\n ## Screenshot: Code Snippet - Mathematical Theorem Proofs ### Overview The image is a screenshot of a code editor displaying two mathematical theorem proofs, likely written in a formal proof assistant language (possibly Coq or similar). The background is dark, and the text is light-colored, typical of many code editors. There are three colored circles at the top-left corner, likely status indicators. ### Components/Axes There are no axes or traditional chart components. The key elements are: * **Colored Circles:** Three circles at the top-left: Red, Yellow, Green. Their meaning is not explicitly stated but likely indicate status (e.g., compilation status, error status). * **Theorem 1:** `theorem my_lemma3 : ∀ {x y ∈ ℝ}, 0 < ε → ε ≤ 1 → |x| < ε → |y| < ε → |x * y| < ε := by apply C03S01.my_lemma` * **Theorem 2:** `theorem abs_lt : |x| < y ↔ y < x ∧ x < y := by cases x exact abs_lt` ### Detailed Analysis or Content Details **Theorem 1: `my_lemma3`** * **Theorem Statement:** `∀ {x y ∈ ℝ}, 0 < ε → ε ≤ 1 → |x| < ε → |y| < ε → |x * y| < ε` * This states: For all real numbers x and y, if epsilon (ε) is greater than 0 and less than or equal to 1, and the absolute values of x and y are both less than epsilon, then the absolute value of their product (x * y) is also less than epsilon. * **Proof Step:** `apply C03S01.my_lemma` * This indicates that the theorem is being proven by applying a previously defined lemma named `my_lemma` located in the module `C03S01`. **Theorem 2: `abs_lt`** * **Theorem Statement:** `|x| < y ↔ y < x ∧ x < y` * This states: The absolute value of x is less than y if and only if y is greater than x and x is greater than y. This appears to be a definition or a basic property of absolute values. * **Proof Step:** `cases x` * This indicates that the proof proceeds by considering different cases based on the value of x. * **Proof Step:** `exact abs_lt` * This indicates that the current case is directly solved by the lemma or definition `abs_lt`. ### Key Observations * The theorems involve real numbers (ℝ) and epsilon-delta arguments, common in mathematical analysis. * The proofs are concise, relying on applying existing lemmas or definitions. * The code uses a formal syntax, suggesting a proof assistant environment. ### Interpretation The image demonstrates a snippet of formal mathematical reasoning. The theorems and their proofs are likely part of a larger mathematical development within a proof assistant. The use of lemmas and case analysis are standard techniques in formal verification. The `my_lemma3` theorem likely relates to the continuity of multiplication, while `abs_lt` defines the relationship between absolute values and inequalities. The colored circles at the top-left likely indicate the status of the code (e.g., compilation success, errors). The overall context suggests a rigorous mathematical environment where proofs are constructed and verified with high precision.