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<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="relat_2.html#D1">RELAT_2:def 1</a>,<a class="ref" href="relat_2.html#D9">RELAT_2:def 9</a><br/>

<b>assume </b><a NAME="E1:37_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K5">*</a> <font color="Maroon">R</font></span>)</span> <a href="relat_1.html#K5">*</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K2">"</a> </span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:37_1_1"/><b>then </b><i><font color="Green">E37</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K5">*</a> <font color="Maroon">R</font></span>)</span> <a href="relat_1.html#K5">*</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K2">"</a> </span>)</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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K5">*</a> <font color="Maroon">R</font></span>)</span> <a href="relat_1.html#K5">*</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K2">"</a> </span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_1.html#D6">RELAT_1:def 6</a></i>;<br/>
<a NAME="E3:37_1_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">R</font> <a href="relat_2.html#R1">is_reflexive_in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font>
 
<b>by </b><i><a class="ref" href="relat_2.html#D9">RELAT_2:def 9</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:37_1_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K5">*</a> <font color="Maroon">R</font></span>)</span> <a href="relat_1.html#K5">*</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K2">"</a> </span>)</span></span>)</span> or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K5">*</a> <font color="Maroon">R</font></span>)</span> <a href="relat_1.html#K5">*</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K2">"</a> </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 class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_1_1"><b>suppose </b></a><a NAME="E1:37_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K5">*</a> <font color="Maroon">R</font></span>)</span> <a href="relat_1.html#K5">*</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K2">"</a> </span>)</span></span>)</span>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:37_1_1_1_1"/><i><font color="Green">E40</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K5">*</a> <font color="Maroon">R</font></span>)</span> <a href="relat_1.html#K5">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K2">"</a> </span>)</span>
 <b>by </b><i><a class="ref" href="relat_1.html#D4">RELAT_1:def 4</a></i>;<br/><a NAME="E4:37_1_1_1_1"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> )
 <b>by </b><i>, </i>;<br/><a NAME="E5:37_1_1_1_1"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font>
 <b>by </b><i><a class="ref" href="relat_1.html#T30">RELAT_1:30</a></i>;<br/><a NAME="E6:37_1_1_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 <b>by </b><i>, <a class="ref" href="relat_2.html#D1">RELAT_2:def 1</a></i>;<br/><b>hence </b><a NAME="E7:37_1_1_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K5">*</a> <font color="Maroon">R</font></span>)</span> <a href="relat_1.html#K5">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K2">"</a> </span>)</span>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_1_2"><b>suppose </b></a><a NAME="E1:37_1_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K5">*</a> <font color="Maroon">R</font></span>)</span> <a href="relat_1.html#K5">*</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K2">"</a> </span>)</span></span>)</span>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:37_1_1_1_2"/><i><font color="Green">E43</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K5">*</a> <font color="Maroon">R</font></span>)</span> <a href="relat_1.html#K5">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K2">"</a> </span>)</span>
 <b>by </b><i><a class="ref" href="relat_1.html#D5">RELAT_1:def 5</a></i>;<br/><a NAME="E4:37_1_1_1_2"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> )
 <b>by </b><i>, </i>;<br/><a NAME="E5:37_1_1_1_2"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font>
 <b>by </b><i><a class="ref" href="relat_1.html#T30">RELAT_1:30</a></i>;<br/><a NAME="E6:37_1_1_1_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 <b>by </b><i>, <a class="ref" href="relat_2.html#D1">RELAT_2:def 1</a></i>;<br/><b>hence </b><a NAME="E7:37_1_1_1_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K5">*</a> <font color="Maroon">R</font></span>)</span> <a href="relat_1.html#K5">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K2">"</a> </span>)</span>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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