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<b>let </b><font color="Maroon" title="c1">A</font> be  <a href="orders_2.html#V3" title="ORDERS_2:attr.3">reflexive</a> <a href="orders_2.html#V5" title="ORDERS_2:attr.5">antisymmetric</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> ;<br/><b>let </b><font color="Maroon" title="c2">a1</font>, <font color="Maroon" title="c3">a2</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">A</font>;<br/>



<b>thus </b><a NAME="E1:34"/>
(  ex <font color="Olive" title="b1">C</font> being  <a href="orders_2.html#NM2" title="ORDERS_2:NM.2">Chain</a> of <font color="Maroon" title="c1">A</font> st <br/>( <font color="Maroon" title="c2">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> &amp; <font color="Maroon" title="c3">a2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> ) implies ( <font color="Maroon" title="c2">a1</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Maroon" title="c3">a2</font> iff not <font color="Maroon" title="c3">a2</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">a1</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1"><b>proof </b></a><div class="add">

<b>given </b><font color="Maroon" title="c4">C</font> being   <a href="orders_2.html#NM2" title="ORDERS_2:NM.2">Chain</a> of <font color="Maroon" title="c1">A</font><b> such that </b><a NAME="E2:34_1"/><i><font color="Green" title="E20">A1</font></i>: 
( <font color="Maroon" title="c2">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font> &amp; <font color="Maroon" title="c3">a2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font> )
 ;<br/>

<b>thus </b><a NAME="E3:34_1"/>
( <font color="Maroon" title="c2">a1</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Maroon" title="c3">a2</font> implies  not <font color="Maroon" title="c3">a2</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">a1</font> )
 <b>by </b><i><a class="ref" href="orders_2.html#T30" target="_self" title="ORDERS_2:th.30">Th30</a></i>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E4:34_1"/><i><font color="Green" title="E21">A2</font></i>: 
not <font color="Maroon" title="c3">a2</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">a1</font>
 <b>and </b><br/><a NAME="E5:34_1"/><i><font color="Green" title="E22">A3</font></i>: 
not <font color="Maroon" title="c2">a1</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Maroon" title="c3">a2</font>
 ;<br/>

<a NAME="E6:34_1"/>
(  not <font color="Maroon" title="c2">a1</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c3">a2</font> or <font color="Maroon" title="c2">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a2</font> )
 
<b>by </b><i><a class="txt" href="orders_2.html#E5:34_1"><i><font color="Green" title="E22">A3</font></i></a>, <a class="ref" href="orders_2.html#D10" target="_self" title="ORDERS_2:def.10">Def10</a></i>;<br/>
<b>hence </b><a NAME="E7:34_1"/>
contradiction
 <b>by </b><i><a class="txt" href="orders_2.html#E2:34_1"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="orders_2.html#E4:34_1"><i><font color="Green" title="E21">A2</font></i></a>, <a class="ref" href="orders_2.html#T38" target="_self" title="ORDERS_2:th.38">Th38</a></i>;<br/>


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

<b>assume </b><a NAME="E2:34"/>
( <font color="Maroon" title="c2">a1</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Maroon" title="c3">a2</font> iff not <font color="Maroon" title="c3">a2</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">a1</font> )
 ;<br/>

<a NAME="E3:34"/><b>then </b>
( <font color="Maroon" title="c2">a1</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c3">a2</font> or <font color="Maroon" title="c3">a2</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">a1</font> )
 
<b>by </b><i><a class="ref" href="orders_2.html#D10" target="_self" title="ORDERS_2:def.10">Def10</a></i>;<br/>
<b>hence </b><a NAME="E4:34"/>
 ex <font color="Olive" title="b1">C</font> being  <a href="orders_2.html#NM2" title="ORDERS_2:NM.2">Chain</a> of <font color="Maroon" title="c1">A</font> st <br/>( <font color="Maroon" title="c2">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> &amp; <font color="Maroon" title="c3">a2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> )
 <b>by </b><i><a class="ref" href="orders_2.html#T38" target="_self" title="ORDERS_2:th.38">Th38</a></i>;<br/>


</div>
