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 /] = [/ r \] iff not [\ r /] + 1 = [/ r \]
proof
  set Diff = [/ r \] + (- [\ r /]);
  now
    assume not r is Integer;
    then [\ r /] < [/ r \] by Th35;
    then [\ r /] + (- [\ r /]) < Diff by XREAL_1:6;
    then
A2: 1 <= Diff by Lm4;
    r - 1 < [\ r /] by Def6;
    then
A3: - [\ r /] < - (r - 1) by XREAL_1:24;
    [/ r \] < r + 1 by Def7;
    then Diff < r + 1 + (- (r - 1)) by A3,XREAL_1:8;
    then Diff + 1 + (- 1) <= 1 + 1 + (- 1) by Th7;
    then [\ r /] + 1 = [\ r /] + Diff by A2,XXREAL_0:1;
    hence [\ r /] + 1 = [/ r \] & [\ r /] <> [/ r \];
  end;
  hence thesis;
end;
