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

<b>let </b><font color="Maroon" title="c1">R</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> ;<br/>

<b>set </b><font color="Maroon" title="c2">Y</font> =  <a href="necklace.html#K4" title="NECKLACE:func.4">Necklace</a> 4;<br/>
<b>assume </b><a NAME="E1:33_1_1"/>
( <font color="Maroon" title="c1">R</font> is <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a> &amp; <font color="Maroon" title="c1">R</font> is <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a> )
 ;<br/>

<b>then consider </b><font color="Maroon" title="c3">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:33_1_1"/><i><font color="Green" title="E18">A1</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="group_6.html#D2" title="GROUP_6:def.2">GROUP_6:def 2</a></i>;<br/>
<b>assume </b><a NAME="E4:33_1_1"/>
not <font color="Maroon" title="c1">R</font> is <a href="neckla_2.html#V1" title="NECKLA_2:attr.1">N-free</a>
 ;<br/>

<a NAME="E5:33_1_1"/><b>then </b>
<font color="Maroon" title="c1">R</font> <a href="necklace.html#R2" title="NECKLACE:pred.2">embeds</a>  <a href="necklace.html#K4" title="NECKLACE:func.4">Necklace</a> 4
 
<b>by </b><i><a class="ref" href="neckla_2.html#D1" title="NECKLA_2:def.1">NECKLA_2:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c4">f</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4" title="NECKLACE:func.4">Necklace</a> 4</span>)</span>,<font color="Maroon" title="c1">R</font><b> such that </b><br/><a NAME="E7:33_1_1"/><i><font color="Green" title="E19">A2</font></i>: 
( <font color="Maroon" title="c4">f</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp; ( for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4" title="NECKLACE:func.4">Necklace</a> 4</span>)</span> holds <br/> ( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</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#K4" title="NECKLACE:func.4">Necklace</a> 4</span>)</span> iff <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">y</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</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="necklace.html#D2" title="NECKLACE:def.2">NECKLACE:def 2</a></i>;<br/>
<i><font color="Green" title="E20">A3</font></i>:  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">f</font> = 
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#K4" title="NECKLACE:func.4">Necklace</a> 4</span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>
<br/>.= 
<span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>
<b>by </b><i><a class="ref" href="necklace.html#T2" title="NECKLACE:th.2">NECKLACE:2</a>, <a class="ref" href="necklace.html#T21" title="NECKLACE:th.21">NECKLACE:21</a></i>
;<br/>
<a NAME="E9:33_1_1"/><b>then </b><i><font color="Green" title="E21">A4</font></i>: 
0 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">f</font>
 
<b>by </b><i><a class="ref" href="enumset1.html#D2" title="ENUMSET1:def.2">ENUMSET1:def 2</a></i>;<br/>
<a NAME="E10:33_1_1"/><i><font color="Green" title="E22">A5</font></i>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">f</font>
 
<b>by </b><i><a class="txt" href="neckla_3.html#E8:33_1_1"><i><font color="Green" title="E20">A3</font></i></a>, <a class="ref" href="enumset1.html#D2" title="ENUMSET1:def.2">ENUMSET1:def 2</a></i>;<br/>
<a NAME="E11:33_1_1"/><i><font color="Green" title="E23">A6</font></i>: 
<font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 0 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="txt" href="neckla_3.html#E3:33_1_1"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="neckla_3.html#E9:33_1_1"><i><font color="Green" title="E21">A4</font></i></a>, <a class="ref" href="partfun1.html#T27" title="PARTFUN1:th.27">PARTFUN1:27</a></i>;<br/>
<a NAME="E12:33_1_1"/>
<font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="txt" href="neckla_3.html#E3:33_1_1"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="neckla_3.html#E10:33_1_1"><i><font color="Green" title="E22">A5</font></i></a>, <a class="ref" href="partfun1.html#T27" title="PARTFUN1:th.27">PARTFUN1:27</a></i>;<br/>
<a NAME="E13:33_1_1"/><b>then </b>
( <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">y</font> &amp; <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">y</font> )
 
<b>by </b><i><a class="txt" href="neckla_3.html#E11:33_1_1"><i><font color="Green" title="E23">A6</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E14:33_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="neckla_3.html#E7:33_1_1"><i><font color="Green" title="E19">A2</font></i></a>, <a class="txt" href="neckla_3.html#E9:33_1_1"><i><font color="Green" title="E21">A4</font></i></a>, <a class="txt" href="neckla_3.html#E10:33_1_1"><i><font color="Green" title="E22">A5</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/>


</div>
