<?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> be    <a href="yellow_6.html#M4" target="_self">Convergence-Class</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>set </b><font color="Maroon">c<sub>3</sub></font> =  <a href="yellow_6.html#K14" target="_self">ConvergenceSpace</a> <font color="Maroon">c<sub>2</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>4</sub></font> =  <a href="yellow_6.html#K13" target="_self">Convergence</a> <span class="p1">(<span class="default"><a href="yellow_6.html#K14" target="_self">ConvergenceSpace</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>;<br/>
<a NAME="E1:121"/><i><font color="Green">E73</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K14" target="_self">ConvergenceSpace</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="yellow_6.html#D27" target="_self">Def27</a></i>;<br/>
<a NAME="E2:121"/><i><font color="Green">E74</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Olive">b<sub>1</sub></font> is   <a href="waybel_0.html#NM2">net</a> of  <a href="yellow_6.html#K14" target="_self">ConvergenceSpace</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="yellow_6.html#D27" target="_self">Def27</a></i>;<br/>
<a NAME="E3:121"/><i><font color="Green">E75</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">c<sub>1</sub></font><br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="waybel_0.html#NM2">net</a> of  <a href="yellow_6.html#K14" target="_self">ConvergenceSpace</a> <font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> holds <br/>for <font color="Olive">b<sub>3</sub></font> being   <a href="yellow_6.html#M2" target="_self">subnet</a> of <font color="Olive">b<sub>1</sub></font> holds <font color="Olive">b<sub>3</sub></font> is    <a href="yellow_6.html#M2" target="_self">subnet</a> of <font color="Olive">b<sub>2</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="121_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</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>6</sub></font> be   <a href="waybel_0.html#NM2">net</a> of  <a href="yellow_6.html#K14" target="_self">ConvergenceSpace</a> <font color="Maroon">c<sub>2</sub></font>;<br/>



<b>assume </b><a NAME="E1:121_1"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>7</sub></font> be    <a href="yellow_6.html#M2" target="_self">subnet</a> of <font color="Maroon">c<sub>5</sub></font>;<br/>

<b>consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>5</sub></font><b> such that </b><br/><a NAME="E3:121_1"/><i><font color="Green">E77</font></i>: 
( the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#R2">=</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">c<sub>5</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>8</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>5</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>7</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>7</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>8</sub></font> <a href="yellow_6.html#K3" target="_self">.</a> <font color="Olive">b<sub>3</sub></font> ) )
 <b>by </b><i><a class="ref" href="yellow_6.html#D12" target="_self">Def12</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>7</sub></font> as   <a href="waybel_0.html#NM2">net</a> of  <a href="yellow_6.html#K14" target="_self">ConvergenceSpace</a> <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="yellow_6.html#D27" target="_self">Def27</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>8</sub></font> as   <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>6</sub></font> <b>by </b><i><a class="txt" href="yellow_6.html#E1:121_1"><i><font color="Green">E76</font></i></a></i>;<br/>
<a NAME="E6:121_1"/>
the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">c<sub>9</sub></font> <a href="funct_2.html#R2">=</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">c<sub>6</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="txt" href="yellow_6.html#E1:121_1"><i><font color="Green">E76</font></i></a>, <a class="txt" href="yellow_6.html#E3:121_1"><i><font color="Green">E77</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:121_1"/>
<font color="Maroon">c<sub>7</sub></font> is    <a href="yellow_6.html#M2" target="_self">subnet</a> of <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="yellow_6.html#E1:121_1"><i><font color="Green">E76</font></i></a>, <a class="txt" href="yellow_6.html#E3:121_1"><i><font color="Green">E77</font></i></a>, <a class="ref" href="yellow_6.html#D12" target="_self">Def12</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:121"/><i><font color="Green">E76</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">c<sub>1</sub></font><br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="waybel_0.html#NM2">net</a> of  <a href="yellow_6.html#K14" target="_self">ConvergenceSpace</a> <font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> holds <br/>for <font color="Olive">b<sub>3</sub></font> being   <a href="yellow_6.html#M2" target="_self">subnet</a> of <font color="Olive">b<sub>2</sub></font> holds <font color="Olive">b<sub>3</sub></font> is    <a href="yellow_6.html#M2" target="_self">subnet</a> of <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="121_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</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>6</sub></font> be   <a href="waybel_0.html#NM2">net</a> of  <a href="yellow_6.html#K14" target="_self">ConvergenceSpace</a> <font color="Maroon">c<sub>2</sub></font>;<br/>



<b>assume </b><a NAME="E1:121_2"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>7</sub></font> be    <a href="yellow_6.html#M2" target="_self">subnet</a> of <font color="Maroon">c<sub>6</sub></font>;<br/>

<b>consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>6</sub></font><b> such that </b><br/><a NAME="E3:121_2"/><i><font color="Green">E78</font></i>: 
( the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#R2">=</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">c<sub>6</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>8</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>6</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>7</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>7</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>8</sub></font> <a href="yellow_6.html#K3" target="_self">.</a> <font color="Olive">b<sub>3</sub></font> ) )
 <b>by </b><i><a class="ref" href="yellow_6.html#D12" target="_self">Def12</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>7</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="ref" href="yellow_6.html#D27" target="_self">Def27</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>8</sub></font> as   <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>5</sub></font> <b>by </b><i><a class="txt" href="yellow_6.html#E1:121_2"><i><font color="Green">E77</font></i></a></i>;<br/>
<a NAME="E6:121_2"/>
the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">c<sub>9</sub></font> <a href="funct_2.html#R2">=</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">c<sub>5</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="txt" href="yellow_6.html#E1:121_2"><i><font color="Green">E77</font></i></a>, <a class="txt" href="yellow_6.html#E3:121_2"><i><font color="Green">E78</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:121_2"/>
<font color="Maroon">c<sub>7</sub></font> is    <a href="yellow_6.html#M2" target="_self">subnet</a> of <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="yellow_6.html#E1:121_2"><i><font color="Green">E77</font></i></a>, <a class="txt" href="yellow_6.html#E3:121_2"><i><font color="Green">E78</font></i></a>, <a class="ref" href="yellow_6.html#D12" target="_self">Def12</a></i>;<br/>


</div><b>end;</b></div>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:121_3"/><i><font color="Green">E77</font></i>: 
 <a href="yellow_6.html#K13" target="_self">Convergence</a> <span class="p1">(<span class="default"><a href="yellow_6.html#K14" target="_self">ConvergenceSpace</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><b>thus </b><a NAME="E2:121_3"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="yellow_6.html#V8" target="_self">topological</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="121_3_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:121_3_1"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="yellow_6.html#V4" target="_self">(CONSTANTS)</a>
  <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_6.html#D28" target="_self">YELLOW_6:def 28</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="121_3_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font> be  <a href="yellow_6.html#V1" target="_self">constant</a> <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">c<sub>1</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_6.html#D23" target="_self">YELLOW_6:def 23</a><br/>

<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> as   <a href="waybel_0.html#NM2">net</a> of  <a href="yellow_6.html#K14" target="_self">ConvergenceSpace</a> <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="yellow_6.html#D27" target="_self">Def27</a></i>;<br/>
<a NAME="E2:121_3_1_1"/>
the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">c<sub>5</sub></font> is <a href="funct_1.html#V3">constant</a>
 
;<br/>
<a NAME="E3:121_3_1_1"/><b>then </b>
the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">c<sub>6</sub></font> is <a href="funct_1.html#V3">constant</a>
 
;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font> as  <a href="yellow_6.html#V1" target="_self">constant</a> <a href="waybel_0.html#NM2">net</a> of  <a href="yellow_6.html#K14" target="_self">ConvergenceSpace</a> <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="yellow_6.html#D6" target="_self">Def6</a></i>;<br/>
<i><font color="Green">E78</font></i>:  <a href="yellow_6.html#K5" target="_self">the_value_of</a> <font color="Maroon">c<sub>7</sub></font> = 
 <a href="yellow_6.html#K1" target="_self">the_value_of</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="ref" href="yellow_6.html#D10" target="_self">Def10</a></i>
<br/>.= 
 <a href="yellow_6.html#K1" target="_self">the_value_of</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">c<sub>5</sub></font>

<br/>.= 
 <a href="yellow_6.html#K5" target="_self">the_value_of</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="ref" href="yellow_6.html#D10" target="_self">Def10</a></i>
;<br/>
<b>assume </b><a NAME="E6:121_3_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K7" target="_self">NetUniv</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<a NAME="E7:121_3_1_1"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K7" target="_self">NetUniv</a> <span class="p1">(<span class="default"><a href="yellow_6.html#K14" target="_self">ConvergenceSpace</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="yellow_6.html#E1:121"><i><font color="Green">E73</font></i></a>, <a class="txt" href="yellow_6.html#E53"><i><font color="Green">Lemma41</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:121_3_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<span class="p2">(<span class="default"><a href="yellow_6.html#K5" target="_self">the_value_of</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="yellow_6.html#E1:121_3"><i><font color="Green">E77</font></i></a>, <a class="txt" href="yellow_6.html#E5:121_3_1_1"><i><font color="Green">E78</font></i></a>, <a class="ref" href="yellow_6.html#D23" target="_self">Def23</a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E2:121_3_1"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="yellow_6.html#V5" target="_self">(SUBNETS)</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="121_3_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">c<sub>1</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_6.html#D24" target="_self">YELLOW_6:def 24</a><br/><b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="yellow_6.html#M2" target="_self">subnet</a> of <font color="Maroon">c<sub>5</sub></font>;<br/>



<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>5</sub></font> as   <a href="waybel_0.html#NM2">net</a> of  <a href="yellow_6.html#K14" target="_self">ConvergenceSpace</a> <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="yellow_6.html#D27" target="_self">Def27</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>6</sub></font> as    <a href="yellow_6.html#M2" target="_self">subnet</a> of <font color="Maroon">c<sub>7</sub></font> <b>by </b><i><a class="txt" href="yellow_6.html#E3:121"><i><font color="Green">E75</font></i></a></i>;<br/>
<b>assume </b><a NAME="E3:121_3_1_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K7" target="_self">NetUniv</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<a NAME="E4:121_3_1_2"/><b>then </b><i><font color="Green">E78</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K7" target="_self">NetUniv</a> <span class="p1">(<span class="default"><a href="yellow_6.html#K14" target="_self">ConvergenceSpace</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="yellow_6.html#E1:121"><i><font color="Green">E73</font></i></a>, <a class="txt" href="yellow_6.html#E53"><i><font color="Green">Lemma41</font></i></a></i>;<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>1</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 <span class="p1">(<span class="default"><a href="yellow_6.html#K14" target="_self">ConvergenceSpace</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <b>by </b><i><a class="ref" href="yellow_6.html#D27" target="_self">Def27</a></i>;<br/>
<b>assume </b><a NAME="E6:121_3_1_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>5</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> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E7:121_3_1_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K13" target="_self">Convergence</a> <span class="p1">(<span class="default"><a href="yellow_6.html#K14" target="_self">ConvergenceSpace</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="yellow_6.html#E1:121_3"><i><font color="Green">E77</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:121_3_1_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>6</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> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="yellow_6.html#E1:121_3"><i><font color="Green">E77</font></i></a>, <a class="txt" href="yellow_6.html#E4:121_3_1_2"><i><font color="Green">E78</font></i></a>, <a class="ref" href="yellow_6.html#D24" target="_self">Def24</a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E3:121_3_1"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="yellow_6.html#V6" target="_self">(DIVERGENCE)</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="121_3_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">c<sub>1</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_6.html#D25" target="_self">YELLOW_6:def 25</a><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>1</sub></font>;<br/>



<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>5</sub></font> as   <a href="waybel_0.html#NM2">net</a> of  <a href="yellow_6.html#K14" target="_self">ConvergenceSpace</a> <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="yellow_6.html#D27" target="_self">Def27</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>6</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K14" target="_self">ConvergenceSpace</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <b>by </b><i><a class="ref" href="yellow_6.html#D27" target="_self">Def27</a></i>;<br/>
<b>assume </b><a NAME="E3:121_3_1_3"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K7" target="_self">NetUniv</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<a NAME="E4:121_3_1_3"/><b>then </b><i><font color="Green">E78</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K7" target="_self">NetUniv</a> <span class="p1">(<span class="default"><a href="yellow_6.html#K14" target="_self">ConvergenceSpace</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="yellow_6.html#E1:121"><i><font color="Green">E73</font></i></a>, <a class="txt" href="yellow_6.html#E53"><i><font color="Green">Lemma41</font></i></a></i>;<br/>
<b>assume </b><a NAME="E5:121_3_1_3"/>
not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>9</sub></font> being    <a href="yellow_6.html#M2" target="_self">subnet</a> of <font color="Maroon">c<sub>7</sub></font><b> such that </b><br/><a NAME="E7:121_3_1_3"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K7" target="_self">NetUniv</a> <span class="p1">(<span class="default"><a href="yellow_6.html#K14" target="_self">ConvergenceSpace</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E8:121_3_1_3"/><i><font color="Green">E80</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="yellow_6.html#M2" target="_self">subnet</a> of <font color="Maroon">c<sub>9</sub></font> holds not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K13" target="_self">Convergence</a> <span class="p1">(<span class="default"><a href="yellow_6.html#K14" target="_self">ConvergenceSpace</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="yellow_6.html#E1:121_3"><i><font color="Green">E77</font></i></a>, <a class="txt" href="yellow_6.html#E4:121_3_1_3"><i><font color="Green">E78</font></i></a>, <a class="ref" href="yellow_6.html#D25" target="_self">Def25</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="yellow_6.html#M2" target="_self">subnet</a> of <font color="Maroon">c<sub>5</sub></font> <b>by </b><i><a class="txt" href="yellow_6.html#E4:121"><i><font color="Green">E76</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>10</sub></font>
;<br/>

<b>thus </b><a NAME="E10:121_3_1_3"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K7" target="_self">NetUniv</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="yellow_6.html#E1:121"><i><font color="Green">E73</font></i></a>, <a class="txt" href="yellow_6.html#E7:121_3_1_3"><i><font color="Green">E79</font></i></a>, <a class="txt" href="yellow_6.html#E53"><i><font color="Green">Lemma41</font></i></a></i>;<br/>

<b>let </b><font color="Maroon">c<sub>11</sub></font> be    <a href="yellow_6.html#M2" target="_self">subnet</a> of <font color="Maroon">c<sub>10</sub></font>;<br/>

<b>reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>11</sub></font> as    <a href="yellow_6.html#M2" target="_self">subnet</a> of <font color="Maroon">c<sub>9</sub></font> <b>by </b><i><a class="txt" href="yellow_6.html#E3:121"><i><font color="Green">E75</font></i></a></i>;<br/>
<a NAME="E12:121_3_1_3"/>
not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K13" target="_self">Convergence</a> <span class="p1">(<span class="default"><a href="yellow_6.html#K14" target="_self">ConvergenceSpace</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="yellow_6.html#E8:121_3_1_3"><i><font color="Green">E80</font></i></a></i>;<br/>
<b>hence </b><a NAME="E13:121_3_1_3"/>
not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="yellow_6.html#E1:121_3"><i><font color="Green">E77</font></i></a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E4:121_3_1"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="yellow_6.html#V7" target="_self">(ITERATED_LIMITS)</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="121_3_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">c<sub>1</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_6.html#D26" target="_self">YELLOW_6:def 26</a><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>1</sub></font>;<br/>



<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>5</sub></font> as   <a href="waybel_0.html#NM2">net</a> of  <a href="yellow_6.html#K14" target="_self">ConvergenceSpace</a> <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="yellow_6.html#D27" target="_self">Def27</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>6</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K14" target="_self">ConvergenceSpace</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <b>by </b><i><a class="ref" href="yellow_6.html#D27" target="_self">Def27</a></i>;<br/>
<b>assume </b><a NAME="E3:121_3_1_4"/><i><font color="Green">E78</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>9</sub></font> be    <a href="yellow_6.html#M3" target="_self">net_set</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>1</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="pboole.html#M1">ManySortedSet</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>7</sub></font> ;<br/>
<a NAME="E5:121_3_1_4"/>
<font color="Maroon">c<sub>10</sub></font> is    <a href="yellow_6.html#M3" target="_self">net_set</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>7</sub></font>, <a href="yellow_6.html#K14" target="_self">ConvergenceSpace</a> <font color="Maroon">c<sub>2</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="121_3_1_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>11</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_6.html#D15" target="_self">YELLOW_6:def 15</a><br/>

<b>assume </b><a NAME="E1:121_3_1_4_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/>

<a NAME="E2:121_3_1_4_1"/><b>then </b>
<font color="Maroon">c<sub>11</sub></font> is   <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="yellow_6.html#D15" target="_self">Def15</a></i>;<br/>
<b>hence </b><a NAME="E3:121_3_1_4_1"/>
<font color="Maroon">c<sub>11</sub></font> is   <a href="waybel_0.html#NM2">net</a> of  <a href="yellow_6.html#K14" target="_self">ConvergenceSpace</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="yellow_6.html#E2:121"><i><font color="Green">E74</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>9</sub></font> as    <a href="yellow_6.html#M3" target="_self">net_set</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>7</sub></font>, <a href="yellow_6.html#K14" target="_self">ConvergenceSpace</a> <font color="Maroon">c<sub>2</sub></font> ;<br/>
<b>assume </b><a NAME="E7:121_3_1_4"/><i><font color="Green">E79</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>5</sub></font> holds <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="yellow_6.html#K8" target="_self">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="waybel_0.html#K2">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="121_3_1_4_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>12</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>7</sub></font>;<br/><b>reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>12</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>5</sub></font> ;<br/><font color="Maroon">c<sub>5</sub></font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">c<sub>13</sub></font> = 
the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font>

<br/>.= 
the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font>

<br/>.= 
<font color="Maroon">c<sub>7</sub></font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">c<sub>12</sub></font>

;<br/><b>hence </b><a NAME="E3:121_3_1_4_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="yellow_6.html#K8" target="_self">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K13" target="_self">Convergence</a> <span class="p1">(<span class="default"><a href="yellow_6.html#K14" target="_self">ConvergenceSpace</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="yellow_6.html#E1:121_3"><i><font color="Green">E77</font></i></a>, <a class="txt" href="yellow_6.html#E7:121_3_1_4"><i><font color="Green">E79</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E9:121_3_1_4"/><b>then </b><i><font color="Green">E80</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_6.html#K9" target="_self">Iterated</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>,<font color="Maroon">c<sub>8</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K13" target="_self">Convergence</a> <span class="p1">(<span class="default"><a href="yellow_6.html#K14" target="_self">ConvergenceSpace</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="yellow_6.html#E1:121_3"><i><font color="Green">E77</font></i></a>, <a class="txt" href="yellow_6.html#E3:121_3_1_4"><i><font color="Green">E78</font></i></a>, <a class="ref" href="yellow_6.html#D26" target="_self">Def26</a></i>;<br/>
<i><font color="Green">E81</font></i>:  <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9" target="_self">Iterated</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>,the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9" target="_self">Iterated</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> #) = 
<span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<span class="p2">(<span class="default"><a href="yellow_1.html#K5">product</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span><a href="yellow_3.html#K3">:]</a></span>
<b>by </b><i><a class="ref" href="yellow_6.html#D16" target="_self">Def16</a></i>
<br/>.= 
 <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9" target="_self">Iterated</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>,the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9" target="_self">Iterated</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> #)
<b>by </b><i><a class="ref" href="yellow_6.html#D16" target="_self">Def16</a></i>
;<br/>
<a NAME="E11:121_3_1_4"/><b>then </b><i><font color="Green">E82</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9" target="_self">Iterated</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9" target="_self">Iterated</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
 
;<br/>
<a NAME="E12:121_3_1_4"/><i><font color="Green">E83</font></i>: 
the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9" target="_self">Iterated</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9" target="_self">Iterated</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="yellow_6.html#E10:121_3_1_4"><i><font color="Green">E81</font></i></a></i>;<br/>
<a NAME="E13:121_3_1_4"/>
 <a href="relset_1.html#K4">dom</a> the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9" target="_self">Iterated</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9" target="_self">Iterated</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E14:121_3_1_4"/><b>then </b><i><font color="Green">E84</font></i>: 
 <a href="relset_1.html#K4">dom</a> the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9" target="_self">Iterated</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9" target="_self">Iterated</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="yellow_6.html#E10:121_3_1_4"><i><font color="Green">E81</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="121_3_1_4_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>12</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><a NAME="E1:121_3_1_4_4"/><i><font color="Green">E85</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9" target="_self">Iterated</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>7</sub></font>,<span class="p2">(<span class="default"><a href="card_3.html#K4">product</a> <span class="p3">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="ref" href="yellow_6.html#T35" target="_self">Th35</a></i>;<br/><a NAME="E2:121_3_1_4_4"/><i><font color="Green">E86</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9" target="_self">Iterated</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>5</sub></font>,<span class="p2">(<span class="default"><a href="card_3.html#K4">product</a> <span class="p3">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="ref" href="yellow_6.html#T35" target="_self">Th35</a></i>;<br/><b>assume </b><a NAME="E3:121_3_1_4_4"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9" target="_self">Iterated</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>12</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9" target="_self">Iterated</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> ;<br/><b>consider </b><font color="Maroon">c<sub>14</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>15</sub></font> being    <a href="subset_1.html#M1">Element</a> of  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">c<sub>11</sub></font></span>)</span><b> such that </b><br/><a NAME="E6:121_3_1_4_4"/><i><font color="Green">E87</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="txt" href="yellow_6.html#E1:121_3_1_4_4"><i><font color="Green">E85</font></i></a>, <a class="ref" href="domain_1.html#T9">DOMAIN_1:9</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>15</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <b>by </b><i><a class="ref" href="yellow_1.html#D4">YELLOW_1:def 4</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>14</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>5</sub></font> ;<br/><b>set </b><font color="Maroon">c<sub>18</sub></font> = <font color="Maroon">c<sub>16</sub></font>;<br/><b>reconsider </b><font color="Maroon">c<sub>19</sub></font> = <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>16</sub></font></span><a href="yellow_3.html#K7">]</a></span> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9" target="_self">Iterated</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <b>by </b><i><a class="txt" href="yellow_6.html#E2:121_3_1_4_4"><i><font color="Green">E86</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><b>thus </b>the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9" target="_self">Iterated</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> = 
<span class="p1">(<span class="default"><a href="yellow_6.html#K9" target="_self">Iterated</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="waybel_0.html#K2">.</a> <font color="Maroon">c<sub>13</sub></font>

<br/>.= 
the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="yellow_6.html#K8" target="_self">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="waybel_3.html#K4">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="yellow_6.html#E6:121_3_1_4_4"><i><font color="Green">E87</font></i></a>, <a class="ref" href="yellow_6.html#T36" target="_self">Th36</a></i>
<br/>.= 
the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="yellow_6.html#K8" target="_self">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><a href="yellow_6.html#K9" target="_self">Iterated</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="waybel_0.html#K2">.</a> <font color="Maroon">c<sub>19</sub></font>
<b>by </b><i><a class="ref" href="yellow_6.html#T36" target="_self">Th36</a></i>
<br/>.= 
the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9" target="_self">Iterated</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font>
<b>by </b><i><a class="txt" href="yellow_6.html#E6:121_3_1_4_4"><i><font color="Green">E87</font></i></a></i>
;<br/></div><b>end;</b></div>
<a NAME="E16:121_3_1_4"/><b>then </b><i><font color="Green">E85</font></i>: 
the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9" target="_self">Iterated</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="funct_2.html#R1">=</a> the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9" target="_self">Iterated</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="yellow_6.html#E14:121_3_1_4"><i><font color="Green">E84</font></i></a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>
 <a href="yellow_6.html#K9" target="_self">Iterated</a> <font color="Maroon">c<sub>11</sub></font> = 
 <a href="waybel_0.html#G1">NetStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9" target="_self">Iterated</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>,the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9" target="_self">Iterated</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>,the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9" target="_self">Iterated</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> #)

<br/>.= 
 <a href="waybel_0.html#G1">NetStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9" target="_self">Iterated</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>,the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9" target="_self">Iterated</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>,the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9" target="_self">Iterated</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> #)
<b>by </b><i><a class="txt" href="yellow_6.html#E11:121_3_1_4"><i><font color="Green">E82</font></i></a>, <a class="txt" href="yellow_6.html#E12:121_3_1_4"><i><font color="Green">E83</font></i></a>, <a class="txt" href="yellow_6.html#E16:121_3_1_4"><i><font color="Green">E85</font></i></a></i>
<br/>.= 
 <a href="yellow_6.html#K9" target="_self">Iterated</a> <font color="Maroon">c<sub>9</sub></font>

;<br/>
<b>hence </b><a NAME="E18:121_3_1_4"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_6.html#K9" target="_self">Iterated</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="yellow_6.html#E1:121_3"><i><font color="Green">E77</font></i></a>, <a class="txt" href="yellow_6.html#E9:121_3_1_4"><i><font color="Green">E80</font></i></a></i>;<br/>


</div><b>end;</b></div>


</div><b>end;</b></div></div>
<b>end;</b></div>

<b>assume </b><a NAME="E6:121"/><i><font color="Green">E77</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> is <a href="yellow_6.html#V4" target="_self">(CONSTANTS)</a> &amp; <font color="Maroon">c<sub>2</sub></font> is <a href="yellow_6.html#V5" target="_self">(SUBNETS)</a> &amp; <font color="Maroon">c<sub>2</sub></font> is <a href="yellow_6.html#V6" target="_self">(DIVERGENCE)</a> &amp; <font color="Maroon">c<sub>2</sub></font> is <a href="yellow_6.html#V7" target="_self">(ITERATED_LIMITS)</a> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_6.html#D28" target="_self">YELLOW_6:def 28</a><br/>

<b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>2</sub></font> as  <a href="yellow_6.html#V8" target="_self">topological</a>  <a href="yellow_6.html#M4" target="_self">Convergence-Class</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="yellow_6.html#D28" target="_self">Def28</a></i>;<br/>
<a NAME="E8:121"/><i><font color="Green">E78</font></i>: 
 <a href="yellow_6.html#K13" target="_self">Convergence</a> <span class="p1">(<span class="default"><a href="yellow_6.html#K14" target="_self">ConvergenceSpace</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="121_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="relat_1.html#D3">RELAT_1:def 3</a><br/><b>let </b><font color="Maroon">c<sub>7</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>



<a NAME="E1:121_4"/><i><font color="Green">E79</font></i>: 
 <a href="yellow_6.html#K13" target="_self">Convergence</a> <span class="p1">(<span class="default"><a href="yellow_6.html#K14" target="_self">ConvergenceSpace</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_6.html#K7" target="_self">NetUniv</a> <span class="p3">(<span class="default"><a href="yellow_6.html#K14" target="_self">ConvergenceSpace</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="yellow_6.html#K14" target="_self">ConvergenceSpace</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="yellow_6.html#D21" target="_self">Def21</a></i>;<br/>
<b>assume </b><a NAME="E2:121_4"/><i><font color="Green">E80</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K13" target="_self">Convergence</a> <span class="p1">(<span class="default"><a href="yellow_6.html#K14" target="_self">ConvergenceSpace</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>8</sub></font> being    <a href="subset_1.html#M1">Element</a> of  <a href="yellow_6.html#K7" target="_self">NetUniv</a> <span class="p1">(<span class="default"><a href="yellow_6.html#K14" target="_self">ConvergenceSpace</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>, <font color="Maroon">c<sub>9</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K14" target="_self">ConvergenceSpace</a> <font color="Maroon">c<sub>2</sub></font></span>)</span><b> such that </b><br/><a NAME="E4:121_4"/><i><font color="Green">E81</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <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>
 <b>by </b><i><a class="txt" href="yellow_6.html#E1:121_4"><i><font color="Green">E79</font></i></a>, <a class="ref" href="domain_1.html#T9">DOMAIN_1:9</a></i>;<br/>
<a NAME="E5:121_4"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K7" target="_self">NetUniv</a> <span class="p1">(<span class="default"><a href="yellow_6.html#K14" target="_self">ConvergenceSpace</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
;<br/>
<a NAME="E6:121_4"/>
ex <font color="Olive">b<sub>1</sub></font> being <a href="waybel_0.html#V6">strict</a> <a href="waybel_0.html#NM2">net</a> of  <a href="yellow_6.html#K14" target="_self">ConvergenceSpace</a> <font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K2" target="_self">the_universe_of</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K14" target="_self">ConvergenceSpace</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> )
 
<b>by </b><i><a class="ref" href="yellow_6.html#D14" target="_self">Def14</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>8</sub></font> as   <a href="waybel_0.html#NM2">net</a> of  <a href="yellow_6.html#K14" target="_self">ConvergenceSpace</a> <font color="Maroon">c<sub>2</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="waybel_0.html#NM2">net</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="yellow_6.html#D27" target="_self">Def27</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>9</sub></font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="yellow_6.html#D27" target="_self">Def27</a></i>;<br/>
<a NAME="E10:121_4"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K7" target="_self">NetUniv</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="yellow_6.html#E1:121"><i><font color="Green">E73</font></i></a>, <a class="txt" href="yellow_6.html#E5:121_4"><i><font color="Green">E82</font></i></a>, <a class="txt" href="yellow_6.html#E53"><i><font color="Green">Lemma41</font></i></a></i>;<br/>
<b>assume </b><a NAME="E11:121_4"/>
not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>13</sub></font> being    <a href="yellow_6.html#M2" target="_self">subnet</a> of <font color="Maroon">c<sub>11</sub></font><b> such that </b><br/><a NAME="E13:121_4"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K7" target="_self">NetUniv</a> <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E14:121_4"/><i><font color="Green">E85</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="yellow_6.html#M2" target="_self">subnet</a> of <font color="Maroon">c<sub>13</sub></font> holds not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="yellow_6.html#E6:121"><i><font color="Green">E77</font></i></a>, <a class="txt" href="yellow_6.html#E4:121_4"><i><font color="Green">E81</font></i></a>, <a class="txt" href="yellow_6.html#E10:121_4"><i><font color="Green">E83</font></i></a>, <a class="ref" href="yellow_6.html#D25" target="_self">Def25</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>14</sub></font> =  <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>13</sub></font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>13</sub></font> #) as  non <a href="struct_0.html#V3">empty</a> <a href="orders_2.html#V3">transitive</a> <a href="waybel_0.html#V7">directed</a>  <a href="orders_2.html#L1">RelStr</a>  <b>by </b><i><a class="txt" href="yellow_6.html#E16"><i><font color="Green">Lemma9</font></i></a>, <a class="txt" href="yellow_6.html#E17"><i><font color="Green">Lemma10</font></i></a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>14</sub></font> <a href="yellow_6.html#K4" target="_self">--&gt;</a> <font color="Maroon">c<sub>12</sub></font>;<br/>
<a NAME="E16:121_4"/><i><font color="Green">E86</font></i>: 
 <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">c<sub>14</sub></font> <a href="yellow_6.html#K4" target="_self">--&gt;</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>,the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="yellow_6.html#K4" target="_self">--&gt;</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> #) <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font>
 
<b>by </b><i><a class="ref" href="yellow_6.html#D7" target="_self">Def7</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>14</sub></font> <a href="yellow_6.html#K4" target="_self">--&gt;</a> <font color="Maroon">c<sub>12</sub></font> as  non <a href="struct_0.html#V3">empty</a> <a href="waybel_0.html#V6">strict</a> <a href="yellow_6.html#V1" target="_self">constant</a> <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E18:121_4"/><i><font color="Green">E87</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being <a href="waybel_0.html#V6">strict</a> <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K2" target="_self">the_universe_of</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="txt" href="yellow_6.html#E13:121_4"><i><font color="Green">E84</font></i></a>, <a class="ref" href="yellow_6.html#D14" target="_self">Def14</a></i>;<br/>
<a NAME="E19:121_4"/>
 <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>16</sub></font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>16</sub></font> #) <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">c<sub>13</sub></font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>13</sub></font> #)
 
<b>by </b><i><a class="ref" href="yellow_6.html#D7" target="_self">Def7</a></i>;<br/>
<a NAME="E20:121_4"/><b>then </b><i><font color="Green">E88</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K7" target="_self">NetUniv</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="yellow_6.html#E18:121_4"><i><font color="Green">E87</font></i></a>, <a class="ref" href="yellow_6.html#D14" target="_self">Def14</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>16</sub></font> as   <a href="waybel_0.html#NM2">net</a> of  <a href="yellow_6.html#K14" target="_self">ConvergenceSpace</a> <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="yellow_6.html#D27" target="_self">Def27</a></i>;<br/>
<a NAME="E22:121_4"/>
the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">c<sub>16</sub></font> is <a href="funct_1.html#V3">constant</a>
 
;<br/>
<a NAME="E23:121_4"/><b>then </b>
the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">c<sub>17</sub></font> is <a href="funct_1.html#V3">constant</a>
 
;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>18</sub></font> = <font color="Maroon">c<sub>17</sub></font> as  <a href="yellow_6.html#V1" target="_self">constant</a> <a href="waybel_0.html#NM2">net</a> of  <a href="yellow_6.html#K14" target="_self">ConvergenceSpace</a> <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="yellow_6.html#D6" target="_self">Def6</a></i>;<br/>
<a NAME="E25:121_4"/>
 <a href="yellow_6.html#K5" target="_self">the_value_of</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font>
 
<b>by </b><i><a class="ref" href="yellow_6.html#T22" target="_self">Th22</a></i>;<br/>
<a NAME="E26:121_4"/><b>then </b><i><font color="Green">E89</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="yellow_6.html#E6:121"><i><font color="Green">E77</font></i></a>, <a class="txt" href="yellow_6.html#E20:121_4"><i><font color="Green">E88</font></i></a>, <a class="ref" href="yellow_6.html#D23" target="_self">Def23</a></i>;<br/>
<a NAME="E27:121_4"/><i><font color="Green">E90</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K11" target="_self">Lim</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="txt" href="yellow_6.html#E2:121_4"><i><font color="Green">E80</font></i></a>, <a class="txt" href="yellow_6.html#E4:121_4"><i><font color="Green">E81</font></i></a>, <a class="ref" href="yellow_6.html#D22" target="_self">Def22</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>19</sub></font> = <font color="Maroon">c<sub>13</sub></font> as    <a href="yellow_6.html#M2" target="_self">subnet</a> of <font color="Maroon">c<sub>10</sub></font> <b>by </b><i><a class="txt" href="yellow_6.html#E3:121"><i><font color="Green">E75</font></i></a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] means ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>13</sub></font>ex <font color="Olive">b<sub>2</sub></font> being  <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>2</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>2</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> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Olive">b<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="waybel_0.html#K2">.</a> <font color="Olive">b<sub>3</sub></font></span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>13</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">b<sub>3</sub></font> </span>}</span>  );<br/>
<a NAME="E29:121_4"/><i><font color="Green">E91</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>16</sub></font> holds <br/>ex <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="121_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>20</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:121_4_1"/>
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>16</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>21</sub></font> = <font color="Maroon">c<sub>20</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>19</sub></font> <b>by </b><i><a class="ref" href="yellow_6.html#T13" target="_self">Th13</a></i>;<br/>
<a NAME="E3:121_4_1"/>
 <a href="yellow_6.html#K11" target="_self">Lim</a> <font color="Maroon">c<sub>10</sub></font> <a href="tarski.html#R1">c=</a>  <a href="yellow_6.html#K11" target="_self">Lim</a> <font color="Maroon">c<sub>19</sub></font>
 
<b>by </b><i><a class="ref" href="yellow_6.html#T41" target="_self">Th41</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>22</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K14" target="_self">ConvergenceSpace</a> <font color="Maroon">c<sub>2</sub></font></span>)</span><b> such that </b><br/><a NAME="E5:121_4_1"/><i><font color="Green">E92</font></i>: 
<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="waybel_0.html#K2">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>19</sub></font> : <font color="Maroon">c<sub>21</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> </span>}</span> 
 <b>and </b><br/><a NAME="E6:121_4_1"/><i><font color="Green">E93</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>22</sub></font>
 <b>by </b><i><a class="txt" href="yellow_6.html#E27:121_4"><i><font color="Green">E90</font></i></a>, <a class="ref" href="yellow_6.html#T43" target="_self">Th43</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>23</sub></font> being   <a href="waybel_0.html#NM2">net</a> of  <a href="yellow_6.html#K14" target="_self">ConvergenceSpace</a> <font color="Maroon">c<sub>5</sub></font><b> such that </b><br/><a NAME="E8:121_4_1"/><i><font color="Green">E94</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>23</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> <font color="Maroon">c<sub>5</sub></font>
 <b>and </b><br/><a NAME="E9:121_4_1"/><i><font color="Green">E95</font></i>: 
 <a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">c<sub>23</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>22</sub></font>
 <b>and </b><br/><a NAME="E10:121_4_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K11" target="_self">Lim</a> <font color="Maroon">c<sub>23</sub></font>
 <b>by </b><i><a class="txt" href="yellow_6.html#E6:121_4_1"><i><font color="Green">E93</font></i></a>, <a class="ref" href="yellow_6.html#T52" target="_self">Th52</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>24</sub></font> = <font color="Maroon">c<sub>23</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="ref" href="yellow_6.html#D27" target="_self">Def27</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>25</sub></font> = <font color="Maroon">c<sub>21</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>13</sub></font> ;<br/>
<b>take </b>
<font color="Maroon">c<sub>24</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>25</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>24</sub></font>
;<br/>

<b>thus </b><a NAME="E13:121_4_1"/>
( <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>25</sub></font> &amp; <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>24</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>24</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> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="yellow_6.html#E8:121_4_1"><i><font color="Green">E94</font></i></a></i>;<br/>

<b>let </b><font color="Maroon">c<sub>26</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E14:121_4_1"/>
<font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">c<sub>24</sub></font>
 ;<br/>

<a NAME="E15:121_4_1"/><b>then </b>
<font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>22</sub></font>
 
<b>by </b><i><a class="txt" href="yellow_6.html#E9:121_4_1"><i><font color="Green">E95</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>27</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>19</sub></font><b> such that </b><br/><a NAME="E17:121_4_1"/><i><font color="Green">E96</font></i>: 
<font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">c<sub>27</sub></font>
 <b>and </b><br/><a NAME="E18:121_4_1"/><i><font color="Green">E97</font></i>: 
<font color="Maroon">c<sub>21</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>27</sub></font>
 <b>by </b><i><a class="txt" href="yellow_6.html#E5:121_4_1"><i><font color="Green">E92</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>28</sub></font> = <font color="Maroon">c<sub>27</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>13</sub></font> ;<br/>
<font color="Maroon">c<sub>26</sub></font> = 
the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">c<sub>19</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>27</sub></font>
<b>by </b><i><a class="txt" href="yellow_6.html#E17:121_4_1"><i><font color="Green">E96</font></i></a></i>
<br/>.= 
the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">c<sub>13</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>28</sub></font>

<br/>.= 
<font color="Maroon">c<sub>13</sub></font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">c<sub>28</sub></font>

;<br/>
<b>hence </b><a NAME="E21:121_4_1"/>
<font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="waybel_0.html#K2">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>13</sub></font> : <font color="Maroon">c<sub>25</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> </span>}</span> 
 <b>by </b><i><a class="txt" href="yellow_6.html#E18:121_4_1"><i><font color="Green">E97</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>20</sub></font> being    <a href="pboole.html#M1">ManySortedSet</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>16</sub></font><b> such that </b><br/><a NAME="E31:121_4"/><i><font color="Green">E92</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>16</sub></font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>20</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="pboole.html#S3">PBOOLE:sch 3</a>(<a class="txt" href="yellow_6.html#E29:121_4"><i><font color="Green">E91</font></i></a>);<br/></i>
<div><i><font color="Green">E93</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="121_4_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>21</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:121_4_2"/>
<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>20</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>22</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:121_4_2"/><i><font color="Green">E94</font></i>: 
<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>20</sub></font>
 <b>and </b><br/><a NAME="E4:121_4_2"/><i><font color="Green">E95</font></i>: 
<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E5:121_4_2"/>
<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="txt" href="yellow_6.html#E3:121_4_2"><i><font color="Green">E94</font></i></a>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><a NAME="E6:121_4_2"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>13</sub></font>ex <font color="Olive">b<sub>2</sub></font> being  <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>2</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> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Olive">b<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="waybel_0.html#K2">.</a> <font color="Olive">b<sub>3</sub></font></span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>13</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">b<sub>3</sub></font> </span>}</span>  )
 <b>by </b><i><a class="txt" href="yellow_6.html#E31:121_4"><i><font color="Green">E92</font></i></a></i>;<br/><b>hence </b><a NAME="E7:121_4_2"/>
<font color="Maroon">c<sub>21</sub></font> is  non <a href="struct_0.html#V3">empty</a>  <a href="struct_0.html#L1">1-sorted</a> 
 <b>by </b><i><a class="txt" href="yellow_6.html#E4:121_4_2"><i><font color="Green">E95</font></i></a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E94</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="121_4_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>21</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:121_4_3"/>
<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>16</sub></font>
 ;<br/><a NAME="E2:121_4_3"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>13</sub></font>ex <font color="Olive">b<sub>2</sub></font> being  <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>2</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> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Olive">b<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="waybel_0.html#K2">.</a> <font color="Olive">b<sub>3</sub></font></span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>13</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">b<sub>3</sub></font> </span>}</span>  )
 <b>by </b><i><a class="txt" href="yellow_6.html#E31:121_4"><i><font color="Green">E92</font></i></a></i>;<br/><b>hence </b><a NAME="E3:121_4_3"/>
<font color="Maroon">c<sub>20</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font> is   <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">c<sub>1</sub></font>
 ;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="121_4_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>21</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:121_4_4"/>
<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>20</sub></font>
 ;<br/><a NAME="E2:121_4_4"/><b>then </b>
<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><a NAME="E3:121_4_4"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>13</sub></font>ex <font color="Olive">b<sub>2</sub></font> being  <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>2</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> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Olive">b<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="waybel_0.html#K2">.</a> <font color="Olive">b<sub>3</sub></font></span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>13</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">b<sub>3</sub></font> </span>}</span>  )
 <b>by </b><i><a class="txt" href="yellow_6.html#E31:121_4"><i><font color="Green">E92</font></i></a></i>;<br/><b>hence </b><a NAME="E4:121_4_4"/>
<font color="Maroon">c<sub>20</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font> is    <a href="struct_0.html#L1">1-sorted</a> 
 ;<br/></div><b>end;</b></div>
<a NAME="E35:121_4"/><b>then </b>
<font color="Maroon">c<sub>20</sub></font> is <a href="pralg_1.html#V2">1-sorted-yielding</a>
 
<b>by </b><i><a class="ref" href="pralg_1.html#D11">PRALG_1:def 11</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>21</sub></font> = <font color="Maroon">c<sub>20</sub></font> as  <a href="waybel_3.html#V4">yielding_non-empty_carriers</a>  <a href="yellow_6.html#M3" target="_self">net_set</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="yellow_6.html#E32:121_4"><i><font color="Green">E93</font></i></a>, <a class="txt" href="yellow_6.html#E33:121_4"><i><font color="Green">E94</font></i></a>, <a class="ref" href="yellow_6.html#D4" target="_self">Def4</a>, <a class="ref" href="yellow_6.html#T33" target="_self">Th33</a></i>;<br/>
<a NAME="E37:121_4"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>16</sub></font> holds <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="yellow_6.html#K8" target="_self">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="waybel_0.html#K2">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="121_4_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>22</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>16</sub></font>;<br/>

<a NAME="E1:121_4_5"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>13</sub></font>ex <font color="Olive">b<sub>2</sub></font> being  <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> <a href="yellow_6.html#K8" target="_self">.</a> <font color="Maroon">c<sub>22</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>2</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> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Olive">b<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="waybel_0.html#K2">.</a> <font color="Olive">b<sub>3</sub></font></span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>13</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">b<sub>3</sub></font> </span>}</span>  )
 
<b>by </b><i><a class="txt" href="yellow_6.html#E31:121_4"><i><font color="Green">E92</font></i></a></i>;<br/>
<b>hence </b><a NAME="E2:121_4_5"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="yellow_6.html#K8" target="_self">.</a> <font color="Maroon">c<sub>22</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">c<sub>22</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="yellow_6.html#T14" target="_self">Th14</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E38:121_4"/><b>then </b><i><font color="Green">E95</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_6.html#K9" target="_self">Iterated</a> <font color="Maroon">c<sub>21</sub></font></span>)</span>,<font color="Maroon">c<sub>12</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="yellow_6.html#E6:121"><i><font color="Green">E77</font></i></a>, <a class="txt" href="yellow_6.html#E26:121_4"><i><font color="Green">E89</font></i></a>, <a class="ref" href="yellow_6.html#D26" target="_self">Def26</a></i>;<br/>
<a NAME="E39:121_4"/><i><font color="Green">E96</font></i>: 
 <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9" target="_self">Iterated</a> <font color="Maroon">c<sub>21</sub></font></span>)</span>,the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9" target="_self">Iterated</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> #) <a href="hidden.html#R1">=</a> <span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<span class="p2">(<span class="default"><a href="yellow_1.html#K5">product</a> <font color="Maroon">c<sub>21</sub></font></span>)</span></span><a href="yellow_3.html#K3">:]</a></span>
 
<b>by </b><i><a class="ref" href="yellow_6.html#D16" target="_self">Def16</a></i>;<br/>
<a NAME="E40:121_4"/><b>then </b><i><font color="Green">E97</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9" target="_self">Iterated</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>16</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="yellow_1.html#K5">product</a> <font color="Maroon">c<sub>21</sub></font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="yellow_3.html#D2">YELLOW_3:def 2</a></i>;<br/>
<a NAME="E41:121_4"/>
 <a href="yellow_6.html#K9" target="_self">Iterated</a> <font color="Maroon">c<sub>21</sub></font> is    <a href="yellow_6.html#M2" target="_self">subnet</a> of <font color="Maroon">c<sub>13</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="121_4_6"><b>proof </b></a><div class="add">

<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>13</sub></font>) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>13</sub></font> : <font color="Maroon">a<sub>1</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> </span>}</span> ;<br/>
<b>consider </b><font color="Maroon">c<sub>22</sub></font> being    <a href="pboole.html#M1">ManySortedSet</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>13</sub></font><b> such that </b><br/><a NAME="E2:121_4_6"/><i><font color="Green">E98</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>13</sub></font> holds <font color="Maroon">c<sub>22</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>)
 <b>from </b><i><a class="ref" href="pboole.html#S5">PBOOLE:sch 5</a>();<br/></i>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="121_4_6_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:121_4_6_1"/>
 <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>22</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>23</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:121_4_6_1"/><i><font color="Green">E99</font></i>: 
<font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>22</sub></font>
 <b>and </b><br/><a NAME="E4:121_4_6_1"/><i><font color="Green">E100</font></i>: 
<font color="Maroon">c<sub>22</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>24</sub></font> = <font color="Maroon">c<sub>23</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>13</sub></font> <b>by </b><i><a class="txt" href="yellow_6.html#E3:121_4_6_1"><i><font color="Green">E99</font></i></a>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>25</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>13</sub></font><b> such that </b><br/><a NAME="E7:121_4_6_1"/><i><font color="Green">E101</font></i>: 
<font color="Maroon">c<sub>24</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>25</sub></font>
 <b>and </b><br/><a NAME="E8:121_4_6_1"/>
<font color="Maroon">c<sub>24</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>25</sub></font>
 <b>by </b><i><a class="ref" href="yellow_6.html#D5" target="_self">Def5</a></i>;<br/><a NAME="E9:121_4_6_1"/>
<font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>13</sub></font> : <font color="Maroon">c<sub>24</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> </span>}</span> 
 <b>by </b><i><a class="txt" href="yellow_6.html#E7:121_4_6_1"><i><font color="Green">E101</font></i></a></i>;<br/><b>hence </b><a NAME="E10:121_4_6_1"/>
contradiction
 <b>by </b><i><a class="txt" href="yellow_6.html#E2:121_4_6"><i><font color="Green">E98</font></i></a>, <a class="txt" href="yellow_6.html#E4:121_4_6_1"><i><font color="Green">E100</font></i></a></i>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>23</sub></font> = <font color="Maroon">c<sub>22</sub></font> as  <a href="relat_1.html#V2">V3</a>  <a href="pboole.html#M1">ManySortedSet</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>13</sub></font> <b>by </b><i><a class="ref" href="relat_1.html#D9">RELAT_1:def 9</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>24</sub></font> = <font color="Maroon">c<sub>23</sub></font> as  <a href="relat_1.html#V2">V3</a>  <a href="pboole.html#M1">ManySortedSet</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>16</sub></font> <b>by </b><i><a class="txt" href="yellow_6.html#E16:121_4"><i><font color="Green">E86</font></i></a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>16</sub></font>) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="yellow_6.html#K8" target="_self">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>;<br/>
<b>consider </b><font color="Maroon">c<sub>25</sub></font> being    <a href="pboole.html#M1">ManySortedSet</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>16</sub></font><b> such that </b><br/><a NAME="E7:121_4_6"/><i><font color="Green">E99</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>16</sub></font> holds <font color="Maroon">c<sub>25</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b<sub>1</sub></font>)
 <b>from </b><i><a class="ref" href="pboole.html#S5">PBOOLE:sch 5</a>();<br/></i>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] means ex <font color="Olive">b<sub>1</sub></font> being  <a href="funct_1.html#NM1">Function</a>ex <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>16</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>3</sub></font> &amp; the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="yellow_6.html#K8" target="_self">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> );<br/>
<a NAME="E8:121_4_6"/><i><font color="Green">E100</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>16</sub></font> holds <br/>for <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>25</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> holds <br/>ex <font color="Olive">b<sub>3</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>24</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="121_4_6_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>26</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:121_4_6_2"/><i><font color="Green">E101</font></i>: 
<font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>16</sub></font>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>27</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E2:121_4_6_2"/><i><font color="Green">E102</font></i>: 
<font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>25</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>26</sub></font>
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>28</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>13</sub></font>, <font color="Maroon">c<sub>29</sub></font> being   <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E4:121_4_6_2"/><i><font color="Green">E103</font></i>: 
<font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>28</sub></font>
 <b>and </b><br/><a NAME="E5:121_4_6_2"/><i><font color="Green">E104</font></i>: 
<font color="Maroon">c<sub>29</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>26</sub></font>
 <b>and </b><br/><a NAME="E6:121_4_6_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>29</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> <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E7:121_4_6_2"/><i><font color="Green">E105</font></i>: 
 <a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">c<sub>29</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="waybel_0.html#K2">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>13</sub></font> : <font color="Maroon">c<sub>28</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> </span>}</span> 
 <b>by </b><i><a class="txt" href="yellow_6.html#E31:121_4"><i><font color="Green">E92</font></i></a>, <a class="txt" href="yellow_6.html#E1:121_4_6_2"><i><font color="Green">E101</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>30</sub></font> = <font color="Maroon">c<sub>26</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>16</sub></font> <b>by </b><i><a class="txt" href="yellow_6.html#E1:121_4_6_2"><i><font color="Green">E101</font></i></a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] means the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">c<sub>29</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>2</sub></font>;<br/>
<a NAME="E9:121_4_6_2"/><i><font color="Green">E106</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>29</sub></font>ex <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>24</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>30</sub></font> st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="121_4_6_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>31</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>29</sub></font>;<br/>

<a NAME="E1:121_4_6_2_1"/>
<font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>29</sub></font>
 
;<br/>
<a NAME="E2:121_4_6_2_1"/><b>then </b>
<font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">c<sub>29</sub></font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E3:121_4_6_2_1"/><b>then </b>
the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">c<sub>29</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">c<sub>29</sub></font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E4:121_4_6_2_1"/><b>then </b>
the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">c<sub>29</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="waybel_0.html#K2">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>13</sub></font> : <font color="Maroon">c<sub>28</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="yellow_6.html#E7:121_4_6_2"><i><font color="Green">E105</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>32</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>13</sub></font><b> such that </b><br/><a NAME="E6:121_4_6_2_1"/><i><font color="Green">E107</font></i>: 
the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">c<sub>29</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">c<sub>32</sub></font>
 <b>and </b><br/><a NAME="E7:121_4_6_2_1"/><i><font color="Green">E108</font></i>: 
<font color="Maroon">c<sub>28</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>32</sub></font>
 ;<br/>
<a NAME="E8:121_4_6_2_1"/>
<font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>13</sub></font> : <font color="Maroon">c<sub>28</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="yellow_6.html#E7:121_4_6_2_1"><i><font color="Green">E108</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>33</sub></font> = <font color="Maroon">c<sub>32</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>24</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>30</sub></font> <b>by </b><i><a class="txt" href="yellow_6.html#E2:121_4_6"><i><font color="Green">E98</font></i></a>, <a class="txt" href="yellow_6.html#E4:121_4_6_2"><i><font color="Green">E103</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>33</sub></font>
;<br/>

<b>thus </b><a NAME="E10:121_4_6_2_1"/>
the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">c<sub>29</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R1">=</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>33</sub></font>
 <b>by </b><i><a class="txt" href="yellow_6.html#E6:121_4_6_2_1"><i><font color="Green">E107</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>31</sub></font> being   <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>29</sub></font>,<font color="Maroon">c<sub>24</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>30</sub></font><b> such that </b><br/><a NAME="E11:121_4_6_2"/><i><font color="Green">E107</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>29</sub></font> holds <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>31</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</a>(<a class="txt" href="yellow_6.html#E9:121_4_6_2"><i><font color="Green">E106</font></i></a>);<br/></i>
<b>reconsider </b><font color="Maroon">c<sub>32</sub></font> = <font color="Maroon">c<sub>31</sub></font> as   <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>29</sub></font>,<font color="Maroon">c<sub>23</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>26</sub></font> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>33</sub></font> = <font color="Maroon">c<sub>27</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>29</sub></font> <b>by </b><i><a class="txt" href="yellow_6.html#E7:121_4_6"><i><font color="Green">E99</font></i></a>, <a class="txt" href="yellow_6.html#E1:121_4_6_2"><i><font color="Green">E101</font></i></a>, <a class="txt" href="yellow_6.html#E2:121_4_6_2"><i><font color="Green">E102</font></i></a>, <a class="txt" href="yellow_6.html#E5:121_4_6_2"><i><font color="Green">E104</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>32</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>27</sub></font>
;<br/>

<a NAME="E14:121_4_6_2"/>
<font color="Maroon">c<sub>32</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>23</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>26</sub></font>
 
;<br/>
<b>hence </b><a NAME="E15:121_4_6_2"/>
<font color="Maroon">c<sub>32</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>24</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>26</sub></font>
 ;<br/>

<b>take </b>
<font color="Maroon">c<sub>32</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>30</sub></font>
;<br/>

<b>thus </b><a NAME="E16:121_4_6_2"/>
( <font color="Maroon">c<sub>32</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>32</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>27</sub></font> &amp; <font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>26</sub></font> )
 ;<br/>

<b>thus </b>the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="yellow_6.html#K8" target="_self">.</a> <font color="Maroon">c<sub>30</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>27</sub></font> = 
the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">c<sub>29</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>33</sub></font>
<b>by </b><i><a class="txt" href="yellow_6.html#E5:121_4_6_2"><i><font color="Green">E104</font></i></a></i>
<br/>.= 
the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>32</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>27</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="yellow_6.html#E11:121_4_6_2"><i><font color="Green">E107</font></i></a></i>
;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>26</sub></font> being    <a href="pboole.html#M3">ManySortedFunction</a> of <font color="Maroon">c<sub>25</sub></font>,<font color="Maroon">c<sub>24</sub></font><b> such that </b><br/><a NAME="E10:121_4_6"/><i><font color="Green">E101</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>16</sub></font> holds <br/>ex <font color="Olive">b<sub>2</sub></font> being  <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>25</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>24</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>26</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>3</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>25</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font>] ) )
 <b>from </b><i><a class="ref" href="mssubfam.html#S1">MSSUBFAM:sch 1</a>(<a class="txt" href="yellow_6.html#E8:121_4_6"><i><font color="Green">E100</font></i></a>);<br/></i>
<a NAME="E11:121_4_6"/><i><font color="Green">E102</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>16</sub></font><br/>for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>25</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> holds the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="yellow_6.html#K8" target="_self">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>26</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="121_4_6_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>27</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>16</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>28</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>25</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>27</sub></font>;<br/>



<b>consider </b><font color="Maroon">c<sub>29</sub></font> being   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>25</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>27</sub></font>,<font color="Maroon">c<sub>24</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>27</sub></font><b> such that </b><br/><a NAME="E2:121_4_6_3"/><i><font color="Green">E103</font></i>: 
<font color="Maroon">c<sub>29</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>26</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>27</sub></font>
 <b>and </b><br/><a NAME="E3:121_4_6_3"/><i><font color="Green">E104</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>25</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>27</sub></font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>29</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>27</sub></font>]
 <b>by </b><i><a class="txt" href="yellow_6.html#E10:121_4_6"><i><font color="Green">E101</font></i></a></i>;<br/>
<a NAME="E4:121_4_6_3"/>
<font color="Maroon">c<sub>25</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="yellow_6.html#K8" target="_self">.</a> <font color="Maroon">c<sub>27</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="yellow_6.html#E7:121_4_6"><i><font color="Green">E99</font></i></a></i>;<br/>
<a NAME="E5:121_4_6_3"/><b>then </b>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>29</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>28</sub></font>,<font color="Maroon">c<sub>28</sub></font>,<font color="Maroon">c<sub>27</sub></font>]
 
<b>by </b><i><a class="txt" href="yellow_6.html#E3:121_4_6_3"><i><font color="Green">E104</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:121_4_6_3"/>
the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="yellow_6.html#K8" target="_self">.</a> <font color="Maroon">c<sub>27</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R1">=</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>26</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>27</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>28</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="yellow_6.html#E2:121_4_6_3"><i><font color="Green">E103</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>deffunc </b><font color="Maroon">H<sub>3</sub></font>(  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>16</sub></font>,  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <font color="Maroon">c<sub>21</sub></font></span>)</span>) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <span class="p1">(<span class="default"><font color="Maroon">c<sub>26</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">a<sub>2</sub></font> <a href="waybel_3.html#K4">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>;<br/>
<a NAME="E12:121_4_6"/><i><font color="Green">E103</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>16</sub></font><br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> holds <font color="Maroon">H<sub>3</sub></font>(<font color="Olive">b<sub>1</sub></font>,<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>13</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="121_4_6_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>27</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>16</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>28</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <font color="Maroon">c<sub>21</sub></font></span>)</span>;<br/>



<b>reconsider </b><font color="Maroon">c<sub>29</sub></font> = <font color="Maroon">c<sub>27</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>13</sub></font> <b>by </b><i><a class="txt" href="yellow_6.html#E16:121_4"><i><font color="Green">E86</font></i></a></i>;<br/>
<a NAME="E2:121_4_6_4"/><i><font color="Green">E104</font></i>: 
<font color="Maroon">c<sub>26</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>29</sub></font> is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>25</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>29</sub></font>,<font color="Maroon">c<sub>23</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>29</sub></font>
 
<b>by </b><i><a class="ref" href="pboole.html#D18">PBOOLE:def 18</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>13</sub></font>] means <font color="Maroon">c<sub>29</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>30</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>13</sub></font> : <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span> ;<br/>
<a NAME="E3:121_4_6_4"/><i><font color="Green">E105</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>13</sub></font> : <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span>  is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>13</sub></font>
 
<b>from </b><i><a class="ref" href="domain_1.html#S7">DOMAIN_1:sch 7</a>();<br/></i>
<a NAME="E4:121_4_6_4"/><i><font color="Green">E106</font></i>: 
<font color="Maroon">c<sub>23</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>29</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>13</sub></font> : <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span> 
 
<b>by </b><i><a class="txt" href="yellow_6.html#E2:121_4_6"><i><font color="Green">E98</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>31</sub></font> = <font color="Maroon">c<sub>27</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>18</sub></font> ;<br/>
<a NAME="E6:121_4_6_4"/>
<font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <font color="Maroon">c<sub>21</sub></font></span>)</span>
 
;<br/>
<a NAME="E7:121_4_6_4"/><b>then </b><i><font color="Green">E107</font></i>: 
<font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">c<sub>21</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_1.html#D4">YELLOW_1:def 4</a></i>;<br/>
<a NAME="E8:121_4_6_4"/>
<font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>18</sub></font>
 
;<br/>
<a NAME="E9:121_4_6_4"/><b>then </b>
<font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">c<sub>21</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E10:121_4_6_4"/><b>then </b>
<font color="Maroon">c<sub>28</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>31</sub></font>
 
<b>by </b><i><a class="txt" href="yellow_6.html#E7:121_4_6_4"><i><font color="Green">E107</font></i></a>, <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/>
<a NAME="E11:121_4_6_4"/><b>then </b>
<font color="Maroon">c<sub>28</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="yellow_6.html#K8" target="_self">.</a> <font color="Maroon">c<sub>27</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_6.html#T9" target="_self">Th9</a></i>;<br/>
<a NAME="E12:121_4_6_4"/><b>then </b>
<font color="Maroon">c<sub>28</sub></font> <a href="waybel_3.html#K4">.</a> <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>25</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>29</sub></font>
 
<b>by </b><i><a class="txt" href="yellow_6.html#E7:121_4_6"><i><font color="Green">E99</font></i></a></i>;<br/>
<a NAME="E13:121_4_6_4"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>26</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>29</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>28</sub></font> <a href="waybel_3.html#K4">.</a> <font color="Maroon">c<sub>27</sub></font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>23</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>29</sub></font>
 
<b>by </b><i><a class="txt" href="yellow_6.html#E2:121_4_6_4"><i><font color="Green">E104</font></i></a>, <a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/>
<b>hence </b><a NAME="E14:121_4_6_4"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>26</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>27</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>28</sub></font> <a href="waybel_3.html#K4">.</a> <font color="Maroon">c<sub>27</sub></font></span>)</span> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="yellow_6.html#E3:121_4_6_4"><i><font color="Green">E105</font></i></a>, <a class="txt" href="yellow_6.html#E4:121_4_6_4"><i><font color="Green">E106</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>27</sub></font> being   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>16</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="yellow_1.html#K5">product</a> <font color="Maroon">c<sub>21</sub></font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>13</sub></font><b> such that </b><br/><a NAME="E14:121_4_6"/><i><font color="Green">E104</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>16</sub></font><br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> holds <font color="Maroon">c<sub>27</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="yellow_3.html#K7">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>)
 <b>from </b><i><a class="ref" href="funct_7.html#S1">FUNCT_7:sch 1</a>(<a class="txt" href="yellow_6.html#E12:121_4_6"><i><font color="Green">E103</font></i></a>);<br/></i>
<b>reconsider </b><font color="Maroon">c<sub>28</sub></font> = <font color="Maroon">c<sub>27</sub></font> as   <a href="struct_0.html#NM7">Function</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9" target="_self">Iterated</a> <font color="Maroon">c<sub>21</sub></font></span>)</span>,<font color="Maroon">c<sub>13</sub></font> <b>by </b><i><a class="txt" href="yellow_6.html#E40:121_4"><i><font color="Green">E97</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>28</sub></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/>

<b>set </b><font color="Maroon">c<sub>29</sub></font> = the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9" target="_self">Iterated</a> <font color="Maroon">c<sub>21</sub></font></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>30</sub></font> = the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">c<sub>19</sub></font>;<br/>
<a NAME="E16:121_4_6"/><i><font color="Green">E105</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9" target="_self">Iterated</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> iff ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>28</sub></font> &amp; <font color="Maroon">c<sub>28</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">c<sub>19</sub></font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="121_4_6_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>31</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:121_4_6_5_1"/>
<font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9" target="_self">Iterated</a> <font color="Maroon">c<sub>21</sub></font></span>)</span>
 ;<br/><a NAME="E2:121_4_6_5_1"/><b>then </b>
<font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9" target="_self">Iterated</a> <font color="Maroon">c<sub>21</sub></font></span>)</span>
 ;<br/><a NAME="E3:121_4_6_5_1"/><b>then </b>
<font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>18</sub></font>,<span class="p2">(<span class="default"><a href="card_3.html#K4">product</a> <span class="p3">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">c<sub>21</sub></font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="ref" href="yellow_6.html#T35" target="_self">Th35</a></i>;<br/><a NAME="E4:121_4_6_5_1"/><b>then </b><i><font color="Green">E106</font></i>: 
<font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>18</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="yellow_1.html#K5">product</a> <font color="Maroon">c<sub>21</sub></font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="ref" href="yellow_1.html#D4">YELLOW_1:def 4</a></i>;<br/><b>hence </b><a NAME="E5:121_4_6_5_1"/>
<font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>28</sub></font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E6:121_4_6_5_1"/>
<font color="Maroon">c<sub>28</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="yellow_6.html#E4:121_4_6_5_1"><i><font color="Green">E106</font></i></a>, <a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/><b>hence </b><a NAME="E7:121_4_6_5_1"/>
<font color="Maroon">c<sub>28</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">c<sub>19</sub></font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:121_4_6_5"/>
( <font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>28</sub></font> &amp; <font color="Maroon">c<sub>28</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">c<sub>19</sub></font> )
 ;<br/>

<a NAME="E3:121_4_6_5"/><b>then </b>
<font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>18</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="yellow_1.html#K5">product</a> <font color="Maroon">c<sub>21</sub></font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E4:121_4_6_5"/><b>then </b>
<font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>18</sub></font>,<span class="p2">(<span class="default"><a href="card_3.html#K4">product</a> <span class="p3">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">c<sub>21</sub></font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="yellow_1.html#D4">YELLOW_1:def 4</a></i>;<br/>
<a NAME="E5:121_4_6_5"/><b>then </b>
<font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9" target="_self">Iterated</a> <font color="Maroon">c<sub>21</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_6.html#T35" target="_self">Th35</a></i>;<br/>
<b>hence </b><a NAME="E6:121_4_6_5"/>
<font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9" target="_self">Iterated</a> <font color="Maroon">c<sub>21</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E17:121_4_6"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9" target="_self">Iterated</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> holds <br/>the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9" target="_self">Iterated</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">c<sub>19</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>28</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="121_4_6_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>31</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:121_4_6_6"/>
<font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9" target="_self">Iterated</a> <font color="Maroon">c<sub>21</sub></font></span>)</span>
 ;<br/>

<a NAME="E2:121_4_6_6"/><b>then </b>
<font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9" target="_self">Iterated</a> <font color="Maroon">c<sub>21</sub></font></span>)</span>
 
;<br/>
<a NAME="E3:121_4_6_6"/><b>then </b>
<font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>18</sub></font>,<span class="p2">(<span class="default"><a href="card_3.html#K4">product</a> <span class="p3">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">c<sub>21</sub></font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="yellow_6.html#T35" target="_self">Th35</a></i>;<br/>
<a NAME="E4:121_4_6_6"/><b>then </b>
<font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>18</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="yellow_1.html#K5">product</a> <font color="Maroon">c<sub>21</sub></font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="yellow_1.html#D4">YELLOW_1:def 4</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>32</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>18</sub></font>, <font color="Maroon">c<sub>33</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <font color="Maroon">c<sub>21</sub></font></span>)</span><b> such that </b><br/><a NAME="E6:121_4_6_6"/><i><font color="Green">E106</font></i>: 
<font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">c<sub>32</sub></font>,<font color="Maroon">c<sub>33</sub></font></span><a href="yellow_3.html#K7">]</a></span>
 <b>by </b><i><a class="ref" href="domain_1.html#T9">DOMAIN_1:9</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>34</sub></font> = <font color="Maroon">c<sub>32</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>16</sub></font> ;<br/>
<a NAME="E8:121_4_6_6"/><i><font color="Green">E107</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>18</sub></font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E9:121_4_6_6"/>
<font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <font color="Maroon">c<sub>21</sub></font></span>)</span>
 
;<br/>
<a NAME="E10:121_4_6_6"/><b>then </b><i><font color="Green">E108</font></i>: 
<font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">c<sub>21</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_1.html#D4">YELLOW_1:def 4</a></i>;<br/>
<a NAME="E11:121_4_6_6"/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="yellow_6.html#K8" target="_self">.</a> <font color="Maroon">c<sub>34</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>32</sub></font>
 
<b>by </b><i><a class="ref" href="yellow_6.html#T9" target="_self">Th9</a></i>;<br/>
<a NAME="E12:121_4_6_6"/><b>then </b>
<font color="Maroon">c<sub>33</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="yellow_6.html#K8" target="_self">.</a> <font color="Maroon">c<sub>34</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="yellow_6.html#E8:121_4_6_6"><i><font color="Green">E107</font></i></a>, <a class="txt" href="yellow_6.html#E10:121_4_6_6"><i><font color="Green">E108</font></i></a>, <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>35</sub></font> = <font color="Maroon">c<sub>33</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>32</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>25</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>34</sub></font> <b>by </b><i><a class="txt" href="yellow_6.html#E7:121_4_6"><i><font color="Green">E99</font></i></a></i>;<br/>
<b>thus </b>the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9" target="_self">Iterated</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>31</sub></font> = 
the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9" target="_self">Iterated</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>32</sub></font>,<font color="Maroon">c<sub>33</sub></font>
<b>by </b><i><a class="txt" href="yellow_6.html#E6:121_4_6_6"><i><font color="Green">E106</font></i></a></i>
<br/>.= 
the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="yellow_6.html#K8" target="_self">.</a> <font color="Maroon">c<sub>34</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>33</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>32</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="yellow_6.html#D16" target="_self">Def16</a></i>
<br/>.= 
the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">c<sub>19</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>26</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>34</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>35</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="yellow_6.html#E11:121_4_6"><i><font color="Green">E102</font></i></a></i>
<br/>.= 
the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">c<sub>19</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>28</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>31</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="yellow_6.html#E14:121_4_6"><i><font color="Green">E104</font></i></a>, <a class="txt" href="yellow_6.html#E6:121_4_6_6"><i><font color="Green">E106</font></i></a></i>
;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E18:121_4_6"/>
the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9" target="_self">Iterated</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="funct_2.html#R2">=</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">c<sub>13</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>28</sub></font>
 <b>by </b><i><a class="txt" href="yellow_6.html#E16:121_4_6"><i><font color="Green">E105</font></i></a>, <a class="ref" href="funct_1.html#T20">FUNCT_1:20</a></i>;<br/>

<b>let </b><font color="Maroon">c<sub>31</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>13</sub></font>;<br/>

<b>consider </b><font color="Maroon">c<sub>32</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <font color="Maroon">c<sub>21</sub></font></span>)</span>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>33</sub></font> = <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">c<sub>31</sub></font>,<font color="Maroon">c<sub>32</sub></font></span><a href="yellow_3.html#K7">]</a></span> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9" target="_self">Iterated</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <b>by </b><i><a class="txt" href="yellow_6.html#E16:121_4"><i><font color="Green">E86</font></i></a>, <a class="txt" href="yellow_6.html#E40:121_4"><i><font color="Green">E97</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>34</sub></font> = <font color="Maroon">c<sub>32</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>35</sub></font> = <font color="Maroon">c<sub>31</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>16</sub></font> <b>by </b><i><a class="txt" href="yellow_6.html#E16:121_4"><i><font color="Green">E86</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>33</sub></font>
;<br/>

<b>let </b><font color="Maroon">c<sub>36</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9" target="_self">Iterated</a> <font color="Maroon">c<sub>21</sub></font></span>)</span>;<br/>

<b>consider </b><font color="Maroon">c<sub>37</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>16</sub></font>, <font color="Maroon">c<sub>38</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <font color="Maroon">c<sub>21</sub></font></span>)</span><b> such that </b><br/><a NAME="E24:121_4_6"/><i><font color="Green">E106</font></i>: 
<font color="Maroon">c<sub>36</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">c<sub>37</sub></font>,<font color="Maroon">c<sub>38</sub></font></span><a href="yellow_3.html#K7">]</a></span>
 <b>by </b><i><a class="txt" href="yellow_6.html#E40:121_4"><i><font color="Green">E97</font></i></a>, <a class="ref" href="domain_1.html#T9">DOMAIN_1:9</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>39</sub></font> = <font color="Maroon">c<sub>37</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>13</sub></font> <b>by </b><i><a class="txt" href="yellow_6.html#E16:121_4"><i><font color="Green">E86</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>40</sub></font> = <font color="Maroon">c<sub>37</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>18</sub></font> ;<br/>
<a NAME="E27:121_4_6"/><i><font color="Green">E107</font></i>: 
<font color="Maroon">c<sub>28</sub></font> <a href="yellow_6.html#K3" target="_self">.</a> <font color="Maroon">c<sub>36</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>26</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>39</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>38</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>39</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="yellow_6.html#E14:121_4_6"><i><font color="Green">E104</font></i></a>, <a class="txt" href="yellow_6.html#E24:121_4_6"><i><font color="Green">E106</font></i></a></i>;<br/>
<a NAME="E28:121_4_6"/><i><font color="Green">E108</font></i>: 
<font color="Maroon">c<sub>26</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>39</sub></font> is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>25</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>39</sub></font>,<font color="Maroon">c<sub>23</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>39</sub></font>
 
<b>by </b><i><a class="ref" href="pboole.html#D18">PBOOLE:def 18</a></i>;<br/>
<b>then </b><i><font color="Green">E109</font></i>:  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>26</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>39</sub></font></span>)</span> = 
<font color="Maroon">c<sub>25</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>39</sub></font>
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
<br/>.= 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="yellow_6.html#K8" target="_self">.</a> <font color="Maroon">c<sub>37</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="yellow_6.html#E7:121_4_6"><i><font color="Green">E99</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>40</sub></font>
<b>by </b><i><a class="ref" href="yellow_6.html#T9" target="_self">Th9</a></i>
;<br/>
<a NAME="E30:121_4_6"/><i><font color="Green">E110</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>18</sub></font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E31:121_4_6"/>
<font color="Maroon">c<sub>38</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <font color="Maroon">c<sub>21</sub></font></span>)</span>
 
;<br/>
<a NAME="E32:121_4_6"/><b>then </b>
<font color="Maroon">c<sub>38</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">c<sub>21</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_1.html#D4">YELLOW_1:def 4</a></i>;<br/>
<a NAME="E33:121_4_6"/><b>then </b>
<font color="Maroon">c<sub>38</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>40</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>26</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>39</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="yellow_6.html#E29:121_4_6"><i><font color="Green">E109</font></i></a>, <a class="txt" href="yellow_6.html#E30:121_4_6"><i><font color="Green">E110</font></i></a>, <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/>
<a NAME="E34:121_4_6"/><b>then </b><i><font color="Green">E111</font></i>: 
<font color="Maroon">c<sub>28</sub></font> <a href="yellow_6.html#K3" target="_self">.</a> <font color="Maroon">c<sub>36</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>26</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>39</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="yellow_6.html#E27:121_4_6"><i><font color="Green">E107</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E35:121_4_6"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>26</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>39</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>23</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>39</sub></font>
 
<b>by </b><i><a class="txt" href="yellow_6.html#E28:121_4_6"><i><font color="Green">E108</font></i></a>, <a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/>
<a NAME="E36:121_4_6"/><b>then </b>
<font color="Maroon">c<sub>28</sub></font> <a href="yellow_6.html#K3" target="_self">.</a> <font color="Maroon">c<sub>36</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>23</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>39</sub></font>
 
<b>by </b><i><a class="txt" href="yellow_6.html#E34:121_4_6"><i><font color="Green">E111</font></i></a></i>;<br/>
<a NAME="E37:121_4_6"/><b>then </b>
<font color="Maroon">c<sub>28</sub></font> <a href="yellow_6.html#K3" target="_self">.</a> <font color="Maroon">c<sub>36</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>13</sub></font> : <font color="Maroon">c<sub>39</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="yellow_6.html#E2:121_4_6"><i><font color="Green">E98</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>41</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>13</sub></font><b> such that </b><br/><a NAME="E39:121_4_6"/><i><font color="Green">E112</font></i>: 
<font color="Maroon">c<sub>41</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>28</sub></font> <a href="yellow_6.html#K3" target="_self">.</a> <font color="Maroon">c<sub>36</sub></font>
 <b>and </b><br/><a NAME="E40:121_4_6"/><i><font color="Green">E113</font></i>: 
<font color="Maroon">c<sub>39</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>41</sub></font>
 ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>42</sub></font> = <font color="Maroon">c<sub>38</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>43</sub></font> = <font color="Maroon">c<sub>39</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>16</sub></font> ;<br/>
<b>assume </b><a NAME="E43:121_4_6"/>
<font color="Maroon">c<sub>33</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>36</sub></font>
 ;<br/>

<a NAME="E44:121_4_6"/><b>then </b>
<span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">c<sub>35</sub></font>,<font color="Maroon">c<sub>34</sub></font></span><a href="yellow_3.html#K7">]</a></span> <a href="orders_2.html#R1">&lt;=</a> <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">c<sub>43</sub></font>,<font color="Maroon">c<sub>42</sub></font></span><a href="yellow_3.html#K7">]</a></span>
 
<b>by </b><i><a class="txt" href="yellow_6.html#E39:121_4"><i><font color="Green">E96</font></i></a>, <a class="txt" href="yellow_6.html#E24:121_4_6"><i><font color="Green">E106</font></i></a>, <a class="txt" href="yellow_6.html#E20"><i><font color="Green">Lemma13</font></i></a></i>;<br/>
<a NAME="E45:121_4_6"/><b>then </b>
<font color="Maroon">c<sub>35</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>43</sub></font>
 
<b>by </b><i><a class="ref" href="yellow_3.html#T11">YELLOW_3:11</a></i>;<br/>
<a NAME="E46:121_4_6"/><b>then </b>
<font color="Maroon">c<sub>31</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>39</sub></font>
 
<b>by </b><i><a class="txt" href="yellow_6.html#E16:121_4"><i><font color="Green">E86</font></i></a>, <a class="txt" href="yellow_6.html#E20"><i><font color="Green">Lemma13</font></i></a></i>;<br/>
<b>hence </b><a NAME="E47:121_4_6"/>
<font color="Maroon">c<sub>31</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>28</sub></font> <a href="yellow_6.html#K3" target="_self">.</a> <font color="Maroon">c<sub>36</sub></font>
 <b>by </b><i><a class="txt" href="yellow_6.html#E39:121_4_6"><i><font color="Green">E112</font></i></a>, <a class="txt" href="yellow_6.html#E40:121_4_6"><i><font color="Green">E113</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="E42:121_4"/>
contradiction
 <b>by </b><i><a class="txt" href="yellow_6.html#E14:121_4"><i><font color="Green">E85</font></i></a>, <a class="txt" href="yellow_6.html#E38:121_4"><i><font color="Green">E95</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:121"/>
<font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a>  <a href="yellow_6.html#K13" target="_self">Convergence</a> <span class="p1">(<span class="default"><a href="yellow_6.html#K14" target="_self">ConvergenceSpace</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_6.html#T49" target="_self">Th49</a></i>;<br/>
<b>hence </b><a NAME="E10:121"/>
 <a href="yellow_6.html#K13" target="_self">Convergence</a> <span class="p1">(<span class="default"><a href="yellow_6.html#K14" target="_self">ConvergenceSpace</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="yellow_6.html#E8:121"><i><font color="Green">E78</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div>
