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<b>set </b><font color="Maroon">c<sub>1</sub></font> = <span class="p1"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span>;<br/>
<b>set </b><font color="Maroon">c<sub>2</sub></font> =  <a href="necklace.html#K4">Necklace</a> 4;<br/>
<b>reconsider </b><font color="Maroon">c<sub>3</sub></font> =  <a href="relat_1.html#K6">id</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span> as   <a href="relset_1.html#NM3">Relation</a> of <span class="p1"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span> <b>by </b><i><a class="ref" href="relset_1.html#T29">RELSET_1:29</a></i>;<br/>
<b>take </b><font color="Maroon">c<sub>4</sub></font> =  <a href="orders_2.html#G1">RelStr</a>(# <span class="p1"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span>,<font color="Maroon">c<sub>3</sub></font> #);<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="struct_0.html#NM7">Function</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span>,<font color="Maroon">c<sub>4</sub></font>;<br/><b>assume </b><a NAME="E1:8_1_1_1"/><i><font color="Green">E5</font></i>: 
<font color="Maroon">c<sub>5</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 ;<br/><i><font color="Green">E6</font></i>:  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>5</sub></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="E3:8_1_1_1"/><b>then </b><i><font color="Green">E7</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="enumset1.html#D2">ENUMSET1:def 2</a></i>;<br/><a NAME="E4:8_1_1_1"/><i><font color="Green">E8</font></i>: 
2 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="neckla_2.html#E2:8_1_1_1"><i><font color="Green">E6</font></i></a>, <a class="ref" href="enumset1.html#D2">ENUMSET1:def 2</a></i>;<br/><a NAME="E5:8_1_1_1"/><i><font color="Green">E9</font></i>: 
3 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="neckla_2.html#E2:8_1_1_1"><i><font color="Green">E6</font></i></a>, <a class="ref" href="enumset1.html#D2">ENUMSET1:def 2</a></i>;<br/><a NAME="E6:8_1_1_1"/><b>then </b><i><font color="Green">E10</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> 3
 <b>by </b><i><a class="txt" href="neckla_2.html#E1:8_1_1_1"><i><font color="Green">E5</font></i></a>, <a class="txt" href="neckla_2.html#E3:8_1_1_1"><i><font color="Green">E7</font></i></a>, <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/><a NAME="E7:8_1_1_1"/><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> 3
 <b>by </b><i><a class="txt" href="neckla_2.html#E1:8_1_1_1"><i><font color="Green">E5</font></i></a>, <a class="txt" href="neckla_2.html#E4:8_1_1_1"><i><font color="Green">E8</font></i></a>, <a class="txt" href="neckla_2.html#E5:8_1_1_1"><i><font color="Green">E9</font></i></a>, <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/><a NAME="E8:8_1_1_1"/><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span>
 <b>by </b><i><a class="txt" href="neckla_2.html#E3:8_1_1_1"><i><font color="Green">E7</font></i></a>, <a class="ref" href="partfun1.html#T27">PARTFUN1:27</a></i>;<br/><a NAME="E9:8_1_1_1"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span>
 <b>by </b><i><a class="txt" href="neckla_2.html#E4:8_1_1_1"><i><font color="Green">E8</font></i></a>, <a class="ref" href="partfun1.html#T27">PARTFUN1:27</a></i>;<br/><a NAME="E10:8_1_1_1"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> 3 <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span>
 <b>by </b><i><a class="txt" href="neckla_2.html#E5:8_1_1_1"><i><font color="Green">E9</font></i></a>, <a class="ref" href="partfun1.html#T27">PARTFUN1:27</a></i>;<br/><a NAME="E11:8_1_1_1"/><i><font color="Green">E15</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i><a class="txt" href="neckla_2.html#E8:8_1_1_1"><i><font color="Green">E12</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><a NAME="E12:8_1_1_1"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i><a class="txt" href="neckla_2.html#E9:8_1_1_1"><i><font color="Green">E13</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><b>hence </b><a NAME="E13:8_1_1_1"/>
not for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></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="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</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 <span class="p1">(<span class="default"><a href="necklace.html#K4">Necklace</a> 4</span>)</span> iff <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span></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>4</sub></font> )
 <b>by </b><i><a class="txt" href="neckla_2.html#E1:8_1_1_1"><i><font color="Green">E5</font></i></a>, <a class="txt" href="neckla_2.html#E3:8_1_1_1"><i><font color="Green">E7</font></i></a>, <a class="txt" href="neckla_2.html#E4:8_1_1_1"><i><font color="Green">E8</font></i></a>, <a class="txt" href="neckla_2.html#E6:8_1_1_1"><i><font color="Green">E10</font></i></a>, <a class="txt" href="neckla_2.html#E7:8_1_1_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="neckla_2.html#E10:8_1_1_1"><i><font color="Green">E14</font></i></a>, <a class="txt" href="neckla_2.html#E11:8_1_1_1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/></div><b>end;</b></div>
<a NAME="E3:8_1_1"/><b>then </b>
not <font color="Maroon">c<sub>4</sub></font> <a href="necklace.html#R2">embeds</a>  <a href="necklace.html#K4">Necklace</a> 4
 
<b>by </b><i><a class="ref" href="necklace.html#D2">NECKLACE:def 2</a></i>;<br/>
<b>hence </b><a NAME="E4:8_1_1"/>
( <font color="Maroon">c<sub>4</sub></font> is <a href="orders_2.html#V1">strict</a> &amp; <font color="Maroon">c<sub>4</sub></font> is <a href="group_1.html#V6">finite</a> &amp; <font color="Maroon">c<sub>4</sub></font> is <a href="neckla_2.html#V1" target="_self">N-free</a> )
 <b>by </b><i><a class="ref" href="neckla_2.html#D1" target="_self">Def1</a>, <a class="ref" href="group_1.html#D14">GROUP_1:def 14</a></i>;<br/>


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