<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c7">x</font>, <font color="Maroon" title="c8">y</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#D3" title="RELAT_2:def.3">RELAT_2:def 3</a>,<a class="ref" href="relat_2.html#D11" title="RELAT_2:def.11">RELAT_2:def 11</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  not <font color="Maroon" title="c8">y</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  not <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> 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="E1:16_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> &amp; <font color="Maroon" title="c8">y</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"> (  not <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> 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="E2:16_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="c8">y</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>
 ; <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="c9">p</font>, <font color="Maroon" title="c10">q</font>, <font color="Maroon" title="c11">s</font>, <font color="Maroon" title="c12">t</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:16_1_1"/><i><font color="Green" title="E6">A1</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="c9">p</font>,<font color="Maroon" title="c10">q</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c8">y</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="c11">s</font>,<font color="Maroon" title="c12">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <b>and </b><br/><a NAME="E5:16_1_1"/><i><font color="Green" title="E7">A2</font></i>: 
( <font color="Maroon" title="c9">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">R1</font> &amp; <font color="Maroon" title="c10">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">S1</font> &amp; <font color="Maroon" title="c11">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">R1</font> &amp; <font color="Maroon" title="c12">t</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="E6:16_1_1"/><i><font color="Green" title="E8">A3</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">p</font>,<font color="Maroon" title="c11">s</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="c5">R</font> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">q</font>,<font color="Maroon" title="c12">t</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="ref" href="pcs_0.html#D2" target="_self" title="PCS_0:def.2">Def2</a></i>;<br/>
<a NAME="E7:16_1_1"/><i><font color="Green" title="E9">A4</font></i>: 
( <font color="Maroon" title="c5">R</font> <a href="relat_2.html#R3" title="RELAT_2:pred.3">is_symmetric_in</a>  <a href="pcs_0.html#K1" title="PCS_0:func.1">field</a> <font color="Maroon" title="c5">R</font> &amp; <font color="Maroon" title="c6">S</font> <a href="relat_2.html#R3" title="RELAT_2:pred.3">is_symmetric_in</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="relat_2.html#D11" title="RELAT_2:def.11">RELAT_2:def 11</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:16_1_1_1"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">p</font>,<font color="Maroon" title="c11">s</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="c5">R</font> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">q</font>,<font color="Maroon" title="c12">t</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> )
 </span><b>by </b><i><a class="txt" href="pcs_0.html#E6:16_1_1"><i><font color="Green" title="E8">A3</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1_1_1"><b>suppose </b></a><a NAME="E1:16_1_1_1_1"/><i><font color="Green" title="E10">A5</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">p</font>,<font color="Maroon" title="c11">s</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="c5">R</font>
 ; <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/><div class="add"><a NAME="E2:16_1_1_1_1"/><b>then </b>
( <font color="Maroon" title="c9">p</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> <font color="Maroon" title="c5">R</font> &amp; <font color="Maroon" title="c11">s</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> <font color="Maroon" title="c5">R</font> )
 <b>by </b><i><a class="ref" href="relat_1.html#T30" title="RELAT_1:th.30">RELAT_1:30</a></i>;<br/><a NAME="E3:16_1_1_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="c11">s</font>,<font color="Maroon" title="c9">p</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="c5">R</font>
 <b>by </b><i><a class="txt" href="pcs_0.html#E7:16_1_1"><i><font color="Green" title="E9">A4</font></i></a>, <a class="txt" href="pcs_0.html#E1:16_1_1_1_1"><i><font color="Green" title="E10">A5</font></i></a>, <a class="ref" href="relat_2.html#D3" title="RELAT_2:def.3">RELAT_2:def 3</a></i>;<br/><b>hence </b><a NAME="E4:16_1_1_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">y</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="pcs_0.html#E4:16_1_1"><i><font color="Green" title="E6">A1</font></i></a>, <a class="txt" href="pcs_0.html#E5:16_1_1"><i><font color="Green" title="E7">A2</font></i></a>, <a class="ref" href="pcs_0.html#D2" target="_self" title="PCS_0:def.2">Def2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1_1_2"><b>suppose </b></a><a NAME="E1:16_1_1_1_2"/><i><font color="Green" title="E10">A6</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">q</font>,<font color="Maroon" title="c12">t</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>
 ; <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/><div class="add"><a NAME="E2:16_1_1_1_2"/><b>then </b>
( <font color="Maroon" title="c10">q</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> <font color="Maroon" title="c6">S</font> &amp; <font color="Maroon" title="c12">t</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> <font color="Maroon" title="c6">S</font> )
 <b>by </b><i><a class="ref" href="relat_1.html#T30" title="RELAT_1:th.30">RELAT_1:30</a></i>;<br/><a NAME="E3:16_1_1_1_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">t</font>,<font color="Maroon" title="c10">q</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="pcs_0.html#E7:16_1_1"><i><font color="Green" title="E9">A4</font></i></a>, <a class="txt" href="pcs_0.html#E1:16_1_1_1_2"><i><font color="Green" title="E10">A6</font></i></a>, <a class="ref" href="relat_2.html#D3" title="RELAT_2:def.3">RELAT_2:def 3</a></i>;<br/><b>hence </b><a NAME="E4:16_1_1_1_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">y</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="pcs_0.html#E4:16_1_1"><i><font color="Green" title="E6">A1</font></i></a>, <a class="txt" href="pcs_0.html#E5:16_1_1"><i><font color="Green" title="E7">A2</font></i></a>, <a class="ref" href="pcs_0.html#D2" target="_self" title="PCS_0:def.2">Def2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
