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<div class="add">

<a NAME="E1:8_1_1"/>
0 <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>1</sub></font> = <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default">0,0</span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span> as   <a href="relset_1.html#NM3">Relation</a> of <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span> <b>by </b><i><a class="ref" href="relset_1.html#T8">RELSET_1:8</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>2</sub></font> =  <a href="orders_2.html#G1">RelStr</a>(# <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">c<sub>1</sub></font> #) as  non <a href="struct_0.html#V3">empty</a>  <a href="orders_2.html#L1">RelStr</a>  ;<br/>
<b>take </b>
<font color="Maroon">c<sub>2</sub></font>
;<br/>

<b>thus </b><a NAME="E4:8_1_1"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="realset2.html#V3">trivial</a>
 ;<br/>

<b>thus </b><a NAME="E5:8_1_1"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="orders_2.html#V2">reflexive</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_0.html#D1">YELLOW_0:def 1</a><br/>

<a NAME="E1:8_1_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E2:8_1_1_1"/><b>then </b>
<span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="yellow_3.html#K7">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default">0,0</span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:8_1_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/>


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

<b>then reconsider </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>2</sub></font> as  non <a href="struct_0.html#V3">empty</a> <a href="realset2.html#V3">trivial</a> <a href="orders_2.html#V2">reflexive</a>  <a href="orders_2.html#L1">RelStr</a>  ;<br/>
<a NAME="E7:8_1_1"/>
<font color="Maroon">c<sub>3</sub></font> is <a href="orders_2.html#V3">transitive</a>
 
;<br/>
<b>hence </b><a NAME="E8:8_1_1"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="orders_2.html#V3">transitive</a>
 ;<br/>

<b>thus </b><a NAME="E9:8_1_1"/>
not <font color="Maroon">c<sub>2</sub></font> is <a href="struct_0.html#V3">empty</a>
 ;<br/>

<a NAME="E10:8_1_1"/>
<font color="Maroon">c<sub>3</sub></font> is <a href="orders_2.html#V4">antisymmetric</a>
 
;<br/>
<b>hence </b><a NAME="E11:8_1_1"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="orders_2.html#V4">antisymmetric</a>
 ;<br/>

<a NAME="E12:8_1_1"/>
<font color="Maroon">c<sub>3</sub></font> is <a href="lattice3.html#V1">with_suprema</a>
 
;<br/>
<b>hence </b><a NAME="E13:8_1_1"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="lattice3.html#V1">with_suprema</a>
 ;<br/>

<a NAME="E14:8_1_1"/>
<font color="Maroon">c<sub>3</sub></font> is <a href="lattice3.html#V2">with_infima</a>
 
;<br/>
<b>hence </b><a NAME="E15:8_1_1"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="lattice3.html#V2">with_infima</a>
 ;<br/>

<b>thus </b><a NAME="E16:8_1_1"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="group_1.html#V6">finite</a>
 <b>by </b><i><a class="ref" href="group_1.html#D14">GROUP_1:def 14</a></i>;<br/>

<b>thus </b><a NAME="E17:8_1_1"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="orders_2.html#V1">strict</a>
 ;<br/>


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