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

<b>thus </b><a NAME="E1:40_2_1"/>
( <font color="Maroon">c<sub>1</sub></font> is <a href="necklace.html#V3">irreflexive</a> implies the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> <a href="relat_2.html#R2">is_irreflexive_in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:40_2_1_1"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="necklace.html#V3">irreflexive</a>
 ;<br/>

<a NAME="E2:40_2_1_1"/><b>then </b>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/> not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="necklace.html#D6">NECKLACE:def 6</a></i>;<br/>
<b>hence </b><a NAME="E3:40_2_1_1"/>
the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> <a href="relat_2.html#R2">is_irreflexive_in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="relat_2.html#D2">RELAT_2:def 2</a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:40_2_1"/>
the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> <a href="relat_2.html#R2">is_irreflexive_in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<a NAME="E3:40_2_1"/><b>then </b>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/> not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="relat_2.html#D2">RELAT_2:def 2</a></i>;<br/>
<b>hence </b><a NAME="E4:40_2_1"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="necklace.html#V3">irreflexive</a>
 <b>by </b><i><a class="ref" href="necklace.html#D6">NECKLACE:def 6</a></i>;<br/>


</div>
