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 Th31:
  a>0 & n>=1 implies n -Root a - 1 <= (a-1)/n
proof
  assume that
A1: a>0 and
A2: n>=1;
  n -Root a > 0 by A1,A2,Def2;
  then n -Root a - 1 > 0 - 1 by XREAL_1:9;
  then (1 + (n -Root a - 1)) |^ n >= 1 + n*(n -Root a - 1) by Th16;
  then a >= 1 + n*(n -Root a - 1) by A1,A2,Lm2;
  then a - 1 >= n*(n -Root a - 1) by XREAL_1:19;
  then (a - 1)/n >= n*(n -Root a - 1)/n by XREAL_1:72;
  hence thesis by A2,XCMPLX_1:89;
end;
