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<b>let </b><font color="Maroon">L</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="struct_0.html#L1">1-sorted</a> ;<br/><b>let </b><font color="Maroon">N</font> be   <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">L</font>, <font color="Maroon">M</font> be   <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">L</font>;<br/>



<b>assume </b><a NAME="E1:18"/><i><font color="Green">E25</font></i>: 
 <a href="waybel_0.html#G1">NetStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">N</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">N</font>,the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">N</font> #) <a href="hidden.html#R1">=</a>  <a href="waybel_0.html#G1">NetStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">M</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">M</font>,the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">M</font> #)
 ;<br/>

<a NAME="E2:18"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">N</font> is    <a href="yellow_6.html#M2">subnet</a> of <font color="Maroon">N</font>
 
<b>by </b><i><a class="ref" href="yellow_6.html#T23">YELLOW_6:23</a></i>;<br/>
<a NAME="E3:18"/>
 ex <font color="Olive">f</font> being  <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">M</font>,<font color="Maroon">N</font> st <br/>( the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">M</font> <a href="funct_2.html#R2">=</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">N</font> <a href="partfun1.html#K1">*</a> <font color="Olive">f</font> &amp; ( for <font color="Olive">m</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font>  ex <font color="Olive">n</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font> st <br/>for <font color="Olive">p</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font>  st <font color="Olive">n</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">p</font> holds <br/><font color="Olive">m</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">f</font> <a href="yellow_6.html#K3">.</a> <font color="Olive">p</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">f</font> being   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">N</font>,<font color="Maroon">N</font><b> such that </b><br/><a NAME="E2:18_1"/><i><font color="Green">E29</font></i>: 
the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">N</font> <a href="funct_2.html#R2">=</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">N</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E3:18_1"/><i><font color="Green">E30</font></i>: 
for <font color="Olive">m</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font>  ex <font color="Olive">n</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> st <br/>for <font color="Olive">p</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font>  st <font color="Olive">n</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">p</font> holds <br/><font color="Olive">m</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">f</font> <a href="yellow_6.html#K3">.</a> <font color="Olive">p</font>
 <b>by </b><i>, <a class="ref" href="yellow_6.html#D12">YELLOW_6:def 12</a></i>;<br/>
<b>reconsider </b><font color="Maroon">f2</font> = <font color="Maroon">f</font> as   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">M</font>,<font color="Maroon">N</font> <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">f2</font>
;<br/>

<b>thus </b><a NAME="E5:18_1"/>
the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">M</font> <a href="funct_2.html#R2">=</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">N</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f2</font>
 <b>by </b><i>, </i>;<br/>

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

<b>consider </b><font color="Maroon">n</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font><b> such that </b><br/><a NAME="E7:18_1"/><i><font color="Green">E31</font></i>: 
for <font color="Olive">p</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font>  st <font color="Maroon">n</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">p</font> holds <br/><font color="Maroon">m</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">f</font> <a href="yellow_6.html#K3">.</a> <font color="Olive">p</font>
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">n2</font> = <font color="Maroon">n</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font> <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">n2</font>
;<br/>

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

<b>reconsider </b><font color="Maroon">p1</font> = <font color="Maroon">p</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> <b>by </b><i/>;<br/>
<b>assume </b><a NAME="E10:18_1"/>
<font color="Maroon">n2</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">p</font>
 ;<br/>

<a NAME="E11:18_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">n2</font>,<font color="Maroon">p</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">M</font>
 
<b>by </b><i><a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/>
<a NAME="E12:18_1"/><b>then </b>
<font color="Maroon">n</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">p1</font>
 
<b>by </b><i>, <a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/>
<b>hence </b><a NAME="E13:18_1"/>
<font color="Maroon">m</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">f2</font> <a href="yellow_6.html#K3">.</a> <font color="Maroon">p</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:18"/>
<font color="Maroon">M</font> is    <a href="yellow_6.html#M2">subnet</a> of <font color="Maroon">N</font>
 <b>by </b><i><a class="ref" href="yellow_6.html#D12">YELLOW_6:def 12</a></i>;<br/>


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