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<b>let </b><font color="Maroon">N</font> be  non <a href="struct_0.html#V3">empty</a> <a href="waybel_0.html#V7">directed</a>  <a href="orders_2.html#L1">RelStr</a> ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b>for <font color="Olive">n</font>, <font color="Olive">m</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font>  st <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">n</font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">m</font> holds <br/><font color="Olive">n</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">m</font>;<br/>
<a NAME="E1:7"/><i><font color="Green">E25</font></i>: 
for <font color="Olive">e</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">e</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">N</font> holds <br/> ex <font color="Olive">u</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">u</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">N</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">e</font>,<font color="Olive">u</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:7_1"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">N</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">e'</font> = <font color="Maroon">e</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> ;<br/>
<b>consider </b><font color="Maroon">u'</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font><b> such that </b><br/><a NAME="E4:7_1"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">e'</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">u'</font> &amp; <font color="Maroon">e'</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">u'</font> )
 <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">u'</font>
;<br/>

<b>thus </b><a NAME="E5:7_1"/>
<font color="Maroon">u'</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">N</font>
 ;<br/>

<b>let </b><font color="Maroon">n</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font>, <font color="Maroon">m</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font>;<br/>

<b>assume </b><a NAME="E6:7_1"/>
( <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">u'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> )
 ;<br/>

<b>hence </b><a NAME="E7:7_1"/>
<font color="Maroon">n</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">m</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">p</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E3:7"/><i><font color="Green">E29</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">N</font>
 <b>and </b><br/><a NAME="E4:7"/><i><font color="Green">E30</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">N</font>
 <b>and </b><br/><a NAME="E5:7"/><i><font color="Green">E31</font></i>: 
for <font color="Olive">e</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">e</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">N</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">e</font>,<font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Olive">e</font>]
 <b>from </b><i><a class="ref" href="wellord2.html#S1">WELLORD2:sch 1</a>();<br/></i>
<b>reconsider </b><font color="Maroon">p</font> = <font color="Maroon">p</font> as   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">N</font>,<font color="Maroon">N</font> <b>by </b><i>, , <a class="ref" href="funct_2.html#T4">FUNCT_2:4</a></i>;<br/>
<b>take </b>
<font color="Maroon">p</font>
;<br/>

<b>let </b><font color="Maroon">j</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel28.html#D1" target="_self">WAYBEL28:def 1</a><br/>

<b>reconsider </b><font color="Maroon">j'</font> = <font color="Maroon">j</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> ;<br/>
<b>thus </b><a NAME="E8:7"/>
<font color="Maroon">j</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">p</font> <a href="yellow_6.html#K3">.</a> <font color="Maroon">j</font>
 <b>by </b><i/>;<br/>


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