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

<b>let </b><font color="Maroon">X</font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">Y</font> be   <a href="t_0topsp.html#NM1">T_0-TopSpace</a>;<br/><b>let </b><font color="Maroon">N</font> be   <a href="waybel_0.html#NM2">net</a> of  <a href="waybel24.html#K3">ContMaps</a> <font color="Maroon">X</font>,<span class="p1">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">Y</font></span>)</span>;<br/>





<b>assume </b><a NAME="E1:54"/><i><font color="Green">E42</font></i>: 
for <font color="Olive">x</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font> holds   <a href="waybel_9.html#R1">ex_sup_of</a>  <a href="waybel25.html#K2" target="_self">commute</a> <font color="Maroon">N</font>,<font color="Olive">x</font>,<span class="p1">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">Y</font></span>)</span>
 ;<br/>

<b>set </b><font color="Maroon">n</font> = the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">N</font>;<br/>
<b>set </b><font color="Maroon">L</font> = <span class="p1">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">Y</font></span>)</span> <a href="yellow_1.html#K6">|^</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">Y</font></span>)</span> =  <a href="waybel_2.html#K1">sup</a> <span class="p1">(<span class="default"><a href="waybel25.html#K2" target="_self">commute</a> <font color="Maroon">N</font>,<font color="Maroon">a<sub>1</sub></font>,<span class="p2">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">Y</font></span>)</span></span>)</span>;<br/>
<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">X</font>,the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">Y</font></span>)</span><b> such that </b><br/><a NAME="E3:54"/><i><font color="Green">E43</font></i>: 
for <font color="Olive">u</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font> holds  <font color="Maroon">f</font> <a href="yellow_6.html#K3">.</a> <font color="Olive">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">u</font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S4">FUNCT_2:sch 4</a>();<br/></i>
<b>reconsider </b><font color="Maroon">a</font> = <font color="Maroon">f</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">Y</font></span>)</span> <a href="yellow_1.html#K6">|^</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font></span>)</span> <b>by </b><i><a class="ref" href="waybel24.html#T19">WAYBEL24:19</a></i>;<br/>
<a NAME="E5:54"/>
 ex <font color="Olive">a</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">Y</font></span>)</span> <a href="yellow_1.html#K6">|^</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font></span>)</span> st <br/>(  <a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">N</font> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Olive">a</font> &amp; ( for <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">Y</font></span>)</span> <a href="yellow_1.html#K6">|^</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font></span>)</span>  st  <a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">N</font> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Olive">b</font> holds <br/><font color="Olive">a</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">b</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1"><b>proof </b></a><div class="add">

<a NAME="E1:54_1"/><i><font color="Green">E44</font></i>: 
 <a href="relset_1.html#K4">dom</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">N</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">N</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E2:54_1"/><i><font color="Green">E47</font></i>: 
<span class="p1">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">Y</font></span>)</span> <a href="yellow_1.html#K6">|^</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="yellow_1.html#K5">product</a> <span class="p1">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p2">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">Y</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_1.html#D5">YELLOW_1:def 5</a></i>;<br/>
<b>take </b>
<font color="Maroon">a</font>
;<br/>

<b>thus </b><a NAME="E3:54_1"/>
 <a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">N</font> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Maroon">a</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">Y</font></span>)</span> <a href="yellow_1.html#K6">|^</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D9">LATTICE3:def 9</a><br/>

<b>assume </b><a NAME="E1:54_1_1"/>
<font color="Maroon">k</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">N</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="E3:54_1_1"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">i</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">N</font>
 <b>and </b><br/><a NAME="E4:54_1_1"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">N</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <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="pre_topc.html#NM2">Point</a> of <font color="Maroon">N</font> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">k'</font> = <font color="Maroon">k</font>, <font color="Maroon">a'</font> = <font color="Maroon">a</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> <span class="p2">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p3">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">Y</font></span>)</span></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="yellow_1.html#D5">YELLOW_1:def 5</a></i>;<br/>
<a NAME="E7:54_1_1"/>
for <font color="Olive">u</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font> holds  <font color="Maroon">k'</font> <a href="waybel_3.html#K7">.</a> <font color="Olive">u</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">a'</font> <a href="waybel_3.html#K7">.</a> <font color="Olive">u</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1_1_1"><b>proof </b></a><div class="add">

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

<a NAME="E1:54_1_1_1"/><i><font color="Green">E64</font></i>: 
 <a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p2">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">Y</font></span>)</span></span>)</span> <a href="yellow_1.html#K4">.</a> <font color="Maroon">u</font>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/>
<a NAME="E2:54_1_1_1"/>
 <a href="waybel_9.html#R1">ex_sup_of</a>  <a href="waybel25.html#K2" target="_self">commute</a> <font color="Maroon">N</font>,<font color="Maroon">u</font>,<span class="p1">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">Y</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E3:54_1_1_1"/><b>then </b><i><font color="Green">E65</font></i>: 
 <a href="yellow_0.html#R1">ex_sup_of</a>  <a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K2" target="_self">commute</a> <font color="Maroon">N</font>,<font color="Maroon">u</font>,<span class="p2">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">Y</font></span>)</span></span>)</span>, <a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">Y</font>
 
<b>by </b><i><a class="ref" href="waybel_9.html#D3">WAYBEL_9:def 3</a></i>;<br/>
<a NAME="E4:54_1_1_1"/><i><font color="Green">E68</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="waybel25.html#K2" target="_self">commute</a> <font color="Maroon">N</font>,<font color="Maroon">u</font>,<span class="p2">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">Y</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">N</font>
 
<b>by </b><i>, </i>;<br/>
<i><font color="Green">E69</font></i>: <font color="Maroon">a'</font> <a href="waybel_3.html#K7">.</a> <font color="Maroon">u</font> = 
 <a href="waybel_2.html#K1">sup</a> <span class="p1">(<span class="default"><a href="waybel25.html#K2" target="_self">commute</a> <font color="Maroon">N</font>,<font color="Maroon">u</font>,<span class="p2">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">Y</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="yellow_2.html#NK6" target="_self">Sup</a> 
<b>by </b><i><a class="ref" href="waybel_2.html#D1">WAYBEL_2:def 1</a></i>
<br/>.= 
 <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <span class="p2">(<span class="default"><a href="waybel25.html#K2" target="_self">commute</a> <font color="Maroon">N</font>,<font color="Maroon">u</font>,<span class="p3">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">Y</font></span>)</span></span>)</span></span>)</span>

;<br/>
<a NAME="E6:54_1_1_1"/>
<font color="Maroon">k'</font> <a href="waybel_3.html#K7">.</a> <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K2" target="_self">commute</a> <font color="Maroon">N</font>,<font color="Maroon">u</font>,<span class="p2">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E7:54_1_1_1"/><b>then </b>
<font color="Maroon">k'</font> <a href="waybel_3.html#K7">.</a> <font color="Maroon">u</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 <span class="p1">(<span class="default"><a href="waybel25.html#K2" target="_self">commute</a> <font color="Maroon">N</font>,<font color="Maroon">u</font>,<span class="p2">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">Y</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E8:54_1_1_1"/>
<font color="Maroon">k'</font> <a href="waybel_3.html#K7">.</a> <font color="Maroon">u</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">a'</font> <a href="waybel_3.html#K7">.</a> <font color="Maroon">u</font>
 <b>by </b><i>, , , <a class="ref" href="yellow_4.html#T1">YELLOW_4:1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:54_1_1"/>
<font color="Maroon">k</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">a</font>
 <b>by </b><i>, <a class="ref" href="waybel_3.html#T28">WAYBEL_3:28</a></i>;<br/>


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

<b>let </b><font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">Y</font></span>)</span> <a href="yellow_1.html#K6">|^</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font></span>)</span>;<br/>

<b>assume </b><a NAME="E4:54_1"/><i><font color="Green">E71</font></i>: 
for <font color="Olive">k</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">Y</font></span>)</span> <a href="yellow_1.html#K6">|^</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font></span>)</span>  st <font color="Olive">k</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">N</font> holds <br/><font color="Olive">k</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">b</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D9">LATTICE3:def 9</a><br/>

<b>reconsider </b><font color="Maroon">a'</font> = <font color="Maroon">a</font>, <font color="Maroon">b'</font> = <font color="Maroon">b</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> <span class="p2">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p3">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">Y</font></span>)</span></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="yellow_1.html#D5">YELLOW_1:def 5</a></i>;<br/>
<a NAME="E6:54_1"/>
for <font color="Olive">u</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font> holds  <font color="Maroon">a'</font> <a href="waybel_3.html#K7">.</a> <font color="Olive">u</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">b'</font> <a href="waybel_3.html#K7">.</a> <font color="Olive">u</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1_2"><b>proof </b></a><div class="add">

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

<a NAME="E1:54_1_2"/><i><font color="Green">E72</font></i>: 
 <a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p2">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">Y</font></span>)</span></span>)</span> <a href="yellow_1.html#K4">.</a> <font color="Maroon">u</font>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/>
<i><font color="Green">E73</font></i>: <font color="Maroon">a'</font> <a href="waybel_3.html#K7">.</a> <font color="Maroon">u</font> = 
 <a href="waybel_2.html#K1">sup</a> <span class="p1">(<span class="default"><a href="waybel25.html#K2" target="_self">commute</a> <font color="Maroon">N</font>,<font color="Maroon">u</font>,<span class="p2">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">Y</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="yellow_2.html#NK6" target="_self">Sup</a> 
<b>by </b><i><a class="ref" href="waybel_2.html#D1">WAYBEL_2:def 1</a></i>
<br/>.= 
 <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <span class="p2">(<span class="default"><a href="waybel25.html#K2" target="_self">commute</a> <font color="Maroon">N</font>,<font color="Maroon">u</font>,<span class="p3">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">Y</font></span>)</span></span>)</span></span>)</span>

;<br/>
<a NAME="E3:54_1_2"/>
 <a href="waybel_9.html#R1">ex_sup_of</a>  <a href="waybel25.html#K2" target="_self">commute</a> <font color="Maroon">N</font>,<font color="Maroon">u</font>,<span class="p1">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">Y</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E4:54_1_2"/><b>then </b><i><font color="Green">E74</font></i>: 
 <a href="yellow_0.html#R1">ex_sup_of</a>  <a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K2" target="_self">commute</a> <font color="Maroon">N</font>,<font color="Maroon">u</font>,<span class="p2">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">Y</font></span>)</span></span>)</span>, <a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">Y</font>
 
<b>by </b><i><a class="ref" href="waybel_9.html#D3">WAYBEL_9:def 3</a></i>;<br/>
<a NAME="E5:54_1_2"/>
 <a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K2" target="_self">commute</a> <font color="Maroon">N</font>,<font color="Maroon">u</font>,<span class="p2">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">Y</font></span>)</span></span>)</span> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Maroon">b</font> <a href="yellow14.html#K1">.</a> <font color="Maroon">u</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">s</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">Y</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D9">LATTICE3:def 9</a><br/>

<b>assume </b><a NAME="E1:54_1_2_2"/>
<font color="Maroon">s</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 <span class="p1">(<span class="default"><a href="waybel25.html#K2" target="_self">commute</a> <font color="Maroon">N</font>,<font color="Maroon">u</font>,<span class="p2">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">Y</font></span>)</span></span>)</span>
 ;<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="E3:54_1_2_2"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">i</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="waybel25.html#K2" target="_self">commute</a> <font color="Maroon">N</font>,<font color="Maroon">u</font>,<span class="p2">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">Y</font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E4:54_1_2_2"/><i><font color="Green">E76</font></i>: 
the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K2" target="_self">commute</a> <font color="Maroon">N</font>,<font color="Maroon">u</font>,<span class="p2">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">Y</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font>
 <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="pre_topc.html#NM2">Point</a> of <font color="Maroon">N</font> <b>by </b><i>, , </i>;<br/>
<a NAME="E6:54_1_2_2"/><i><font color="Green">E90</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">N</font> <a href="yellow_6.html#K3">.</a> <font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E7:54_1_2_2"/><i><font color="Green">E91</font></i>: 
the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">N</font> <a href="yellow_6.html#K3">.</a> <font color="Maroon">i</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">N</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E8:54_1_2_2"/>
the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">N</font> <a href="yellow_6.html#K3">.</a> <font color="Maroon">i</font> is   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X</font>,<span class="p1">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">Y</font></span>)</span>
 
<b>by </b><i><a class="ref" href="waybel24.html#T21">WAYBEL24:21</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">k</font> = the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">N</font> <a href="yellow_6.html#K3">.</a> <font color="Maroon">i</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">Y</font></span>)</span> <a href="yellow_1.html#K6">|^</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font></span>)</span> <b>by </b><i><a class="ref" href="waybel24.html#T19">WAYBEL24:19</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 <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <span class="p2">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p3">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">Y</font></span>)</span></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="yellow_1.html#D5">YELLOW_1:def 5</a></i>;<br/>
<a NAME="E11:54_1_2_2"/>
<font color="Maroon">k</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">b</font>
 
<b>by </b><i>, <a class="txt" href="waybel25.html#E1:54"><i><font color="Green">E42</font></i></a></i>;<br/>
<a NAME="E12:54_1_2_2"/><b>then </b>
<font color="Maroon">k'</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">b'</font>
 
<b>by </b><i><a class="ref" href="yellow_1.html#D5">YELLOW_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E13:54_1_2_2"/>
<font color="Maroon">s</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">b</font> <a href="yellow14.html#K1">.</a> <font color="Maroon">u</font>
 <b>by </b><i>, , <a class="ref" href="waybel_3.html#T28">WAYBEL_3:28</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:54_1_2"/>
<font color="Maroon">a'</font> <a href="waybel_3.html#K7">.</a> <font color="Maroon">u</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">b'</font> <a href="waybel_3.html#K7">.</a> <font color="Maroon">u</font>
 <b>by </b><i>, , , <a class="ref" href="yellow_0.html#T30">YELLOW_0:30</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:54_1"/>
<font color="Maroon">a</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">b</font>
 <b>by </b><i>, <a class="ref" href="waybel_3.html#T28">WAYBEL_3:28</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:54"/>
 <a href="yellow_0.html#R1">ex_sup_of</a>  <a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">N</font>,<span class="p1">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">Y</font></span>)</span> <a href="yellow_1.html#K6">|^</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T15">YELLOW_0:15</a></i>;<br/>


</div>
