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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>;<br/><b>let </b><font color="Maroon">p</font> be  <a href="waybel28.html#V1" target="_self">greater_or_equal_to_id</a> <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">N</font>,<font color="Maroon">N</font>;<br/>





<a NAME="E1:20"/>
 ex <font color="Olive">f</font> being  <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><font color="Maroon">N</font> <a href="waybel28.html#K1" target="_self">*</a> <font color="Maroon">p</font></span>)</span>,<font color="Maroon">N</font> st <br/>( the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon">N</font> <a href="waybel28.html#K1" target="_self">*</a> <font color="Maroon">p</font></span>)</span> <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 <span class="p1">(<span class="default"><font color="Maroon">N</font> <a href="waybel28.html#K1" target="_self">*</a> <font color="Maroon">p</font></span>)</span> st <br/>for <font color="Olive">k</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">N</font> <a href="waybel28.html#K1" target="_self">*</a> <font color="Maroon">p</font></span>)</span>  st <font color="Olive">n</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">k</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">k</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1"><b>proof </b></a><div class="add">

<a NAME="E1:20_1"/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">N</font> <a href="waybel28.html#K1" target="_self">*</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">N</font>
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">f</font> = <font color="Maroon">p</font> as   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><font color="Maroon">N</font> <a href="waybel28.html#K1" target="_self">*</a> <font color="Maroon">p</font></span>)</span>,<font color="Maroon">N</font> ;<br/>
<b>take </b>
<font color="Maroon">f</font>
;<br/>

<b>thus </b><a NAME="E3:20_1"/>
the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon">N</font> <a href="waybel28.html#K1" target="_self">*</a> <font color="Maroon">p</font></span>)</span> <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>by </b><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>reconsider </b><font color="Maroon">n</font> = <font color="Maroon">m</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">N</font> <a href="waybel28.html#K1" target="_self">*</a> <font color="Maroon">p</font></span>)</span> <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">n</font>
;<br/>

<b>let </b><font color="Maroon">k</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">N</font> <a href="waybel28.html#K1" target="_self">*</a> <font color="Maroon">p</font></span>)</span>;<br/>

<b>assume </b><a NAME="E5:20_1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">n</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">k</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">k1</font> = <font color="Maroon">k</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> <b>by </b><i/>;<br/>
<a NAME="E7:20_1"/>
 <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">N</font> <a href="waybel28.html#K1" target="_self">*</a> <font color="Maroon">p</font></span>)</span>,the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><font color="Maroon">N</font> <a href="waybel28.html#K1" target="_self">*</a> <font color="Maroon">p</font></span>)</span> #) <a href="hidden.html#R1">=</a>  <a href="orders_2.html#G1">RelStr</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> #)
 
<b>by </b><i/>;<br/>
<a NAME="E8:20_1"/><b>then </b><i><font color="Green">E28</font></i>: 
<font color="Maroon">m</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">k1</font>
 
<b>by </b><i>, <a class="ref" href="yellow_0.html#T1">YELLOW_0:1</a></i>;<br/>
<a NAME="E9:20_1"/>
<font color="Maroon">k1</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">p</font> <a href="yellow_6.html#K3">.</a> <font color="Maroon">k1</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E10:20_1"/>
<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="Maroon">k</font>
 <b>by </b><i>, <a class="ref" href="yellow_0.html#D2">YELLOW_0:def 2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:20"/>
<font color="Maroon">N</font> <a href="waybel28.html#K1" target="_self">*</a> <font color="Maroon">p</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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