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<b>let </b><font color="Maroon">f</font> be   <a href="funct_1.html#NM1">Function</a>;<br/><b>let </b><font color="Maroon">d</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">e</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">i</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">j</font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><a NAME="E1:47"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">j</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:47_1"/>
( ( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> ) or ( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> &amp; not <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> ) or ( not <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> ) or ( not <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> &amp; not <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_1_1"><b>suppose </b></a><b>that </b><a NAME="E1:47_1_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E2:47_1_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 ;<br/><div class="add"><a NAME="E3:47_1_1"/>
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">d</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i</font></span><a href="tarski.html#K1">}</a></span> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">e</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">j</font></span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i><a class="ref" href="cqc_lang.html#T5">CQC_LANG:5</a></i>;<br/><a NAME="E4:47_1_1"/><b>then </b><i><font color="Green">E33</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">d</font></span>)</span> <a href="subset_1.html#R1">misses</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">e</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T17">ZFMISC_1:17</a></i>;<br/><a NAME="E5:47_1_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_7.html#K2" target="_self">+*</a> <font color="Maroon">j</font>,<font color="Maroon">e</font></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E6:47_1_1"/>
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_7.html#K2" target="_self">+*</a> <font color="Maroon">i</font>,<font color="Maroon">d</font></span>)</span>
 <b>by </b><i>, </i>;<br/><b>hence </b><span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_7.html#K2" target="_self">+*</a> <font color="Maroon">i</font>,<font color="Maroon">d</font></span>)</span> <a href="funct_7.html#K2" target="_self">+*</a> <font color="Maroon">j</font>,<font color="Maroon">e</font> = 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_7.html#K2" target="_self">+*</a> <font color="Maroon">i</font>,<font color="Maroon">d</font></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">e</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">d</font></span>)</span></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">e</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">f</font> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">d</font></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">e</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_4.html#T15">FUNCT_4:15</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">e</font></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">d</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="funct_4.html#T36">FUNCT_4:36</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">e</font></span>)</span></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">d</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_4.html#T15">FUNCT_4:15</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_7.html#K2" target="_self">+*</a> <font color="Maroon">j</font>,<font color="Maroon">e</font></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">d</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_7.html#K2" target="_self">+*</a> <font color="Maroon">j</font>,<font color="Maroon">e</font></span>)</span> <a href="funct_7.html#K2" target="_self">+*</a> <font color="Maroon">i</font>,<font color="Maroon">d</font>
<b>by </b><i>, </i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_1_2"><b>suppose </b></a><b>that </b><a NAME="E1:47_1_2"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E2:47_1_2"/><i><font color="Green">E45</font></i>: 
not <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 ;<br/><div class="add"><a NAME="E3:47_1_2"/>
not <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_7.html#K2" target="_self">+*</a> <font color="Maroon">i</font>,<font color="Maroon">d</font></span>)</span>
 <b>by </b><i>, </i>;<br/><b>hence </b><span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_7.html#K2" target="_self">+*</a> <font color="Maroon">i</font>,<font color="Maroon">d</font></span>)</span> <a href="funct_7.html#K2" target="_self">+*</a> <font color="Maroon">j</font>,<font color="Maroon">e</font> = 
<font color="Maroon">f</font> <a href="funct_7.html#K2" target="_self">+*</a> <font color="Maroon">i</font>,<font color="Maroon">d</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_7.html#K2" target="_self">+*</a> <font color="Maroon">j</font>,<font color="Maroon">e</font></span>)</span> <a href="funct_7.html#K2" target="_self">+*</a> <font color="Maroon">i</font>,<font color="Maroon">d</font>
<b>by </b><i>, </i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_1_3"><b>suppose </b></a><b>that </b><a NAME="E1:47_1_3"/><i><font color="Green">E46</font></i>: 
not <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E2:47_1_3"/>
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 ;<br/><div class="add"><a NAME="E3:47_1_3"/><i><font color="Green">E47</font></i>: 
not <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_7.html#K2" target="_self">+*</a> <font color="Maroon">j</font>,<font color="Maroon">e</font></span>)</span>
 <b>by </b><i>, </i>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_7.html#K2" target="_self">+*</a> <font color="Maroon">i</font>,<font color="Maroon">d</font></span>)</span> <a href="funct_7.html#K2" target="_self">+*</a> <font color="Maroon">j</font>,<font color="Maroon">e</font> = 
<font color="Maroon">f</font> <a href="funct_7.html#K2" target="_self">+*</a> <font color="Maroon">j</font>,<font color="Maroon">e</font>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_7.html#K2" target="_self">+*</a> <font color="Maroon">j</font>,<font color="Maroon">e</font></span>)</span> <a href="funct_7.html#K2" target="_self">+*</a> <font color="Maroon">i</font>,<font color="Maroon">d</font>
<b>by </b><i>, </i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_1_4"><b>suppose </b></a><b>that </b><a NAME="E1:47_1_4"/><i><font color="Green">E48</font></i>: 
not <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E2:47_1_4"/><i><font color="Green">E53</font></i>: 
not <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 ;<br/><div class="add"><a NAME="E3:47_1_4"/><i><font color="Green">E54</font></i>: 
not <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_7.html#K2" target="_self">+*</a> <font color="Maroon">j</font>,<font color="Maroon">e</font></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E4:47_1_4"/>
not <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_7.html#K2" target="_self">+*</a> <font color="Maroon">i</font>,<font color="Maroon">d</font></span>)</span>
 <b>by </b><i>, </i>;<br/><b>hence </b><span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_7.html#K2" target="_self">+*</a> <font color="Maroon">i</font>,<font color="Maroon">d</font></span>)</span> <a href="funct_7.html#K2" target="_self">+*</a> <font color="Maroon">j</font>,<font color="Maroon">e</font> = 
<font color="Maroon">f</font> <a href="funct_7.html#K2" target="_self">+*</a> <font color="Maroon">i</font>,<font color="Maroon">d</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">f</font>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">f</font> <a href="funct_7.html#K2" target="_self">+*</a> <font color="Maroon">j</font>,<font color="Maroon">e</font>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_7.html#K2" target="_self">+*</a> <font color="Maroon">j</font>,<font color="Maroon">e</font></span>)</span> <a href="funct_7.html#K2" target="_self">+*</a> <font color="Maroon">i</font>,<font color="Maroon">d</font>
<b>by </b><i>, </i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div>

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