reserve X for set, x,y,z for object,
  k,l,n for Nat,
  r for Real;
reserve i,i0,i1,i2,i3,i4,i5,i8,i9,j for Integer;
reserve r1,p,p1,g,g1,g2 for Real,
  Y for Subset of REAL;

theorem Th41:
  frac r < 1 & 0 <= frac r
proof
  r - 1 < [\ r /] by Def6;
  then frac r + (r - 1) < r - [\ r /] + [\ r /] by XREAL_1:6;
  then frac r + (- 1) + r - r < r - r by XREAL_1:9;
  then
A1: frac r - 1 + 1 < 0 + 1 by XREAL_1:6;
  [\ r /] <= r by Def6;
  then [\ r /] + (r - [\ r /]) <= r + frac r by XREAL_1:6;
  then r - r <= r + frac r - r by XREAL_1:9;
  hence thesis by A1;
end;
