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<b>let </b><font color="Maroon">L</font> be  non <a href="struct_0.html#V3">empty</a> <a href="orders_2.html#V2">reflexive</a>  <a href="orders_2.html#L1">RelStr</a> ;<br/>

<b>assume </b><a NAME="E1:4_2_1"/><i><font color="Green">E24</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> holds  <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">y</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="realset2.html#D7">REALSET2:def 7</a><br/>

<b>consider </b><font color="Maroon">x</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/>
<b>thus </b><a NAME="E3:4_2_1"/>
<font color="Maroon">L</font> is <a href="lattice3.html#V3">complete</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D12">LATTICE3:def 12</a><br/>

<b>take </b>
<font color="Maroon">x</font>
;<br/>

<b>thus </b><a NAME="E1:4_2_1_1"/>
for <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>  st <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> holds <br/><font color="Olive">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">x</font>
 <b>by </b><i/>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D9">LATTICE3:def 9</a><br/>

<b>let </b><font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/>

<a NAME="E2:4_2_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:4_2_1_1"/>
(  not <font color="Maroon">X</font> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Maroon">y</font> or <font color="Maroon">x</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">y</font> )
 <b>by </b><i><a class="ref" href="orders_2.html#T24">ORDERS_2:24</a></i>;<br/>


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

<b>thus </b><a NAME="E4:4_2_1"/>
<font color="Maroon">L</font> is <a href="orders_2.html#V3">transitive</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_2_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">z</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_0.html#D2" target="_self">YELLOW_0:def 2</a><br/>

<b>thus </b><a NAME="E1:4_2_1_2"/>
( <font color="Maroon">x</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">y</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">z</font> implies <font color="Maroon">x</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">z</font> )
 <b>by </b><i/>;<br/>


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

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_0.html#D3" target="_self">YELLOW_0:def 3</a><br/>

<b>thus </b><a NAME="E5:4_2_1"/>
for <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>  st <font color="Maroon">x</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">y</font> &amp; <font color="Olive">y</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">x</font> holds <br/><font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">y</font>
 <b>by </b><i/>;<br/>


</div>
