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<b>let </b><font color="Maroon">x</font> be   <a href="zf_lang.html#NM1">Variable</a>;<br/><b>let </b><font color="Maroon">y</font> be   <a href="zf_lang.html#NM1">Variable</a>;<br/>



<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="101_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>thus </b><a NAME="E1:101_1"/>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="zf_lang.html#K29" target="_self">Subformulae</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="zf_lang.html#K5" target="_self">'in'</a> <font color="Maroon">y</font></span>)</span> implies <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="zf_lang.html#K5" target="_self">'in'</a> <font color="Maroon">y</font></span>)</span></span><a href="tarski.html#K1">}</a></span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="101_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:101_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="zf_lang.html#K29" target="_self">Subformulae</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="zf_lang.html#K5" target="_self">'in'</a> <font color="Maroon">y</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">F</font> being   <a href="zf_lang.html#NM2">ZF-formula</a><b> such that </b><br/><a NAME="E3:101_1_1"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">F</font> <a href="zf_lang.html#R2" target="_self">is_subformula_of</a> <font color="Maroon">x</font> <a href="zf_lang.html#K5" target="_self">'in'</a> <font color="Maroon">y</font> )
 <b>by </b><i><a class="ref" href="zf_lang.html#T34" target="_self">Th34</a></i>;<br/>
<a NAME="E4:101_1_1"/>
<font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="zf_lang.html#K5" target="_self">'in'</a> <font color="Maroon">y</font>
 
<b>by </b><i><a class="ref" href="zf_lang.html#T38" target="_self">Th38</a>, <a class="ref" href="zf_lang.html#T33" target="_self">Th33</a></i>;<br/>
<b>hence </b><a NAME="E5:101_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="zf_lang.html#K5" target="_self">'in'</a> <font color="Maroon">y</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="zf_lang.html#T38" target="_self">Th38</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div><b>assume </b><a NAME="E2:101_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="zf_lang.html#K5" target="_self">'in'</a> <font color="Maroon">y</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 ;<br/><a NAME="E3:101_1"/><b>then </b>
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="zf_lang.html#K5" target="_self">'in'</a> <font color="Maroon">y</font> &amp; <font color="Maroon">x</font> <a href="zf_lang.html#K5" target="_self">'in'</a> <font color="Maroon">y</font> <a href="zf_lang.html#R2" target="_self">is_subformula_of</a> <font color="Maroon">x</font> <a href="zf_lang.html#K5" target="_self">'in'</a> <font color="Maroon">y</font> )
 <b>by </b><i><a class="ref" href="zf_lang.html#D13" target="_self">Def13</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E4:101_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="zf_lang.html#K29" target="_self">Subformulae</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="zf_lang.html#K5" target="_self">'in'</a> <font color="Maroon">y</font></span>)</span>
 <b>by </b><i><a class="ref" href="zf_lang.html#T34" target="_self">Th34</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:101"/>
 <a href="zf_lang.html#K29" target="_self">Subformulae</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="zf_lang.html#K5" target="_self">'in'</a> <font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="zf_lang.html#K5" target="_self">'in'</a> <font color="Maroon">y</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>


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