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<b>let </b><font color="Maroon">A</font> be  <a href="orders_2.html#V3" target="_self">transitive</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">a1</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font>, <font color="Maroon">a2</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font>, <font color="Maroon">a3</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font>;<br/>



<b>assume </b><a NAME="E1:19"/><i><font color="Green">E18</font></i>: 
( ( <font color="Maroon">a1</font> <a href="orders_2.html#R2" target="_self">&lt;</a> <font color="Maroon">a2</font> &amp; <font color="Maroon">a2</font> <a href="orders_2.html#R1" target="_self">&lt;=</a> <font color="Maroon">a3</font> ) or ( <font color="Maroon">a1</font> <a href="orders_2.html#R1" target="_self">&lt;=</a> <font color="Maroon">a2</font> &amp; <font color="Maroon">a2</font> <a href="orders_2.html#R2" target="_self">&lt;</a> <font color="Maroon">a3</font> ) )
 ;<br/>

<div><i><font color="Green">E19</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:19_1"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">a1</font> <a href="orders_2.html#R2" target="_self">&lt;</a> <font color="Maroon">a2</font> &amp; <font color="Maroon">a2</font> <a href="orders_2.html#R1" target="_self">&lt;=</a> <font color="Maroon">a3</font> )
 ;<br/><a NAME="E2:19_1"/><b>then </b><i><font color="Green">E40</font></i>: 
( <font color="Maroon">a1</font> <a href="orders_2.html#R1" target="_self">&lt;=</a> <font color="Maroon">a2</font> &amp; <font color="Maroon">a2</font> <a href="orders_2.html#R1" target="_self">&lt;=</a> <font color="Maroon">a3</font> )
 <b>by </b><i/>;<br/><a NAME="E3:19_1"/><b>then </b><i><font color="Green">E41</font></i>: 
<font color="Maroon">a1</font> <a href="orders_2.html#R1" target="_self">&lt;=</a> <font color="Maroon">a3</font>
 <b>by </b><i/>;<br/><a NAME="E4:19_1"/>
<font color="Maroon">a1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a3</font>
 <b>by </b><i><a class="ref" href="orders_2.html#D6" target="_self">Def6</a>, , </i>;<br/><b>hence </b><a NAME="E5:19_1"/>
<font color="Maroon">a1</font> <a href="orders_2.html#R2" target="_self">&lt;</a> <font color="Maroon">a3</font>
 <b>by </b><i><a class="txt" href="orders_2.html#E1:19"><i><font color="Green">E18</font></i></a>, </i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:19_2"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">a1</font> <a href="orders_2.html#R1" target="_self">&lt;=</a> <font color="Maroon">a2</font> &amp; <font color="Maroon">a2</font> <a href="orders_2.html#R2" target="_self">&lt;</a> <font color="Maroon">a3</font> )
 ;<br/><a NAME="E2:19_2"/><b>then </b><i><font color="Green">E43</font></i>: 
( <font color="Maroon">a1</font> <a href="orders_2.html#R1" target="_self">&lt;=</a> <font color="Maroon">a2</font> &amp; <font color="Maroon">a2</font> <a href="orders_2.html#R1" target="_self">&lt;=</a> <font color="Maroon">a3</font> )
 <b>by </b><i/>;<br/><a NAME="E3:19_2"/><b>then </b><i><font color="Green">E44</font></i>: 
<font color="Maroon">a1</font> <a href="orders_2.html#R1" target="_self">&lt;=</a> <font color="Maroon">a3</font>
 <b>by </b><i/>;<br/><a NAME="E4:19_2"/>
<font color="Maroon">a1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a3</font>
 <b>by </b><i><a class="ref" href="orders_2.html#D6" target="_self">Def6</a>, , </i>;<br/><b>hence </b><a NAME="E5:19_2"/>
<font color="Maroon">a1</font> <a href="orders_2.html#R2" target="_self">&lt;</a> <font color="Maroon">a3</font>
 <b>by </b><i><a class="txt" href="orders_2.html#E1:19"><i><font color="Green">E18</font></i></a>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:19"/>
<font color="Maroon">a1</font> <a href="orders_2.html#R2" target="_self">&lt;</a> <font color="Maroon">a3</font>
 <b>by </b><i><a class="ref" href="orders_2.html#D4" target="_self">Def4</a>, <a class="ref" href="orders_2.html#D5" target="_self">Def5</a></i>;<br/>


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