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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="orders_2.html#V2" target="_self">reflexive</a> <a href="orders_2.html#V4" target="_self">antisymmetric</a>  <a href="orders_2.html#L1" target="_self">RelStr</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>thus </b><a NAME="E1:28"/>
( ex <font color="Olive">b<sub>1</sub></font> being  <a href="orders_2.html#NM2">Chain</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> ) implies ( <font color="Maroon">c<sub>2</sub></font> <a href="orders_2.html#R2" target="_self">&lt;</a> <font color="Maroon">c<sub>3</sub></font> iff not <font color="Maroon">c<sub>3</sub></font> <a href="orders_2.html#R1" target="_self">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1"><b>proof </b></a><div class="add">

<b>given </b><font color="Maroon">c<sub>4</sub></font> being   <a href="orders_2.html#NM2">Chain</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><a NAME="E1:28_1"/><i><font color="Green">E19</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> )
 ;<br/>

<b>thus </b><a NAME="E2:28_1"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="orders_2.html#R2" target="_self">&lt;</a> <font color="Maroon">c<sub>3</sub></font> implies  not <font color="Maroon">c<sub>3</sub></font> <a href="orders_2.html#R1" target="_self">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="ref" href="orders_2.html#T30" target="_self">Th30</a></i>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E3:28_1"/><i><font color="Green">E20</font></i>: 
not <font color="Maroon">c<sub>3</sub></font> <a href="orders_2.html#R1" target="_self">&lt;=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E4:28_1"/><i><font color="Green">E21</font></i>: 
not <font color="Maroon">c<sub>2</sub></font> <a href="orders_2.html#R2" target="_self">&lt;</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<a NAME="E5:28_1"/>
(  not <font color="Maroon">c<sub>2</sub></font> <a href="orders_2.html#R1" target="_self">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> or <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="txt" href="orders_2.html#E4:28_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="orders_2.html#D10" target="_self">Def10</a></i>;<br/>
<b>hence </b><a NAME="E6:28_1"/>
contradiction
 <b>by </b><i><a class="txt" href="orders_2.html#E1:28_1"><i><font color="Green">E19</font></i></a>, <a class="txt" href="orders_2.html#E3:28_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="orders_2.html#T38" target="_self">Th38</a></i>;<br/>


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

<b>assume </b><a NAME="E2:28"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="orders_2.html#R2" target="_self">&lt;</a> <font color="Maroon">c<sub>3</sub></font> iff not <font color="Maroon">c<sub>3</sub></font> <a href="orders_2.html#R1" target="_self">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> )
 ;<br/>

<a NAME="E3:28"/><b>then </b>
( <font color="Maroon">c<sub>2</sub></font> <a href="orders_2.html#R1" target="_self">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> or <font color="Maroon">c<sub>3</sub></font> <a href="orders_2.html#R1" target="_self">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="ref" href="orders_2.html#D10" target="_self">Def10</a></i>;<br/>
<b>hence </b><a NAME="E4:28"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="orders_2.html#NM2">Chain</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> )
 <b>by </b><i><a class="ref" href="orders_2.html#T38" target="_self">Th38</a></i>;<br/>


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