
theorem NDIFF825: :::: preliminaries
  for n be Nat, r be Real st 0 < r
  ex s be Real st 0 < s & s < r & sqrt((s*s) * n) < r
  proof
    let n be Nat, r be Real;
    assume
    A1: 0 < r;
    per cases;
    suppose
      A2: 0 = n;
      set s = r/2;
      take s;
      thus 0 < s & s < r & sqrt((s*s)*n) < r
        by A1,A2,XREAL_1:216;
    end;
    suppose
      A3: 0 <> n;
      set s = r/(n+1);
      take s;
      set s = r/(n+1);
      A4: n + 0 < n + 1 by XREAL_1:8;
      thus 0 < s by A1;
      0+1 <= n by A3,NAT_1:13; then
      1*n <= n*n by XREAL_1:66; then
      A6: (s*s)*n <= s*s*(n*n) by XREAL_1:66;
      0 < 1 & 0 + 1 < n + 1 by A3,XREAL_1:8; then
      1/(n+1) < 1 by XREAL_1:191; then
      1/(n+1) * r < r*1 by A1,XREAL_1:68;
      hence s < r;
      sqrt((s*s)*n) <= sqrt((s*n)^2) by A6,SQUARE_1:26; then
      A8: sqrt((s*s)*n) <= s*n by A1,SQUARE_1:22;
      n/(n+1) < 1 by A4,XREAL_1:191; then
      n/(n+1) * r < r*1 by A1,XREAL_1:68;
      hence sqrt((s*s)*n) < r by A8,XXREAL_0:2;
    end;
  end;
