# Properties - Theorem 7.1.4: - IF: $X$ is an $n$ type - IF: $X \rightarrow Y$ is a retraction (has a left-inverse) - THEN: $Y$ is an $n$ type - Corollary 7.1.5: - IF: $X \simeq Y$ - IF: $X$ is an $n$ type - THEN: $Y$ is an $n$ type - Theorem 7.1.7: - IF: $X$ is an $n$ type - THEN: it is also an $(n + 1)$ type - Theorem 7.1.8: - IF: $A$ is an $n$ type - IF: $B(a)$ is an $n$ type for all $a : A$ - THEN: $\sum_{(x : A)} B(x)$ is an $n$ type ## -2: Contractible ## -1: Mere props - If $A$ and $B$ are mere props, so is $A \times B$ - If $B(a)$ is a prop for any $a:A$, then $\prod_{(x:A)} B(x)$ is a prop -