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  "body": "\n## Proof\n\nBy the definition of free product, it suffices to check that a given (nonempty) reduced word represents a nontrivial element of $G$. Let $w$ be such a word of length $m\\geq 2$, and let\n\n$$w = \\prod_{i=1}^m w_i,$$\n\nwhere $w_i \\in H_{\\alpha_i}$ for some $\\alpha_i \\in \\{1,\\dots,k\\}$. Since $w$ is reduced, we have $\\alpha_i \\neq \\alpha_{i+1}$ for any $i = 1, \\dots, m-1$ and each $w_i$ is distinct from the [[identity element|07WYQPB8E3]] of $H_{\\alpha_i}$. We then let $w$ [[act|0WCY42EV9V]] on an element of one of the sets $X_i$. As we assume that at least one subgroup $H_i$ has order at least 3, without loss of generality we may assume that $H_1$ has order at least 3. We first make the assumption that $\\alpha_1$and $\\alpha_m$ are both 1 (which implies $m \\geq 3$). From here we consider $w$ acting on $X_2$. We get the following chain of containments:\n\n$$w(X_2) \\subseteq \\prod_{i=1}^{m-1} w_i(X_1) \\subseteq \\prod_{i=1}^{m-2} w_i(X_{\\alpha_{m-1}}) \\subseteq \\dots \\subseteq w_1(X_{\\alpha_2}) \\subseteq X_1.$$\n\nBy the assumption that different $X_i$'s are disjoint, we conclude that $w$ acts nontrivially on some element of $X_2$, thus $w$ represents a nontrivial element of $G$.\n\nTo finish the proof we must consider the three cases:\n\n- if $\\alpha_1 = 1,\\,\\alpha_m \\neq 1$, then let $h\\in H_1\\setminus \\{w_1^{-1},1\\}$ (such an $h$ exists since by assumption $H_1$ has order at least 3);\n- if $\\alpha_1 \\neq 1,\\,\\alpha_m=1$, then let $h\\in H_1\\setminus \\{w_m,1\\}$;\n- and if $\\alpha_1\\neq 1,\\,\\alpha_m\\neq 1$, then let $h\\in H_1\\setminus \\{1\\}$.\n\nIn each case, $hwh^{-1}$ after reduction becomes a reduced word with its first and last letter in $H_1$. Finally, $hwh^{-1}$ represents a nontrivial element of $G$, and so does $w$. This proves the claim.\n",
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