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 \] + i0 = [/ r + i0 \]
proof
  [/ r \] < r + 1 by Def7;
  then [/ r \] + i0 < r + 1 + i0 by XREAL_1:6;
  then
A1: [/ r \] + i0 < r + i0 + 1;
  r <= [/ r \] by Def7;
  then r + i0 <= [/ r \] + i0 by XREAL_1:6;
  hence thesis by A1,Def7;
end;
