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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#NM1">Poset</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<a NAME="E1:78"/><i><font color="Green">E50</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K3">field</a> the <a href="orders_2.html#U1" target="_self">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="orders_1.html#T100">ORDERS_1:100</a></i>;<br/>
<b>thus </b><a NAME="E2:78"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="orders_1.html#R9">is_inferior_of</a> the <a href="orders_2.html#U1" target="_self">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> implies for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>2</sub></font> <a href="orders_2.html#R2" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="78_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:78_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="orders_1.html#R9">is_inferior_of</a> the <a href="orders_2.html#U1" target="_self">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<b>let </b><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>assume </b><a NAME="E2:78_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<a NAME="E3:78_1"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1" target="_self">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="orders_2.html#E1:78"><i><font color="Green">E50</font></i></a>, <a class="txt" href="orders_2.html#E1:78_1"><i><font color="Green">E51</font></i></a>, <a class="ref" href="orders_1.html#D14">ORDERS_1:def 14</a></i>;<br/>
<a NAME="E4:78_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="orders_2.html#R3" target="_self">&lt;=</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="orders_2.html#D9" target="_self">Def9</a></i>;<br/>
<b>hence </b><a NAME="E5:78_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>
 <b>by </b><i><a class="txt" href="orders_2.html#E2:78_1"><i><font color="Green">E52</font></i></a>, <a class="ref" href="orders_2.html#D10" target="_self">Def10</a></i>;<br/>


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

<b>assume </b><a NAME="E3:78"/><i><font color="Green">E51</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>2</sub></font> <a href="orders_2.html#R2" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font>
 ;<br/>

<b>thus </b><a NAME="E4:78"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> the <a href="orders_2.html#U1" target="_self">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="orders_2.html#E1:78"><i><font color="Green">E50</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="orders_1.html#D14">ORDERS_1:def 14</a><br/>

<b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E5:78"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> the <a href="orders_2.html#U1" target="_self">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="orders_1.html#T100">ORDERS_1:100</a></i>;<br/>
<b>assume </b><a NAME="E7:78"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E8:78"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="orders_2.html#R2" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="orders_2.html#E3:78"><i><font color="Green">E51</font></i></a></i>;<br/>
<a NAME="E9:78"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="orders_2.html#R3" target="_self">&lt;=</a> <font color="Maroon">c<sub>4</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="E10:78"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1" target="_self">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="orders_2.html#D9" target="_self">Def9</a></i>;<br/>


</div>
