<?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="pre_topc.html#NM1">TopSpace</a>;<br/><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">p</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font>;<br/>





<b>assume </b><a NAME="E1:105"/><i><font color="Green">E29</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">N</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">N</font>,<font color="Maroon">T</font>;<br/>

<b>assume </b><a NAME="E2:105"/><i><font color="Green">E32</font></i>: 
for <font color="Olive">i</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> holds  <font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K11" target="_self">Lim</a> <span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="yellow_6.html#K8" target="_self">.</a> <font color="Olive">i</font></span>)</span>
 ;<br/>

<a NAME="E3:105"/>
for <font color="Olive">V</font> being   <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">p</font> holds   <a href="yellow_6.html#K9" target="_self">Iterated</a> <font color="Maroon">J</font> <a href="waybel_0.html#R1">is_eventually_in</a> <font color="Olive">V</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">V</font> be    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">p</font>;<br/>

<a NAME="E1:105_1"/><i><font color="Green">E41</font></i>: 
 <a href="tops_1.html#K1">Int</a> <font color="Maroon">V</font> <a href="hidden.html#R1">=</a>  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">V</font></span>)</span>
 
<b>by </b><i><a class="ref" href="tops_1.html#T45">TOPS_1:45</a></i>;<br/>
<a NAME="E2:105_1"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">V</font></span>)</span>
 
<b>by </b><i><a class="ref" href="connsp_2.html#D1">CONNSP_2:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">W</font> =  <a href="tops_1.html#K1">Int</a> <font color="Maroon">V</font> as    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">p</font> <b>by </b><i><a class="ref" href="connsp_2.html#D1">CONNSP_2:def 1</a></i>;<br/>
<a NAME="E4:105_1"/>
<font color="Maroon">N</font> <a href="waybel_0.html#R1">is_eventually_in</a> <font color="Maroon">W</font>
 
<b>by </b><i>, </i>;<br/>
<b>then consider </b><font color="Maroon">i</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font><b> such that </b><br/><a NAME="E6:105_1"/><i><font color="Green">E42</font></i>: 
for <font color="Olive">j</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font>  st <font color="Maroon">i</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">j</font> holds <br/><font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font>
 <b>by </b><i><a class="ref" href="waybel_0.html#D11">WAYBEL_0:def 11</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font>,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">m</font> being  <a href="struct_0.html#NM1">Element</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> st <br/>( <font color="Olive">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>2</sub></font> &amp; ( <font color="Maroon">i</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">a<sub>1</sub></font> implies for <font color="Olive">n</font> being  <a href="struct_0.html#NM1">Element</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>  st <font color="Olive">m</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">n</font> holds <br/><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> <a href="waybel_0.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> ) );<br/>
<a NAME="E7:105_1"/><i><font color="Green">E55</font></i>: 
for <font color="Olive">j</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font>  ex <font color="Olive">e</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">j</font>,<font color="Olive">e</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_1"><b>proof </b></a><div class="add">

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

<b>reconsider </b><font color="Maroon">j'</font> = <font color="Maroon">j</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> ;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="105_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:105_1_1_1"/>
( <font color="Maroon">i</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">j</font> or  not <font color="Maroon">i</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">j</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="105_1_1_1_1"><b>suppose </b></a><a NAME="E1:105_1_1_1_1"/>
<font color="Maroon">i</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">j</font>
 ;<br/><div class="add"><a NAME="E2:105_1_1_1_1"/><b>then </b><i><font color="Green">E56</font></i>: 
<font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">j'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font>
 <b>by </b><i><a class="txt" href="yellow_6.html#E17"><i><font color="Green">Lemma64</font></i></a></i>;<br/><a NAME="E3:105_1_1_1_1"/><i><font color="Green">E117</font></i>: 
<font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K11" target="_self">Lim</a> <span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="yellow_6.html#K8" target="_self">.</a> <font color="Maroon">j</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E4:105_1_1_1_1"/>
<font color="Maroon">W</font> is    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">j</font>
 <b>by </b><i>, , <a class="ref" href="connsp_2.html#D1">CONNSP_2:def 1</a></i>;<br/><a NAME="E5:105_1_1_1_1"/><b>then </b>
<font color="Maroon">J</font> <a href="yellow_6.html#K8" target="_self">.</a> <font color="Maroon">j</font> <a href="waybel_0.html#R1">is_eventually_in</a> <font color="Maroon">W</font>
 <b>by </b><i><a class="ref" href="yellow_6.html#T11" target="_self">Th11</a>, </i>;<br/><b>then consider </b><font color="Maroon">e</font> being   <a href="struct_0.html#NM1">Element</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">j</font></span>)</span><b> such that </b><br/><a NAME="E7:105_1_1_1_1"/><i><font color="Green">E118</font></i>: 
for <font color="Olive">u</font> being  <a href="struct_0.html#NM1">Element</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">j</font></span>)</span>  st <font color="Maroon">e</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">u</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="yellow_6.html#K8" target="_self">.</a> <font color="Maroon">j</font></span>)</span> <a href="waybel_0.html#K2">.</a> <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font>
 <b>by </b><i><a class="ref" href="waybel_0.html#D11">WAYBEL_0:def 11</a></i>;<br/><b>take </b>
<font color="Maroon">e</font>
;<br/><b>take </b>
<font color="Maroon">e</font>
;<br/><b>thus </b><a NAME="E8:105_1_1_1_1"/>
<font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e</font>
 ;<br/><b>assume </b><a NAME="E9:105_1_1_1_1"/>
<font color="Maroon">i</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">j</font>
 ;<br/><b>thus </b><a NAME="E10:105_1_1_1_1"/>
for <font color="Olive">n</font> being  <a href="struct_0.html#NM1">Element</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">j</font></span>)</span>  st <font color="Maroon">e</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">n</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="yellow_6.html#K8" target="_self">.</a> <font color="Maroon">j</font></span>)</span> <a href="waybel_0.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="105_1_1_1_2"><b>suppose </b></a><a NAME="E1:105_1_1_1_2"/><i><font color="Green">E119</font></i>: 
not <font color="Maroon">i</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">j</font>
 ;<br/><div class="add"><b>consider </b><font color="Maroon">e</font> being   <a href="struct_0.html#NM1">Element</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">j</font></span>)</span>;<br/><b>take </b>
<font color="Maroon">e</font>
;<br/><b>take </b>
<font color="Maroon">e</font>
;<br/><b>thus </b><a NAME="E3:105_1_1_1_2"/>
( <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e</font> &amp; ( <font color="Maroon">i</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">j</font> implies for <font color="Olive">n</font> being  <a href="struct_0.html#NM1">Element</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">j</font></span>)</span>  st <font color="Maroon">e</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">n</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="yellow_6.html#K8" target="_self">.</a> <font color="Maroon">j</font></span>)</span> <a href="waybel_0.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> ) )
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>consider </b><font color="Maroon">f</font> being    <a href="pboole.html#M1">ManySortedSet</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">N</font><b> such that </b><br/><a NAME="E9:105_1"/><i><font color="Green">E127</font></i>: 
for <font color="Olive">j</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">j</font>,<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font>]
 <b>from </b><i><a class="ref" href="msualg_9.html#S1">MSUALG_9:sch 1</a>();<br/></i>
<a NAME="E10:105_1"/><i><font color="Green">E128</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">N</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="E11:105_1"/>
 <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">N</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E12:105_1"/><b>then </b><i><font color="Green">E129</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</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="E13:105_1"/>
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="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> holds <br/><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">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="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_2"><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="E1:105_1_2"/>
<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>
 ;<br/>

<b>then reconsider </b><font color="Maroon">j</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E3:105_1_2"/>
 ex <font color="Olive">m</font> being  <a href="struct_0.html#NM1">Element</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">j</font></span>)</span> st <br/>( <font color="Olive">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> &amp; ( <font color="Maroon">i</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">j</font> implies for <font color="Olive">n</font> being  <a href="struct_0.html#NM1">Element</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">j</font></span>)</span>  st <font color="Olive">m</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">n</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="yellow_6.html#K8" target="_self">.</a> <font color="Maroon">j</font></span>)</span> <a href="waybel_0.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> ) )
 
<b>by </b><i/>;<br/>
<a NAME="E4:105_1_2"/><b>then </b>
<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 <span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="yellow_6.html#K8" target="_self">.</a> <font color="Maroon">j</font></span>)</span>
 
;<br/>
<b>hence </b><a NAME="E5:105_1_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> <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/>;<br/>


</div><b>end;</b></div>
<a NAME="E14:105_1"/><b>then </b><i><font color="Green">E147</font></i>: 
<font color="Maroon">f</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="card_3.html#T18">CARD_3:18</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">x</font> = <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">f</font></span><a href="tarski.html#K4">]</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#T11" target="_self">Th11</a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>take </b>
<font color="Maroon">x</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D11">WAYBEL_0:def 11</a><br/>

<b>let </b><font color="Maroon">j</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>assume </b><a NAME="E16:105_1"/><i><font color="Green">E149</font></i>: 
<font color="Maroon">x</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">j</font>
 ;<br/>

<b>consider </b><font color="Maroon">j1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</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">J</font></span>)</span><b> such that </b><br/><a NAME="E18:105_1"/><i><font color="Green">E181</font></i>: 
<font color="Maroon">j</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#T11" target="_self">Th11</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>, <font color="Maroon">i2</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> <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">i</font>, <font color="Maroon">j1</font> = <font color="Maroon">j1</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> ;<br/>
<a NAME="E21:105_1"/>
 <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">N</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="E22:105_1"/><b>then </b><i><font color="Green">E182</font></i>: 
<span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">i1</font>,<font color="Maroon">i2</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">j1</font>,<font color="Maroon">j2</font></span><a href="yellow_3.html#K7">]</a></span>
 
<b>by </b><i>, , <a class="ref" href="yellow_0.html#T1">YELLOW_0:1</a></i>;<br/>
<a NAME="E23:105_1"/><b>then </b><i><font color="Green">E183</font></i>: 
( <font color="Maroon">i1</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">j1</font> &amp; <font color="Maroon">i2</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">j2</font> )
 
<b>by </b><i><a class="ref" href="yellow_3.html#T11">YELLOW_3:11</a></i>;<br/>
<b>consider </b><font color="Maroon">m</font> being   <a href="struct_0.html#NM1">Element</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">j1</font></span>)</span><b> such that </b><br/><a NAME="E25:105_1"/><i><font color="Green">E184</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j1</font>
 <b>and </b><br/><a NAME="E26:105_1"/><i><font color="Green">E185</font></i>: 
( <font color="Maroon">i</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">j1</font> implies for <font color="Olive">n</font> being  <a href="struct_0.html#NM1">Element</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">j1</font></span>)</span>  st <font color="Maroon">m</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">n</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="yellow_6.html#K8" target="_self">.</a> <font color="Maroon">j1</font></span>)</span> <a href="waybel_0.html#K2">.</a> <font color="Olive">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> )
 <b>by </b><i/>;<br/>
<a NAME="E27:105_1"/>
<font color="Maroon">i2</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="E28:105_1"/><b>then </b>
<font color="Maroon">i2</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="E29:105_1"/><b>then </b>
 ex <font color="Olive">f</font>, <font color="Olive">g</font> being  <a href="funct_1.html#NM1">Function</a> st <br/>( <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Olive">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> &amp; ( 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">N</font> holds <br/> ex <font color="Olive">R</font> being   <a href="orders_2.html#L1">RelStr</a>  ex <font color="Olive">xi</font>, <font color="Olive">yi</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Olive">R</font> st <br/>( <font color="Olive">R</font> <a href="hidden.html#R1">=</a> <font color="Maroon">J</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> &amp; <font color="Olive">xi</font> <a href="hidden.html#R1">=</a> <font color="Olive">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> &amp; <font color="Olive">yi</font> <a href="hidden.html#R1">=</a> <font color="Olive">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> &amp; <font color="Olive">xi</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">yi</font> ) ) )
 
<b>by </b><i>, <a class="ref" href="yellow_1.html#D4">YELLOW_1:def 4</a></i>;<br/>
<a NAME="E30:105_1"/><b>then </b>
 ex <font color="Olive">R</font> being   <a href="orders_2.html#L1">RelStr</a>  ex <font color="Olive">xi</font>, <font color="Olive">yi</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Olive">R</font> st <br/>( <font color="Olive">R</font> <a href="hidden.html#R1">=</a> <font color="Maroon">J</font> <a href="yellow_6.html#K8" target="_self">.</a> <font color="Maroon">j1</font> &amp; <font color="Olive">xi</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> <a href="waybel_3.html#K4">.</a> <font color="Maroon">j1</font> &amp; <font color="Olive">yi</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> <a href="waybel_3.html#K4">.</a> <font color="Maroon">j1</font> &amp; <font color="Olive">xi</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">yi</font> )
 
;<br/>
<a NAME="E31:105_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="yellow_6.html#K8" target="_self">.</a> <font color="Maroon">j1</font></span>)</span> <a href="waybel_0.html#K2">.</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> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font>
 
<b>by </b><i>, <a class="ref" href="yellow_6.html#T12" target="_self">Th12</a>, <a class="txt" href="yellow_6.html#E20"><i><font color="Green">Lemma75</font></i></a>, <a class="ref" href="yellow_3.html#T11">YELLOW_3:11</a></i>;<br/>
<a NAME="E32:105_1"/><b>then </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">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> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font>
 
;<br/>
<a NAME="E33:105_1"/><b>then </b><i><font color="Green">E186</font></i>: 
<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">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E34:105_1"/>
<font color="Maroon">W</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font>
 
<b>by </b><i><a class="ref" href="tops_1.html#T44">TOPS_1:44</a></i>;<br/>
<b>hence </b><a NAME="E35:105_1"/>
<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">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:105"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K11" target="_self">Lim</a> <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/>


</div>
