## Screenshot: Theorem Definition ### Overview The image is a screenshot of a code snippet defining a theorem named `abs_add`. The theorem states that the absolute value of the sum of two real numbers x and y is less than or equal to the sum of their absolute values. The code snippet also includes a command to apply a lemma named `abs_add_le`. ### Components/Axes The image contains the following text: - `theorem abs_add (x y : R) : |x + y| ≤ |x| + |y| := by` - `apply abs_add_le` The top-left corner of the window contains three circles: red, yellow, and green. ### Detailed Analysis or ### Content Details The code defines a theorem named `abs_add`. - The theorem takes two real numbers, `x` and `y`, as input. The type of `x` and `y` is `R`, which represents the set of real numbers. - The theorem states that the absolute value of the sum of `x` and `y` is less than or equal to the sum of the absolute values of `x` and `y`. This is represented by the expression `|x + y| ≤ |x| + |y|`. - The `:= by` part indicates that the theorem is being defined by a proof. - The `apply abs_add_le` command applies a lemma named `abs_add_le` to prove the theorem. ### Key Observations The code snippet defines a fundamental theorem related to absolute values and real numbers. The theorem is proven by applying a lemma. ### Interpretation The code snippet demonstrates the definition and proof of a mathematical theorem within a formal system. The theorem `abs_add` is a well-known result in real analysis. The use of `apply abs_add_le` suggests that the proof relies on a previously established lemma, `abs_add_le`, which likely provides a more specific or foundational result related to absolute values.