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
  [\ r /] - 1 < r & [\ r /] < r + 1
proof
  [\ r /] <= r by Def6;
  then
A1: [\ r /] + 0 < r + 1 by XREAL_1:8;
  then [\ r /] + (- 1) < r + 1 + (- 1) by XREAL_1:6;
  hence thesis by A1;
end;
