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

<b>let </b><font color="Maroon" title="c1">T</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>;<br/><b>let </b><font color="Maroon" title="c2">N</font> be   <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of <font color="Maroon" title="c1">T</font>;<br/><b>let </b><font color="Maroon" title="c3">p</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font>;<br/>





<b>assume </b><a NAME="E1:105"/><i><font color="Green" title="E61">A1</font></i>: 
<font color="Maroon" title="c3">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_6.html#K11" title="YELLOW_6:func.11">Lim</a> <font color="Maroon" title="c2">N</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c4">J</font> be    <a href="yellow_6.html#M3" title="YELLOW_6:mode.3">net_set</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">N</font>,<font color="Maroon" title="c1">T</font>;<br/>

<b>assume </b><a NAME="E2:105"/><i><font color="Green" title="E62">A2</font></i>: 
for <font color="Olive" title="b1">i</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">N</font> holds  <font color="Maroon" title="c2">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_6.html#K11" title="YELLOW_6:func.11">Lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">J</font> <a href="yellow_6.html#K8" title="YELLOW_6:func.8">.</a> <font color="Olive" title="b1">i</font></span>)</span>
 ;<br/>

<a NAME="E3:105"/>
for <font color="Olive" title="b1">V</font> being   <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c3">p</font> holds   <a href="yellow_6.html#K9" title="YELLOW_6:func.9">Iterated</a> <font color="Maroon" title="c4">J</font> <a href="waybel_0.html#R1" title="WAYBEL_0:pred.1">is_eventually_in</a> <font color="Olive" title="b1">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" title="c5">V</font> be    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c3">p</font>;<br/>

<a NAME="E1:105_1"/><i><font color="Green" title="E63">A3</font></i>: 
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c5">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c5">V</font></span>)</span>
 
<b>by </b><i><a class="ref" href="tops_1.html#T45" title="TOPS_1:th.45">TOPS_1:45</a></i>;<br/>
<a NAME="E2:105_1"/><b>then </b>
<font color="Maroon" title="c3">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c5">V</font></span>)</span>
 
<b>by </b><i><a class="ref" href="connsp_2.html#D1" title="CONNSP_2:def.1">CONNSP_2:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c6">W</font> =  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c5">V</font> as    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c3">p</font> <b>by </b><i><a class="ref" href="connsp_2.html#D1" title="CONNSP_2:def.1">CONNSP_2:def 1</a></i>;<br/>
<a NAME="E4:105_1"/>
<font color="Maroon" title="c2">N</font> <a href="waybel_0.html#R1" title="WAYBEL_0:pred.1">is_eventually_in</a> <font color="Maroon" title="c6">W</font>
 
<b>by </b><i><a class="txt" href="yellow_6.html#E1:105"><i><font color="Green" title="E61">A1</font></i></a>, <a class="ref" href="yellow_6.html#D18" target="_self" title="YELLOW_6:def.18">Def18</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c7">i</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">N</font><b> such that </b><br/><a NAME="E6:105_1"/><i><font color="Green" title="E64">A4</font></i>: 
for <font color="Olive" title="b1">j</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">N</font>  st <font color="Maroon" title="c7">i</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b1">j</font> holds <br/><font color="Maroon" title="c2">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">W</font>
 <b>by </b><i><a class="ref" href="waybel_0.html#D11" title="WAYBEL_0:def.11">WAYBEL_0:def 11</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">N</font>,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <font color="Olive" title="b1">m</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">J</font> <a href="yellow_6.html#K8" title="YELLOW_6:func.8">.</a> $1</span>)</span> st <br/>( <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $2 &amp; ( <font color="Maroon" title="c7">i</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> $1 implies for <font color="Olive" title="b2">n</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">J</font> <a href="yellow_6.html#K8" title="YELLOW_6:func.8">.</a> $1</span>)</span>  st <font color="Olive" title="b1">m</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b2">n</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c4">J</font> <a href="yellow_6.html#K8" title="YELLOW_6:func.8">.</a> $1</span>)</span> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b2">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">W</font> ) );<br/>
<a NAME="E7:105_1"/><i><font color="Green" title="E65">A5</font></i>: 
for <font color="Olive" title="b1">j</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">N</font>  ex <font color="Olive" title="b2">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">j</font>,<font color="Olive" title="b2">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" title="c8">j</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">N</font>;<br/>

<b>reconsider </b><font color="Maroon" title="c9">j'</font> = <font color="Maroon" title="c8">j</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">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" title="c7">i</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c8">j</font> or  not <font color="Maroon" title="c7">i</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c8">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" title="c7">i</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c8">j</font>
 ;<br/><div class="add"><a NAME="E2:105_1_1_1_1"/><b>then </b><i><font color="Green" title="E66">A6</font></i>: 
<font color="Maroon" title="c2">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c9">j'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">W</font>
 <b>by </b><i><a class="txt" href="yellow_6.html#E6:105_1"><i><font color="Green" title="E64">A4</font></i></a></i>;<br/><a NAME="E3:105_1_1_1_1"/><i><font color="Green" title="E67">A7</font></i>: 
<font color="Maroon" title="c2">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c8">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_6.html#K11" title="YELLOW_6:func.11">Lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">J</font> <a href="yellow_6.html#K8" title="YELLOW_6:func.8">.</a> <font color="Maroon" title="c8">j</font></span>)</span>
 <b>by </b><i><a class="txt" href="yellow_6.html#E2:105"><i><font color="Green" title="E62">A2</font></i></a></i>;<br/><a NAME="E4:105_1_1_1_1"/>
<font color="Maroon" title="c6">W</font> is    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c2">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c8">j</font>
 <b>by </b><i><a class="txt" href="yellow_6.html#E1:105_1"><i><font color="Green" title="E63">A3</font></i></a>, <a class="txt" href="yellow_6.html#E2:105_1_1_1_1"><i><font color="Green" title="E66">A6</font></i></a>, <a class="ref" href="connsp_2.html#D1" title="CONNSP_2:def.1">CONNSP_2:def 1</a></i>;<br/><a NAME="E5:105_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c4">J</font> <a href="yellow_6.html#K8" title="YELLOW_6:func.8">.</a> <font color="Maroon" title="c8">j</font> <a href="waybel_0.html#R1" title="WAYBEL_0:pred.1">is_eventually_in</a> <font color="Maroon" title="c6">W</font>
 <b>by </b><i><a class="txt" href="yellow_6.html#E3:105_1_1_1_1"><i><font color="Green" title="E67">A7</font></i></a>, <a class="ref" href="yellow_6.html#D18" target="_self" title="YELLOW_6:def.18">Def18</a></i>;<br/><b>then consider </b><font color="Maroon" title="c10">e</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">J</font> <a href="yellow_6.html#K8" title="YELLOW_6:func.8">.</a> <font color="Maroon" title="c8">j</font></span>)</span><b> such that </b><br/><a NAME="E7:105_1_1_1_1"/><i><font color="Green" title="E68">A8</font></i>: 
for <font color="Olive" title="b1">u</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">J</font> <a href="yellow_6.html#K8" title="YELLOW_6:func.8">.</a> <font color="Maroon" title="c8">j</font></span>)</span>  st <font color="Maroon" title="c10">e</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b1">u</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c4">J</font> <a href="yellow_6.html#K8" title="YELLOW_6:func.8">.</a> <font color="Maroon" title="c8">j</font></span>)</span> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b1">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">W</font>
 <b>by </b><i><a class="ref" href="waybel_0.html#D11" title="WAYBEL_0:def.11">WAYBEL_0:def 11</a></i>;<br/><b>take </b>
<font color="Maroon" title="c10">e</font>
;<br/><b>take </b>
<font color="Maroon" title="c10">e</font>
;<br/><b>thus </b><a NAME="E8:105_1_1_1_1"/>
<font color="Maroon" title="c10">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">e</font>
 ;<br/><b>assume </b><a NAME="E9:105_1_1_1_1"/>
<font color="Maroon" title="c7">i</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c8">j</font>
 ;<br/><b>thus </b><a NAME="E10:105_1_1_1_1"/>
for <font color="Olive" title="b1">n</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">J</font> <a href="yellow_6.html#K8" title="YELLOW_6:func.8">.</a> <font color="Maroon" title="c8">j</font></span>)</span>  st <font color="Maroon" title="c10">e</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c4">J</font> <a href="yellow_6.html#K8" title="YELLOW_6:func.8">.</a> <font color="Maroon" title="c8">j</font></span>)</span> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">W</font>
 <b>by </b><i><a class="txt" href="yellow_6.html#E7:105_1_1_1_1"><i><font color="Green" title="E68">A8</font></i></a></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" title="E66">A9</font></i>: 
not <font color="Maroon" title="c7">i</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c8">j</font>
 ;<br/><div class="add"><b>consider </b><font color="Maroon" title="c10">e</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">J</font> <a href="yellow_6.html#K8" title="YELLOW_6:func.8">.</a> <font color="Maroon" title="c8">j</font></span>)</span>;<br/><b>take </b>
<font color="Maroon" title="c10">e</font>
;<br/><b>take </b>
<font color="Maroon" title="c10">e</font>
;<br/><b>thus </b><a NAME="E3:105_1_1_1_2"/>
( <font color="Maroon" title="c10">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">e</font> &amp; ( <font color="Maroon" title="c7">i</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c8">j</font> implies for <font color="Olive" title="b1">n</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">J</font> <a href="yellow_6.html#K8" title="YELLOW_6:func.8">.</a> <font color="Maroon" title="c8">j</font></span>)</span>  st <font color="Maroon" title="c10">e</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c4">J</font> <a href="yellow_6.html#K8" title="YELLOW_6:func.8">.</a> <font color="Maroon" title="c8">j</font></span>)</span> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">W</font> ) )
 <b>by </b><i><a class="txt" href="yellow_6.html#E1:105_1_1_1_2"><i><font color="Green" title="E66">A9</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c8">f</font> being    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">N</font><b> such that </b><br/><a NAME="E9:105_1"/><i><font color="Green" title="E66">A10</font></i>: 
for <font color="Olive" title="b1">j</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">N</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">j</font>,<font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font>]
 <b>from </b><i><a class="ref" href="msualg_9.html#S1" title="MSUALG_9:sch.1">MSUALG_9:sch 1</a>(<a class="txt" href="yellow_6.html#E7:105_1"><i><font color="Green" title="E65">A5</font></i></a>);<br/></i>
<a NAME="E10:105_1"/><i><font color="Green" title="E67">A11</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9" title="YELLOW_6:func.9">Iterated</a> <font color="Maroon" title="c4">J</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">N</font>,<span class="p2">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p3">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c4">J</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="yellow_6.html#T35" target="_self" title="YELLOW_6:th.35">Th35</a></i>;<br/>
<a NAME="E11:105_1"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c4">J</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">N</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E12:105_1"/><b>then </b><i><font color="Green" title="E68">A12</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c8">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c4">J</font></span>)</span>
 
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E13:105_1"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c4">J</font></span>)</span> holds <br/><font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c4">J</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">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" title="c9">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:105_1_2"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c4">J</font></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c10">j</font> = <font color="Maroon" title="c9">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">N</font> <b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E3:105_1_2"/>
 ex <font color="Olive" title="b1">m</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">J</font> <a href="yellow_6.html#K8" title="YELLOW_6:func.8">.</a> <font color="Maroon" title="c10">j</font></span>)</span> st <br/>( <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">j</font> &amp; ( <font color="Maroon" title="c7">i</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c10">j</font> implies for <font color="Olive" title="b2">n</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">J</font> <a href="yellow_6.html#K8" title="YELLOW_6:func.8">.</a> <font color="Maroon" title="c10">j</font></span>)</span>  st <font color="Olive" title="b1">m</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b2">n</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c4">J</font> <a href="yellow_6.html#K8" title="YELLOW_6:func.8">.</a> <font color="Maroon" title="c10">j</font></span>)</span> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b2">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">W</font> ) )
 
<b>by </b><i><a class="txt" href="yellow_6.html#E9:105_1"><i><font color="Green" title="E66">A10</font></i></a></i>;<br/>
<a NAME="E4:105_1_2"/><b>then </b>
<font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">J</font> <a href="yellow_6.html#K8" title="YELLOW_6:func.8">.</a> <font color="Maroon" title="c10">j</font></span>)</span>
 
;<br/>
<b>hence </b><a NAME="E5:105_1_2"/>
<font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c4">J</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font>
 <b>by </b><i><a class="ref" href="yellow_6.html#T9" target="_self" title="YELLOW_6:th.9">Th9</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E14:105_1"/><b>then </b><i><font color="Green" title="E69">A13</font></i>: 
<font color="Maroon" title="c8">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c4">J</font></span>)</span>
 
<b>by </b><i><a class="txt" href="yellow_6.html#E12:105_1"><i><font color="Green" title="E68">A12</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c9">x</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c8">f</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9" title="YELLOW_6:func.9">Iterated</a> <font color="Maroon" title="c4">J</font></span>)</span> <b>by </b><i><a class="txt" href="yellow_6.html#E10:105_1"><i><font color="Green" title="E67">A11</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c9">x</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D11" title="WAYBEL_0:def.11">WAYBEL_0:def 11</a><br/>

<b>let </b><font color="Maroon" title="c10">j</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9" title="YELLOW_6:func.9">Iterated</a> <font color="Maroon" title="c4">J</font></span>)</span>;<br/>

<b>assume </b><a NAME="E16:105_1"/><i><font color="Green" title="E70">A14</font></i>: 
<font color="Maroon" title="c9">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c10">j</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c11">j1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">N</font>, <font color="Maroon" title="c12">j2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c4">J</font></span>)</span><b> such that </b><br/><a NAME="E18:105_1"/><i><font color="Green" title="E71">A15</font></i>: 
<font color="Maroon" title="c10">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">j1</font>,<font color="Maroon" title="c12">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="txt" href="yellow_6.html#E10:105_1"><i><font color="Green" title="E67">A11</font></i></a>, <a class="ref" href="domain_1.html#T9" title="DOMAIN_1:th.9">DOMAIN_1:9</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c13">j2</font> = <font color="Maroon" title="c12">j2</font>, <font color="Maroon" title="c14">i2</font> = <font color="Maroon" title="c8">f</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <font color="Maroon" title="c4">J</font></span>)</span> <b>by </b><i><a class="txt" href="yellow_6.html#E14:105_1"><i><font color="Green" title="E69">A13</font></i></a>, <a class="ref" href="yellow_1.html#D4" title="YELLOW_1:def.4">YELLOW_1:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c15">i1</font> = <font color="Maroon" title="c7">i</font>, <font color="Maroon" title="c16">j1</font> = <font color="Maroon" title="c11">j1</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">N</font> ;<br/>
<a NAME="E21:105_1"/>
 <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9" title="YELLOW_6:func.9">Iterated</a> <font color="Maroon" title="c4">J</font></span>)</span>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K9" title="YELLOW_6:func.9">Iterated</a> <font color="Maroon" title="c4">J</font></span>)</span> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="yellow_3.html#K3" title="YELLOW_3:func.3">[:</a><span class="default"><font color="Maroon" title="c2">N</font>,<span class="p2">(<span class="default"><a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <font color="Maroon" title="c4">J</font></span>)</span></span><a href="yellow_3.html#K3" title="YELLOW_3:func.3">:]</a></span>
 
<b>by </b><i><a class="ref" href="yellow_6.html#D16" target="_self" title="YELLOW_6:def.16">Def16</a></i>;<br/>
<a NAME="E22:105_1"/><b>then </b><i><font color="Green" title="E72">A16</font></i>: 
<span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><font color="Maroon" title="c15">i1</font>,<font color="Maroon" title="c14">i2</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><font color="Maroon" title="c16">j1</font>,<font color="Maroon" title="c13">j2</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span>
 
<b>by </b><i><a class="txt" href="yellow_6.html#E16:105_1"><i><font color="Green" title="E70">A14</font></i></a>, <a class="txt" href="yellow_6.html#E18:105_1"><i><font color="Green" title="E71">A15</font></i></a>, <a class="ref" href="yellow_0.html#T1" title="YELLOW_0:th.1">YELLOW_0:1</a></i>;<br/>
<a NAME="E23:105_1"/><b>then </b><i><font color="Green" title="E73">A17</font></i>: 
( <font color="Maroon" title="c15">i1</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c16">j1</font> &amp; <font color="Maroon" title="c14">i2</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c13">j2</font> )
 
<b>by </b><i><a class="ref" href="yellow_3.html#T11" title="YELLOW_3:th.11">YELLOW_3:11</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c17">m</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">J</font> <a href="yellow_6.html#K8" title="YELLOW_6:func.8">.</a> <font color="Maroon" title="c16">j1</font></span>)</span><b> such that </b><br/><a NAME="E25:105_1"/><i><font color="Green" title="E74">A18</font></i>: 
<font color="Maroon" title="c17">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">j1</font>
 <b>and </b><br/><a NAME="E26:105_1"/><i><font color="Green" title="E75">A19</font></i>: 
( <font color="Maroon" title="c7">i</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c16">j1</font> implies for <font color="Olive" title="b1">n</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">J</font> <a href="yellow_6.html#K8" title="YELLOW_6:func.8">.</a> <font color="Maroon" title="c16">j1</font></span>)</span>  st <font color="Maroon" title="c17">m</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c4">J</font> <a href="yellow_6.html#K8" title="YELLOW_6:func.8">.</a> <font color="Maroon" title="c16">j1</font></span>)</span> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">W</font> )
 <b>by </b><i><a class="txt" href="yellow_6.html#E9:105_1"><i><font color="Green" title="E66">A10</font></i></a></i>;<br/>
<a NAME="E27:105_1"/>
<font color="Maroon" title="c14">i2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <font color="Maroon" title="c4">J</font></span>)</span>
 
;<br/>
<a NAME="E28:105_1"/><b>then </b>
<font color="Maroon" title="c14">i2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c4">J</font></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_1.html#D4" title="YELLOW_1:def.4">YELLOW_1:def 4</a></i>;<br/>
<a NAME="E29:105_1"/><b>then </b>
 ex <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>( <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">i2</font> &amp; <font color="Olive" title="b2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">j2</font> &amp; ( for <font color="Olive" title="b3">i</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b3">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">N</font> holds <br/> ex <font color="Olive" title="b4">R</font> being   <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a>  ex <font color="Olive" title="b5">xi</font>, <font color="Olive" title="b6">yi</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b4">R</font> st <br/>( <font color="Olive" title="b4">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">J</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">i</font> &amp; <font color="Olive" title="b5">xi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">i</font> &amp; <font color="Olive" title="b6">yi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">i</font> &amp; <font color="Olive" title="b5">xi</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b6">yi</font> ) ) )
 
<b>by </b><i><a class="txt" href="yellow_6.html#E23:105_1"><i><font color="Green" title="E73">A17</font></i></a>, <a class="ref" href="yellow_1.html#D4" title="YELLOW_1:def.4">YELLOW_1:def 4</a></i>;<br/>
<a NAME="E30:105_1"/><b>then </b>
 ex <font color="Olive" title="b1">R</font> being   <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a>  ex <font color="Olive" title="b2">xi</font>, <font color="Olive" title="b3">yi</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b1">R</font> st <br/>( <font color="Olive" title="b1">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">J</font> <a href="yellow_6.html#K8" title="YELLOW_6:func.8">.</a> <font color="Maroon" title="c16">j1</font> &amp; <font color="Olive" title="b2">xi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">i2</font> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Maroon" title="c16">j1</font> &amp; <font color="Olive" title="b3">yi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">j2</font> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Maroon" title="c16">j1</font> &amp; <font color="Olive" title="b2">xi</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b3">yi</font> )
 
;<br/>
<a NAME="E31:105_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">J</font> <a href="yellow_6.html#K8" title="YELLOW_6:func.8">.</a> <font color="Maroon" title="c16">j1</font></span>)</span> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">j2</font> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Maroon" title="c16">j1</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">W</font>
 
<b>by </b><i><a class="txt" href="yellow_6.html#E22:105_1"><i><font color="Green" title="E72">A16</font></i></a>, <a class="txt" href="yellow_6.html#E25:105_1"><i><font color="Green" title="E74">A18</font></i></a>, <a class="txt" href="yellow_6.html#E26:105_1"><i><font color="Green" title="E75">A19</font></i></a>, <a class="ref" href="yellow_3.html#T11" title="YELLOW_3:th.11">YELLOW_3:11</a></i>;<br/>
<a NAME="E32:105_1"/><b>then </b>
the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">J</font> <a href="yellow_6.html#K8" title="YELLOW_6:func.8">.</a> <font color="Maroon" title="c16">j1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">j2</font> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Maroon" title="c16">j1</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">W</font>
 
;<br/>
<a NAME="E33:105_1"/><b>then </b><i><font color="Green" title="E76">A20</font></i>: 
<span class="p1">(<span class="default"><a href="yellow_6.html#K9" title="YELLOW_6:func.9">Iterated</a> <font color="Maroon" title="c4">J</font></span>)</span> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c10">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">W</font>
 
<b>by </b><i><a class="txt" href="yellow_6.html#E18:105_1"><i><font color="Green" title="E71">A15</font></i></a>, <a class="ref" href="yellow_6.html#T36" target="_self" title="YELLOW_6:th.36">Th36</a></i>;<br/>
<a NAME="E34:105_1"/>
<font color="Maroon" title="c6">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">V</font>
 
<b>by </b><i><a class="ref" href="tops_1.html#T44" title="TOPS_1:th.44">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" title="YELLOW_6:func.9">Iterated</a> <font color="Maroon" title="c4">J</font></span>)</span> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c10">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">V</font>
 <b>by </b><i><a class="txt" href="yellow_6.html#E33:105_1"><i><font color="Green" title="E76">A20</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:105"/>
<font color="Maroon" title="c3">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_6.html#K11" title="YELLOW_6:func.11">Lim</a> <span class="p1">(<span class="default"><a href="yellow_6.html#K9" title="YELLOW_6:func.9">Iterated</a> <font color="Maroon" title="c4">J</font></span>)</span>
 <b>by </b><i><a class="ref" href="yellow_6.html#D18" target="_self" title="YELLOW_6:def.18">Def18</a></i>;<br/>


</div>
