<?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">x</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="E1:86"/><i><font color="Green" title="E39">A1</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="waybel_9.html#R3" title="WAYBEL_9:pred.3">is_a_cluster_point_of</a> <font color="Maroon" title="c2">N</font>
 ;<br/>

<b>set </b><font color="Maroon" title="c4">n</font> = the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c2">N</font>;<br/>
<b>set </b><font color="Maroon" title="c5">S'</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">s</font>,<font color="Olive" title="b2">O</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> where <font color="Olive" title="b1">s</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">N</font>, <font color="Olive" title="b2">O</font> is   <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#K10" title="YELLOW_6:func.10">OpenNeighborhoods</a> <font color="Maroon" title="c3">x</font></span>)</span> : <font color="Maroon" title="c2">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">O</font> </span>}</span> ;<br/>
<a NAME="E2:86"/>
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="ref" href="tops_1.html#T53" title="TOPS_1:th.53">TOPS_1:53</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c6">O</font> =  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T</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_6.html#K10" title="YELLOW_6:func.10">OpenNeighborhoods</a> <font color="Maroon" title="c3">x</font></span>)</span> <b>by </b><i><a class="ref" href="yellow_6.html#T39" title="YELLOW_6:th.39">YELLOW_6:39</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c7">q</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">N</font>;<br/>
<a NAME="E5:86"/>
<font color="Maroon" title="c2">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c7">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T</font>
 
;<br/>
<a NAME="E6:86"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c7">q</font>,<font color="Maroon" title="c6">O</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">s</font>,<font color="Olive" title="b2">O</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> where <font color="Olive" title="b1">s</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">N</font>, <font color="Olive" title="b2">O</font> is   <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#K10" title="YELLOW_6:func.10">OpenNeighborhoods</a> <font color="Maroon" title="c3">x</font></span>)</span> : <font color="Maroon" title="c2">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">O</font> </span>}</span> 
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c8">S'</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">s</font>,<font color="Olive" title="b2">O</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> where <font color="Olive" title="b1">s</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">N</font>, <font color="Olive" title="b2">O</font> is   <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#K10" title="YELLOW_6:func.10">OpenNeighborhoods</a> <font color="Maroon" title="c3">x</font></span>)</span> : <font color="Maroon" title="c2">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">O</font> </span>}</span>  as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <font color="Olive" title="b1">s1</font>, <font color="Olive" title="b2">s2</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">N</font> st <br/>( <font color="Olive" title="b1">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  &amp; <font color="Olive" title="b2">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $2 <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  &amp; <font color="Olive" title="b1">s1</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b2">s2</font> &amp; $2 <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> $1 <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  );<br/>
<b>consider </b><font color="Maroon" title="c9">L</font> being  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> <b> such that </b><br/><a NAME="E9:86"/><i><font color="Green" title="E40">A2</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c9">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">S'</font>
 <b>and </b><br/><a NAME="E10:86"/><i><font color="Green" title="E41">A3</font></i>: 
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c9">L</font> holds <br/> ( <font color="Olive" title="b1">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b2">b</font> iff <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>] )
 <b>from </b><i><a class="ref" href="yellow_0.html#S1" title="YELLOW_0:sch.1">YELLOW_0:sch 1</a>();<br/></i>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c9">L</font>) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c2">N</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">$1 <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span>;<br/>
<a NAME="E11:86"/><i><font color="Green" title="E42">A4</font></i>: 
for <font color="Olive" title="b1">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c9">L</font> holds  <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">a</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="c1">T</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="86_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c9">L</font>;<br/>

<a NAME="E1:86_1"/>
<font color="Maroon" title="c10">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">S'</font>
 
<b>by </b><i><a class="txt" href="waybel_9.html#E9:86"><i><font color="Green" title="E40">A2</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c11">s</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">O</font> being   <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#K10" title="YELLOW_6:func.10">OpenNeighborhoods</a> <font color="Maroon" title="c3">x</font></span>)</span><b> such that </b><br/><a NAME="E3:86_1"/><i><font color="Green" title="E43">A5</font></i>: 
<font color="Maroon" title="c10">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c11">s</font>,<font color="Maroon" title="c12">O</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 <b>and </b><br/><a NAME="E4:86_1"/>
<font color="Maroon" title="c2">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c11">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">O</font>
 ;<br/>
<a NAME="E5:86_1"/>
the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c2">N</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">a</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c2">N</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">s</font>
 
<b>by </b><i><a class="txt" href="waybel_9.html#E3:86_1"><i><font color="Green" title="E43">A5</font></i></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>;<br/>
<b>hence </b><a NAME="E6:86_1"/>
the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c2">N</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">a</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span> <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="c1">T</font>
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c10">g</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c9">L</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font><b> such that </b><br/><a NAME="E13:86"/><i><font color="Green" title="E43">A6</font></i>: 
for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c9">L</font> holds  <font color="Maroon" title="c10">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">x</font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S8" title="FUNCT_2:sch.8">FUNCT_2:sch 8</a>(<a class="txt" href="waybel_9.html#E11:86"><i><font color="Green" title="E42">A4</font></i></a>);<br/></i>
<b>set </b><font color="Maroon" title="c11">M</font> =  <a href="waybel_0.html#G1" title="WAYBEL_0:aggr.1">NetStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c9">L</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c9">L</font>,<font color="Maroon" title="c10">g</font> #);<br/>
<a NAME="E14:86"/>
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="waybel_0.html#G1" title="WAYBEL_0:aggr.1">NetStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c9">L</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c9">L</font>,<font color="Maroon" title="c10">g</font> #)  ex <font color="Olive" title="b3">z</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="waybel_0.html#G1" title="WAYBEL_0:aggr.1">NetStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c9">L</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c9">L</font>,<font color="Maroon" title="c10">g</font> #) st <br/>( <font color="Olive" title="b1">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b3">z</font> &amp; <font color="Olive" title="b2">b</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b3">z</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="86_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c12">a</font>, <font color="Maroon" title="c13">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="waybel_0.html#G1" title="WAYBEL_0:aggr.1">NetStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c9">L</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c9">L</font>,<font color="Maroon" title="c10">g</font> #);<br/>

<a NAME="E1:86_2"/>
<font color="Maroon" title="c12">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">S'</font>
 
<b>by </b><i><a class="txt" href="waybel_9.html#E9:86"><i><font color="Green" title="E40">A2</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c14">aa</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="c15">Oa</font> being   <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#K10" title="YELLOW_6:func.10">OpenNeighborhoods</a> <font color="Maroon" title="c3">x</font></span>)</span><b> such that </b><br/><a NAME="E3:86_2"/><i><font color="Green" title="E44">A7</font></i>: 
<font color="Maroon" title="c12">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c14">aa</font>,<font color="Maroon" title="c15">Oa</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 <b>and </b><br/><a NAME="E4:86_2"/>
<font color="Maroon" title="c2">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c14">aa</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">Oa</font>
 ;<br/>
<a NAME="E5:86_2"/>
<font color="Maroon" title="c13">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">S'</font>
 
<b>by </b><i><a class="txt" href="waybel_9.html#E9:86"><i><font color="Green" title="E40">A2</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c16">bb</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="c17">Ob</font> being   <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#K10" title="YELLOW_6:func.10">OpenNeighborhoods</a> <font color="Maroon" title="c3">x</font></span>)</span><b> such that </b><br/><a NAME="E7:86_2"/><i><font color="Green" title="E45">A8</font></i>: 
<font color="Maroon" title="c13">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c16">bb</font>,<font color="Maroon" title="c17">Ob</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 <b>and </b><br/><a NAME="E8:86_2"/>
<font color="Maroon" title="c2">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c16">bb</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">Ob</font>
 ;<br/>
<b>consider </b><font color="Maroon" title="c18">z1</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="E10:86_2"/><i><font color="Green" title="E46">A9</font></i>: 
( <font color="Maroon" title="c14">aa</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c18">z1</font> &amp; <font color="Maroon" title="c16">bb</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c18">z1</font> )
 <b>by </b><i><a class="ref" href="yellow_6.html#D5" title="YELLOW_6:def.5">YELLOW_6:def 5</a></i>;<br/>
<a NAME="E11:86_2"/>
( <font color="Maroon" title="c15">Oa</font> is    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c3">x</font> &amp; <font color="Maroon" title="c17">Ob</font> is    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c3">x</font> )
 
<b>by </b><i><a class="ref" href="waybel_9.html#T21" target="_self" title="WAYBEL_9:th.21">Th21</a></i>;<br/>
<a NAME="E12:86_2"/><b>then </b>
<font color="Maroon" title="c15">Oa</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c17">Ob</font> is    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c3">x</font>
 
<b>by </b><i><a class="ref" href="connsp_2.html#T4" title="CONNSP_2:th.4">CONNSP_2:4</a></i>;<br/>
<a NAME="E13:86_2"/><b>then </b>
<font color="Maroon" title="c2">N</font> <a href="waybel_0.html#R2" title="WAYBEL_0:pred.2">is_often_in</a> <font color="Maroon" title="c15">Oa</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c17">Ob</font>
 
<b>by </b><i><a class="txt" href="waybel_9.html#E1:86"><i><font color="Green" title="E39">A1</font></i></a>, <a class="ref" href="waybel_9.html#D9" target="_self" title="WAYBEL_9:def.9">Def9</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c19">d</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="E15:86_2"/><i><font color="Green" title="E47">A10</font></i>: 
<font color="Maroon" title="c18">z1</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c19">d</font>
 <b>and </b><br/><a NAME="E16:86_2"/><i><font color="Green" title="E48">A11</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="c19">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">Oa</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c17">Ob</font>
 <b>by </b><i><a class="ref" href="waybel_0.html#D12" title="WAYBEL_0:def.12">WAYBEL_0:def 12</a></i>;<br/>
<a NAME="E17:86_2"/>
<font color="Maroon" title="c15">Oa</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c17">Ob</font> is   <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#K10" title="YELLOW_6:func.10">OpenNeighborhoods</a> <font color="Maroon" title="c3">x</font></span>)</span>
 
<b>by </b><i><a class="ref" href="waybel_9.html#T22" target="_self" title="WAYBEL_9:th.22">Th22</a></i>;<br/>
<a NAME="E18:86_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c19">d</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c15">Oa</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c17">Ob</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">S'</font>
 
<b>by </b><i><a class="txt" href="waybel_9.html#E16:86_2"><i><font color="Green" title="E48">A11</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c20">z</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c19">d</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c15">Oa</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c17">Ob</font></span>)</span></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 <a href="waybel_0.html#G1" title="WAYBEL_0:aggr.1">NetStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c9">L</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c9">L</font>,<font color="Maroon" title="c10">g</font> #) <b>by </b><i><a class="txt" href="waybel_9.html#E9:86"><i><font color="Green" title="E40">A2</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c20">z</font>
;<br/>

<b>reconsider </b><font color="Maroon" title="c21">A1</font> = <font color="Maroon" title="c12">a</font>, <font color="Maroon" title="c22">C7</font> = <font color="Maroon" title="c13">b</font>, <font color="Maroon" title="c23">z2</font> = <font color="Maroon" title="c20">z</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c9">L</font> ;<br/>
<a NAME="E21:86_2"/><i><font color="Green" title="E49">A12</font></i>: 
( <font color="Maroon" title="c15">Oa</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c17">Ob</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c15">Oa</font> &amp; <font color="Maroon" title="c15">Oa</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c17">Ob</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c17">Ob</font> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E22:86_2"/><i><font color="Green" title="E50">A13</font></i>: 
( <font color="Maroon" title="c14">aa</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c19">d</font> &amp; <font color="Maroon" title="c16">bb</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c19">d</font> )
 
<b>by </b><i><a class="txt" href="waybel_9.html#E10:86_2"><i><font color="Green" title="E46">A9</font></i></a>, <a class="txt" href="waybel_9.html#E15:86_2"><i><font color="Green" title="E47">A10</font></i></a>, <a class="ref" href="yellow_0.html#D2" title="YELLOW_0:def.2">YELLOW_0:def 2</a></i>;<br/>
<a NAME="E23:86_2"/>
( <font color="Maroon" title="c21">A1</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">aa</font> &amp; <font color="Maroon" title="c22">C7</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">bb</font> &amp; <font color="Maroon" title="c23">z2</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">d</font> &amp; <font color="Maroon" title="c21">A1</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">Oa</font> &amp; <font color="Maroon" title="c22">C7</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">Ob</font> &amp; <font color="Maroon" title="c23">z2</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">Oa</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c17">Ob</font> )
 
<b>by </b><i><a class="txt" href="waybel_9.html#E3:86_2"><i><font color="Green" title="E44">A7</font></i></a>, <a class="txt" href="waybel_9.html#E7:86_2"><i><font color="Green" title="E45">A8</font></i></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>;<br/>
<a NAME="E24:86_2"/><b>then </b>
( <font color="Maroon" title="c21">A1</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c23">z2</font> &amp; <font color="Maroon" title="c22">C7</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c23">z2</font> )
 
<b>by </b><i><a class="txt" href="waybel_9.html#E10:86"><i><font color="Green" title="E41">A3</font></i></a>, <a class="txt" href="waybel_9.html#E21:86_2"><i><font color="Green" title="E49">A12</font></i></a>, <a class="txt" href="waybel_9.html#E22:86_2"><i><font color="Green" title="E50">A13</font></i></a></i>;<br/>
<b>hence </b><a NAME="E25:86_2"/>
( <font color="Maroon" title="c12">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c20">z</font> &amp; <font color="Maroon" title="c13">b</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c20">z</font> )
 <b>by </b><i><a class="ref" href="yellow_0.html#T1" title="YELLOW_0:th.1">YELLOW_0:1</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c12">M</font> =  <a href="waybel_0.html#G1" title="WAYBEL_0:aggr.1">NetStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c9">L</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c9">L</font>,<font color="Maroon" title="c10">g</font> #) as   <a href="waybel_0.html#NM1" title="WAYBEL_0:NM.1">prenet</a> of <font color="Maroon" title="c1">T</font> <b>by </b><i><a class="ref" href="yellow_6.html#D5" title="YELLOW_6:def.5">YELLOW_6:def 5</a></i>;<br/>
<a NAME="E16:86"/>
<font color="Maroon" title="c12">M</font> is <a href="orders_2.html#V3" title="ORDERS_2:attr.3">transitive</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="86_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">x</font>, <font color="Maroon" title="c14">y</font>, <font color="Maroon" title="c15">z</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c12">M</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_0.html#D2" title="YELLOW_0:def.2">YELLOW_0:def 2</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:86_3"/><i><font color="Green" title="E44">A14</font></i>: 
<font color="Maroon" title="c13">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c14">y</font>
 <b>and </b><br/><a NAME="E2:86_3"/><i><font color="Green" title="E45">A15</font></i>: 
<font color="Maroon" title="c14">y</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c15">z</font>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c16">x1</font> = <font color="Maroon" title="c13">x</font>, <font color="Maroon" title="c17">y1</font> = <font color="Maroon" title="c14">y</font>, <font color="Maroon" title="c18">z1</font> = <font color="Maroon" title="c15">z</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c9">L</font> ;<br/>
<a NAME="E4:86_3"/>
<font color="Maroon" title="c16">x1</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c17">y1</font>
 
<b>by </b><i><a class="txt" href="waybel_9.html#E1:86_3"><i><font color="Green" title="E44">A14</font></i></a>, <a class="ref" href="yellow_0.html#T1" title="YELLOW_0:th.1">YELLOW_0:1</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c19">sx</font>, <font color="Maroon" title="c20">sy</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="E6:86_3"/><i><font color="Green" title="E46">A16</font></i>: 
( <font color="Maroon" title="c19">sx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">x1</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  &amp; <font color="Maroon" title="c20">sy</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">y1</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  &amp; <font color="Maroon" title="c19">sx</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c20">sy</font> &amp; <font color="Maroon" title="c17">y1</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c16">x1</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  )
 <b>by </b><i><a class="txt" href="waybel_9.html#E10:86"><i><font color="Green" title="E41">A3</font></i></a></i>;<br/>
<a NAME="E7:86_3"/>
<font color="Maroon" title="c17">y1</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c18">z1</font>
 
<b>by </b><i><a class="txt" href="waybel_9.html#E2:86_3"><i><font color="Green" title="E45">A15</font></i></a>, <a class="ref" href="yellow_0.html#T1" title="YELLOW_0:th.1">YELLOW_0:1</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c21">s1</font>, <font color="Maroon" title="c22">s2</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="E9:86_3"/><i><font color="Green" title="E47">A17</font></i>: 
( <font color="Maroon" title="c21">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">y1</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  &amp; <font color="Maroon" title="c22">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">z1</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  &amp; <font color="Maroon" title="c21">s1</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c22">s2</font> &amp; <font color="Maroon" title="c18">z1</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c17">y1</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  )
 <b>by </b><i><a class="txt" href="waybel_9.html#E10:86"><i><font color="Green" title="E41">A3</font></i></a></i>;<br/>
<a NAME="E10:86_3"/><i><font color="Green" title="E48">A18</font></i>: 
<font color="Maroon" title="c19">sx</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c22">s2</font>
 
<b>by </b><i><a class="txt" href="waybel_9.html#E6:86_3"><i><font color="Green" title="E46">A16</font></i></a>, <a class="txt" href="waybel_9.html#E9:86_3"><i><font color="Green" title="E47">A17</font></i></a>, <a class="ref" href="yellow_0.html#D2" title="YELLOW_0:def.2">YELLOW_0:def 2</a></i>;<br/>
<a NAME="E11:86_3"/>
<font color="Maroon" title="c18">z1</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c16">x1</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
 
<b>by </b><i><a class="txt" href="waybel_9.html#E6:86_3"><i><font color="Green" title="E46">A16</font></i></a>, <a class="txt" href="waybel_9.html#E9:86_3"><i><font color="Green" title="E47">A17</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E12:86_3"/><b>then </b>
<font color="Maroon" title="c16">x1</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c18">z1</font>
 
<b>by </b><i><a class="txt" href="waybel_9.html#E10:86"><i><font color="Green" title="E41">A3</font></i></a>, <a class="txt" href="waybel_9.html#E6:86_3"><i><font color="Green" title="E46">A16</font></i></a>, <a class="txt" href="waybel_9.html#E9:86_3"><i><font color="Green" title="E47">A17</font></i></a>, <a class="txt" href="waybel_9.html#E10:86_3"><i><font color="Green" title="E48">A18</font></i></a></i>;<br/>
<b>hence </b><a NAME="E13:86_3"/>
<font color="Maroon" title="c13">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c15">z</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T1" title="YELLOW_0:th.1">YELLOW_0:1</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c13">M</font> = <font color="Maroon" title="c12">M</font> as   <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of <font color="Maroon" title="c1">T</font> ;<br/>
<a NAME="E18:86"/>
<font color="Maroon" title="c13">M</font> is    <a href="yellow_6.html#M2" title="YELLOW_6:mode.2">subnet</a> of <font color="Maroon" title="c2">N</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="86_4"><b>proof </b></a><div class="add">

<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c9">L</font>) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = $1 <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> ;<br/>
<a NAME="E1:86_4"/><i><font color="Green" title="E44">A19</font></i>: 
for <font color="Olive" title="b1">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c9">L</font> holds  <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">a</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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="86_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c14">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c9">L</font>;<br/>

<a NAME="E1:86_4_1"/>
<font color="Maroon" title="c14">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">S'</font>
 
<b>by </b><i><a class="txt" href="waybel_9.html#E9:86"><i><font color="Green" title="E40">A2</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c15">s</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="c16">O</font> being   <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#K10" title="YELLOW_6:func.10">OpenNeighborhoods</a> <font color="Maroon" title="c3">x</font></span>)</span><b> such that </b><br/><a NAME="E3:86_4_1"/><i><font color="Green" title="E45">A20</font></i>: 
<font color="Maroon" title="c14">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c15">s</font>,<font color="Maroon" title="c16">O</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 <b>and </b><br/><a NAME="E4:86_4_1"/>
<font color="Maroon" title="c2">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c15">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">O</font>
 ;<br/>
<a NAME="E5:86_4_1"/>
<font color="Maroon" title="c14">a</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">s</font>
 
<b>by </b><i><a class="txt" href="waybel_9.html#E3:86_4_1"><i><font color="Green" title="E45">A20</font></i></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>;<br/>
<b>hence </b><a NAME="E6:86_4_1"/>
<font color="Maroon" title="c14">a</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <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>
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c14">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c9">L</font>,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="E3:86_4"/><i><font color="Green" title="E45">A21</font></i>: 
for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c9">L</font> holds  <font color="Maroon" title="c14">f</font> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">x</font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S8" title="FUNCT_2:sch.8">FUNCT_2:sch 8</a>(<a class="txt" href="waybel_9.html#E1:86_4"><i><font color="Green" title="E44">A19</font></i></a>);<br/></i>
<b>reconsider </b><font color="Maroon" title="c15">f</font> = <font color="Maroon" title="c14">f</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c13">M</font>,<font color="Maroon" title="c2">N</font> ;<br/>
<b>take </b>
<font color="Maroon" title="c15">f</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_6.html#D12" title="YELLOW_6:def.12">YELLOW_6:def 12</a><br/>

<a NAME="E5:86_4"/>
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> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c13">M</font> holds <br/><font color="Maroon" title="c10">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c2">N</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c15">f</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="86_4_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c16">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:86_4_2"/><i><font color="Green" title="E46">A22</font></i>: 
<font color="Maroon" title="c16">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 <font color="Maroon" title="c13">M</font>
 ;<br/>

<b>hence </b><font color="Maroon" title="c10">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">x</font> = 
the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c2">N</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">x</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span>
<b>by </b><i><a class="txt" href="waybel_9.html#E13:86"><i><font color="Green" title="E43">A6</font></i></a></i>
<br/>.= 
the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c2">N</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">x</font></span>)</span>
<b>by </b><i><a class="txt" href="waybel_9.html#E3:86_4"><i><font color="Green" title="E45">A21</font></i></a>, <a class="txt" href="waybel_9.html#E1:86_4_2"><i><font color="Green" title="E46">A22</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default">the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c2">N</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c15">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">x</font>
<b>by </b><i><a class="txt" href="waybel_9.html#E1:86_4_2"><i><font color="Green" title="E46">A22</font></i></a>, <a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></i>
;<br/><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:86_4"/>
the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c13">M</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c2">N</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c15">f</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T18" title="FUNCT_2:th.18">FUNCT_2:18</a></i>;<br/>

<b>let </b><font color="Maroon" title="c16">m</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">N</font>;<br/>

<a NAME="E7:86_4"/>
<font color="Maroon" title="c2">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c16">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T</font>
 
;<br/>
<a NAME="E8:86_4"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c16">m</font>,<font color="Maroon" title="c6">O</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">S'</font>
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c17">n</font> = <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c16">m</font>,<font color="Maroon" title="c6">O</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c13">M</font> <b>by </b><i><a class="txt" href="waybel_9.html#E9:86"><i><font color="Green" title="E40">A2</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c17">n</font>
;<br/>

<b>let </b><font color="Maroon" title="c18">p</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c13">M</font>;<br/>

<b>assume </b><a NAME="E10:86_4"/><i><font color="Green" title="E46">A23</font></i>: 
<font color="Maroon" title="c17">n</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c18">p</font>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c19">n1</font> = <font color="Maroon" title="c17">n</font>, <font color="Maroon" title="c20">p1</font> = <font color="Maroon" title="c18">p</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c9">L</font> ;<br/>
<a NAME="E12:86_4"/>
<font color="Maroon" title="c19">n1</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c20">p1</font>
 
<b>by </b><i><a class="txt" href="waybel_9.html#E10:86_4"><i><font color="Green" title="E46">A23</font></i></a>, <a class="ref" href="yellow_0.html#T1" title="YELLOW_0:th.1">YELLOW_0:1</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c21">s1</font>, <font color="Maroon" title="c22">s2</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="E14:86_4"/><i><font color="Green" title="E47">A24</font></i>: 
( <font color="Maroon" title="c21">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">n1</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  &amp; <font color="Maroon" title="c22">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">p1</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  &amp; <font color="Maroon" title="c21">s1</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c22">s2</font> )
 <b>and </b><br/><a NAME="E15:86_4"/>
<font color="Maroon" title="c20">p1</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c19">n1</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
 <b>by </b><i><a class="txt" href="waybel_9.html#E10:86"><i><font color="Green" title="E41">A3</font></i></a></i>;<br/>
<a NAME="E16:86_4"/>
<font color="Maroon" title="c15">f</font> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <font color="Maroon" title="c18">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">p</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
 
<b>by </b><i><a class="txt" href="waybel_9.html#E3:86_4"><i><font color="Green" title="E45">A21</font></i></a></i>;<br/>
<b>hence </b><a NAME="E17:86_4"/>
<font color="Maroon" title="c16">m</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c15">f</font> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <font color="Maroon" title="c18">p</font>
 <b>by </b><i><a class="txt" href="waybel_9.html#E14:86_4"><i><font color="Green" title="E47">A24</font></i></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c14">M</font> = <font color="Maroon" title="c13">M</font> as    <a href="yellow_6.html#M2" title="YELLOW_6:mode.2">subnet</a> of <font color="Maroon" title="c2">N</font> ;<br/>
<b>take </b>
<font color="Maroon" title="c14">M</font>
;<br/>

<a NAME="E20:86"/>
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">x</font> holds  <font color="Maroon" title="c14">M</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="86_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c15">V</font> be    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c3">x</font>;<br/>

<b>consider </b><font color="Maroon" title="c16">i</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">N</font>;<br/>
<a NAME="E2:86_5"/><i><font color="Green" title="E44">A25</font></i>: 
<font color="Maroon" title="c3">x</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> <font color="Maroon" title="c15">V</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="E3:86_5"/>
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c15">V</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="ref" href="tops_1.html#T51" title="TOPS_1:th.51">TOPS_1:51</a></i>;<br/>
<a NAME="E4:86_5"/><b>then </b><i><font color="Green" title="E45">A26</font></i>: 
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c15">V</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_6.html#K10" title="YELLOW_6:func.10">OpenNeighborhoods</a> <font color="Maroon" title="c3">x</font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel_9.html#E2:86_5"><i><font color="Green" title="E44">A25</font></i></a>, <a class="ref" href="yellow_6.html#T39" title="YELLOW_6:th.39">YELLOW_6:39</a></i>;<br/>
<a NAME="E5:86_5"/><b>then </b>
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c15">V</font> is    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c3">x</font>
 
<b>by </b><i><a class="ref" href="waybel_9.html#T21" target="_self" title="WAYBEL_9:th.21">Th21</a></i>;<br/>
<a NAME="E6:86_5"/><b>then </b>
<font color="Maroon" title="c2">N</font> <a href="waybel_0.html#R2" title="WAYBEL_0:pred.2">is_often_in</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c15">V</font>
 
<b>by </b><i><a class="txt" href="waybel_9.html#E1:86"><i><font color="Green" title="E39">A1</font></i></a>, <a class="ref" href="waybel_9.html#D9" target="_self" title="WAYBEL_9:def.9">Def9</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c17">s</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="E8:86_5"/>
<font color="Maroon" title="c16">i</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c17">s</font>
 <b>and </b><br/><a NAME="E9:86_5"/><i><font color="Green" title="E46">A27</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="c17">s</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> <font color="Maroon" title="c15">V</font>
 <b>by </b><i><a class="ref" href="waybel_0.html#D12" title="WAYBEL_0:def.12">WAYBEL_0:def 12</a></i>;<br/>
<a NAME="E10:86_5"/><i><font color="Green" title="E47">A28</font></i>: 
<font color="Maroon" title="c14">M</font> <a href="waybel_0.html#R1" title="WAYBEL_0:pred.1">is_eventually_in</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c15">V</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="86_5_1"><b>proof </b></a><div class="add">

<a NAME="E1:86_5_1"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c17">s</font>,<span class="p2">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c15">V</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">S'</font>
 
<b>by </b><i><a class="txt" href="waybel_9.html#E4:86_5"><i><font color="Green" title="E45">A26</font></i></a>, <a class="txt" href="waybel_9.html#E9:86_5"><i><font color="Green" title="E46">A27</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c18">j</font> = <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c17">s</font>,<span class="p2">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c15">V</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c14">M</font> <b>by </b><i><a class="txt" href="waybel_9.html#E9:86"><i><font color="Green" title="E40">A2</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c18">j</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="c19">s'</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c14">M</font>;<br/>

<b>assume </b><a NAME="E3:86_5_1"/><i><font color="Green" title="E48">A29</font></i>: 
<font color="Maroon" title="c18">j</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c19">s'</font>
 ;<br/>

<a NAME="E4:86_5_1"/>
<font color="Maroon" title="c19">s'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">S'</font>
 
<b>by </b><i><a class="txt" href="waybel_9.html#E9:86"><i><font color="Green" title="E40">A2</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c20">ss</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="c21">OO</font> being   <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#K10" title="YELLOW_6:func.10">OpenNeighborhoods</a> <font color="Maroon" title="c3">x</font></span>)</span><b> such that </b><br/><a NAME="E6:86_5_1"/><i><font color="Green" title="E49">A30</font></i>: 
<font color="Maroon" title="c19">s'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c20">ss</font>,<font color="Maroon" title="c21">OO</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 <b>and </b><br/><a NAME="E7:86_5_1"/><i><font color="Green" title="E50">A31</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="c20">ss</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c21">OO</font>
 ;<br/>
<b>reconsider </b><font color="Maroon" title="c22">j1</font> = <font color="Maroon" title="c18">j</font>, <font color="Maroon" title="c23">s1</font> = <font color="Maroon" title="c19">s'</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c9">L</font> ;<br/>
<a NAME="E9:86_5_1"/>
<font color="Maroon" title="c22">j1</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c23">s1</font>
 
<b>by </b><i><a class="txt" href="waybel_9.html#E3:86_5_1"><i><font color="Green" title="E48">A29</font></i></a>, <a class="ref" href="yellow_0.html#T1" title="YELLOW_0:th.1">YELLOW_0:1</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c24">s1</font>, <font color="Maroon" title="c25">s2</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="E11:86_5_1"/>
( <font color="Maroon" title="c24">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">j</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  &amp; <font color="Maroon" title="c25">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">s'</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  &amp; <font color="Maroon" title="c24">s1</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c25">s2</font> )
 <b>and </b><br/><a NAME="E12:86_5_1"/><i><font color="Green" title="E51">A32</font></i>: 
<font color="Maroon" title="c19">s'</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c18">j</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
 <b>by </b><i><a class="txt" href="waybel_9.html#E10:86"><i><font color="Green" title="E41">A3</font></i></a></i>;<br/>
<a NAME="E13:86_5_1"/><i><font color="Green" title="E52">A33</font></i>: 
<font color="Maroon" title="c18">j</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c15">V</font>
 
<b>by </b><i><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>;<br/>
<a NAME="E14:86_5_1"/><i><font color="Green" title="E53">A34</font></i>: 
<font color="Maroon" title="c19">s'</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">OO</font>
 
<b>by </b><i><a class="txt" href="waybel_9.html#E6:86_5_1"><i><font color="Green" title="E49">A30</font></i></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>;<br/>
<font color="Maroon" title="c14">M</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c19">s'</font> = 
the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c2">N</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">s'</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span>
<b>by </b><i><a class="txt" href="waybel_9.html#E13:86"><i><font color="Green" title="E43">A6</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c2">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c20">ss</font>
<b>by </b><i><a class="txt" href="waybel_9.html#E6:86_5_1"><i><font color="Green" title="E49">A30</font></i></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>
;<br/>
<b>hence </b><a NAME="E16:86_5_1"/>
<font color="Maroon" title="c14">M</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c19">s'</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> <font color="Maroon" title="c15">V</font>
 <b>by </b><i><a class="txt" href="waybel_9.html#E7:86_5_1"><i><font color="Green" title="E50">A31</font></i></a>, <a class="txt" href="waybel_9.html#E12:86_5_1"><i><font color="Green" title="E51">A32</font></i></a>, <a class="txt" href="waybel_9.html#E13:86_5_1"><i><font color="Green" title="E52">A33</font></i></a>, <a class="txt" href="waybel_9.html#E14:86_5_1"><i><font color="Green" title="E53">A34</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:86_5"/>
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c15">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c15">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="E12:86_5"/>
<font color="Maroon" title="c14">M</font> <a href="waybel_0.html#R1" title="WAYBEL_0:pred.1">is_eventually_in</a> <font color="Maroon" title="c15">V</font>
 <b>by </b><i><a class="txt" href="waybel_9.html#E10:86_5"><i><font color="Green" title="E47">A28</font></i></a>, <a class="ref" href="waybel_0.html#T8" title="WAYBEL_0:th.8">WAYBEL_0:8</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E21:86"/>
<font color="Maroon" title="c3">x</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="c14">M</font>
 <b>by </b><i><a class="ref" href="yellow_6.html#D18" title="YELLOW_6:def.18">YELLOW_6:def 18</a></i>;<br/>


</div>
