Positive.coe_add

theorem Positive.coe_add : ∀ {M : Type u_1} [inst : AddMonoid M] [inst_1 : Preorder M] [inst_2 : CovariantClass M M (fun x x_1 => x + x_1) fun x x_1 => x < x_1] (x y : { x // 0 < x }), ↑(x + y) = ↑x + ↑y

The theorem `Positive.coe_add` states that for any type `M` that forms an additive monoid and a preorder, and where addition (`fun x x_1 => x + x_1`) is a covariant function (meaning if `x < x_1` then `x + something < x_1 + something`), the sum of two strictly positive elements `x` and `y` (represented as `{ x // 0 < x }`), when coerced (or converted) back to `M`, is equivalent to the sum of the coerced elements `x` and `y`. Essentially, this theorem is establishing the property that the coercion respects addition in this context.

More concisely: For any additive monoid and preorder type `M` with covariant addition, and for `x` and `y` being strictly positive elements of `M`, the coercion of `x + y` is equal to the coercion of `x` plus the coercion of `y`.