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(** *    Fractional part                                 *)
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Definition Int_part (r:R) : Z := (up r - 1)%Z.(**********)
Definition frac_part (r:R) : R := r - IZR (Int_part r).

Lemma plus_Int_part2 : forall r1 r2:R, frac_part r1 + frac_part r2 < 1 -> Int_part (r1 + r2) = (Int_part r1 + Int_part r2)%Z.