<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></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">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be   <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>assume </b><a NAME="E1:18"/><i><font color="Green">E14</font></i>: 
 <a href="waybel_0.html#G1">NetStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">c<sub>2</sub></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">c<sub>3</sub></font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>3</sub></font>,the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">c<sub>3</sub></font> #)
 ;<br/>

<a NAME="E2:18"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is    <a href="yellow_6.html#M2">subnet</a> of <font color="Maroon">c<sub>2</sub></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">b<sub>1</sub></font> being  <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font> st <br/>( the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#R2">=</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">c<sub>2</sub></font> <a href="partfun1.html#K1">*</a> <font color="Olive">b<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font>ex <font color="Olive">b<sub>3</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>3</sub></font> st <br/>for <font color="Olive">b<sub>4</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>3</sub></font> st <font color="Olive">b<sub>3</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">b<sub>4</sub></font> holds <br/><font color="Olive">b<sub>2</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> <a href="yellow_6.html#K3">.</a> <font color="Olive">b<sub>4</sub></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">c<sub>4</sub></font> being   <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E2:18_1"/><i><font color="Green">E16</font></i>: 
the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#R2">=</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">c<sub>2</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font>
 <b>and </b><br/><a NAME="E3:18_1"/><i><font color="Green">E17</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>2</sub></font>ex <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font> st <br/>for <font color="Olive">b<sub>3</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>2</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">b<sub>3</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="yellow_6.html#K3">.</a> <font color="Olive">b<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="waybel28.html#E2:18"><i><font color="Green">E15</font></i></a>, <a class="ref" href="yellow_6.html#D12">YELLOW_6:def 12</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="waybel28.html#E1:18"><i><font color="Green">E14</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>5</sub></font>
;<br/>

<b>thus </b><a NAME="E5:18_1"/>
the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#R2">=</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">c<sub>2</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="waybel28.html#E1:18"><i><font color="Green">E14</font></i></a>, <a class="txt" href="waybel28.html#E2:18_1"><i><font color="Green">E16</font></i></a></i>;<br/>

<b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>

<b>consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E7:18_1"/><i><font color="Green">E18</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>2</sub></font> st <font color="Maroon">c<sub>7</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>6</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="yellow_6.html#K3">.</a> <font color="Olive">b<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="waybel28.html#E3:18_1"><i><font color="Green">E17</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="txt" href="waybel28.html#E1:18"><i><font color="Green">E14</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>8</sub></font>
;<br/>

<b>let </b><font color="Maroon">c<sub>9</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>3</sub></font>;<br/>

<b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>9</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="waybel28.html#E1:18"><i><font color="Green">E14</font></i></a></i>;<br/>
<b>assume </b><a NAME="E10:18_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>9</sub></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">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></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">c<sub>3</sub></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">c<sub>7</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="txt" href="waybel28.html#E1:18"><i><font color="Green">E14</font></i></a>, <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">c<sub>6</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> <a href="yellow_6.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="waybel28.html#E7:18_1"><i><font color="Green">E18</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:18"/>
<font color="Maroon">c<sub>3</sub></font> is    <a href="yellow_6.html#M2">subnet</a> of <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="yellow_6.html#D12">YELLOW_6:def 12</a></i>;<br/>


</div>
