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





<b>assume </b><a NAME="E1:67"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">N</font> <a href="waybel_0.html#R2">is_often_in</a> <font color="Maroon">X</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">M</font> = <font color="Maroon">N</font> <a href="yellow_6.html#K6" target="_self">"</a> <font color="Maroon">X</font> as   <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">S</font> <b>by </b><i/>;<br/>
<a NAME="E3:67"/>
<font color="Maroon">M</font> is    <a href="yellow_6.html#M2" target="_self">subnet</a> of <font color="Maroon">N</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">f</font> =  <a href="grcat_1.html#K7">id</a> <font color="Maroon">M</font>;<br/>
<a NAME="E1:67_1"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">M</font> <a href="tarski.html#R1">c=</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> =  <a href="grcat_1.html#K7">id</a> <font color="Maroon">M</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><a class="ref" href="funct_2.html#T9">FUNCT_2:9</a></i>;<br/>
<b>take </b>
<font color="Maroon">f</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_6.html#D12" target="_self">YELLOW_6:def 12</a><br/>

<a NAME="E3:67_1"/><i><font color="Green">E32</font></i>: 
 <a href="grcat_1.html#K7">id</a> <font color="Maroon">M</font> <a href="funct_2.html#R2">=</a>  <a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">M</font>
 
<b>by </b><i><a class="ref" href="grcat_1.html#D11">GRCAT_1:def 11</a></i>;<br/>
<a NAME="E4:67_1"/>
the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">M</font> <a href="hidden.html#R1">=</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">N</font> <a href="partfun1.html#K2">|</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">M</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E5:67_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">f</font>
 <b>by </b><i>, <a class="ref" href="relat_1.html#T94">RELAT_1:94</a></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">j</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font><b> such that </b><br/><a NAME="E7:67_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">m</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">j</font>
 <b>and </b><br/><a NAME="E8:67_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i>, <a class="ref" href="waybel_0.html#D12">WAYBEL_0:def 12</a></i>;<br/>
<a NAME="E9:67_1"/>
the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">N</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="txt" href="yellow_6.html#E7:67_1"><i><font color="Green">E41</font></i></a></i>;<br/>
<a NAME="E10:67_1"/><b>then </b>
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">N</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T46">FUNCT_2:46</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">n</font> = <font color="Maroon">j</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">n</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>assume </b><a NAME="E12:67_1"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">n</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">p</font>
 ;<br/>

<a NAME="E13:67_1"/>
<font color="Maroon">f</font> <a href="yellow_6.html#K3" target="_self">.</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font>
 
<b>by </b><i><a class="ref" href="tmap_1.html#T91">TMAP_1:91</a></i>;<br/>
<a NAME="E14:67_1"/><b>then </b>
<font color="Maroon">j</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">f</font> <a href="yellow_6.html#K3" target="_self">.</a> <font color="Maroon">p</font>
 
<b>by </b><i><a class="txt" href="yellow_6.html#E8:67_1"><i><font color="Green">E42</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E15:67_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" target="_self">.</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="ref" href="yellow_6.html#T5" target="_self">Th5</a>, <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="E4:67"/>
<font color="Maroon">N</font> <a href="yellow_6.html#K6" target="_self">"</a> <font color="Maroon">X</font> is    <a href="yellow_6.html#M2" target="_self">subnet</a> of <font color="Maroon">N</font>
 ;<br/>


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