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<b>let </b><font color="Maroon">A</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">B</font> be    <a href="hidden.html#M1">set</a> ;<br/>





<b>assume </b><a NAME="E1:20"/><i><font color="Green">E12</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:20_1_1"/>
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_1_1"><b>suppose </b></a><a NAME="E1:20_1_1_1"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:20_1_1_1"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="xboole_0.html#K1">{}</a>  <a href="funcop_1.html#K2" target="_self">--&gt;</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><a NAME="E3:20_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="funcop_1.html#K2" target="_self">--&gt;</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i/>;<br/><a NAME="E4:20_1_1_1"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="funcop_1.html#K2" target="_self">--&gt;</a> <font color="Maroon">x</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><b>hence </b><a NAME="E5:20_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="funcop_1.html#K2" target="_self">--&gt;</a> <font color="Maroon">x</font></span>)</span> <a href="relat_1.html#K10">"</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font>
 <b>by </b><i>, <a class="ref" href="relat_1.html#T173">RELAT_1:173</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_1_2"><b>suppose </b></a><a NAME="E1:20_1_1_2"/>
<font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:20_1_1_2"/><b>then </b><i><font color="Green">E14</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="funcop_1.html#K2" target="_self">--&gt;</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</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/>;<br/><a NAME="E3:20_1_1_2"/>
<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="tarski.html#R1">c=</a> <font color="Maroon">B</font>
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/><a NAME="E4:20_1_1_2"/><b>then </b>
<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#K3">/\</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</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_1.html#T28">XBOOLE_1:28</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="funcop_1.html#K2" target="_self">--&gt;</a> <font color="Maroon">x</font></span>)</span> <a href="relat_1.html#K10">"</a> <font color="Maroon">B</font> = 
<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="funcop_1.html#K2" target="_self">--&gt;</a> <font color="Maroon">x</font></span>)</span> <a href="relat_1.html#K10">"</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="relat_1.html#T168">RELAT_1:168</a></i>
<br/>.= 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="funcop_1.html#K2" target="_self">--&gt;</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i>, <a class="ref" href="relat_1.html#T169">RELAT_1:169</a></i>
<br/>.= 
<font color="Maroon">A</font>
<b>by </b><i><a class="txt" href="funcop_1.html#E2:20_1_1_2"><i><font color="Green">E14</font></i></a></i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:20"/>
<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="funcop_1.html#K2" target="_self">--&gt;</a> <font color="Maroon">x</font></span>)</span> <a href="relat_1.html#K10">"</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font>
 ;<br/>


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