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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="orders_2.html#V2">reflexive</a>  <a href="oposet_1.html#L1">OrthoRelStr</a> ;<br/>

<b>assume </b><a NAME="E1:66_1_1"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="realset2.html#V3">trivial</a>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>2</sub></font> = <font color="Maroon">c<sub>1</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="oposet_1.html#L1">OrthoRelStr</a>  ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>3</sub></font> = the <a href="robbins1.html#U1">Compl</a> of <font color="Maroon">c<sub>2</sub></font> as   <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font> ;<br/>
<a NAME="E4:66_1_1"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> is <a href="realset1.html#V1">trivial</a>
 
<b>by </b><i><a class="ref" href="realset2.html#D5">REALSET2:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>4</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:66_1_1"/><i><font color="Green">E27</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="realset1.html#D4">REALSET1:def 4</a></i>;<br/>
<a NAME="E7:66_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#R1">=</a>  <a href="partfun1.html#K6">id</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="txt" href="robbins3.html#E6:66_1_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="funct_2.html#T66">FUNCT_2:66</a></i>;<br/>
<a NAME="E8:66_1_1"/><b>then </b><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="oposet_1.html#V2">dneg</a>
 
<b>by </b><i><a class="txt" href="robbins3.html#E6:66_1_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="oposet_1.html#T20">OPOSET_1:20</a></i>;<br/>
<a NAME="E9:66_1_1"/>
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 <font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">b<sub>2</sub></font> holds <br/><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>2</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>

<b>assume </b><a NAME="E1:66_1_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/>

<a NAME="E2:66_1_1_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="robbins3.html#E6:66_1_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="funct_2.html#T65">FUNCT_2:65</a></i>;<br/>
<a NAME="E3:66_1_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="robbins3.html#E6:66_1_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="funct_2.html#T65">FUNCT_2:65</a></i>;<br/>
<b>hence </b><a NAME="E4:66_1_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="robbins3.html#E2:66_1_1_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="yellow_0.html#D1">YELLOW_0:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:66_1_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> is <a href="waybel_0.html#V5">antitone</a>
 
<b>by </b><i><a class="ref" href="waybel_9.html#D1">WAYBEL_9:def 1</a></i>;<br/>
<a NAME="E11:66_1_1"/><b>then </b><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="oposet_1.html#V18">Orderinvolutive</a>
 
<b>by </b><i><a class="txt" href="robbins3.html#E8:66_1_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="oposet_1.html#D32">OPOSET_1:def 32</a></i>;<br/>
<a NAME="E12:66_1_1"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="oposet_1.html#V3">Dneg</a>
 
<b>by </b><i><a class="txt" href="robbins3.html#E8:66_1_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="oposet_1.html#D6">OPOSET_1:def 6</a></i>;<br/>
<b>hence </b><a NAME="E13:66_1_1"/>
( <font color="Maroon">c<sub>1</sub></font> is <a href="oposet_1.html#V19">OrderInvolutive</a> &amp; <font color="Maroon">c<sub>1</sub></font> is <a href="oposet_1.html#V15">Pure</a> &amp; <font color="Maroon">c<sub>1</sub></font> is <a href="oposet_1.html#V14">PartialOrdered</a> )
 <b>by </b><i><a class="txt" href="robbins3.html#E11:66_1_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="oposet_1.html#D26">OPOSET_1:def 26</a>, <a class="ref" href="oposet_1.html#D33">OPOSET_1:def 33</a></i>;<br/>


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