<?xml version="1.0"?>
<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">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/>





<b>assume </b><a NAME="E1:9"/><i><font color="Green">E47</font></i>: 
( not <font color="Maroon">x</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>  <a href="zf_lang1.html#K4">variables_in</a> <font color="Maroon">H</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="zf_lang.html#K2">x.</a> 0 &amp; <font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="zf_lang.html#K2">x.</a> 0 &amp; <font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="zf_lang.html#K2">x.</a> 4 )
 ;<br/>

<a NAME="E2:9"/><i><font color="Green">E48</font></i>: 
(  <a href="zf_lang.html#K2">x.</a> 0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="zf_lang.html#K2">x.</a> 3 &amp;  <a href="zf_lang.html#K2">x.</a> 0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="zf_lang.html#K2">x.</a> 4 )
 
<b>by </b><i><a class="ref" href="zf_lang1.html#T86">ZF_LANG1:86</a></i>;<br/>
<b>set </b><font color="Maroon">G</font> = <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="zf_lang1.html#K6">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<font color="Maroon">y</font></span>)</span> <a href="zf_lang1.html#K6">/</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>;<br/>
<i><font color="Green">E49</font></i>:  <a href="zf_fund1.html#K6">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K13">Ex</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span> <a href="zf_lang.html#K5">'in'</a> <font color="Maroon">x</font></span>)</span> <a href="zf_lang.html#K7">'&amp;'</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">H</font> <a href="zf_lang1.html#K6">/</a> <span class="p5">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<font color="Maroon">y</font></span>)</span> <a href="zf_lang1.html#K6">/</a> <span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="zf_fund1.html#K6">Free</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span> <a href="zf_lang.html#K5">'in'</a> <font color="Maroon">x</font></span>)</span> <a href="zf_lang.html#K7">'&amp;'</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">H</font> <a href="zf_lang1.html#K6">/</a> <span class="p5">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<font color="Maroon">y</font></span>)</span> <a href="zf_lang1.html#K6">/</a> <span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i><a class="ref" href="zf_lang1.html#T71">ZF_LANG1:71</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_fund1.html#K6">Free</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span> <a href="zf_lang.html#K5">'in'</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="zf_fund1.html#K6">Free</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">H</font> <a href="zf_lang1.html#K6">/</a> <span class="p5">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<font color="Maroon">y</font></span>)</span> <a href="zf_lang1.html#K6">/</a> <span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i><a class="ref" href="zf_lang1.html#T66">ZF_LANG1:66</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K2">{</a><span class="default"><span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<font color="Maroon">x</font></span><a href="tarski.html#K2">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><a href="zf_fund1.html#K6">Free</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">H</font> <a href="zf_lang1.html#K6">/</a> <span class="p5">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<font color="Maroon">y</font></span>)</span> <a href="zf_lang1.html#K6">/</a> <span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i><a class="ref" href="zf_lang1.html#T64">ZF_LANG1:64</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K2">{</a><span class="default"><span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<font color="Maroon">x</font></span><a href="tarski.html#K2">}</a></span> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span></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">(<span class="default"><a href="zf_fund1.html#K6">Free</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">H</font> <a href="zf_lang1.html#K6">/</a> <span class="p5">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<font color="Maroon">y</font></span>)</span> <a href="zf_lang1.html#K6">/</a> <span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T42">XBOOLE_1:42</a></i>
;<br/>
<div><i><font color="Green">E50</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="9_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:9_2"/><i><font color="Green">E51</font></i>: 
 <a href="zf_lang.html#K2">x.</a> 4 <a href="hidden.html#R2">in</a>  <a href="zf_fund1.html#K6">Free</a> <font color="Maroon">H</font>
 ;<br/><a NAME="E2:9_2"/><i><font color="Green">E52</font></i>: 
 <a href="zf_lang.html#K2">x.</a> 4 <a href="hidden.html#R2">in</a>  <a href="zf_fund1.html#K6">Free</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="zf_lang1.html#K6">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<font color="Maroon">y</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_2_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:9_2_1_1_1"/>
(  <a href="zf_lang.html#K2">x.</a> 0 <a href="hidden.html#R2">in</a>  <a href="zf_fund1.html#K6">Free</a> <font color="Maroon">H</font> or  not  <a href="zf_lang.html#K2">x.</a> 0 <a href="hidden.html#R2">in</a>  <a href="zf_fund1.html#K6">Free</a> <font color="Maroon">H</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_2_1_1_1_1"><b>suppose </b></a><a NAME="E1:9_2_1_1_1_1"/>
 <a href="zf_lang.html#K2">x.</a> 0 <a href="hidden.html#R2">in</a>  <a href="zf_fund1.html#K6">Free</a> <font color="Maroon">H</font>
 ;<br/><div class="add"><a NAME="E2:9_2_1_1_1_1"/><b>then </b><i><font color="Green">E53</font></i>: 
 <a href="zf_fund1.html#K6">Free</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="zf_lang1.html#K6">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<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_fund1.html#K6">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"><span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span></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="zfmodel2.html#T2">ZFMODEL2:2</a></i>;<br/><a NAME="E3:9_2_1_1_1_1"/>
not  <a href="zf_lang.html#K2">x.</a> 4 <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"><a href="zf_lang.html#K2">x.</a> 0</span>)</span></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/><a NAME="E4:9_2_1_1_1_1"/><b>then </b>
 <a href="zf_lang.html#K2">x.</a> 4 <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K6">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"><span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span></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="E5:9_2_1_1_1_1"/>
 <a href="zf_lang.html#K2">x.</a> 4 <a href="hidden.html#R2">in</a>  <a href="zf_fund1.html#K6">Free</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="zf_lang1.html#K6">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<font color="Maroon">y</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_2_1_1_1_2"><b>suppose </b></a><a NAME="E1:9_2_1_1_1_2"/>
not  <a href="zf_lang.html#K2">x.</a> 0 <a href="hidden.html#R2">in</a>  <a href="zf_fund1.html#K6">Free</a> <font color="Maroon">H</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:9_2_1_1_1_2"/>
 <a href="zf_lang.html#K2">x.</a> 4 <a href="hidden.html#R2">in</a>  <a href="zf_fund1.html#K6">Free</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="zf_lang1.html#K6">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<font color="Maroon">y</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="zfmodel2.html#T2">ZFMODEL2:2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:9_2_1"/>
 <a href="zf_lang.html#K2">x.</a> 4 <a href="hidden.html#R2">in</a>  <a href="zf_fund1.html#K6">Free</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="zf_lang1.html#K6">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<font color="Maroon">y</font></span>)</span>
 ;<br/>


</div><b>end;</b></div><a NAME="E3:9_2"/>
not  <a href="zf_lang.html#K2">x.</a> 0 <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">H</font> <a href="zf_lang1.html#K6">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<font color="Maroon">y</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="zf_lang1.html#T198">ZF_LANG1:198</a></i>;<br/><a NAME="E4:9_2"/><b>then </b><i><font color="Green">E54</font></i>: 
 <a href="zf_fund1.html#K6">Free</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">H</font> <a href="zf_lang1.html#K6">/</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<font color="Maroon">y</font></span>)</span> <a href="zf_lang1.html#K6">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_fund1.html#K6">Free</a> <span class="p3">(<span class="default"><font color="Maroon">H</font> <a href="zf_lang1.html#K6">/</a> <span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<font color="Maroon">y</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span></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"><span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, <a class="ref" href="zfmodel2.html#T2">ZFMODEL2:2</a></i>;<br/><a NAME="E5:9_2"/>
 <a href="zf_lang.html#K2">x.</a> 0 <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"><a href="zf_lang.html#K2">x.</a> 0</span>)</span></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/><a NAME="E6:9_2"/><b>then </b>
(  <a href="zf_lang.html#K2">x.</a> 0 <a href="hidden.html#R2">in</a>  <a href="zf_fund1.html#K6">Free</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">H</font> <a href="zf_lang1.html#K6">/</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<font color="Maroon">y</font></span>)</span> <a href="zf_lang1.html#K6">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span></span>)</span> &amp; not  <a href="zf_lang.html#K2">x.</a> 0 <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"><a href="zf_lang.html#K2">x.</a> 3</span>)</span></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></i>;<br/><a NAME="E7:9_2"/><b>then </b>
 <a href="zf_lang.html#K2">x.</a> 0 <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K6">Free</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">H</font> <a href="zf_lang1.html#K6">/</a> <span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<font color="Maroon">y</font></span>)</span> <a href="zf_lang1.html#K6">/</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span></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="E8:9_2"/>
 <a href="zf_lang.html#K2">x.</a> 0 <a href="hidden.html#R2">in</a>  <a href="zf_fund1.html#K6">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K13">Ex</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span> <a href="zf_lang.html#K5">'in'</a> <font color="Maroon">x</font></span>)</span> <a href="zf_lang.html#K7">'&amp;'</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">H</font> <a href="zf_lang1.html#K6">/</a> <span class="p5">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<font color="Maroon">y</font></span>)</span> <a href="zf_lang1.html#K6">/</a> <span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span></span>)</span></span>)</span></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>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:9_3"/>
 <a href="zf_lang.html#K2">x.</a> 0 <a href="hidden.html#R2">in</a>  <a href="zf_fund1.html#K6">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K13">Ex</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span> <a href="zf_lang.html#K5">'in'</a> <font color="Maroon">x</font></span>)</span> <a href="zf_lang.html#K7">'&amp;'</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">H</font> <a href="zf_lang1.html#K6">/</a> <span class="p5">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<font color="Maroon">y</font></span>)</span> <a href="zf_lang1.html#K6">/</a> <span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span></span>)</span></span>)</span></span>)</span>
 ;<br/><a NAME="E2:9_3"/><b>then </b>
(  <a href="zf_lang.html#K2">x.</a> 0 <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<font color="Maroon">x</font></span><a href="tarski.html#K2">}</a></span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span></span><a href="tarski.html#K1">}</a></span> or  <a href="zf_lang.html#K2">x.</a> 0 <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K6">Free</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">H</font> <a href="zf_lang1.html#K6">/</a> <span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<font color="Maroon">y</font></span>)</span> <a href="zf_lang1.html#K6">/</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span></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="E3:9_3"/><b>then </b><i><font color="Green">E55</font></i>: 
(  <a href="zf_lang.html#K2">x.</a> 0 <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<font color="Maroon">x</font></span><a href="tarski.html#K2">}</a></span> or  <a href="zf_lang.html#K2">x.</a> 0 <a href="hidden.html#R2">in</a>  <a href="zf_fund1.html#K6">Free</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">H</font> <a href="zf_lang1.html#K6">/</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<font color="Maroon">y</font></span>)</span> <a href="zf_lang1.html#K6">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E4:9_3"/><i><font color="Green">E56</font></i>: 
not  <a href="zf_lang.html#K2">x.</a> 0 <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">H</font> <a href="zf_lang1.html#K6">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<font color="Maroon">y</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="zf_lang1.html#T198">ZF_LANG1:198</a></i>;<br/><div><i><font color="Green">E69</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="9_3_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:9_3_1"/>
not  <a href="zf_lang.html#K2">x.</a> 4 <a href="hidden.html#R2">in</a>  <a href="zf_fund1.html#K6">Free</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="zf_lang1.html#K6">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<font color="Maroon">y</font></span>)</span>
 ;<br/><a NAME="E2:9_3_1"/><b>then </b><i><font color="Green">E71</font></i>: 
 <a href="zf_lang.html#K2">x.</a> 0 <a href="hidden.html#R2">in</a>  <a href="zf_fund1.html#K6">Free</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="zf_lang1.html#K6">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<font color="Maroon">y</font></span>)</span>
 <b>by </b><i>, , , , <a class="ref" href="tarski.html#D2">TARSKI:def 2</a>, <a class="ref" href="zfmodel2.html#T2">ZFMODEL2:2</a></i>;<br/><a NAME="E3:9_3_1"/>
 <a href="zf_fund1.html#K6">Free</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="zf_lang1.html#K6">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<font color="Maroon">y</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"><font color="Maroon">H</font> <a href="zf_lang1.html#K6">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<font color="Maroon">y</font></span>)</span>
 <b>by </b><i><a class="ref" href="zf_lang1.html#T164">ZF_LANG1:164</a></i>;<br/><b>hence </b><a NAME="E4:9_3_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="zf_lang1.html#T198">ZF_LANG1:198</a></i>;<br/></div><b>end;</b></div><a NAME="E6:9_3"/>
 <a href="zf_fund1.html#K6">Free</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="zf_lang1.html#K6">/</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<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_fund1.html#K6">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"><span class="p3">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span></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="zfmodel2.html#T1">ZFMODEL2:1</a></i>;<br/><a NAME="E7:9_3"/><b>then </b>
(  <a href="zf_lang.html#K2">x.</a> 4 <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K6">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"><span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span></span><a href="tarski.html#K1">}</a></span> or  <a href="zf_lang.html#K2">x.</a> 4 <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/><b>hence </b><a NAME="E8:9_3"/>
 <a href="zf_lang.html#K2">x.</a> 4 <a href="hidden.html#R2">in</a>  <a href="zf_fund1.html#K6">Free</a> <font color="Maroon">H</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/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:9"/>
(  <a href="zf_lang.html#K2">x.</a> 4 <a href="hidden.html#R2">in</a>  <a href="zf_fund1.html#K6">Free</a> <font color="Maroon">H</font> iff  <a href="zf_lang.html#K2">x.</a> 0 <a href="hidden.html#R2">in</a>  <a href="zf_fund1.html#K6">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K13">Ex</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> 3</span>)</span> <a href="zf_lang.html#K5">'in'</a> <font color="Maroon">x</font></span>)</span> <a href="zf_lang.html#K7">'&amp;'</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">H</font> <a href="zf_lang1.html#K6">/</a> <span class="p5">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span>,<font color="Maroon">y</font></span>)</span> <a href="zf_lang1.html#K6">/</a> <span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> 4</span>)</span>,<span class="p4">(<span class="default"><a href="zf_lang.html#K2">x.</a> 0</span>)</span></span>)</span></span>)</span></span>)</span> )
 <b>by </b><i/>;<br/>


</div>
