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

<b>let </b><font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c3">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="necklace.html#D4" target="_self" title="NECKLACE:def.4">NECKLACE:def 4</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:31_1_1"/>
( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="necklace.html#K2" title="NECKLACE:func.2">SymRelStr</a> <font color="Maroon" title="c1">R</font></span>)</span> &amp; <font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="necklace.html#K2" title="NECKLACE:func.2">SymRelStr</a> <font color="Maroon" title="c1">R</font></span>)</span> )
 <b>and </b><br/><a NAME="E2:31_1_1"/><i><font color="Green" title="E18">A1</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K2" title="NECKLACE:func.2">SymRelStr</a> <font color="Maroon" title="c1">R</font></span>)</span>
 ;<br/>

<a NAME="E3:31_1_1"/><i><font color="Green" title="E19">A2</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> <a href="relset_1.html#K1" title="RELSET_1:func.1">\/</a> <span class="p1">(<span class="default">the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> <a href="relset_1.html#K6" title="RELSET_1:func.6">~</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="necklace.html#E2:31_1_1"><i><font color="Green" title="E18">A1</font></i></a>, <a class="ref" href="necklace.html#D8" target="_self" title="NECKLACE:def.8">Def8</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:31_1_1_1"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> <a href="relset_1.html#K6" title="RELSET_1:func.6">~</a>  )
 </span><b>by </b><i><a class="txt" href="necklace.html#E3:31_1_1"><i><font color="Green" title="E19">A2</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1_1_1_1"><b>suppose </b></a><a NAME="E1:31_1_1_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font>
 ;<br/><div class="add"><a NAME="E2:31_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="c3">y</font>,<font color="Maroon" title="c2">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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> <a href="relset_1.html#K6" title="RELSET_1:func.6">~</a> 
 <b>by </b><i><a class="ref" href="relat_1.html#D7" title="RELAT_1:def.7">RELAT_1:def 7</a></i>;<br/><a NAME="E3:31_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="c3">y</font>,<font color="Maroon" title="c2">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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> <a href="relset_1.html#K1" title="RELSET_1:func.1">\/</a> <span class="p1">(<span class="default">the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> <a href="relset_1.html#K6" title="RELSET_1:func.6">~</a> </span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><b>hence </b><a NAME="E4:31_1_1_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c2">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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K2" title="NECKLACE:func.2">SymRelStr</a> <font color="Maroon" title="c1">R</font></span>)</span>
 <b>by </b><i><a class="ref" href="necklace.html#D8" target="_self" title="NECKLACE:def.8">Def8</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1_1_1_2"><b>suppose </b></a><a NAME="E1:31_1_1_1_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> <a href="relset_1.html#K6" title="RELSET_1:func.6">~</a> 
 ;<br/><div class="add"><a NAME="E2:31_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="c3">y</font>,<font color="Maroon" title="c2">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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font>
 <b>by </b><i><a class="ref" href="relat_1.html#D7" title="RELAT_1:def.7">RELAT_1:def 7</a></i>;<br/><a NAME="E3:31_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="c3">y</font>,<font color="Maroon" title="c2">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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> <a href="relset_1.html#K1" title="RELSET_1:func.1">\/</a> <span class="p1">(<span class="default">the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> <a href="relset_1.html#K6" title="RELSET_1:func.6">~</a> </span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><b>hence </b><a NAME="E4:31_1_1_1_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c2">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> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K2" title="NECKLACE:func.2">SymRelStr</a> <font color="Maroon" title="c1">R</font></span>)</span>
 <b>by </b><i><a class="ref" href="necklace.html#D8" target="_self" title="NECKLACE:def.8">Def8</a></i>;<br/><b>thus </b><a NAME="E5:31_1_1_1_2"/>
verum
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
