[math-fun] Theorem-dependance: possibly trivial question

Let us say that, if Theorem 1 (T1) can be used to prove Theorem 2 (T2), then T1 --> T2. Are there simple examples of 'equivalence', T1 --> T2 and T2 --> T1 (setting up loop T1 --> T2 --> T1) that could be explained to school children? Are there cases of longer loops T1 --> ... --> Tn --> T1 which cannot, or cannot naturally, be shortened ? Schoolchildren-compatable preferred again ! Thanks - Guy