<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">y</font> be   <a href="zf_lang.html#NM1">Variable</a>;<br/><b>let </b><font color="Maroon">x</font> be   <a href="zf_lang.html#NM1">Variable</a>;<br/><b>let </b><font color="Maroon">H</font> be   <a href="zf_lang.html#NM2">ZF-formula</a>;<br/>





<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="zf_lang.html#NM2">ZF-formula</a>] <b>means </b>( not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="zf_lang1.html#K4">variables_in</a> <font color="Maroon">a<sub>1</sub></font> implies ( ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K2">Free</a> <font color="Maroon">a<sub>1</sub></font> implies  <a href="zf_model.html#K2">Free</a> <span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="zf_lang1.html#K6">/</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_model.html#K2">Free</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span> ) &amp; ( not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K2">Free</a> <font color="Maroon">a<sub>1</sub></font> implies  <a href="zf_model.html#K2">Free</a> <span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="zf_lang1.html#K6">/</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="zf_model.html#K2">Free</a> <font color="Maroon">a<sub>1</sub></font> ) ) );<br/>
<a NAME="E1:2"/><i><font color="Green">E46</font></i>: 
for <font color="Olive">x1</font>, <font color="Olive">x2</font> being  <a href="zf_lang.html#NM1">Variable</a> holds <br/> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x1</font> <a href="zf_lang.html#K4">'='</a> <font color="Olive">x2</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x1</font> <a href="zf_lang.html#K5">'in'</a> <font color="Olive">x2</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x1</font> be   <a href="zf_lang.html#NM1">Variable</a>;<br/><b>let </b><font color="Maroon">x2</font> be   <a href="zf_lang.html#NM1">Variable</a>;<br/>



<a NAME="E1:2_1"/>
( ( <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> or <font color="Maroon">x1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font> ) &amp; ( <font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> or <font color="Maroon">x2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font> ) )
 
;<br/>
<b>then consider </b><font color="Maroon">y1</font> being   <a href="zf_lang.html#NM1">Variable</a>, <font color="Maroon">y2</font> being   <a href="zf_lang.html#NM1">Variable</a><b> such that </b><br/><a NAME="E3:2_1"/><i><font color="Green">E47</font></i>: 
( ( <font color="Maroon">x1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font> &amp; <font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> &amp; <font color="Maroon">y2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font> ) or ( <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font> &amp; <font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">y2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font> ) or ( <font color="Maroon">x1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> &amp; <font color="Maroon">y2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> ) or ( <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">y2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> ) )
 ;<br/>
<a NAME="E4:2_1"/>
( <span class="p1">(<span class="default"><font color="Maroon">x1</font> <a href="zf_lang.html#K4">'='</a> <font color="Maroon">x2</font></span>)</span> <a href="zf_lang1.html#K6">/</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y1</font> <a href="zf_lang.html#K4">'='</a> <font color="Maroon">y2</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">x1</font> <a href="zf_lang.html#K5">'in'</a> <font color="Maroon">x2</font></span>)</span> <a href="zf_lang1.html#K6">/</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y1</font> <a href="zf_lang.html#K5">'in'</a> <font color="Maroon">y2</font> )
 
<b>by </b><i>, <a class="ref" href="zf_lang1.html#T166">ZF_LANG1:166</a>, <a class="ref" href="zf_lang1.html#T168">ZF_LANG1:168</a></i>;<br/>
<a NAME="E5:2_1"/><b>then </b><i><font color="Green">E48</font></i>: 
(  <a href="zf_model.html#K2">Free</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x1</font> <a href="zf_lang.html#K4">'='</a> <font color="Maroon">x2</font></span>)</span> <a href="zf_lang1.html#K6">/</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">y1</font>,<font color="Maroon">y2</font></span><a href="tarski.html#K2">}</a></span> &amp;  <a href="zf_model.html#K2">Free</a> <span class="p1">(<span class="default"><font color="Maroon">x1</font> <a href="zf_lang.html#K4">'='</a> <font color="Maroon">x2</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="tarski.html#K2">}</a></span> &amp;  <a href="zf_model.html#K2">Free</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x1</font> <a href="zf_lang.html#K5">'in'</a> <font color="Maroon">x2</font></span>)</span> <a href="zf_lang1.html#K6">/</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">y1</font>,<font color="Maroon">y2</font></span><a href="tarski.html#K2">}</a></span> &amp;  <a href="zf_model.html#K2">Free</a> <span class="p1">(<span class="default"><font color="Maroon">x1</font> <a href="zf_lang.html#K5">'in'</a> <font color="Maroon">x2</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="tarski.html#K2">}</a></span> )
 
<b>by </b><i><a class="ref" href="zf_lang1.html#T63">ZF_LANG1:63</a>, <a class="ref" href="zf_lang1.html#T64">ZF_LANG1:64</a></i>;<br/>
<a NAME="E6:2_1"/><i><font color="Green">E49</font></i>: 
(  <a href="zf_lang1.html#K4">variables_in</a> <span class="p1">(<span class="default"><font color="Maroon">x1</font> <a href="zf_lang.html#K4">'='</a> <font color="Maroon">x2</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="tarski.html#K2">}</a></span> &amp;  <a href="zf_lang1.html#K4">variables_in</a> <span class="p1">(<span class="default"><font color="Maroon">x1</font> <a href="zf_lang.html#K5">'in'</a> <font color="Maroon">x2</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="tarski.html#K2">}</a></span> )
 
<b>by </b><i><a class="ref" href="zf_lang1.html#T151">ZF_LANG1:151</a>, <a class="ref" href="zf_lang1.html#T152">ZF_LANG1:152</a></i>;<br/>
<b>thus </b><a NAME="E7:2_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x1</font> <a href="zf_lang.html#K4">'='</a> <font color="Maroon">x2</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:2_1_1"/>
not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="zf_lang1.html#K4">variables_in</a> <span class="p1">(<span class="default"><font color="Maroon">x1</font> <a href="zf_lang.html#K4">'='</a> <font color="Maroon">x2</font></span>)</span>
 ;<br/>

<b>thus </b><a NAME="E2:2_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K2">Free</a> <span class="p1">(<span class="default"><font color="Maroon">x1</font> <a href="zf_lang.html#K4">'='</a> <font color="Maroon">x2</font></span>)</span> implies  <a href="zf_model.html#K2">Free</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x1</font> <a href="zf_lang.html#K4">'='</a> <font color="Maroon">x2</font></span>)</span> <a href="zf_lang1.html#K6">/</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_model.html#K2">Free</a> <span class="p3">(<span class="default"><font color="Maroon">x1</font> <a href="zf_lang.html#K4">'='</a> <font color="Maroon">x2</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:2_1_1_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K2">Free</a> <span class="p1">(<span class="default"><font color="Maroon">x1</font> <a href="zf_lang.html#K4">'='</a> <font color="Maroon">x2</font></span>)</span>
 ;<br/>

<b>thus </b><a NAME="E2:2_1_1_1"/>
 <a href="zf_model.html#K2">Free</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x1</font> <a href="zf_lang.html#K4">'='</a> <font color="Maroon">x2</font></span>)</span> <a href="zf_lang1.html#K6">/</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_model.html#K2">Free</a> <span class="p3">(<span class="default"><font color="Maroon">x1</font> <a href="zf_lang.html#K4">'='</a> <font color="Maroon">x2</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i/>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><br/>

<b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E3:2_1_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_model.html#K2">Free</a> <span class="p3">(<span class="default"><font color="Maroon">x1</font> <a href="zf_lang.html#K4">'='</a> <font color="Maroon">x2</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<a NAME="E4:2_1_1_1"/><b>then </b>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="zf_model.html#K2">Free</a> <span class="p2">(<span class="default"><font color="Maroon">x1</font> <a href="zf_lang.html#K4">'='</a> <font color="Maroon">x2</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> or <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E5:2_1_1_1"/><b>then </b>
( ( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K2">Free</a> <span class="p1">(<span class="default"><font color="Maroon">x1</font> <a href="zf_lang.html#K4">'='</a> <font color="Maroon">x2</font></span>)</span> &amp; not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> ) or <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E6:2_1_1_1"/><b>then </b>
( ( ( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> or <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font> ) &amp; <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font> ) or <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> )
 
<b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<b>hence </b><a NAME="E7:2_1_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K2">Free</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x1</font> <a href="zf_lang.html#K4">'='</a> <font color="Maroon">x2</font></span>)</span> <a href="zf_lang1.html#K6">/</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span>
 <b>by </b><i>, , , <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E3:2_1_1"/>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K2">Free</a> <span class="p1">(<span class="default"><font color="Maroon">x1</font> <a href="zf_lang.html#K4">'='</a> <font color="Maroon">x2</font></span>)</span>
 ;<br/>

<b>hence </b><a NAME="E4:2_1_1"/>
 <a href="zf_model.html#K2">Free</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x1</font> <a href="zf_lang.html#K4">'='</a> <font color="Maroon">x2</font></span>)</span> <a href="zf_lang1.html#K6">/</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="zf_model.html#K2">Free</a> <span class="p1">(<span class="default"><font color="Maroon">x1</font> <a href="zf_lang.html#K4">'='</a> <font color="Maroon">x2</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="zf_lang1.html#T196">ZF_LANG1:196</a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E8:2_1"/>
not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="zf_lang1.html#K4">variables_in</a> <span class="p1">(<span class="default"><font color="Maroon">x1</font> <a href="zf_lang.html#K5">'in'</a> <font color="Maroon">x2</font></span>)</span>
 ;<br/>

<b>thus </b><a NAME="E9:2_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K2">Free</a> <span class="p1">(<span class="default"><font color="Maroon">x1</font> <a href="zf_lang.html#K5">'in'</a> <font color="Maroon">x2</font></span>)</span> implies  <a href="zf_model.html#K2">Free</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x1</font> <a href="zf_lang.html#K5">'in'</a> <font color="Maroon">x2</font></span>)</span> <a href="zf_lang1.html#K6">/</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_model.html#K2">Free</a> <span class="p3">(<span class="default"><font color="Maroon">x1</font> <a href="zf_lang.html#K5">'in'</a> <font color="Maroon">x2</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:2_1_2"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K2">Free</a> <span class="p1">(<span class="default"><font color="Maroon">x1</font> <a href="zf_lang.html#K5">'in'</a> <font color="Maroon">x2</font></span>)</span>
 ;<br/>

<b>thus </b><a NAME="E2:2_1_2"/>
 <a href="zf_model.html#K2">Free</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x1</font> <a href="zf_lang.html#K5">'in'</a> <font color="Maroon">x2</font></span>)</span> <a href="zf_lang1.html#K6">/</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_model.html#K2">Free</a> <span class="p3">(<span class="default"><font color="Maroon">x1</font> <a href="zf_lang.html#K5">'in'</a> <font color="Maroon">x2</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i/>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><br/>

<b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E3:2_1_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_model.html#K2">Free</a> <span class="p3">(<span class="default"><font color="Maroon">x1</font> <a href="zf_lang.html#K5">'in'</a> <font color="Maroon">x2</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<a NAME="E4:2_1_2"/><b>then </b>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="zf_model.html#K2">Free</a> <span class="p2">(<span class="default"><font color="Maroon">x1</font> <a href="zf_lang.html#K5">'in'</a> <font color="Maroon">x2</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> or <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E5:2_1_2"/><b>then </b>
( ( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K2">Free</a> <span class="p1">(<span class="default"><font color="Maroon">x1</font> <a href="zf_lang.html#K5">'in'</a> <font color="Maroon">x2</font></span>)</span> &amp; not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> ) or <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E6:2_1_2"/><b>then </b>
( ( ( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> or <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font> ) &amp; <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font> ) or <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> )
 
<b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<b>hence </b><a NAME="E7:2_1_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K2">Free</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x1</font> <a href="zf_lang.html#K5">'in'</a> <font color="Maroon">x2</font></span>)</span> <a href="zf_lang1.html#K6">/</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span>
 <b>by </b><i>, , , <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E10:2_1"/>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K2">Free</a> <span class="p1">(<span class="default"><font color="Maroon">x1</font> <a href="zf_lang.html#K5">'in'</a> <font color="Maroon">x2</font></span>)</span>
 ;<br/>

<b>hence </b><a NAME="E11:2_1"/>
 <a href="zf_model.html#K2">Free</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x1</font> <a href="zf_lang.html#K5">'in'</a> <font color="Maroon">x2</font></span>)</span> <a href="zf_lang1.html#K6">/</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="zf_model.html#K2">Free</a> <span class="p1">(<span class="default"><font color="Maroon">x1</font> <a href="zf_lang.html#K5">'in'</a> <font color="Maroon">x2</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="zf_lang1.html#T196">ZF_LANG1:196</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:2"/><i><font color="Green">E52</font></i>: 
for <font color="Olive">H</font> being  <a href="zf_lang.html#NM2">ZF-formula</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">H</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[ <a href="zf_lang.html#K6">'not'</a> <font color="Olive">H</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">H</font> be   <a href="zf_lang.html#NM2">ZF-formula</a>;<br/>

<a NAME="E1:2_2"/>
(  <a href="zf_model.html#K2">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K6">'not'</a> <span class="p2">(<span class="default"><font color="Maroon">H</font> <a href="zf_lang1.html#K6">/</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="zf_model.html#K2">Free</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="zf_lang1.html#K6">/</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> &amp;  <a href="zf_model.html#K2">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K6">'not'</a> <font color="Maroon">H</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="zf_model.html#K2">Free</a> <font color="Maroon">H</font> &amp;  <a href="zf_lang1.html#K4">variables_in</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K6">'not'</a> <font color="Maroon">H</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="zf_lang1.html#K4">variables_in</a> <font color="Maroon">H</font> &amp; <span class="p1">(<span class="default"><a href="zf_lang.html#K6">'not'</a> <font color="Maroon">H</font></span>)</span> <a href="zf_lang1.html#K6">/</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="zf_lang.html#K6">'not'</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="zf_lang1.html#K6">/</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="zf_lang1.html#T65">ZF_LANG1:65</a>, <a class="ref" href="zf_lang1.html#T153">ZF_LANG1:153</a>, <a class="ref" href="zf_lang1.html#T170">ZF_LANG1:170</a></i>;<br/>
<b>hence </b><a NAME="E2:2_2"/>
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">H</font>] implies <font color="Maroon">S<sub>1</sub></font>[ <a href="zf_lang.html#K6">'not'</a> <font color="Maroon">H</font>] )
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:2"/><i><font color="Green">E53</font></i>: 
for <font color="Olive">H1</font>, <font color="Olive">H2</font> being  <a href="zf_lang.html#NM2">ZF-formula</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">H1</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">H2</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">H1</font> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Olive">H2</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">H1</font> be   <a href="zf_lang.html#NM2">ZF-formula</a>;<br/><b>let </b><font color="Maroon">H2</font> be   <a href="zf_lang.html#NM2">ZF-formula</a>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:2_3"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">H1</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">H2</font>] )
 <b>and </b><br/><a NAME="E2:2_3"/><i><font color="Green">E55</font></i>: 
not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="zf_lang1.html#K4">variables_in</a> <span class="p1">(<span class="default"><font color="Maroon">H1</font> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Maroon">H2</font></span>)</span>
 ;<br/>

<a NAME="E3:2_3"/><i><font color="Green">E57</font></i>: 
(  <a href="zf_model.html#K2">Free</a> <span class="p1">(<span class="default"><font color="Maroon">H1</font> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Maroon">H2</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="zf_model.html#K2">Free</a> <font color="Maroon">H1</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="zf_model.html#K2">Free</a> <font color="Maroon">H2</font></span>)</span> &amp;  <a href="zf_model.html#K2">Free</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">H1</font> <a href="zf_lang1.html#K6">/</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="zf_lang.html#K7">'&amp;'</a> <span class="p2">(<span class="default"><font color="Maroon">H2</font> <a href="zf_lang1.html#K6">/</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="zf_model.html#K2">Free</a> <span class="p2">(<span class="default"><font color="Maroon">H1</font> <a href="zf_lang1.html#K6">/</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="zf_model.html#K2">Free</a> <span class="p2">(<span class="default"><font color="Maroon">H2</font> <a href="zf_lang1.html#K6">/</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span></span>)</span> &amp;  <a href="zf_lang1.html#K4">variables_in</a> <span class="p1">(<span class="default"><font color="Maroon">H1</font> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Maroon">H2</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="zf_lang1.html#K4">variables_in</a> <font color="Maroon">H1</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="zf_lang1.html#K4">variables_in</a> <font color="Maroon">H2</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="zf_lang1.html#T66">ZF_LANG1:66</a>, <a class="ref" href="zf_lang1.html#T154">ZF_LANG1:154</a></i>;<br/>
<b>set </b><font color="Maroon">H</font> = <font color="Maroon">H1</font> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Maroon">H2</font>;<br/>
<a NAME="E4:2_3"/><i><font color="Green">E58</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">H1</font> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Maroon">H2</font></span>)</span> <a href="zf_lang1.html#K6">/</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">H1</font> <a href="zf_lang1.html#K6">/</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="zf_lang.html#K7">'&amp;'</a> <span class="p1">(<span class="default"><font color="Maroon">H2</font> <a href="zf_lang1.html#K6">/</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span>
 
<b>by </b><i><a class="ref" href="zf_lang1.html#T172">ZF_LANG1:172</a></i>;<br/>
<b>thus </b><a NAME="E5:2_3"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K2">Free</a> <span class="p1">(<span class="default"><font color="Maroon">H1</font> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Maroon">H2</font></span>)</span> implies  <a href="zf_model.html#K2">Free</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">H1</font> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Maroon">H2</font></span>)</span> <a href="zf_lang1.html#K6">/</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_model.html#K2">Free</a> <span class="p3">(<span class="default"><font color="Maroon">H1</font> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Maroon">H2</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_3_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:2_3_1"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K2">Free</a> <span class="p1">(<span class="default"><font color="Maroon">H1</font> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Maroon">H2</font></span>)</span>
 ;<br/>

<div><i><font color="Green">E60</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="2_3_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:2_3_1_1"/>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K2">Free</a> <font color="Maroon">H1</font>
 ;<br/><a NAME="E2:2_3_1_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K2">Free</a> <font color="Maroon">H2</font> &amp;  <a href="zf_model.html#K2">Free</a> <span class="p1">(<span class="default"><font color="Maroon">H1</font> <a href="zf_lang1.html#K6">/</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="zf_model.html#K2">Free</a> <font color="Maroon">H1</font> &amp;  <a href="zf_model.html#K2">Free</a> <font color="Maroon">H1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="zf_model.html#K2">Free</a> <font color="Maroon">H1</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i>, , , , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a>, <a class="ref" href="zfmisc_1.html#T65">ZFMISC_1:65</a></i>;<br/><b>hence </b> <a href="zf_model.html#K2">Free</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">H1</font> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Maroon">H2</font></span>)</span> <a href="zf_lang1.html#K6">/</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="zf_model.html#K2">Free</a> <font color="Maroon">H1</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="zf_model.html#K2">Free</a> <font color="Maroon">H2</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i>, , , , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a>, <a class="ref" href="xboole_1.html#T4">XBOOLE_1:4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_model.html#K2">Free</a> <span class="p3">(<span class="default"><font color="Maroon">H1</font> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Maroon">H2</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i>, <a class="ref" href="xboole_1.html#T42">XBOOLE_1:42</a></i>
;<br/><br/></div><b>end;</b></div>
<div><i><font color="Green">E61</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="2_3_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:2_3_1_2"/>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K2">Free</a> <font color="Maroon">H2</font>
 ;<br/><a NAME="E2:2_3_1_2"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K2">Free</a> <font color="Maroon">H1</font> &amp;  <a href="zf_model.html#K2">Free</a> <span class="p1">(<span class="default"><font color="Maroon">H2</font> <a href="zf_lang1.html#K6">/</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="zf_model.html#K2">Free</a> <font color="Maroon">H2</font> &amp;  <a href="zf_model.html#K2">Free</a> <font color="Maroon">H2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="zf_model.html#K2">Free</a> <font color="Maroon">H2</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i>, , , , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a>, <a class="ref" href="zfmisc_1.html#T65">ZFMISC_1:65</a></i>;<br/><b>hence </b> <a href="zf_model.html#K2">Free</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">H1</font> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Maroon">H2</font></span>)</span> <a href="zf_lang1.html#K6">/</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="zf_model.html#K2">Free</a> <font color="Maroon">H1</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="zf_model.html#K2">Free</a> <font color="Maroon">H2</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i>, , , , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a>, <a class="ref" href="xboole_1.html#T4">XBOOLE_1:4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_model.html#K2">Free</a> <span class="p3">(<span class="default"><font color="Maroon">H1</font> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Maroon">H2</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i>, <a class="ref" href="xboole_1.html#T42">XBOOLE_1:42</a></i>
;<br/><br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_3_1_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:2_3_1_3"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K2">Free</a> <font color="Maroon">H1</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K2">Free</a> <font color="Maroon">H2</font> )
 ;<br/><b>hence </b> <a href="zf_model.html#K2">Free</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">H1</font> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Maroon">H2</font></span>)</span> <a href="zf_lang1.html#K6">/</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="zf_model.html#K2">Free</a> <font color="Maroon">H1</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="zf_model.html#K2">Free</a> <font color="Maroon">H2</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i>, , , , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a>, <a class="ref" href="xboole_1.html#T4">XBOOLE_1:4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="zf_model.html#K2">Free</a> <font color="Maroon">H1</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="zf_model.html#K2">Free</a> <font color="Maroon">H2</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i><a class="ref" href="xboole_1.html#T4">XBOOLE_1:4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="zf_model.html#K2">Free</a> <span class="p4">(<span class="default"><font color="Maroon">H1</font> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Maroon">H2</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i>, <a class="ref" href="xboole_1.html#T42">XBOOLE_1:42</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_model.html#K2">Free</a> <span class="p3">(<span class="default"><font color="Maroon">H1</font> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Maroon">H2</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T4">XBOOLE_1:4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_model.html#K2">Free</a> <span class="p3">(<span class="default"><font color="Maroon">H1</font> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Maroon">H2</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span>

;<br/><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:2_3_1"/>
 <a href="zf_model.html#K2">Free</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">H1</font> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Maroon">H2</font></span>)</span> <a href="zf_lang1.html#K6">/</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_model.html#K2">Free</a> <span class="p3">(<span class="default"><font color="Maroon">H1</font> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Maroon">H2</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E6:2_3"/>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K2">Free</a> <span class="p1">(<span class="default"><font color="Maroon">H1</font> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Maroon">H2</font></span>)</span>
 ;<br/>

<b>hence </b><a NAME="E7:2_3"/>
 <a href="zf_model.html#K2">Free</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">H1</font> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Maroon">H2</font></span>)</span> <a href="zf_lang1.html#K6">/</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="zf_model.html#K2">Free</a> <span class="p1">(<span class="default"><font color="Maroon">H1</font> <a href="zf_lang.html#K7">'&amp;'</a> <font color="Maroon">H2</font></span>)</span>
 <b>by </b><i>, , , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a>, <a class="ref" href="zf_lang1.html#T172">ZF_LANG1:172</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:2"/><i><font color="Green">E62</font></i>: 
for <font color="Olive">H</font> being  <a href="zf_lang.html#NM2">ZF-formula</a><br/> for <font color="Olive">z</font> being  <a href="zf_lang.html#NM1">Variable</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">H</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[ <a href="zf_lang.html#K8">All</a> <font color="Olive">z</font>,<font color="Olive">H</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">H</font> be   <a href="zf_lang.html#NM2">ZF-formula</a>;<br/><b>let </b><font color="Maroon">z</font> be   <a href="zf_lang.html#NM1">Variable</a>;<br/>



<a NAME="E1:2_4"/>
( <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> or <font color="Maroon">z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font> )
 
;<br/>
<b>then consider </b><font color="Maroon">s</font> being   <a href="zf_lang.html#NM1">Variable</a><b> such that </b><br/><a NAME="E3:2_4"/><i><font color="Green">E63</font></i>: 
( ( <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> ) or ( <font color="Maroon">z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font> &amp; <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> ) )
 ;<br/>
<b>assume </b><b>that </b><br/><a NAME="E4:2_4"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">H</font>]
 <b>and </b><br/><a NAME="E5:2_4"/><i><font color="Green">E65</font></i>: 
not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="zf_lang1.html#K4">variables_in</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K8">All</a> <font color="Maroon">z</font>,<font color="Maroon">H</font></span>)</span>
 ;<br/>

<b>set </b><font color="Maroon">G</font> = <span class="p1">(<span class="default"><a href="zf_lang.html#K8">All</a> <font color="Maroon">z</font>,<font color="Maroon">H</font></span>)</span> <a href="zf_lang1.html#K6">/</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>;<br/>
<a NAME="E6:2_4"/><i><font color="Green">E67</font></i>: 
(  <a href="zf_model.html#K2">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K8">All</a> <font color="Maroon">z</font>,<font color="Maroon">H</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="zf_model.html#K2">Free</a> <font color="Maroon">H</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">z</font></span><a href="tarski.html#K1">}</a></span> &amp;  <a href="zf_model.html#K2">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K8">All</a> <font color="Maroon">s</font>,<span class="p2">(<span class="default"><font color="Maroon">H</font> <a href="zf_lang1.html#K6">/</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="zf_model.html#K2">Free</a> <span class="p2">(<span class="default"><font color="Maroon">H</font> <a href="zf_lang1.html#K6">/</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">s</font></span><a href="tarski.html#K1">}</a></span> &amp;  <a href="zf_lang1.html#K4">variables_in</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K8">All</a> <font color="Maroon">z</font>,<font color="Maroon">H</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="zf_lang1.html#K4">variables_in</a> <font color="Maroon">H</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">z</font></span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i><a class="ref" href="zf_lang1.html#T67">ZF_LANG1:67</a>, <a class="ref" href="zf_lang1.html#T155">ZF_LANG1:155</a></i>;<br/>
<a NAME="E7:2_4"/><b>then </b>
( not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="zf_lang1.html#K4">variables_in</a> <font color="Maroon">H</font> &amp; not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">z</font></span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E8:2_4"/><b>then </b><i><font color="Green">E68</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">z</font> &amp; <span class="p1">(<span class="default"><a href="zf_lang.html#K8">All</a> <font color="Maroon">z</font>,<font color="Maroon">H</font></span>)</span> <a href="zf_lang1.html#K6">/</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="zf_lang.html#K8">All</a> <font color="Maroon">s</font>,<span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="zf_lang1.html#K6">/</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a>, <a class="ref" href="zf_lang1.html#T173">ZF_LANG1:173</a>, <a class="ref" href="zf_lang1.html#T174">ZF_LANG1:174</a></i>;<br/>
<b>thus </b><a NAME="E9:2_4"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K2">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K8">All</a> <font color="Maroon">z</font>,<font color="Maroon">H</font></span>)</span> implies  <a href="zf_model.html#K2">Free</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K8">All</a> <font color="Maroon">z</font>,<font color="Maroon">H</font></span>)</span> <a href="zf_lang1.html#K6">/</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_model.html#K2">Free</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K8">All</a> <font color="Maroon">z</font>,<font color="Maroon">H</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_4_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:2_4_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K2">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K8">All</a> <font color="Maroon">z</font>,<font color="Maroon">H</font></span>)</span>
 ;<br/>

<a NAME="E2:2_4_1"/><b>then </b><i><font color="Green">E69</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K2">Free</a> <font color="Maroon">H</font> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">z</font></span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>thus </b><a NAME="E3:2_4_1"/>
 <a href="zf_model.html#K2">Free</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K8">All</a> <font color="Maroon">z</font>,<font color="Maroon">H</font></span>)</span> <a href="zf_lang1.html#K6">/</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_model.html#K2">Free</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K8">All</a> <font color="Maroon">z</font>,<font color="Maroon">H</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_4_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:2_4_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K2">Free</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K8">All</a> <font color="Maroon">z</font>,<font color="Maroon">H</font></span>)</span> <a href="zf_lang1.html#K6">/</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span>
 ;<br/>

<a NAME="E2:2_4_1_1"/><b>then </b><i><font color="Green">E70</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_model.html#K2">Free</a> <font color="Maroon">H</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span> &amp; not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">z</font></span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i>, , , , , , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E3:2_4_1_1"/><b>then </b>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="zf_model.html#K2">Free</a> <font color="Maroon">H</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> or <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E4:2_4_1_1"/><b>then </b>
( ( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K2">Free</a> <font color="Maroon">H</font> &amp; not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> ) or <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E5:2_4_1_1"/><b>then </b>
( ( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K2">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K8">All</a> <font color="Maroon">z</font>,<font color="Maroon">H</font></span>)</span> &amp; not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> ) or <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i>, , <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E6:2_4_1_1"/><b>then </b>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="zf_model.html#K2">Free</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K8">All</a> <font color="Maroon">z</font>,<font color="Maroon">H</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> or <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>hence </b><a NAME="E7:2_4_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_model.html#K2">Free</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K8">All</a> <font color="Maroon">z</font>,<font color="Maroon">H</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E4:2_4_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_model.html#K2">Free</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K8">All</a> <font color="Maroon">z</font>,<font color="Maroon">H</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<a NAME="E5:2_4_1"/><b>then </b>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="zf_model.html#K2">Free</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K8">All</a> <font color="Maroon">z</font>,<font color="Maroon">H</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> or <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E6:2_4_1"/><b>then </b>
( ( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K2">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K8">All</a> <font color="Maroon">z</font>,<font color="Maroon">H</font></span>)</span> &amp; not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> ) or ( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> ) )
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E7:2_4_1"/><b>then </b>
( ( ( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K2">Free</a> <font color="Maroon">H</font> &amp; not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> ) or <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span> ) &amp; not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">z</font></span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E8:2_4_1"/><b>then </b>
( ( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="zf_model.html#K2">Free</a> <font color="Maroon">H</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> or <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span> ) &amp; not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">z</font></span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E9:2_4_1"/><b>then </b>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_model.html#K2">Free</a> <font color="Maroon">H</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span> &amp; not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">z</font></span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E10:2_4_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K2">Free</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K8">All</a> <font color="Maroon">z</font>,<font color="Maroon">H</font></span>)</span> <a href="zf_lang1.html#K6">/</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span>
 <b>by </b><i>, , , , , , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E10:2_4"/>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K2">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K8">All</a> <font color="Maroon">z</font>,<font color="Maroon">H</font></span>)</span>
 ;<br/>

<a NAME="E11:2_4"/><b>then </b><i><font color="Green">E71</font></i>: 
(  not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K2">Free</a> <font color="Maroon">H</font> or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">z</font></span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E12:2_4"/><i><font color="Green">E72</font></i>: 
 <a href="zf_model.html#K2">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K8">All</a> <font color="Maroon">z</font>,<font color="Maroon">H</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="zf_lang1.html#K4">variables_in</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K8">All</a> <font color="Maroon">z</font>,<font color="Maroon">H</font></span>)</span>
 
<b>by </b><i><a class="ref" href="zf_lang1.html#T164">ZF_LANG1:164</a></i>;<br/>
<div><i><font color="Green">E73</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="2_4_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:2_4_2"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K2">Free</a> <font color="Maroon">H</font>
 ;<br/><b>thus </b><a NAME="E2:2_4_2"/>
 <a href="zf_model.html#K2">Free</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K8">All</a> <font color="Maroon">z</font>,<font color="Maroon">H</font></span>)</span> <a href="zf_lang1.html#K6">/</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="zf_model.html#K2">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K8">All</a> <font color="Maroon">z</font>,<font color="Maroon">H</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_4_2_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:2_4_2_1"/>
 <a href="zf_model.html#K2">Free</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K8">All</a> <font color="Maroon">z</font>,<font color="Maroon">H</font></span>)</span> <a href="zf_lang1.html#K6">/</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="zf_model.html#K2">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K8">All</a> <font color="Maroon">z</font>,<font color="Maroon">H</font></span>)</span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_4_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:2_4_2_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K2">Free</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K8">All</a> <font color="Maroon">z</font>,<font color="Maroon">H</font></span>)</span> <a href="zf_lang1.html#K6">/</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span>
 ;<br/>

<a NAME="E2:2_4_2_1_1"/><b>then </b>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_model.html#K2">Free</a> <font color="Maroon">H</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">z</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span> &amp; not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i>, , , , , , , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>hence </b><a NAME="E3:2_4_2_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K2">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K8">All</a> <font color="Maroon">z</font>,<font color="Maroon">H</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:2_4_2_1"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K2">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K8">All</a> <font color="Maroon">z</font>,<font color="Maroon">H</font></span>)</span>
 ;<br/>

<a NAME="E3:2_4_2_1"/><b>then </b><i><font color="Green">E76</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_model.html#K2">Free</a> <font color="Maroon">H</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="zf_lang1.html#K4">variables_in</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K8">All</a> <font color="Maroon">z</font>,<font color="Maroon">H</font></span>)</span> )
 
<b>by </b><i>, , , , , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E4:2_4_2_1"/>
<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E5:2_4_2_1"/><b>then </b>
not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E6:2_4_2_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K2">Free</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K8">All</a> <font color="Maroon">z</font>,<font color="Maroon">H</font></span>)</span> <a href="zf_lang1.html#K6">/</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span>
 <b>by </b><i>, , , , , , , , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_4_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:2_4_3"/>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K2">Free</a> <font color="Maroon">H</font>
 ;<br/><a NAME="E2:2_4_3"/><b>then </b>
( (  <a href="zf_model.html#K2">Free</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K8">All</a> <font color="Maroon">z</font>,<font color="Maroon">H</font></span>)</span> <a href="zf_lang1.html#K6">/</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_model.html#K2">Free</a> <font color="Maroon">H</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">z</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span> &amp; not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K2">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K8">All</a> <font color="Maroon">z</font>,<font color="Maroon">H</font></span>)</span> ) or  <a href="zf_model.html#K2">Free</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K8">All</a> <font color="Maroon">z</font>,<font color="Maroon">H</font></span>)</span> <a href="zf_lang1.html#K6">/</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="zf_model.html#K2">Free</a> <font color="Maroon">H</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">z</font></span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i>, , , , , , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a>, <a class="ref" href="zfmisc_1.html#T65">ZFMISC_1:65</a></i>;<br/><b>hence </b><a NAME="E3:2_4_3"/>
 <a href="zf_model.html#K2">Free</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K8">All</a> <font color="Maroon">z</font>,<font color="Maroon">H</font></span>)</span> <a href="zf_lang1.html#K6">/</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="zf_model.html#K2">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K8">All</a> <font color="Maroon">z</font>,<font color="Maroon">H</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T65">ZFMISC_1:65</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E15:2_4"/>
 <a href="zf_model.html#K2">Free</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K8">All</a> <font color="Maroon">z</font>,<font color="Maroon">H</font></span>)</span> <a href="zf_lang1.html#K6">/</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="zf_model.html#K2">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K8">All</a> <font color="Maroon">z</font>,<font color="Maroon">H</font></span>)</span>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E5:2"/>
for <font color="Olive">H</font> being  <a href="zf_lang.html#NM2">ZF-formula</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">H</font>]
 
<b>from </b><i><a class="ref" href="zf_lang1.html#S1">ZF_LANG1:sch 1</a>(, , , );<br/></i>
<b>hence </b><a NAME="E6:2"/>
( not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="zf_lang1.html#K4">variables_in</a> <font color="Maroon">H</font> implies ( ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K2">Free</a> <font color="Maroon">H</font> implies  <a href="zf_model.html#K2">Free</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="zf_lang1.html#K6">/</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_model.html#K2">Free</a> <font color="Maroon">H</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span> ) &amp; ( not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K2">Free</a> <font color="Maroon">H</font> implies  <a href="zf_model.html#K2">Free</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="zf_lang1.html#K6">/</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="zf_model.html#K2">Free</a> <font color="Maroon">H</font> ) ) )
 ;<br/>


</div>
