<?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/>



 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K8">All</a> <font color="Maroon">x</font>,<font color="Maroon">H</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default">4</span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="zf_lang.html#K3">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="zf_lang.html#K3">*&gt;</a></span></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">H</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T44">FINSEQ_1:44</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default">4</span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p3"><a href="zf_lang.html#K3">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="zf_lang.html#K3">*&gt;</a></span></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">H</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T44">FINSEQ_1:44</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">4</span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p3"><a href="zf_lang.html#K3">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="zf_lang.html#K3">*&gt;</a></span></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">H</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">4</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">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"><a href="relat_1.html#K2">rng</a> <font color="Maroon">H</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default">4</span><a href="tarski.html#K1">}</a></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">x</font></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">H</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T4">XBOOLE_1:4</a></i>
;<br/>
<a NAME="E2:137"/><b>then </b><i><font color="Green">E46</font></i>: 
(  <a href="zf_lang1.html#K3" target="_self">variables_in</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K8">All</a> <font color="Maroon">x</font>,<font color="Maroon">H</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">4</span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><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> <a href="xboole_0.html#K2">\/</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">H</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="enumset1.html#K3">{</a><span class="default">0,1,2,3,4</span><a href="enumset1.html#K3">}</a></span> &amp;  <a href="zf_lang1.html#K3" target="_self">variables_in</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">H</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="enumset1.html#K3">{</a><span class="default">0,1,2,3,4</span><a href="enumset1.html#K3">}</a></span> )
 
;<br/>
<b>thus </b><a NAME="E3:137"/>
 <a href="zf_lang1.html#K3" target="_self">variables_in</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K8">All</a> <font color="Maroon">x</font>,<font color="Maroon">H</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="zf_lang1.html#K3" target="_self">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">x</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="137_2"><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="E1:137_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="zf_lang1.html#K3" target="_self">variables_in</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K8">All</a> <font color="Maroon">x</font>,<font color="Maroon">H</font></span>)</span>
 ;<br/>

<a NAME="E2:137_2"/><b>then </b><i><font color="Green">E48</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">4</span><a href="tarski.html#K1">}</a></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">x</font></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">H</font></span>)</span></span>)</span> &amp; not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="enumset1.html#K3">{</a><span class="default">0,1,2,3,4</span><a href="enumset1.html#K3">}</a></span> &amp; <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 4 )
 
<b>by </b><i><a class="txt" href="zf_lang1.html#E2:137_2"><i><font color="Green">E48</font></i></a>, <a class="ref" href="zf_lang1.html#T6" target="_self">Th6</a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E3:137_2"/><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">4</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="E4:137_2"/><b>then </b>
<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> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">H</font></span>)</span>
 
<b>by </b><i><a class="ref" href="zf_lang1.html#T43" target="_self">Th43</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E5:137_2"/><b>then </b>
( ( <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>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">H</font> ) &amp; ( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">H</font> implies <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="zf_lang1.html#K3" target="_self">variables_in</a> <font color="Maroon">H</font> ) )
 
<b>by </b><i><a class="ref" href="zf_lang1.html#T43" target="_self">Th43</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="E6:137_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="zf_lang1.html#K3" target="_self">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">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></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="E4:137"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="zf_lang1.html#K3" target="_self">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">x</font></span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<a NAME="E5:137"/><b>then </b>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="zf_lang1.html#K3" target="_self">variables_in</a> <font color="Maroon">H</font> 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">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></i>;<br/>
<a NAME="E6:137"/><b>then </b>
( ( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</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="enumset1.html#K3">{</a><span class="default">0,1,2,3,4</span><a href="enumset1.html#K3">}</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">x</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</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="E7:137"/><b>then </b><i><font color="Green">E51</font></i>: 
( <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> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <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="enumset1.html#K3">{</a><span class="default">0,1,2,3,4</span><a href="enumset1.html#K3">}</a></span> )
 
<b>by </b><i><a class="ref" href="zf_lang1.html#T4" target="_self">Th4</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E8:137"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">4</span><a href="tarski.html#K1">}</a></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">x</font></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">H</font></span>)</span></span>)</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="E9:137"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="zf_lang1.html#K3" target="_self">variables_in</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K8">All</a> <font color="Maroon">x</font>,<font color="Maroon">H</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="zf_lang1.html#T43" target="_self">Th43</a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>


</div>
