<?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="orders_2.html#L1">RelStr</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="yellow_6.html#V2">full</a>  <a href="yellow_6.html#M1">SubNetStr</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>





<b>assume </b><a NAME="E1:49"/><i><font color="Green">E29</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/>( <font color="Olive">b<sub>2</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font> )
 ;<br/>

<a NAME="E2:49"/><i><font color="Green">E30</font></i>: 
( the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">c<sub>2</sub></font> <a href="partfun1.html#K2">|</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> is  <a href="yellow_0.html#V4">full</a>  <a href="yellow_0.html#M1">SubRelStr</a> of <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="ref" href="yellow_6.html#D8">YELLOW_6:def 8</a>, <a class="ref" href="yellow_6.html#D9">YELLOW_6:def 9</a></i>;<br/>
<a NAME="E3:49"/><b>then </b><i><font color="Green">E31</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#D13">YELLOW_0:def 13</a></i>;<br/>
<a NAME="E4:49"/>
<font color="Maroon">c<sub>3</sub></font> is <a href="waybel_0.html#V7">directed</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>3</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_6.html#D5">YELLOW_6:def 5</a><br/>

<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>5</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="waybel21.html#E2:49"><i><font color="Green">E30</font></i></a>, <a class="ref" href="yellow_0.html#T59">YELLOW_0:59</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>8</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="E3:49_1"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="ref" href="yellow_6.html#D5">YELLOW_6:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>9</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="E5:49_1"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="waybel21.html#E1:49"><i><font color="Green">E29</font></i></a></i>;<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>3</sub></font> <b>by </b><i><a class="txt" href="waybel21.html#E5:49_1"><i><font color="Green">E33</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>10</sub></font>
;<br/>

<a NAME="E7:49_1"/>
( <font color="Maroon">c<sub>9</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>7</sub></font> )
 
<b>by </b><i><a class="txt" href="waybel21.html#E3:49_1"><i><font color="Green">E32</font></i></a>, <a class="txt" href="waybel21.html#E5:49_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="yellow_0.html#D2">YELLOW_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E8:49_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>10</sub></font> )
 <b>by </b><i><a class="ref" href="yellow_6.html#T21">YELLOW_6:21</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> as   <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="waybel21.html#E2:49"><i><font color="Green">E30</font></i></a></i>;<br/>
<div><i><font color="Green">E32</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="49_2"><b>now </b></a><div class="add"><b>set </b><font color="Maroon">c<sub>5</sub></font> =  <a href="yellow_9.html#K1">incl</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/><a NAME="E1:49_2"/><i><font color="Green">E33</font></i>: 
 <a href="yellow_9.html#K1">incl</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="waybel21.html#E3:49"><i><font color="Green">E31</font></i></a>, <a class="ref" href="yellow_9.html#D1">YELLOW_9:def 1</a></i>;<br/><b>hence </b><a NAME="E2:49_2"/>
the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</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> <span class="p1">(<span class="default"><a href="yellow_9.html#K1">incl</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel21.html#E2:49"><i><font color="Green">E30</font></i></a>, <a class="ref" href="relat_1.html#T94">RELAT_1:94</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="E4:49_2"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="waybel21.html#E1:49"><i><font color="Green">E29</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>4</sub></font> <b>by </b><i><a class="txt" href="waybel21.html#E4:49_2"><i><font color="Green">E34</font></i></a></i>;<br/><b>take </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>8</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>10</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>4</sub></font>;<br/><b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>10</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="waybel21.html#E2:49"><i><font color="Green">E30</font></i></a>, <a class="ref" href="yellow_0.html#T59">YELLOW_0:59</a></i>;<br/><b>assume </b><a NAME="E7:49_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/><a NAME="E8:49_2"/><b>then </b>
( <font color="Maroon">c<sub>7</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <span class="p1">(<span class="default"><a href="yellow_9.html#K1">incl</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="yellow_6.html#K3">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> )
 <b>by </b><i><a class="txt" href="waybel21.html#E1:49_2"><i><font color="Green">E33</font></i></a>, <a class="ref" href="funct_1.html#T35">FUNCT_1:35</a>, <a class="ref" href="yellow_6.html#T20">YELLOW_6:20</a></i>;<br/><b>hence </b><a NAME="E9:49_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="orders_2.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="yellow_9.html#K1">incl</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="yellow_6.html#K3">.</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="waybel21.html#E4:49_2"><i><font color="Green">E34</font></i></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="E7:49"/>
<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/>

<b>hence </b><a NAME="E8:49"/>
 <a href="yellow_9.html#K1">incl</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font> is    <a href="waybel21.html#M2" target="_self">Embedding</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="waybel21.html#E6:49"><i><font color="Green">E32</font></i></a>, <a class="ref" href="waybel21.html#D3" target="_self">Def3</a></i>;<br/>


</div>
