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<b>let </b><font color="Maroon" title="c7">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="relat_2.html#D1" title="RELAT_2:def.1">RELAT_2:def 1</a>,<a class="ref" href="relat_2.html#D9" title="RELAT_2:def.9">RELAT_2:def 9</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1"><a href="pcs_0.html#K3" title="PCS_0:func.3">[^</a><span class="default"><font color="Maroon" title="c5">R</font>,<font color="Maroon" title="c6">S</font></span><a href="pcs_0.html#K3" title="PCS_0:func.3">^]</a></span> or <span class="p1"><a href="pcs_0.html#NK13" title="PCS_0:NK.13">[^</a><span class="default">,</span><a href="pcs_0.html#NK13" title="PCS_0:NK.13">^]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="pcs_0.html#K3" title="PCS_0:func.3">[^</a><span class="default"><font color="Maroon" title="c5">R</font>,<font color="Maroon" title="c6">S</font></span><a href="pcs_0.html#K3" title="PCS_0:func.3">^]</a></span> )</span><br/>

<b>assume </b><a NAME="E1:13_1_1"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pcs_0.html#K1" title="PCS_0:func.1">field</a> <span class="p1"><a href="pcs_0.html#K3" title="PCS_0:func.3">[^</a><span class="default"><font color="Maroon" title="c5">R</font>,<font color="Maroon" title="c6">S</font></span><a href="pcs_0.html#K3" title="PCS_0:func.3">^]</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="pcs_0.html#NK13" title="PCS_0:NK.13">[^</a><span class="default">,</span><a href="pcs_0.html#NK13" title="PCS_0:NK.13">^]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="pcs_0.html#K3" title="PCS_0:func.3">[^</a><span class="default"><font color="Maroon" title="c5">R</font>,<font color="Maroon" title="c6">S</font></span><a href="pcs_0.html#K3" title="PCS_0:func.3">^]</a></span></span><br/>

<b>then consider </b><font color="Maroon" title="c8">x1</font>, <font color="Maroon" title="c9">x2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:13_1_1"/><i><font color="Green" title="E6">A1</font></i>: 
<font color="Maroon" title="c8">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">R1</font>
 <b>and </b><br/><a NAME="E4:13_1_1"/><i><font color="Green" title="E7">A2</font></i>: 
<font color="Maroon" title="c9">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">S1</font>
 <b>and </b><br/><a NAME="E5:13_1_1"/><i><font color="Green" title="E8">A3</font></i>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">x1</font>,<font color="Maroon" title="c9">x2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/>
<a NAME="E6:13_1_1"/>
<font color="Maroon" title="c4">S1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pcs_0.html#K1" title="PCS_0:func.1">field</a> <font color="Maroon" title="c6">S</font>
 
<b>by </b><i><a class="ref" href="orders_1.html#T97" title="ORDERS_1:th.97">ORDERS_1:97</a></i>;<br/>
<a NAME="E7:13_1_1"/><b>then </b>
<font color="Maroon" title="c6">S</font> <a href="relat_2.html#R1" title="RELAT_2:pred.1">is_reflexive_in</a> <font color="Maroon" title="c4">S1</font>
 
<b>by </b><i><a class="ref" href="relat_2.html#D9" title="RELAT_2:def.9">RELAT_2:def 9</a></i>;<br/>
<a NAME="E8:13_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">x2</font>,<font color="Maroon" title="c9">x2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">S</font>
 
<b>by </b><i><a class="txt" href="#E4:13_1_1"><i><font color="Green" title="E7">A2</font></i></a>, <a class="ref" href="relat_2.html#D1" title="RELAT_2:def.1">RELAT_2:def 1</a></i>;<br/>
<b>hence </b><a NAME="E9:13_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c7">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="pcs_0.html#K3" title="PCS_0:func.3">[^</a><span class="default"><font color="Maroon" title="c5">R</font>,<font color="Maroon" title="c6">S</font></span><a href="pcs_0.html#K3" title="PCS_0:func.3">^]</a></span>
 <b>by </b><i><a class="txt" href="#E3:13_1_1"><i><font color="Green" title="E6">A1</font></i></a>, <a class="txt" href="#E4:13_1_1"><i><font color="Green" title="E7">A2</font></i></a>, <a class="txt" href="#E5:13_1_1"><i><font color="Green" title="E8">A3</font></i></a>, <a class="ref" href="pcs_0.html#D3" target="_self" title="PCS_0:def.3">Def3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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