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

<b>let </b><font color="Maroon">T</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</font> be    <a href="yellow_6.html#M4" target="_self">Convergence-Class</a> of <font color="Maroon">T</font>;<br/>



<b>set </b><font color="Maroon">CC</font> =  <a href="yellow_6.html#K14" target="_self">ConvergenceSpace</a> <font color="Maroon">C</font>;<br/>
<b>set </b><font color="Maroon">CCC</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</font></span>)</span>;<br/>
<a NAME="E1:121"/><i><font color="Green">E29</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</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>
 
<b>by </b><i><a class="ref" href="yellow_6.html#T11" target="_self">Th11</a></i>;<br/>
<a NAME="E2:121"/><i><font color="Green">E32</font></i>: 
for <font color="Olive">N</font> being  <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">T</font> holds  <font color="Olive">N</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</font>
 
<b>by </b><i><a class="ref" href="yellow_6.html#T11" target="_self">Th11</a></i>;<br/>
<a NAME="E3:121"/><i><font color="Green">E41</font></i>: 
for <font color="Olive">N</font> being  <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">T</font><br/> for <font color="Olive">n</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</font>  st <font color="Olive">N</font> <a href="hidden.html#R1">=</a> <font color="Olive">n</font> holds <br/>for <font color="Olive">X</font> being   <a href="yellow_6.html#M2" target="_self">subnet</a> of <font color="Olive">N</font> holds  <font color="Olive">X</font> is    <a href="yellow_6.html#M2" target="_self">subnet</a> of <font color="Olive">n</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">N</font> be   <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">T</font>;<br/><b>let </b><font color="Maroon">n</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</font>;<br/>



<b>assume </b><a NAME="E1:121_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">N</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 ;<br/>

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

<b>consider </b><font color="Maroon">f</font> being   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X</font>,<font color="Maroon">N</font><b> such that </b><br/><a NAME="E3:121_1"/><i><font color="Green">E55</font></i>: 
( the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">X</font> <a href="funct_2.html#R2">=</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">N</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font> &amp; ( for <font color="Olive">m</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font>  ex <font color="Olive">n</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font> st <br/>for <font color="Olive">p</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font>  st <font color="Olive">n</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">p</font> holds <br/><font color="Olive">m</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">f</font> <a href="yellow_6.html#K3" target="_self">.</a> <font color="Olive">p</font> ) )
 <b>by </b><i><a class="txt" href="yellow_6.html#E1:121"><i><font color="Green">E29</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">X</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</font> <b>by </b><i><a class="ref" href="yellow_6.html#T11" target="_self">Th11</a></i>;<br/>
<b>reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">x</font>,<font color="Maroon">n</font> <b>by </b><i/>;<br/>
<a NAME="E6:121_1"/>
the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">x</font> <a href="funct_2.html#R2">=</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">n</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E7:121_1"/>
<font color="Maroon">X</font> is    <a href="yellow_6.html#M2" target="_self">subnet</a> of <font color="Maroon">n</font>
 <b>by </b><i>, , <a class="txt" href="yellow_6.html#E1:121"><i><font color="Green">E29</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:121"/><i><font color="Green">E56</font></i>: 
for <font color="Olive">N</font> being  <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">T</font><br/> for <font color="Olive">n</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</font>  st <font color="Olive">N</font> <a href="hidden.html#R1">=</a> <font color="Olive">n</font> holds <br/>for <font color="Olive">x</font> being   <a href="yellow_6.html#M2" target="_self">subnet</a> of <font color="Olive">n</font> holds  <font color="Olive">x</font> is    <a href="yellow_6.html#M2" target="_self">subnet</a> of <font color="Olive">N</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">N</font> be   <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">T</font>;<br/><b>let </b><font color="Maroon">n</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</font>;<br/>



<b>assume </b><a NAME="E1:121_2"/><i><font color="Green">E117</font></i>: 
<font color="Maroon">N</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 ;<br/>

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

<b>consider </b><font color="Maroon">f</font> being   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X</font>,<font color="Maroon">n</font><b> such that </b><br/><a NAME="E3:121_2"/><i><font color="Green">E118</font></i>: 
( the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">X</font> <a href="funct_2.html#R2">=</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">n</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font> &amp; ( for <font color="Olive">m</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">n</font>  ex <font color="Olive">n</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font> st <br/>for <font color="Olive">p</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font>  st <font color="Olive">n</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">p</font> holds <br/><font color="Olive">m</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">f</font> <a href="yellow_6.html#K3" target="_self">.</a> <font color="Olive">p</font> ) )
 <b>by </b><i><a class="txt" href="yellow_6.html#E1:121"><i><font color="Green">E29</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">X</font> as   <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">T</font> <b>by </b><i><a class="ref" href="yellow_6.html#T11" target="_self">Th11</a></i>;<br/>
<b>reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">x</font>,<font color="Maroon">N</font> <b>by </b><i/>;<br/>
<a NAME="E6:121_2"/>
the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">x</font> <a href="funct_2.html#R2">=</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">N</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E7:121_2"/>
<font color="Maroon">X</font> is    <a href="yellow_6.html#M2" target="_self">subnet</a> of <font color="Maroon">N</font>
 <b>by </b><i>, , <a class="txt" href="yellow_6.html#E1:121"><i><font color="Green">E29</font></i></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">E119</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</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font>
 ;<br/><b>thus </b><a NAME="E2:121_3"/>
<font color="Maroon">C</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</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">N</font> be  <a href="yellow_6.html#V1" target="_self">constant</a> <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">T</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">M</font> = <font color="Maroon">N</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</font> <b>by </b><i><a class="ref" href="yellow_6.html#T11" target="_self">Th11</a></i>;<br/>
<a NAME="E2:121_3_1_1"/>
the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">N</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">M</font> is <a href="funct_1.html#V3">constant</a>
 
;<br/>
<b>then reconsider </b><font color="Maroon">M</font> = <font color="Maroon">M</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</font> <b>by </b><i/>;<br/>
<i><font color="Green">E127</font></i>:  <a href="yellow_6.html#K5" target="_self">the_value_of</a> <font color="Maroon">M</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">M</font>
<b>by </b><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">N</font>

<br/>.= 
 <a href="yellow_6.html#K5" target="_self">the_value_of</a> <font color="Maroon">N</font>
<b>by </b><i/>
;<br/>
<b>assume </b><a NAME="E6:121_3_1_1"/>
<font color="Maroon">N</font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K7" target="_self">NetUniv</a> <font color="Maroon">T</font>
 ;<br/>

<a NAME="E7:121_3_1_1"/><b>then </b>
<font color="Maroon">M</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</font></span>)</span>
 
<b>by </b><i>, <a class="txt" href="yellow_6.html#E3:121"><i><font color="Green">E41</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">N</font>,<span class="p2">(<span class="default"><a href="yellow_6.html#K5" target="_self">the_value_of</a> <font color="Maroon">N</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 <b>by </b><i>, , </i>;<br/>


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

<b>thus </b><a NAME="E2:121_3_1"/>
<font color="Maroon">C</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">N</font> be   <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">T</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">Y</font> be    <a href="yellow_6.html#M2" target="_self">subnet</a> of <font color="Maroon">N</font>;<br/>



<b>reconsider </b><font color="Maroon">M</font> = <font color="Maroon">N</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</font> <b>by </b><i><a class="ref" href="yellow_6.html#T11" target="_self">Th11</a></i>;<br/>
<b>reconsider </b><font color="Maroon">X</font> = <font color="Maroon">Y</font> as    <a href="yellow_6.html#M2" target="_self">subnet</a> of <font color="Maroon">M</font> <b>by </b><i><a class="txt" href="yellow_6.html#E20"><i><font color="Green">Lemma75</font></i></a></i>;<br/>
<b>assume </b><a NAME="E3:121_3_1_2"/>
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K7" target="_self">NetUniv</a> <font color="Maroon">T</font>
 ;<br/>

<a NAME="E4:121_3_1_2"/><b>then </b><i><font color="Green">E128</font></i>: 
<font color="Maroon">X</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</font></span>)</span>
 
<b>by </b><i>, <a class="txt" href="yellow_6.html#E3:121"><i><font color="Green">E41</font></i></a></i>;<br/>
<b>let </b><font color="Maroon">p</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font>;<br/>

<b>reconsider </b><font color="Maroon">q</font> = <font color="Maroon">p</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</font></span>)</span> <b>by </b><i><a class="ref" href="yellow_6.html#T11" target="_self">Th11</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">N</font>,<font color="Maroon">p</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">C</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">M</font>,<font color="Maroon">q</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</font></span>)</span>
 
<b>by </b><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">Y</font>,<font color="Maroon">p</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 <b>by </b><i>, , <a class="txt" href="yellow_6.html#E17"><i><font color="Green">Lemma64</font></i></a></i>;<br/>


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

<b>thus </b><a NAME="E3:121_3_1"/>
<font color="Maroon">C</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">X</font> be   <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">T</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">p</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font>;<br/>



<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">X</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</font> <b>by </b><i><a class="ref" href="yellow_6.html#T11" target="_self">Th11</a></i>;<br/>
<b>reconsider </b><font color="Maroon">q</font> = <font color="Maroon">p</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</font></span>)</span> <b>by </b><i><a class="ref" href="yellow_6.html#T11" target="_self">Th11</a></i>;<br/>
<b>assume </b><a NAME="E3:121_3_1_3"/>
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K7" target="_self">NetUniv</a> <font color="Maroon">T</font>
 ;<br/>

<a NAME="E4:121_3_1_3"/><b>then </b><i><font color="Green">E129</font></i>: 
<font color="Maroon">x</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</font></span>)</span>
 
<b>by </b><i>, <a class="txt" href="yellow_6.html#E3:121"><i><font color="Green">E41</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">X</font>,<font color="Maroon">p</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 ;<br/>

<b>then consider </b><font color="Maroon">y</font> being    <a href="yellow_6.html#M2" target="_self">subnet</a> of <font color="Maroon">x</font><b> such that </b><br/><a NAME="E7:121_3_1_3"/><i><font color="Green">E147</font></i>: 
<font color="Maroon">y</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</font></span>)</span>
 <b>and </b><br/><a NAME="E8:121_3_1_3"/><i><font color="Green">E149</font></i>: 
for <font color="Olive">z</font> being   <a href="yellow_6.html#M2" target="_self">subnet</a> of <font color="Maroon">y</font>  holds not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">z</font>,<font color="Maroon">q</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</font></span>)</span>
 <b>by </b><i>, , </i>;<br/>
<b>reconsider </b><font color="Maroon">Y</font> = <font color="Maroon">y</font> as    <a href="yellow_6.html#M2" target="_self">subnet</a> of <font color="Maroon">X</font> <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">Y</font>
;<br/>

<b>thus </b><a NAME="E10:121_3_1_3"/>
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K7" target="_self">NetUniv</a> <font color="Maroon">T</font>
 <b>by </b><i>, <a class="ref" href="yellow_6.html#D6" target="_self">Def6</a>, <a class="txt" href="yellow_6.html#E3:121"><i><font color="Green">E41</font></i></a></i>;<br/>

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

<b>reconsider </b><font color="Maroon">z</font> = <font color="Maroon">Z</font> as    <a href="yellow_6.html#M2" target="_self">subnet</a> of <font color="Maroon">y</font> <b>by </b><i><a class="txt" href="yellow_6.html#E20"><i><font color="Green">Lemma75</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">z</font>,<font color="Maroon">q</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</font></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_6.html#D7" target="_self">Def7</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">Z</font>,<font color="Maroon">p</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 <b>by </b><i/>;<br/>


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

<b>thus </b><a NAME="E4:121_3_1"/>
<font color="Maroon">C</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">X</font> be   <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">T</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">p</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font>;<br/>



<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">X</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</font> <b>by </b><i><a class="ref" href="yellow_6.html#T11" target="_self">Th11</a></i>;<br/>
<b>reconsider </b><font color="Maroon">q</font> = <font color="Maroon">p</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</font></span>)</span> <b>by </b><i><a class="ref" href="yellow_6.html#T11" target="_self">Th11</a></i>;<br/>
<b>assume </b><a NAME="E3:121_3_1_4"/><i><font color="Green">E181</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">X</font>,<font color="Maroon">p</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 ;<br/>

<b>let </b><font color="Maroon">J</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">X</font>,<font color="Maroon">T</font>;<br/>

<b>reconsider </b><font color="Maroon">I</font> = <font color="Maroon">J</font> as    <a href="pboole.html#M1">ManySortedSet</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">x</font> ;<br/>
<a NAME="E5:121_3_1_4"/>
<font color="Maroon">I</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">x</font>, <a href="yellow_6.html#K14" target="_self">ConvergenceSpace</a> <font color="Maroon">C</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">i</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">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">I</font>
 ;<br/>

<a NAME="E2:121_3_1_4_1"/><b>then </b>
<font color="Maroon">i</font> is   <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">T</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:121_3_1_4_1"/>
<font color="Maroon">i</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</font>
 <b>by </b><i><a class="ref" href="yellow_6.html#T12" target="_self">Th12</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">I</font> = <font color="Maroon">J</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">x</font>, <a href="yellow_6.html#K14" target="_self">ConvergenceSpace</a> <font color="Maroon">C</font> ;<br/>
<b>assume </b><a NAME="E7:121_3_1_4"/><i><font color="Green">E182</font></i>: 
for <font color="Olive">i</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font> holds  <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">J</font> <a href="yellow_6.html#K8" target="_self">.</a> <font color="Olive">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">i</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">C</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">i</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">x</font>;<br/><b>reconsider </b><font color="Maroon">j</font> = <font color="Maroon">i</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font> ;<br/><a NAME="E2:121_3_1_4_2"/>
<font color="Maroon">X</font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">i</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">I</font> <a href="yellow_6.html#K8" target="_self">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">i</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</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="yellow_6.html#D6" target="_self">Def6</a></i>;<br/></div><b>end;</b></div>
<a NAME="E9:121_3_1_4"/><b>then </b><i><font color="Green">E183</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">I</font></span>)</span>,<font color="Maroon">q</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</font></span>)</span>
 
<b>by </b><i>, , </i>;<br/>
<i><font color="Green">E184</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">I</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">I</font></span>)</span> #) = 
<span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">X</font>,<span class="p2">(<span class="default"><a href="yellow_1.html#K5">product</a> <font color="Maroon">J</font></span>)</span></span><a href="yellow_3.html#K3">:]</a></span>
<b>by </b><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">J</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">J</font></span>)</span> #)
<b>by </b><i/>
;<br/>
<a NAME="E11:121_3_1_4"/><b>then </b><i><font color="Green">E185</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">I</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">J</font></span>)</span>
 
;<br/>
<a NAME="E12:121_3_1_4"/><i><font color="Green">E186</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">I</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">J</font></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_6.html#T13" target="_self">Th13</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">I</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">I</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">E187</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">I</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">J</font></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_6.html#T13" target="_self">Th13</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">j</font> be    <a href="hidden.html#M1">set</a> ;<br/><a NAME="E1:121_3_1_4_4"/><i><font color="Green">E190</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">I</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">x</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">I</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i/>;<br/><a NAME="E2:121_3_1_4_4"/><i><font color="Green">E191</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">J</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">X</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">J</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i/>;<br/><b>assume </b><a NAME="E3:121_3_1_4_4"/>
<font color="Maroon">j</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">I</font></span>)</span>
 ;<br/><b>then reconsider </b><font color="Maroon">jj</font> = <font color="Maroon">j</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">I</font></span>)</span> ;<br/><b>consider </b><font color="Maroon">j1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">x</font>, <font color="Maroon">j2</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">I</font></span>)</span><b> such that </b><br/><a NAME="E6:121_3_1_4_4"/><i><font color="Green">E193</font></i>: 
<font color="Maroon">jj</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">j1</font>,<font color="Maroon">j2</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="yellow_6.html#T16" target="_self">Th16</a>, <a class="ref" href="domain_1.html#T9">DOMAIN_1:9</a></i>;<br/><b>reconsider </b><font color="Maroon">j2</font> = <font color="Maroon">j2</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">I</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">i1</font> = <font color="Maroon">j1</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font> ;<br/><b>set </b><font color="Maroon">i2</font> = <font color="Maroon">j2</font>;<br/><b>reconsider </b><font color="Maroon">i</font> = <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">i1</font>,<font color="Maroon">j2</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">J</font></span>)</span> <b>by </b><i><a class="ref" href="yellow_6.html#D8" target="_self">Def8</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">I</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> = 
<span class="p1">(<span class="default"><a href="yellow_6.html#K9" target="_self">Iterated</a> <font color="Maroon">I</font></span>)</span> <a href="waybel_0.html#K2">.</a> <font color="Maroon">jj</font>

<br/>.= 
the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="yellow_6.html#K8" target="_self">.</a> <font color="Maroon">j1</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">j2</font> <a href="waybel_3.html#K4">.</a> <font color="Maroon">j1</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="yellow_6.html#K8" target="_self">.</a> <font color="Maroon">i1</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">j2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i1</font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><a href="yellow_6.html#K9" target="_self">Iterated</a> <font color="Maroon">J</font></span>)</span> <a href="waybel_0.html#K2">.</a> <font color="Maroon">i</font>
<b>by </b><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">J</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
<b>by </b><i/>
;<br/></div><b>end;</b></div>
<a NAME="E16:121_3_1_4"/><b>then </b><i><font color="Green">E195</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">I</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">J</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>
<a NAME="E17:121_3_1_4"/>
 <a href="yellow_6.html#K9" target="_self">Iterated</a> <font color="Maroon">I</font> <a href="hidden.html#R1">=</a>  <a href="yellow_6.html#K9" target="_self">Iterated</a> <font color="Maroon">J</font>
 
<b>by </b><i><a class="ref" href="yellow_6.html#T14" target="_self">Th14</a>, <a class="ref" href="yellow_6.html#T15" target="_self">Th15</a>, <a class="ref" href="yellow_6.html#T16" target="_self">Th16</a></i>;<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">J</font></span>)</span>,<font color="Maroon">p</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 <b>by </b><i>, <a class="ref" href="yellow_6.html#D7" target="_self">Def7</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">E196</font></i>: 
( <font color="Maroon">C</font> is <a href="yellow_6.html#V4" target="_self">(CONSTANTS)</a> &amp; <font color="Maroon">C</font> is <a href="yellow_6.html#V5" target="_self">(SUBNETS)</a> &amp; <font color="Maroon">C</font> is <a href="yellow_6.html#V6" target="_self">(DIVERGENCE)</a> &amp; <font color="Maroon">C</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'</font> = <font color="Maroon">C</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">T</font> <b>by </b><i/>;<br/>
<a NAME="E8:121"/><i><font color="Green">E197</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</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">C</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">x</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">y</font> be    <a href="hidden.html#M1">set</a> ;<br/>



<a NAME="E1:121_4"/><i><font color="Green">E198</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</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</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</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i/>;<br/>
<b>assume </b><a NAME="E2:121_4"/><i><font color="Green">E199</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</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</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">M</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</font></span>)</span>, <font color="Maroon">p</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</font></span>)</span><b> such that </b><br/><a NAME="E4:121_4"/><i><font color="Green">E200</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</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">M</font>,<font color="Maroon">p</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="yellow_6.html#D6" target="_self">Def6</a>, <a class="ref" href="domain_1.html#T9">DOMAIN_1:9</a></i>;<br/>
<a NAME="E5:121_4"/><i><font color="Green">E201</font></i>: 
<font color="Maroon">M</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</font></span>)</span>
 
;<br/>
<a NAME="E6:121_4"/>
 ex <font color="Olive">N</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</font> st <br/>( <font color="Olive">N</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">N</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</font></span>)</span> )
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">M</font> = <font color="Maroon">M</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</font> ;<br/>
<b>reconsider </b><font color="Maroon">N</font> = <font color="Maroon">M</font> as   <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">T</font> <b>by </b><i><a class="ref" href="yellow_6.html#T11" target="_self">Th11</a></i>;<br/>
<b>reconsider </b><font color="Maroon">q</font> = <font color="Maroon">p</font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font> <b>by </b><i><a class="ref" href="yellow_6.html#T11" target="_self">Th11</a></i>;<br/>
<a NAME="E10:121_4"/><i><font color="Green">E202</font></i>: 
<font color="Maroon">N</font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K7" target="_self">NetUniv</a> <font color="Maroon">T</font>
 
<b>by </b><i>, <a class="ref" href="yellow_6.html#T14" target="_self">Th14</a>, <a class="txt" href="yellow_6.html#E3:121"><i><font color="Green">E41</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">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 ;<br/>

<b>then consider </b><font color="Maroon">Y</font> being    <a href="yellow_6.html#M2" target="_self">subnet</a> of <font color="Maroon">N</font><b> such that </b><br/><a NAME="E13:121_4"/><i><font color="Green">E203</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K7" target="_self">NetUniv</a> <font color="Maroon">T</font>
 <b>and </b><br/><a NAME="E14:121_4"/><i><font color="Green">E204</font></i>: 
for <font color="Olive">Z</font> being   <a href="yellow_6.html#M2" target="_self">subnet</a> of <font color="Maroon">Y</font>  holds not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">Z</font>,<font color="Maroon">q</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 <b>by </b><i>, <a class="ref" href="yellow_6.html#T13" target="_self">Th13</a>, <a class="ref" href="yellow_6.html#T15" target="_self">Th15</a>, </i>;<br/>
<b>reconsider </b><font color="Maroon">YY</font> =  <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">Y</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>, </i>;<br/>
<b>set </b><font color="Maroon">X</font> = <font color="Maroon">YY</font> <a href="yellow_6.html#K4" target="_self">--&gt;</a> <font color="Maroon">q</font>;<br/>
<a NAME="E16:121_4"/><i><font color="Green">E205</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">YY</font> <a href="yellow_6.html#K4" target="_self">--&gt;</a> <font color="Maroon">q</font></span>)</span>,the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><font color="Maroon">YY</font> <a href="yellow_6.html#K4" target="_self">--&gt;</a> <font color="Maroon">q</font></span>)</span> #) <a href="hidden.html#R1">=</a> <font color="Maroon">YY</font>
 
<b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">X</font> = <font color="Maroon">YY</font> <a href="yellow_6.html#K4" target="_self">--&gt;</a> <font color="Maroon">q</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">T</font> ;<br/>
<a NAME="E18:121_4"/><i><font color="Green">E206</font></i>: 
 ex <font color="Olive">N0</font> being <a href="waybel_0.html#V6">strict</a> <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">T</font> st <br/>( <font color="Olive">N0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</font> &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">N0</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">T</font> )
 
<b>by </b><i>, </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">X</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">X</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">Y</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">Y</font> #)
 
<b>by </b><i/>;<br/>
<a NAME="E20:121_4"/><b>then </b><i><font color="Green">E207</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K7" target="_self">NetUniv</a> <font color="Maroon">T</font>
 
<b>by </b><i>, </i>;<br/>
<b>reconsider </b><font color="Maroon">X'</font> = <font color="Maroon">X</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</font> <b>by </b><i><a class="ref" href="yellow_6.html#T11" target="_self">Th11</a></i>;<br/>
<a NAME="E22:121_4"/>
the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">X</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">X'</font> is <a href="funct_1.html#V3">constant</a>
 
;<br/>
<b>then reconsider </b><font color="Maroon">X'</font> = <font color="Maroon">X'</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</font> <b>by </b><i/>;<br/>
<a NAME="E25:121_4"/>
 <a href="yellow_6.html#K5" target="_self">the_value_of</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font>
 
<b>by </b><i/>;<br/>
<a NAME="E26:121_4"/><b>then </b><i><font color="Green">E208</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">X</font>,<font color="Maroon">q</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 
<b>by </b><i>, <a class="txt" href="yellow_6.html#E30"><i><font color="Green">Lemma88</font></i></a>, </i>;<br/>
<a NAME="E27:121_4"/><i><font color="Green">E239</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K11" target="_self">Lim</a> <font color="Maroon">M</font>
 
<b>by </b><i><a class="ref" href="yellow_6.html#D7" target="_self">Def7</a>, <a class="ref" href="yellow_6.html#T13" target="_self">Th13</a>, </i>;<br/>
<b>reconsider </b><font color="Maroon">Y'</font> = <font color="Maroon">Y</font> as    <a href="yellow_6.html#M2" target="_self">subnet</a> of <font color="Maroon">M</font> <b>by </b><i><a class="txt" href="yellow_6.html#E20"><i><font color="Green">Lemma75</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> ] <b>means </b> ex <font color="Olive">i</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Y</font> ex <font color="Olive">Ji</font> being  <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">T</font> st <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">i</font> &amp; <font color="Olive">Ji</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">Ji</font>,<font color="Maroon">p</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> &amp;  <a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Olive">Ji</font> <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">Y</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">c</font></span>)</span> where <font color="Olive">c</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Y</font> : <font color="Olive">i</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">c</font> </span>}</span>  );<br/>
<a NAME="E29:121_4"/><i><font color="Green">E242</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> holds <br/> ex <font color="Olive">j</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">j</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">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

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

<b>then reconsider </b><font color="Maroon">i'</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Y'</font> <b>by </b><i/>;<br/>
<a NAME="E3:121_4_1"/>
 <a href="yellow_6.html#K11" target="_self">Lim</a> <font color="Maroon">M</font> <a href="tarski.html#R1">c=</a>  <a href="yellow_6.html#K11" target="_self">Lim</a> <font color="Maroon">Y'</font>
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">S</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</font></span>)</span><b> such that </b><br/><a NAME="E5:121_4_1"/><i><font color="Green">E243</font></i>: 
<font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">Y'</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">c</font></span>)</span> where <font color="Olive">c</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Y'</font> : <font color="Maroon">i'</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">c</font> </span>}</span> 
 <b>and </b><br/><a NAME="E6:121_4_1"/><i><font color="Green">E244</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">S</font>
 <b>by </b><i><a class="txt" href="yellow_6.html#E32"><i><font color="Green">Lemma90</font></i></a>, </i>;<br/>
<b>consider </b><font color="Maroon">Go</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'</font><b> such that </b><br/><a NAME="E8:121_4_1"/><i><font color="Green">E246</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">Go</font>,<font color="Maroon">p</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">C'</font>
 <b>and </b><br/><a NAME="E9:121_4_1"/><i><font color="Green">E247</font></i>: 
 <a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">Go</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">S</font>
 <b>and </b><br/><a NAME="E10:121_4_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K11" target="_self">Lim</a> <font color="Maroon">Go</font>
 <b>by </b><i>, </i>;<br/>
<b>reconsider </b><font color="Maroon">Ji</font> = <font color="Maroon">Go</font> as   <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">T</font> <b>by </b><i><a class="ref" href="yellow_6.html#T11" target="_self">Th11</a></i>;<br/>
<b>reconsider </b><font color="Maroon">i</font> = <font color="Maroon">i'</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Y</font> ;<br/>
<b>take </b>
<font color="Maroon">Ji</font>
;<br/>

<b>take </b>
<font color="Maroon">i</font>
;<br/>

<b>take </b>
<font color="Maroon">Ji</font>
;<br/>

<b>thus </b><a NAME="E13:121_4_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">Ji</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Ji</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">Ji</font>,<font color="Maroon">p</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> )
 <b>by </b><i/>;<br/>

<b>let </b><font color="Maroon">e</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">e</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">Ji</font>
 ;<br/>

<a NAME="E15:121_4_1"/><b>then </b>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">c'</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Y'</font><b> such that </b><br/><a NAME="E17:121_4_1"/><i><font color="Green">E249</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y'</font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">c'</font>
 <b>and </b><br/><a NAME="E18:121_4_1"/><i><font color="Green">E250</font></i>: 
<font color="Maroon">i'</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c'</font>
 <b>by </b><i><a class="ref" href="yellow_6.html#T20" target="_self">Th20</a></i>;<br/>
<b>reconsider </b><font color="Maroon">cc</font> = <font color="Maroon">c'</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Y</font> ;<br/>
<a NAME="E20:121_4_1"/>
<font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">cc</font>
 
<b>by </b><i><a class="ref" href="yellow_6.html#T21" target="_self">Th21</a></i>;<br/>
<b>hence </b><a NAME="E21:121_4_1"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">Y</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">c</font></span>)</span> where <font color="Olive">c</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Y</font> : <font color="Maroon">i</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">c</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="yellow_6.html#D10" target="_self">Def10</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">J</font> being    <a href="pboole.html#M1">ManySortedSet</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E31:121_4"/><i><font color="Green">E252</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Maroon">J</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>]
 <b>from </b><i><a class="ref" href="pboole.html#S3">PBOOLE:sch 3</a>(<a class="ref" href="yellow_6.html#T19" target="_self">Th19</a>);<br/></i>
<div><i><font color="Green">E253</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">y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:121_4_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">J</font>
 ;<br/><b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:121_4_2"/><i><font color="Green">E254</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">J</font>
 <b>and </b><br/><a NAME="E4:121_4_2"/><i><font color="Green">E255</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">J</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</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">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>
 <b>by </b><i>, <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">i</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Y</font> ex <font color="Olive">Ji</font> being  <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">T</font> st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">i</font> &amp; <font color="Olive">Ji</font> <a href="hidden.html#R1">=</a> <font color="Maroon">J</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">Ji</font>,<font color="Maroon">p</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> &amp;  <a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Olive">Ji</font> <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">Y</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">c</font></span>)</span> where <font color="Olive">c</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Y</font> : <font color="Olive">i</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">c</font> </span>}</span>  )
 <b>by </b><i><a class="ref" href="yellow_6.html#T20" target="_self">Th20</a></i>;<br/><b>hence </b><a NAME="E7:121_4_2"/>
<font color="Maroon">y</font> is  non <a href="struct_0.html#V3">empty</a>  <a href="struct_0.html#L1">1-sorted</a> 
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E256</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">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:121_4_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>
 ;<br/><a NAME="E2:121_4_3"/><b>then </b>
 ex <font color="Olive">i</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Y</font> ex <font color="Olive">Ji</font> being  <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">T</font> st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">i</font> &amp; <font color="Olive">Ji</font> <a href="hidden.html#R1">=</a> <font color="Maroon">J</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">Ji</font>,<font color="Maroon">p</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> &amp;  <a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Olive">Ji</font> <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">Y</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">c</font></span>)</span> where <font color="Olive">c</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Y</font> : <font color="Olive">i</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">c</font> </span>}</span>  )
 <b>by </b><i><a class="ref" href="yellow_6.html#T20" target="_self">Th20</a></i>;<br/><b>hence </b><a NAME="E3:121_4_3"/>
<font color="Maroon">J</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> is   <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">T</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">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:121_4_4"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">J</font>
 ;<br/><a NAME="E2:121_4_4"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</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">i</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Y</font> ex <font color="Olive">Ji</font> being  <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">T</font> st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">i</font> &amp; <font color="Olive">Ji</font> <a href="hidden.html#R1">=</a> <font color="Maroon">J</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">Ji</font>,<font color="Maroon">p</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> &amp;  <a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Olive">Ji</font> <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">Y</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">c</font></span>)</span> where <font color="Olive">c</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Y</font> : <font color="Olive">i</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">c</font> </span>}</span>  )
 <b>by </b><i><a class="ref" href="yellow_6.html#T20" target="_self">Th20</a></i>;<br/><b>hence </b><a NAME="E4:121_4_4"/>
<font color="Maroon">J</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</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">J</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">J</font> = <font color="Maroon">J</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">X</font>,<font color="Maroon">T</font> <b>by </b><i>, , , <a class="ref" href="yellow_6.html#T6" target="_self">Th6</a></i>;<br/>
<a NAME="E37:121_4"/>
for <font color="Olive">i</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font> holds  <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">J</font> <a href="yellow_6.html#K8" target="_self">.</a> <font color="Olive">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">i</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">C</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">i</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font>;<br/>

<a NAME="E1:121_4_5"/>
 ex <font color="Olive">ii</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Y</font> ex <font color="Olive">Ji</font> being  <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">T</font> st <br/>( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Olive">ii</font> &amp; <font color="Olive">Ji</font> <a href="hidden.html#R1">=</a> <font color="Maroon">J</font> <a href="yellow_6.html#K8" target="_self">.</a> <font color="Maroon">i</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">Ji</font>,<font color="Maroon">p</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> &amp;  <a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Olive">Ji</font> <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">Y</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">c</font></span>)</span> where <font color="Olive">c</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Y</font> : <font color="Olive">ii</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">c</font> </span>}</span>  )
 
<b>by </b><i><a class="ref" href="yellow_6.html#T20" target="_self">Th20</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">J</font> <a href="yellow_6.html#K8" target="_self">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">i</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E38:121_4"/><b>then </b><i><font color="Green">E257</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">J</font></span>)</span>,<font color="Maroon">q</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 
<b>by </b><i>, <a class="ref" href="yellow_6.html#D9" target="_self">Def9</a>, </i>;<br/>
<a NAME="E39:121_4"/><i><font color="Green">E258</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">J</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">J</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">X</font>,<span class="p2">(<span class="default"><a href="yellow_1.html#K5">product</a> <font color="Maroon">J</font></span>)</span></span><a href="yellow_3.html#K3">:]</a></span>
 
<b>by </b><i/>;<br/>
<a NAME="E40:121_4"/><b>then </b><i><font color="Green">E259</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">J</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">X</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">J</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">J</font> is    <a href="yellow_6.html#M2" target="_self">subnet</a> of <font color="Maroon">Y</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">Y</font>) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <span class="p1">{<span class="default"> <font color="Olive">c</font> where <font color="Olive">c</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Y</font> : <font color="Maroon">a<sub>1</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">c</font> </span>}</span> ;<br/>
<b>consider </b><font color="Maroon">B</font> being    <a href="pboole.html#M1">ManySortedSet</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font><b> such that </b><br/><a NAME="E2:121_4_6"/><i><font color="Green">E261</font></i>: 
for <font color="Olive">i</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Y</font> holds  <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">i</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">B</font>
 ;<br/><b>then consider </b><font color="Maroon">i</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">E262</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">B</font>
 <b>and </b><br/><a NAME="E4:121_4_6_1"/><i><font color="Green">E263</font></i>: 
<font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</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">i</font> = <font color="Maroon">i</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Y</font> <b>by </b><i><a class="ref" href="yellow_6.html#D12" target="_self">Def12</a>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><b>consider </b><font color="Maroon">j</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Y</font><b> such that </b><br/><a NAME="E7:121_4_6_1"/><i><font color="Green">E264</font></i>: 
<font color="Maroon">i</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">j</font>
 <b>and </b><br/><a NAME="E8:121_4_6_1"/>
<font color="Maroon">i</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">j</font>
 <b>by </b><i/>;<br/><a NAME="E9:121_4_6_1"/>
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">c</font> where <font color="Olive">c</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Y</font> : <font color="Maroon">i</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">c</font> </span>}</span> 
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E10:121_4_6_1"/>
contradiction
 <b>by </b><i><a class="ref" href="yellow_6.html#T22" target="_self">Th22</a>, </i>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">B</font> = <font color="Maroon">B</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">Y</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">B'</font> = <font color="Maroon">B</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">X</font> <b>by </b><i><a class="ref" href="yellow_6.html#D8" target="_self">Def8</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">X</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">J</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">M</font> being    <a href="pboole.html#M1">ManySortedSet</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E7:121_4_6"/><i><font color="Green">E266</font></i>: 
for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font> holds  <font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">x</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> ] <b>means </b> ex <font color="Olive">f</font> being  <a href="funct_1.html#NM1">Function</a> ex <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font> st <br/>( <font color="Olive">f</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">x</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">J</font> <a href="yellow_6.html#K8" target="_self">.</a> <font color="Olive">x</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">Y</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">E267</font></i>: 
for <font color="Olive">i</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> holds <br/>for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> holds <br/> ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B'</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">y</font>,<font color="Olive">x</font>,<font color="Olive">i</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">i</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:121_4_6_2"/><i><font color="Green">E268</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>
 ;<br/>

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

<b>assume </b><a NAME="E2:121_4_6_2"/><i><font color="Green">E269</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 ;<br/>

<b>consider </b><font color="Maroon">e</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Y</font>, <font color="Maroon">Ji</font> being   <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E4:121_4_6_2"/><i><font color="Green">E270</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e</font>
 <b>and </b><br/><a NAME="E5:121_4_6_2"/><i><font color="Green">E271</font></i>: 
<font color="Maroon">Ji</font> <a href="hidden.html#R1">=</a> <font color="Maroon">J</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</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">Ji</font>,<font color="Maroon">p</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 <b>and </b><br/><a NAME="E7:121_4_6_2"/><i><font color="Green">E272</font></i>: 
 <a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">Ji</font> <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">Y</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">c</font></span>)</span> where <font color="Olive">c</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Y</font> : <font color="Maroon">e</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">c</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="yellow_6.html#T20" target="_self">Th20</a>, </i>;<br/>
<b>reconsider </b><font color="Maroon">i'</font> = <font color="Maroon">i</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font> <b>by </b><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> ] <b>means </b>the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">Ji</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">Y</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">E273</font></i>: 
for <font color="Olive">ji</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Ji</font>  ex <font color="Olive">u</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">B'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i'</font> st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">ji</font>,<font color="Olive">u</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">ji</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Ji</font>;<br/>

<a NAME="E1:121_4_6_2_1"/>
<font color="Maroon">ji</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Ji</font>
 
;<br/>
<a NAME="E2:121_4_6_2_1"/><b>then </b>
<font color="Maroon">ji</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">Ji</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">Ji</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">ji</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">Ji</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">Ji</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">ji</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">Y</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">c</font></span>)</span> where <font color="Olive">c</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Y</font> : <font color="Maroon">e</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">c</font> </span>}</span> 
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">c</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Y</font><b> such that </b><br/><a NAME="E6:121_4_6_2_1"/><i><font color="Green">E275</font></i>: 
the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">Ji</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">ji</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">c</font>
 <b>and </b><br/><a NAME="E7:121_4_6_2_1"/><i><font color="Green">E276</font></i>: 
<font color="Maroon">e</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c</font>
 ;<br/>
<a NAME="E8:121_4_6_2_1"/>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">cc</font> where <font color="Olive">cc</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Y</font> : <font color="Maroon">e</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">cc</font> </span>}</span> 
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">c</font> = <font color="Maroon">c</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">B'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i'</font> <b>by </b><i><a class="ref" href="yellow_6.html#T22" target="_self">Th22</a>, </i>;<br/>
<b>take </b>
<font color="Maroon">c</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">Ji</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">ji</font> <a href="hidden.html#R1">=</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">Y</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c</font>
 <b>by </b><i><a class="ref" href="yellow_6.html#T26" target="_self">Th26</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">f</font> being   <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Ji</font>,<font color="Maroon">B'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i'</font><b> such that </b><br/><a NAME="E11:121_4_6_2"/><i><font color="Green">E277</font></i>: 
for <font color="Olive">ji</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Ji</font> holds  <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">ji</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">ji</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</a>(<a class="ref" href="yellow_6.html#T25" target="_self">Th25</a>);<br/></i>
<b>reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as   <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Ji</font>,<font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> ;<br/>
<b>reconsider </b><font color="Maroon">ji</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Ji</font> <b>by </b><i><a class="ref" href="yellow_6.html#D12" target="_self">Def12</a>, , , </i>;<br/>
<b>take </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
;<br/>

<a NAME="E14:121_4_6_2"/>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">ji</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 
;<br/>
<b>hence </b><a NAME="E15:121_4_6_2"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 ;<br/>

<b>take </b>
<font color="Maroon">f</font>
;<br/>

<b>take </b>
<font color="Maroon">i'</font>
;<br/>

<b>thus </b><a NAME="E16:121_4_6_2"/>
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> &amp; <font color="Maroon">i'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> )
 ;<br/>

<b>thus </b>the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="yellow_6.html#K8" target="_self">.</a> <font color="Maroon">i'</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">Ji</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">ji</font>
<b>by </b><i/>
<br/>.= 
the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">Y</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="ref" href="yellow_6.html#T26" target="_self">Th26</a></i>
;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">u</font> being    <a href="pboole.html#M3">ManySortedFunction</a> of <font color="Maroon">M</font>,<font color="Maroon">B'</font><b> such that </b><br/><a NAME="E10:121_4_6"/><i><font color="Green">E278</font></i>: 
for <font color="Olive">i</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> holds <br/> ex <font color="Olive">f</font> being  <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font>,<font color="Maroon">B'</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> st <br/>( <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> &amp; ( for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>,<font color="Olive">x</font>,<font color="Olive">i</font>] ) )
 <b>from </b><i><a class="ref" href="mssubfam.html#S1">MSSUBFAM:sch 1</a>();<br/></i>
<a NAME="E11:121_4_6"/><i><font color="Green">E279</font></i>: 
for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font><br/> for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> holds  the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="yellow_6.html#K8" target="_self">.</a> <font color="Olive">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font> <a href="hidden.html#R1">=</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">Y</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">u</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">j</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">i</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">j</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>;<br/>



<b>consider </b><font color="Maroon">f</font> being   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>,<font color="Maroon">B'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font><b> such that </b><br/><a NAME="E2:121_4_6_3"/><i><font color="Green">E280</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>and </b><br/><a NAME="E3:121_4_6_3"/><i><font color="Green">E281</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>,<font color="Olive">x</font>,<font color="Maroon">i</font>]
 <b>by </b><i/>;<br/>
<a NAME="E4:121_4_6_3"/>
<font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</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">J</font> <a href="yellow_6.html#K8" target="_self">.</a> <font color="Maroon">i</font></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_6.html#D12" target="_self">Def12</a></i>;<br/>
<a NAME="E5:121_4_6_3"/><b>then </b>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>,<font color="Maroon">j</font>,<font color="Maroon">i</font>]
 
<b>by </b><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">J</font> <a href="yellow_6.html#K8" target="_self">.</a> <font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">Y</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">u</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span>
 <b>by </b><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">X</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">J</font></span>)</span>) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <span class="p1">(<span class="default"><font color="Maroon">u</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">E282</font></i>: 
for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font><br/> for <font color="Olive">y</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">J</font></span>)</span> holds  <font color="Maroon">H<sub>3</sub></font>(<font color="Olive">x</font>,<font color="Olive">y</font>) <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</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">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">y</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">J</font></span>)</span>;<br/>



<b>reconsider </b><font color="Maroon">k</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Y</font> <b>by </b><i><a class="ref" href="yellow_6.html#D8" target="_self">Def8</a></i>;<br/>
<a NAME="E2:121_4_6_4"/><i><font color="Green">E283</font></i>: 
<font color="Maroon">u</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>,<font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</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">Y</font>] <b>means </b><font color="Maroon">k</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<b>set </b><font color="Maroon">ZZ</font> = <span class="p1">{<span class="default"> <font color="Olive">c</font> where <font color="Olive">c</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Y</font> : <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">c</font>] </span>}</span> ;<br/>
<a NAME="E3:121_4_6_4"/><i><font color="Green">E285</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">c</font> where <font color="Olive">c</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Y</font> : <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">c</font>] </span>}</span>  is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</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">E286</font></i>: 
<font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">c</font> where <font color="Olive">c</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Y</font> : <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">c</font>] </span>}</span> 
 
<b>by </b><i><a class="ref" href="yellow_6.html#T22" target="_self">Th22</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X'</font> ;<br/>
<a NAME="E6:121_4_6_4"/>
<font color="Maroon">y</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">J</font></span>)</span>
 
;<br/>
<a NAME="E7:121_4_6_4"/><b>then </b><i><font color="Green">E287</font></i>: 
<font color="Maroon">y</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">J</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">x'</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X'</font>
 
;<br/>
<a NAME="E9:121_4_6_4"/><b>then </b>
<font color="Maroon">x'</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">J</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">y</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x'</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">J</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x'</font>
 
<b>by </b><i><a class="ref" href="yellow_6.html#T26" target="_self">Th26</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">y</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x'</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">J</font> <a href="yellow_6.html#K8" target="_self">.</a> <font color="Maroon">x</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E12:121_4_6_4"/><b>then </b>
<font color="Maroon">y</font> <a href="waybel_3.html#K4">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
 
<b>by </b><i><a class="ref" href="yellow_6.html#D12" target="_self">Def12</a></i>;<br/>
<a NAME="E13:121_4_6_4"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="waybel_3.html#K4">.</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
 
<b>by </b><i>, <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">u</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="waybel_3.html#K4">.</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>
 <b>by </b><i>, <a class="ref" href="yellow_6.html#T25" target="_self">Th25</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">f</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">X</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">J</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">Y</font><b> such that </b><br/><a NAME="E14:121_4_6"/><i><font color="Green">E288</font></i>: 
for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font><br/> for <font color="Olive">y</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">J</font></span>)</span> holds  <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</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">x</font>,<font color="Olive">y</font>)
 <b>from </b><i><a class="ref" href="funct_7.html#S1">FUNCT_7:sch 1</a>();<br/></i>
<b>reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9" target="_self">Iterated</a> <font color="Maroon">J</font></span>)</span>,<font color="Maroon">Y</font> <b>by </b><i><a class="ref" href="yellow_6.html#D10" target="_self">Def10</a></i>;<br/>
<b>take </b>
<font color="Maroon">f</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_6.html#D12" target="_self">YELLOW_6:def 12</a><br/>

<b>set </b><font color="Maroon">h</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">J</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">g</font> = the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">Y'</font>;<br/>
<a NAME="E16:121_4_6"/><i><font color="Green">E290</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</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">J</font></span>)</span> iff ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</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">Y'</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">x</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">x</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">J</font></span>)</span>
 ;<br/><a NAME="E2:121_4_6_5_1"/><b>then </b>
<font color="Maroon">x</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">J</font></span>)</span>
 ;<br/><a NAME="E3:121_4_6_5_1"/><b>then </b>
<font color="Maroon">x</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">X'</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">J</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i/>;<br/><a NAME="E4:121_4_6_5_1"/><b>then </b><i><font color="Green">E291</font></i>: 
<font color="Maroon">x</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">X'</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">J</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">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</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">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="yellow_6.html#T25" target="_self">Th25</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">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</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">Y'</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">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</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">Y'</font> )
 ;<br/>

<a NAME="E3:121_4_6_5"/><b>then </b>
<font color="Maroon">x</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">X'</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">J</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">x</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">X'</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">J</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">x</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">J</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E6:121_4_6_5"/>
<font color="Maroon">x</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">J</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">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</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">J</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">J</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">Y'</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</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">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:121_4_6_6"/>
<font color="Maroon">x</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">J</font></span>)</span>
 ;<br/>

<a NAME="E2:121_4_6_6"/><b>then </b>
<font color="Maroon">x</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">J</font></span>)</span>
 
;<br/>
<a NAME="E3:121_4_6_6"/><b>then </b>
<font color="Maroon">x</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">X'</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">J</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i/>;<br/>
<a NAME="E4:121_4_6_6"/><b>then </b>
<font color="Maroon">x</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">X'</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">J</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">x1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X'</font>, <font color="Maroon">x2</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">J</font></span>)</span><b> such that </b><br/><a NAME="E6:121_4_6_6"/><i><font color="Green">E294</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</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">x'</font> = <font color="Maroon">x1</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font> ;<br/>
<a NAME="E8:121_4_6_6"/><i><font color="Green">E295</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">J</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X'</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">x2</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">J</font></span>)</span>
 
;<br/>
<a NAME="E10:121_4_6_6"/><b>then </b><i><font color="Green">E296</font></i>: 
<font color="Maroon">x2</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">J</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">J</font> <a href="yellow_6.html#K8" target="_self">.</a> <font color="Maroon">x'</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">J</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font>
 
<b>by </b><i/>;<br/>
<a NAME="E12:121_4_6_6"/><b>then </b>
<font color="Maroon">x2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</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">J</font> <a href="yellow_6.html#K8" target="_self">.</a> <font color="Maroon">x'</font></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_6.html#T26" target="_self">Th26</a>, , <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">j</font> = <font color="Maroon">x2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x'</font> <b>by </b><i><a class="ref" href="yellow_6.html#D12" target="_self">Def12</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">J</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</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">J</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">x1</font>,<font color="Maroon">x2</font>
<b>by </b><i><a class="ref" href="yellow_6.html#T25" target="_self">Th25</a></i>
<br/>.= 
the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="yellow_6.html#K8" target="_self">.</a> <font color="Maroon">x'</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">x2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font></span>)</span>
<b>by </b><i/>
<br/>.= 
the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">Y'</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">u</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x'</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span>
<b>by </b><i/>
<br/>.= 
the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">Y'</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i>, <a class="ref" href="yellow_6.html#T25" target="_self">Th25</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">J</font></span>)</span> <a href="funct_2.html#R2">=</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">Y</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T20">FUNCT_1:20</a></i>;<br/>

<b>let </b><font color="Maroon">m</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Y</font>;<br/>

<b>consider </b><font color="Maroon">F</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">J</font></span>)</span>;<br/>
<b>reconsider </b><font color="Maroon">n</font> = <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">m</font>,<font color="Maroon">F</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">J</font></span>)</span> <b>by </b><i><a class="ref" href="yellow_6.html#D8" target="_self">Def8</a>, <a class="ref" href="yellow_6.html#D10" target="_self">Def10</a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>reconsider </b><font color="Maroon">F</font> = <font color="Maroon">F</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">J</font></span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon">m'</font> = <font color="Maroon">m</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font> <b>by </b><i><a class="ref" href="yellow_6.html#D8" target="_self">Def8</a></i>;<br/>
<b>take </b>
<font color="Maroon">n</font>
;<br/>

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

<b>consider </b><font color="Maroon">k'</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font>, <font color="Maroon">G</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">J</font></span>)</span><b> such that </b><br/><a NAME="E24:121_4_6"/><i><font color="Green">E299</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">k'</font>,<font color="Maroon">G</font></span><a href="yellow_3.html#K7">]</a></span>
 <b>by </b><i><a class="ref" href="yellow_6.html#D10" target="_self">Def10</a>, <a class="ref" href="domain_1.html#T9">DOMAIN_1:9</a></i>;<br/>
<b>reconsider </b><font color="Maroon">k</font> = <font color="Maroon">k'</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Y</font> <b>by </b><i><a class="ref" href="yellow_6.html#D8" target="_self">Def8</a></i>;<br/>
<b>reconsider </b><font color="Maroon">k''</font> = <font color="Maroon">k'</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X'</font> ;<br/>
<a NAME="E27:121_4_6"/><i><font color="Green">E301</font></i>: 
<font color="Maroon">f</font> <a href="yellow_6.html#K3" target="_self">.</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="yellow_6.html#T25" target="_self">Th25</a></i>;<br/>
<a NAME="E28:121_4_6"/><i><font color="Green">E302</font></i>: 
<font color="Maroon">u</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>,<font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</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">E303</font></i>:  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span> = 
<font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</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">J</font> <a href="yellow_6.html#K8" target="_self">.</a> <font color="Maroon">k'</font></span>)</span>
<b>by </b><i><a class="ref" href="yellow_6.html#D12" target="_self">Def12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">J</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k''</font>
<b>by </b><i/>
;<br/>
<a NAME="E30:121_4_6"/><i><font color="Green">E304</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">J</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X'</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">G</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">J</font></span>)</span>
 
;<br/>
<a NAME="E32:121_4_6"/><b>then </b>
<font color="Maroon">G</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">J</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">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k''</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">u</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="yellow_6.html#T27" target="_self">Th27</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">E305</font></i>: 
<font color="Maroon">f</font> <a href="yellow_6.html#K3" target="_self">.</a> <font color="Maroon">p</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">u</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_6.html#T26" target="_self">Th26</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">u</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
 
<b>by </b><i>, <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">f</font> <a href="yellow_6.html#K3" target="_self">.</a> <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
 
<b>by </b><i><a class="ref" href="yellow_6.html#T28" target="_self">Th28</a></i>;<br/>
<a NAME="E37:121_4_6"/><b>then </b>
<font color="Maroon">f</font> <a href="yellow_6.html#K3" target="_self">.</a> <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">c</font> where <font color="Olive">c</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Y</font> : <font color="Maroon">k</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">c</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="yellow_6.html#T22" target="_self">Th22</a></i>;<br/>
<b>then consider </b><font color="Maroon">c</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Y</font><b> such that </b><br/><a NAME="E39:121_4_6"/><i><font color="Green">E306</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="yellow_6.html#K3" target="_self">.</a> <font color="Maroon">p</font>
 <b>and </b><br/><a NAME="E40:121_4_6"/><i><font color="Green">E307</font></i>: 
<font color="Maroon">k</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c</font>
 ;<br/>
<b>reconsider </b><font color="Maroon">G</font> = <font color="Maroon">G</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">J</font></span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon">k'</font> = <font color="Maroon">k</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font> ;<br/>
<b>assume </b><a NAME="E43:121_4_6"/>
<font color="Maroon">n</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">p</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">m'</font>,<font color="Maroon">F</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">k'</font>,<font color="Maroon">G</font></span><a href="yellow_3.html#K7">]</a></span>
 
<b>by </b><i><a class="ref" href="yellow_6.html#T21" target="_self">Th21</a>, <a class="ref" href="yellow_6.html#T25" target="_self">Th25</a>, </i>;<br/>
<a NAME="E45:121_4_6"/><b>then </b>
<font color="Maroon">m'</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">k'</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">m</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">k</font>
 
<b>by </b><i><a class="ref" href="yellow_6.html#D8" target="_self">Def8</a>, </i>;<br/>
<b>hence </b><a NAME="E47:121_4_6"/>
<font color="Maroon">m</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">f</font> <a href="yellow_6.html#K3" target="_self">.</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="ref" href="yellow_6.html#T29" target="_self">Th29</a>, <a class="ref" href="yellow_6.html#D13" target="_self">Def13</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="ref" href="yellow_6.html#T16" target="_self">Th16</a>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:121"/>
<font color="Maroon">C</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</font></span>)</span>
 
<b>by </b><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</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div>
