Definition Rmult_reg_l := RIneq.Rmult_eq_reg_l.

(* Specifications: *)
Lemma Rmult_reg_r : forall r r1 r2 : R, r1 * r = r2 * r -> r <> 0 -> r1 = r2.
Lemma Rmult_Rdiv: forall x y z t : R, z <> R0 -> t <> R0 -> x * t = y * z -> x / z = y / t.
Lemma Rmult_Rdiv_pos_Rle: forall x y z t, (R0 < z)%R -> (R0 < t)%R -> (x * t <= y * z)%R -> (x / z <= y / t)%R.
Lemma Rmult_Rdiv_neg_Rle: forall x y z t, (z < R0)%R -> (t < R0)%R -> (x * t <= y * z)%R -> (x / z <= y / t)%R.
Lemma Rdiv_Rmult_simplify: forall x y z : R, z <> 0%R -> y <> 0%R -> (x * z / (y * z))%R = (x / y)%R.
Lemma Rdiv_Rmult_numerator_denominator: forall x y z t: R, t <> 0%R -> y <> 0%R -> ((x/y)*(z/t))%R=((x*z)/(y*t))%R.
Lemma Rdiv_Rdiv_Rmult_numerator: forall x y z : R, y <> 0 -> z <> 0 -> (x / y / z) = (x / (y * z)).
Lemma Rdiv_Rdiv_Rmult_denominator: forall x y z : R, y <> 0 -> z <> 0 -> (x / (y / z)) = (x*z / y ).
Lemma Rmult_Rdiv_pos_Rlt: forall x y z t, (R0 < z)%R -> (R0 < t)%R -> (x * t < y * z)%R -> (x / z < y / t)%R.
Lemma Rmult_Rdiv_neg_Rlt: forall x y z t, (z < R0)%R -> (t < R0)%R -> (x * t < y * z)%R -> (x / z < y / t)%R.
Lemma Rlinear_non_zero_1:forall a b x y, (y<>0)%R -> (a*x+b*y<>0)%R -> (a*(x/y)+b<>0)%R.
Lemma Rlinear_non_zero_2:forall a b x y, (y<>0)%R -> (a*(x/y)+b<>0)%R -> (a*x+b*y<>0)%R.
Lemma Rlinear_non_zero_3: forall a b x : R, a <> 0 -> x <> -b/a -> a * x + b <> 0.
Lemma Rbilinear_non_zero_2:forall a b c d x y x' y', y<>0 -> y'<>0 -> (a*(x/y)*(x'/y')+b*(x/y)+c*(x'/y')+d<>0)%R -> (a*x*x'+b*x*y'+c*y*x'+d*y*y'<>0)%R.
Lemma Rle_dec_weak:forall (x y:R), {Rle x y}+{(Rle y x)}.
Lemma Rtrichotomy_inf:forall r1 r2 : R, {(r1 < r2)%R} + {r1 = r2} + {(r2<r1)%R}.
Lemma not_O_S_INR: forall n : nat, INR (S n) <> 0%R.
Lemma pos_S_INR: forall n : nat, (0 < INR (S n))%R.
Lemma Req_Rdiv_Rone:forall x y, y<>0 -> x=y -> x/y =1.
Lemma Req_Ropp_Rdiv_minus_Rone:forall x y, y<>0 -> x=-y -> x/y =-1.
Lemma Rmax_involutive: forall q, Rmax q q = q.
Lemma Rabs_Rle: forall q p : R, (- p <= q)%R -> (q <= p)%R -> (Rabs q <= p)%R.
Lemma Rabs_Rle_1: forall q, (- 1 <= q)%R -> (q <= 1)%R -> (Rabs q <= 1)%R.
Lemma pow_Rle_r : forall (x y : R) (n : nat), - x <= y -> y <= x -> pow y n <= pow x n.
Lemma pow_Rle_r_1 : forall (y: R) (n : nat), - 1 <= y -> y <= 1 -> pow y n <= 1.
Lemma pow_even_nonneg:forall (y: R) (n : nat), 0 <= pow y (2*n)%nat.
Lemma pow_Rle_l_1 : forall (y: R) (n : nat), - 1 <= y -> y <= 1 -> -1 <= pow y n.
Lemma conjL_range_l:forall r, -1 <= r -> -1<= (r-1)/(r+3).
Lemma conjL_range_r:forall r, -1<=r -> r <= 1 -> (r-1)/(r+3) <= 0.
Lemma conjL_range_weak:forall r, -1 <= r <= 1-> -1<= (r-1)/(r+3)<=1.
Lemma conjR_range_l:forall r, -1 <= r -> r <= 1 -> 0<= (r+1)/(-r+3).
Lemma conjR_range_r:forall r, r <= 1 -> (r+1)/(-r+3)<=1.
Lemma conjR_range_weak:forall r, -1 <= r <= 1-> -1<= (r+1)/(-r+3)<=1.
Lemma conjM_range_l:forall r, -1 <= r -> -1/3<= r/3.
Lemma conjM_range_r:forall r, r <= 1 -> r/3<=1/3.
Lemma conjM_range_weak:forall r, -1 <= r <= 1-> -1<= r/3 <=1.
Lemma conjLinv_range_r:forall r, r <= 0 -> (3*r+1)/(-r+1)<=1.
Lemma conjLinv_range_l:forall r, -1<=r -> r <= 0 -> -1<=(3*r+1)/(-r+1).
Lemma conjRinv_range_r:forall r, 0<=r-> r <= 1 -> (3*r-1)/(r+1)<=1.
Lemma conjRinv_range_l:forall r, 0<=r -> -1<=(3*r-1)/(r+1).
Lemma conjMinv_range_r:forall r, r <= 1/3 -> 3*r<=1.
Lemma conjMinv_range_l:forall r, -1/3<=r -> -1<=3*r.
Lemma CV_const: forall const, Un_cv (fun i : nat => const) const.
Lemma CV_shift_S' : forall Un l, Un_cv (fun n => Un (S n)) l -> Un_cv Un l.
Lemma CV_shift_S : forall Un l, Un_cv Un l -> Un_cv (fun n => Un (S n)) l.
Lemma CV_extensionality : forall Un Un', (forall n, Un n = Un' n) -> forall l, Un_cv Un l -> Un_cv Un' l.