Notation "0" := rO.Notation "1" := rI.Notation "x + y" := (rplus x y).Notation "x * y " := (rtimes x y).Notation "x - y " := (rminus x y).Notation "- x" := (ropp x).Notation "x == y" := (req x y).Notation "x ~= y" := (~ req x y).Notation "x <= y" := (rle x y).Notation "x < y" := (rlt x y).Add Ring SOR : (SORrt sor).Record SOR : Prop := mk_SOR_theory {
  SORsetoid : Setoid_Theory R req;
  SORplus_wd : forall x1 x2, x1 == x2 -> forall y1 y2, y1 == y2 -> x1 + y1 == x2 + y2;
  SORtimes_wd : forall x1 x2, x1 == x2 -> forall y1 y2, y1 == y2 -> x1 * y1 == x2 * y2;
  SORopp_wd : forall x1 x2, x1 == x2 ->  -x1 == -x2;
  SORle_wd : forall x1 x2, x1 == x2 -> forall y1 y2, y1 == y2 -> (x1 <= y1 <-> x2 <= y2);
  SORlt_wd : forall x1 x2, x1 == x2 -> forall y1 y2, y1 == y2 -> (x1 < y1 <-> x2 < y2);
  SORrt : ring_theory rO rI rplus rtimes rminus ropp req;
  SORle_refl : forall n : R, n <= n;
  SORle_antisymm : forall n m : R, n <= m -> m <= n -> n == m;
  SORle_trans : forall n m p : R, n <= m -> m <= p -> n <= p;
  SORlt_le_neq : forall n m : R, n < m <-> n <= m /\ n ~= m;
  SORlt_trichotomy : forall n m : R,  n < m \/ n == m \/ m < n;
  SORplus_le_mono_l : forall n m p : R, n <= m -> p + n <= p + m;
  SORtimes_pos_pos : forall n m : R, 0 < n -> 0 < m -> 0 < n * m;
  SORneq_0_1 : 0 ~= 1
}.Record SOR : Prop := mk_SOR_theory {
  SORsetoid : Setoid_Theory R req;
  SORplus_wd : forall x1 x2, x1 == x2 -> forall y1 y2, y1 == y2 -> x1 + y1 == x2 + y2;
  SORtimes_wd : forall x1 x2, x1 == x2 -> forall y1 y2, y1 == y2 -> x1 * y1 == x2 * y2;
  SORopp_wd : forall x1 x2, x1 == x2 ->  -x1 == -x2;
  SORle_wd : forall x1 x2, x1 == x2 -> forall y1 y2, y1 == y2 -> (x1 <= y1 <-> x2 <= y2);
  SORlt_wd : forall x1 x2, x1 == x2 -> forall y1 y2, y1 == y2 -> (x1 < y1 <-> x2 < y2);
  SORrt : ring_theory rO rI rplus rtimes rminus ropp req;
  SORle_refl : forall n : R, n <= n;
  SORle_antisymm : forall n m : R, n <= m -> m <= n -> n == m;
  SORle_trans : forall n m p : R, n <= m -> m <= p -> n <= p;
  SORlt_le_neq : forall n m : R, n < m <-> n <= m /\ n ~= m;
  SORlt_trichotomy : forall n m : R,  n < m \/ n == m \/ m < n;
  SORplus_le_mono_l : forall n m p : R, n <= m -> p + n <= p + m;
  SORtimes_pos_pos : forall n m : R, 0 < n -> 0 < m -> 0 < n * m;
  SORneq_0_1 : 0 ~= 1
}.Record SOR : Prop := mk_SOR_theory {
  SORsetoid : Setoid_Theory R req;
  SORplus_wd : forall x1 x2, x1 == x2 -> forall y1 y2, y1 == y2 -> x1 + y1 == x2 + y2;
  SORtimes_wd : forall x1 x2, x1 == x2 -> forall y1 y2, y1 == y2 -> x1 * y1 == x2 * y2;
  SORopp_wd : forall x1 x2, x1 == x2 ->  -x1 == -x2;
  SORle_wd : forall x1 x2, x1 == x2 -> forall y1 y2, y1 == y2 -> (x1 <= y1 <-> x2 <= y2);
  SORlt_wd : forall x1 x2, x1 == x2 -> forall y1 y2, y1 == y2 -> (x1 < y1 <-> x2 < y2);
  SORrt : ring_theory rO rI rplus rtimes rminus ropp req;
  SORle_refl : forall n : R, n <= n;
  SORle_antisymm : forall n m : R, n <= m -> m <= n -> n == m;
  SORle_trans : forall n m p : R, n <= m -> m <= p -> n <= p;
  SORlt_le_neq : forall n m : R, n < m <-> n <= m /\ n ~= m;
  SORlt_trichotomy : forall n m : R,  n < m \/ n == m \/ m < n;
  SORplus_le_mono_l : forall n m p : R, n <= m -> p + n <= p + m;
  SORtimes_pos_pos : forall n m : R, 0 < n -> 0 < m -> 0 < n * m;
  SORneq_0_1 : 0 ~= 1
}.Record SOR : Prop := mk_SOR_theory {
  SORsetoid : Setoid_Theory R req;
  SORplus_wd : forall x1 x2, x1 == x2 -> forall y1 y2, y1 == y2 -> x1 + y1 == x2 + y2;
  SORtimes_wd : forall x1 x2, x1 == x2 -> forall y1 y2, y1 == y2 -> x1 * y1 == x2 * y2;
  SORopp_wd : forall x1 x2, x1 == x2 ->  -x1 == -x2;
  SORle_wd : forall x1 x2, x1 == x2 -> forall y1 y2, y1 == y2 -> (x1 <= y1 <-> x2 <= y2);
  SORlt_wd : forall x1 x2, x1 == x2 -> forall y1 y2, y1 == y2 -> (x1 < y1 <-> x2 < y2);
  SORrt : ring_theory rO rI rplus rtimes rminus ropp req;
  SORle_refl : forall n : R, n <= n;
  SORle_antisymm : forall n m : R, n <= m -> m <= n -> n == m;
  SORle_trans : forall n m p : R, n <= m -> m <= p -> n <= p;
  SORlt_le_neq : forall n m : R, n < m <-> n <= m /\ n ~= m;
  SORlt_trichotomy : forall n m : R,  n < m \/ n == m \/ m < n;
  SORplus_le_mono_l : forall n m p : R, n <= m -> p + n <= p + m;
  SORtimes_pos_pos : forall n m : R, 0 < n -> 0 < m -> 0 < n * m;
  SORneq_0_1 : 0 ~= 1
}.Record SOR : Prop := mk_SOR_theory {
  SORsetoid : Setoid_Theory R req;
  SORplus_wd : forall x1 x2, x1 == x2 -> forall y1 y2, y1 == y2 -> x1 + y1 == x2 + y2;
  SORtimes_wd : forall x1 x2, x1 == x2 -> forall y1 y2, y1 == y2 -> x1 * y1 == x2 * y2;
  SORopp_wd : forall x1 x2, x1 == x2 ->  -x1 == -x2;
  SORle_wd : forall x1 x2, x1 == x2 -> forall y1 y2, y1 == y2 -> (x1 <= y1 <-> x2 <= y2);
  SORlt_wd : forall x1 x2, x1 == x2 -> forall y1 y2, y1 == y2 -> (x1 < y1 <-> x2 < y2);
  SORrt : ring_theory rO rI rplus rtimes rminus ropp req;
  SORle_refl : forall n : R, n <= n;
  SORle_antisymm : forall n m : R, n <= m -> m <= n -> n == m;
  SORle_trans : forall n m p : R, n <= m -> m <= p -> n <= p;
  SORlt_le_neq : forall n m : R, n < m <-> n <= m /\ n ~= m;
  SORlt_trichotomy : forall n m : R,  n < m \/ n == m \/ m < n;
  SORplus_le_mono_l : forall n m p : R, n <= m -> p + n <= p + m;
  SORtimes_pos_pos : forall n m : R, 0 < n -> 0 < m -> 0 < n * m;
  SORneq_0_1 : 0 ~= 1
}.Record SOR : Prop := mk_SOR_theory {
  SORsetoid : Setoid_Theory R req;
  SORplus_wd : forall x1 x2, x1 == x2 -> forall y1 y2, y1 == y2 -> x1 + y1 == x2 + y2;
  SORtimes_wd : forall x1 x2, x1 == x2 -> forall y1 y2, y1 == y2 -> x1 * y1 == x2 * y2;
  SORopp_wd : forall x1 x2, x1 == x2 ->  -x1 == -x2;
  SORle_wd : forall x1 x2, x1 == x2 -> forall y1 y2, y1 == y2 -> (x1 <= y1 <-> x2 <= y2);
  SORlt_wd : forall x1 x2, x1 == x2 -> forall y1 y2, y1 == y2 -> (x1 < y1 <-> x2 < y2);
  SORrt : ring_theory rO rI rplus rtimes rminus ropp req;
  SORle_refl : forall n : R, n <= n;
  SORle_antisymm : forall n m : R, n <= m -> m <= n -> n == m;
  SORle_trans : forall n m p : R, n <= m -> m <= p -> n <= p;
  SORlt_le_neq : forall n m : R, n < m <-> n <= m /\ n ~= m;
  SORlt_trichotomy : forall n m : R,  n < m \/ n == m \/ m < n;
  SORplus_le_mono_l : forall n m p : R, n <= m -> p + n <= p + m;
  SORtimes_pos_pos : forall n m : R, 0 < n -> 0 < m -> 0 < n * m;
  SORneq_0_1 : 0 ~= 1
}.Record SOR : Prop := mk_SOR_theory {
  SORsetoid : Setoid_Theory R req;
  SORplus_wd : forall x1 x2, x1 == x2 -> forall y1 y2, y1 == y2 -> x1 + y1 == x2 + y2;
  SORtimes_wd : forall x1 x2, x1 == x2 -> forall y1 y2, y1 == y2 -> x1 * y1 == x2 * y2;
  SORopp_wd : forall x1 x2, x1 == x2 ->  -x1 == -x2;
  SORle_wd : forall x1 x2, x1 == x2 -> forall y1 y2, y1 == y2 -> (x1 <= y1 <-> x2 <= y2);
  SORlt_wd : forall x1 x2, x1 == x2 -> forall y1 y2, y1 == y2 -> (x1 < y1 <-> x2 < y2);
  SORrt : ring_theory rO rI rplus rtimes rminus ropp req;
  SORle_refl : forall n : R, n <= n;
  SORle_antisymm : forall n m : R, n <= m -> m <= n -> n == m;
  SORle_trans : forall n m p : R, n <= m -> m <= p -> n <= p;
  SORlt_le_neq : forall n m : R, n < m <-> n <= m /\ n ~= m;
  SORlt_trichotomy : forall n m : R,  n < m \/ n == m \/ m < n;
  SORplus_le_mono_l : forall n m p : R, n <= m -> p + n <= p + m;
  SORtimes_pos_pos : forall n m : R, 0 < n -> 0 < m -> 0 < n * m;
  SORneq_0_1 : 0 ~= 1
}.Record SOR : Prop := mk_SOR_theory {
  SORsetoid : Setoid_Theory R req;
  SORplus_wd : forall x1 x2, x1 == x2 -> forall y1 y2, y1 == y2 -> x1 + y1 == x2 + y2;
  SORtimes_wd : forall x1 x2, x1 == x2 -> forall y1 y2, y1 == y2 -> x1 * y1 == x2 * y2;
  SORopp_wd : forall x1 x2, x1 == x2 ->  -x1 == -x2;
  SORle_wd : forall x1 x2, x1 == x2 -> forall y1 y2, y1 == y2 -> (x1 <= y1 <-> x2 <= y2);
  SORlt_wd : forall x1 x2, x1 == x2 -> forall y1 y2, y1 == y2 -> (x1 < y1 <-> x2 < y2);
  SORrt : ring_theory rO rI rplus rtimes rminus ropp req;
  SORle_refl : forall n : R, n <= n;
  SORle_antisymm : forall n m : R, n <= m -> m <= n -> n == m;
  SORle_trans : forall n m p : R, n <= m -> m <= p -> n <= p;
  SORlt_le_neq : forall n m : R, n < m <-> n <= m /\ n ~= m;
  SORlt_trichotomy : forall n m : R,  n < m \/ n == m \/ m < n;
  SORplus_le_mono_l : forall n m p : R, n <= m -> p + n <= p + m;
  SORtimes_pos_pos : forall n m : R, 0 < n -> 0 < m -> 0 < n * m;
  SORneq_0_1 : 0 ~= 1
}.Record SOR : Prop := mk_SOR_theory {
  SORsetoid : Setoid_Theory R req;
  SORplus_wd : forall x1 x2, x1 == x2 -> forall y1 y2, y1 == y2 -> x1 + y1 == x2 + y2;
  SORtimes_wd : forall x1 x2, x1 == x2 -> forall y1 y2, y1 == y2 -> x1 * y1 == x2 * y2;
  SORopp_wd : forall x1 x2, x1 == x2 ->  -x1 == -x2;
  SORle_wd : forall x1 x2, x1 == x2 -> forall y1 y2, y1 == y2 -> (x1 <= y1 <-> x2 <= y2);
  SORlt_wd : forall x1 x2, x1 == x2 -> forall y1 y2, y1 == y2 -> (x1 < y1 <-> x2 < y2);
  SORrt : ring_theory rO rI rplus rtimes rminus ropp req;
  SORle_refl : forall n : R, n <= n;
  SORle_antisymm : forall n m : R, n <= m -> m <= n -> n == m;
  SORle_trans : forall n m p : R, n <= m -> m <= p -> n <= p;
  SORlt_le_neq : forall n m : R, n < m <-> n <= m /\ n ~= m;
  SORlt_trichotomy : forall n m : R,  n < m \/ n == m \/ m < n;
  SORplus_le_mono_l : forall n m p : R, n <= m -> p + n <= p + m;
  SORtimes_pos_pos : forall n m : R, 0 < n -> 0 < m -> 0 < n * m;
  SORneq_0_1 : 0 ~= 1
}.Record SOR : Prop := mk_SOR_theory {
  SORsetoid : Setoid_Theory R req;
  SORplus_wd : forall x1 x2, x1 == x2 -> forall y1 y2, y1 == y2 -> x1 + y1 == x2 + y2;
  SORtimes_wd : forall x1 x2, x1 == x2 -> forall y1 y2, y1 == y2 -> x1 * y1 == x2 * y2;
  SORopp_wd : forall x1 x2, x1 == x2 ->  -x1 == -x2;
  SORle_wd : forall x1 x2, x1 == x2 -> forall y1 y2, y1 == y2 -> (x1 <= y1 <-> x2 <= y2);
  SORlt_wd : forall x1 x2, x1 == x2 -> forall y1 y2, y1 == y2 -> (x1 < y1 <-> x2 < y2);
  SORrt : ring_theory rO rI rplus rtimes rminus ropp req;
  SORle_refl : forall n : R, n <= n;
  SORle_antisymm : forall n m : R, n <= m -> m <= n -> n == m;
  SORle_trans : forall n m p : R, n <= m -> m <= p -> n <= p;
  SORlt_le_neq : forall n m : R, n < m <-> n <= m /\ n ~= m;
  SORlt_trichotomy : forall n m : R,  n < m \/ n == m \/ m < n;
  SORplus_le_mono_l : forall n m p : R, n <= m -> p + n <= p + m;
  SORtimes_pos_pos : forall n m : R, 0 < n -> 0 < m -> 0 < n * m;
  SORneq_0_1 : 0 ~= 1
}.Record SOR : Prop := mk_SOR_theory {
  SORsetoid : Setoid_Theory R req;
  SORplus_wd : forall x1 x2, x1 == x2 -> forall y1 y2, y1 == y2 -> x1 + y1 == x2 + y2;
  SORtimes_wd : forall x1 x2, x1 == x2 -> forall y1 y2, y1 == y2 -> x1 * y1 == x2 * y2;
  SORopp_wd : forall x1 x2, x1 == x2 ->  -x1 == -x2;
  SORle_wd : forall x1 x2, x1 == x2 -> forall y1 y2, y1 == y2 -> (x1 <= y1 <-> x2 <= y2);
  SORlt_wd : forall x1 x2, x1 == x2 -> forall y1 y2, y1 == y2 -> (x1 < y1 <-> x2 < y2);
  SORrt : ring_theory rO rI rplus rtimes rminus ropp req;
  SORle_refl : forall n : R, n <= n;
  SORle_antisymm : forall n m : R, n <= m -> m <= n -> n == m;
  SORle_trans : forall n m p : R, n <= m -> m <= p -> n <= p;
  SORlt_le_neq : forall n m : R, n < m <-> n <= m /\ n ~= m;
  SORlt_trichotomy : forall n m : R,  n < m \/ n == m \/ m < n;
  SORplus_le_mono_l : forall n m p : R, n <= m -> p + n <= p + m;
  SORtimes_pos_pos : forall n m : R, 0 < n -> 0 < m -> 0 < n * m;
  SORneq_0_1 : 0 ~= 1
}.Record SOR : Prop := mk_SOR_theory {
  SORsetoid : Setoid_Theory R req;
  SORplus_wd : forall x1 x2, x1 == x2 -> forall y1 y2, y1 == y2 -> x1 + y1 == x2 + y2;
  SORtimes_wd : forall x1 x2, x1 == x2 -> forall y1 y2, y1 == y2 -> x1 * y1 == x2 * y2;
  SORopp_wd : forall x1 x2, x1 == x2 ->  -x1 == -x2;
  SORle_wd : forall x1 x2, x1 == x2 -> forall y1 y2, y1 == y2 -> (x1 <= y1 <-> x2 <= y2);
  SORlt_wd : forall x1 x2, x1 == x2 -> forall y1 y2, y1 == y2 -> (x1 < y1 <-> x2 < y2);
  SORrt : ring_theory rO rI rplus rtimes rminus ropp req;
  SORle_refl : forall n : R, n <= n;
  SORle_antisymm : forall n m : R, n <= m -> m <= n -> n == m;
  SORle_trans : forall n m p : R, n <= m -> m <= p -> n <= p;
  SORlt_le_neq : forall n m : R, n < m <-> n <= m /\ n ~= m;
  SORlt_trichotomy : forall n m : R,  n < m \/ n == m \/ m < n;
  SORplus_le_mono_l : forall n m p : R, n <= m -> p + n <= p + m;
  SORtimes_pos_pos : forall n m : R, 0 < n -> 0 < m -> 0 < n * m;
  SORneq_0_1 : 0 ~= 1
}.Record SOR : Prop := mk_SOR_theory {
  SORsetoid : Setoid_Theory R req;
  SORplus_wd : forall x1 x2, x1 == x2 -> forall y1 y2, y1 == y2 -> x1 + y1 == x2 + y2;
  SORtimes_wd : forall x1 x2, x1 == x2 -> forall y1 y2, y1 == y2 -> x1 * y1 == x2 * y2;
  SORopp_wd : forall x1 x2, x1 == x2 ->  -x1 == -x2;
  SORle_wd : forall x1 x2, x1 == x2 -> forall y1 y2, y1 == y2 -> (x1 <= y1 <-> x2 <= y2);
  SORlt_wd : forall x1 x2, x1 == x2 -> forall y1 y2, y1 == y2 -> (x1 < y1 <-> x2 < y2);
  SORrt : ring_theory rO rI rplus rtimes rminus ropp req;
  SORle_refl : forall n : R, n <= n;
  SORle_antisymm : forall n m : R, n <= m -> m <= n -> n == m;
  SORle_trans : forall n m p : R, n <= m -> m <= p -> n <= p;
  SORlt_le_neq : forall n m : R, n < m <-> n <= m /\ n ~= m;
  SORlt_trichotomy : forall n m : R,  n < m \/ n == m \/ m < n;
  SORplus_le_mono_l : forall n m p : R, n <= m -> p + n <= p + m;
  SORtimes_pos_pos : forall n m : R, 0 < n -> 0 < m -> 0 < n * m;
  SORneq_0_1 : 0 ~= 1
}.Record SOR : Prop := mk_SOR_theory {
  SORsetoid : Setoid_Theory R req;
  SORplus_wd : forall x1 x2, x1 == x2 -> forall y1 y2, y1 == y2 -> x1 + y1 == x2 + y2;
  SORtimes_wd : forall x1 x2, x1 == x2 -> forall y1 y2, y1 == y2 -> x1 * y1 == x2 * y2;
  SORopp_wd : forall x1 x2, x1 == x2 ->  -x1 == -x2;
  SORle_wd : forall x1 x2, x1 == x2 -> forall y1 y2, y1 == y2 -> (x1 <= y1 <-> x2 <= y2);
  SORlt_wd : forall x1 x2, x1 == x2 -> forall y1 y2, y1 == y2 -> (x1 < y1 <-> x2 < y2);
  SORrt : ring_theory rO rI rplus rtimes rminus ropp req;
  SORle_refl : forall n : R, n <= n;
  SORle_antisymm : forall n m : R, n <= m -> m <= n -> n == m;
  SORle_trans : forall n m p : R, n <= m -> m <= p -> n <= p;
  SORlt_le_neq : forall n m : R, n < m <-> n <= m /\ n ~= m;
  SORlt_trichotomy : forall n m : R,  n < m \/ n == m \/ m < n;
  SORplus_le_mono_l : forall n m p : R, n <= m -> p + n <= p + m;
  SORtimes_pos_pos : forall n m : R, 0 < n -> 0 < m -> 0 < n * m;
  SORneq_0_1 : 0 ~= 1
}.Record SOR : Prop := mk_SOR_theory {
  SORsetoid : Setoid_Theory R req;
  SORplus_wd : forall x1 x2, x1 == x2 -> forall y1 y2, y1 == y2 -> x1 + y1 == x2 + y2;
  SORtimes_wd : forall x1 x2, x1 == x2 -> forall y1 y2, y1 == y2 -> x1 * y1 == x2 * y2;
  SORopp_wd : forall x1 x2, x1 == x2 ->  -x1 == -x2;
  SORle_wd : forall x1 x2, x1 == x2 -> forall y1 y2, y1 == y2 -> (x1 <= y1 <-> x2 <= y2);
  SORlt_wd : forall x1 x2, x1 == x2 -> forall y1 y2, y1 == y2 -> (x1 < y1 <-> x2 < y2);
  SORrt : ring_theory rO rI rplus rtimes rminus ropp req;
  SORle_refl : forall n : R, n <= n;
  SORle_antisymm : forall n m : R, n <= m -> m <= n -> n == m;
  SORle_trans : forall n m p : R, n <= m -> m <= p -> n <= p;
  SORlt_le_neq : forall n m : R, n < m <-> n <= m /\ n ~= m;
  SORlt_trichotomy : forall n m : R,  n < m \/ n == m \/ m < n;
  SORplus_le_mono_l : forall n m p : R, n <= m -> p + n <= p + m;
  SORtimes_pos_pos : forall n m : R, 0 < n -> 0 < m -> 0 < n * m;
  SORneq_0_1 : 0 ~= 1
}.Record SOR : Prop := mk_SOR_theory {
  SORsetoid : Setoid_Theory R req;
  SORplus_wd : forall x1 x2, x1 == x2 -> forall y1 y2, y1 == y2 -> x1 + y1 == x2 + y2;
  SORtimes_wd : forall x1 x2, x1 == x2 -> forall y1 y2, y1 == y2 -> x1 * y1 == x2 * y2;
  SORopp_wd : forall x1 x2, x1 == x2 ->  -x1 == -x2;
  SORle_wd : forall x1 x2, x1 == x2 -> forall y1 y2, y1 == y2 -> (x1 <= y1 <-> x2 <= y2);
  SORlt_wd : forall x1 x2, x1 == x2 -> forall y1 y2, y1 == y2 -> (x1 < y1 <-> x2 < y2);
  SORrt : ring_theory rO rI rplus rtimes rminus ropp req;
  SORle_refl : forall n : R, n <= n;
  SORle_antisymm : forall n m : R, n <= m -> m <= n -> n == m;
  SORle_trans : forall n m p : R, n <= m -> m <= p -> n <= p;
  SORlt_le_neq : forall n m : R, n < m <-> n <= m /\ n ~= m;
  SORlt_trichotomy : forall n m : R,  n < m \/ n == m \/ m < n;
  SORplus_le_mono_l : forall n m p : R, n <= m -> p + n <= p + m;
  SORtimes_pos_pos : forall n m : R, 0 < n -> 0 < m -> 0 < n * m;
  SORneq_0_1 : 0 ~= 1
}.Add Relation R req
  reflexivity proved by (@Equivalence_Reflexive _ _ (SORsetoid sor))
  symmetry proved by (@Equivalence_Symmetric _ _ (SORsetoid sor))
  transitivity proved by (@Equivalence_Transitive _ _ (SORsetoid sor))
as sor_setoid.Add Morphism rplus with signature req ==> req ==> req as rplus_morph.Add Morphism rtimes with signature req ==> req ==> req as rtimes_morph.Add Morphism ropp with signature req ==> req as ropp_morph.Add Morphism rle with signature req ==> req ==> iff as rle_morph.Add Morphism rlt with signature req ==> req ==> iff as rlt_morph.Add Morphism rminus with signature req ==> req ==> req as rminus_morph.

Theorem Rle_lt_eq : forall n m : R, n <= m <-> n < m \/ n == m.