by A1,ABIAN:12,XREAL_1:25;
then A5: n-root (-a) < 1 by A2,Th20;
A6: -a <= n-root (-a) by A2,A4,Th20;
A7: - n-root (-a) > -1 by A5,XREAL_1:24;
 a >= - n-root (-a) by A6,XREAL_1:26;
  hence thesis by A3,A7,Th10;
end;
