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<b>let </b><font color="Maroon">R</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="orders_2.html#L1">RelStr</a> ;<br/>

<b>set </b><font color="Maroon">Y</font> =  <a href="necklace.html#K4">Necklace</a> 4;<br/>
<b>assume </b><a NAME="E1:33_1_1"/>
( <font color="Maroon">R</font> is <a href="realset2.html#V3">trivial</a> &amp; <font color="Maroon">R</font> is <a href="orders_2.html#V1">strict</a> )
 ;<br/>

<b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:33_1_1"/><i><font color="Green">E31</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="group_6.html#D2">GROUP_6:def 2</a></i>;<br/>
<b>assume </b><a NAME="E4:33_1_1"/>
not <font color="Maroon">R</font> is <a href="neckla_2.html#V1">N-free</a>
 ;<br/>

<a NAME="E5:33_1_1"/><b>then </b>
<font color="Maroon">R</font> <a href="necklace.html#R2">embeds</a>  <a href="necklace.html#K4">Necklace</a> 4
 
<b>by </b><i><a class="ref" href="neckla_2.html#D1">NECKLA_2:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">f</font> being   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span>,<font color="Maroon">R</font><b> such that </b><br/><a NAME="E7:33_1_1"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">f</font> is <a href="funct_1.html#V2">one-to-one</a> &amp; ( for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span> holds <br/> ( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span> iff <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">y</font></span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> ) ) )
 <b>by </b><i><a class="ref" href="necklace.html#D2">NECKLACE:def 2</a></i>;<br/>
<i><font color="Green">E35</font></i>:  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> = 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
<br/>.= 
<span class="p1"><a href="enumset1.html#K2">{</a><span class="default">0,1,2,3</span><a href="enumset1.html#K2">}</a></span>
<b>by </b><i><a class="ref" href="necklace.html#T2">NECKLACE:2</a>, <a class="ref" href="necklace.html#T21">NECKLACE:21</a></i>
;<br/>
<a NAME="E9:33_1_1"/><b>then </b><i><font color="Green">E36</font></i>: 
0 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="enumset1.html#D2">ENUMSET1:def 2</a></i>;<br/>
<a NAME="E10:33_1_1"/><i><font color="Green">E39</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i>, <a class="ref" href="enumset1.html#D2">ENUMSET1:def 2</a></i>;<br/>
<a NAME="E11:33_1_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i>, , <a class="ref" href="partfun1.html#T27">PARTFUN1:27</a></i>;<br/>
<a NAME="E12:33_1_1"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i>, , <a class="ref" href="partfun1.html#T27">PARTFUN1:27</a></i>;<br/>
<a NAME="E13:33_1_1"/><b>then </b>
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> )
 
<b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E14:33_1_1"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/>


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