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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>





<b>assume </b><a NAME="E1:86"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="waybel_9.html#R3" target="_self">is_a_cluster_point_of</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>set </b><font color="Maroon">c<sub>4</sub></font> = the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>5</sub></font> = <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="domain_1.html#K1">]</a></span> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font>, B is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K10">OpenNeighborhoods</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> : <font color="Maroon">c<sub>2</sub></font> <a href="waybel_0.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>2</sub></font> </span>}</span> ;<br/>
<a NAME="E2:86"/>
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="ref" href="tops_1.html#T53">TOPS_1:53</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> =  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K10">OpenNeighborhoods</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <b>by </b><i><a class="ref" href="yellow_6.html#T39">YELLOW_6:39</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>
<a NAME="E5:86"/>
<font color="Maroon">c<sub>2</sub></font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font>
 
;<br/>
<a NAME="E6:86"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="domain_1.html#K1">]</a></span> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font>, B is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K10">OpenNeighborhoods</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> : <font color="Maroon">c<sub>2</sub></font> <a href="waybel_0.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>2</sub></font> </span>}</span> 
 
;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>8</sub></font> = <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="domain_1.html#K1">]</a></span> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font>, B is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K10">OpenNeighborhoods</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> : <font color="Maroon">c<sub>2</sub></font> <a href="waybel_0.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>2</sub></font> </span>}</span>  as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] means ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> <a href="mcart_1.html#K1">`1</a>  &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>2</sub></font> <a href="mcart_1.html#K1">`1</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="mcart_1.html#K2">`2</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">a<sub>1</sub></font> <a href="mcart_1.html#K2">`2</a>  );<br/>
<b>consider </b><font color="Maroon">c<sub>9</sub></font> being  non <a href="struct_0.html#V3">empty</a> <a href="orders_2.html#V1">strict</a>  <a href="orders_2.html#L1">RelStr</a> <b> such that </b><br/><a NAME="E9:86"/><i><font color="Green">E40</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font>
 <b>and </b><br/><a NAME="E10:86"/><i><font color="Green">E41</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>9</sub></font> holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> iff <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>] )
 <b>from </b><i><a class="ref" href="yellow_0.html#S1">YELLOW_0:sch 1</a>();<br/></i>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>9</sub></font>) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="mcart_1.html#K1">`1</a> </span>)</span>;<br/>
<a NAME="E11:86"/><i><font color="Green">E42</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>9</sub></font> holds <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>) <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></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">c<sub>10</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>9</sub></font>;<br/>

<a NAME="E1:86_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="waybel_9.html#E9:86"><i><font color="Green">E40</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>11</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>12</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K10">OpenNeighborhoods</a> <font color="Maroon">c<sub>3</sub></font></span>)</span><b> such that </b><br/><a NAME="E3:86_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="domain_1.html#K1">]</a></span>
 <b>and </b><br/><a NAME="E4:86_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font>
 ;<br/>
<a NAME="E5:86_1"/>
the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="mcart_1.html#K1">`1</a> </span>)</span> <a href="hidden.html#R1">=</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="txt" href="waybel_9.html#E3:86_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/>
<b>hence </b><a NAME="E6:86_1"/>
the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="mcart_1.html#K1">`1</a> </span>)</span> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>10</sub></font> being   <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>9</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E13:86"/><i><font color="Green">E43</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>9</sub></font> holds <font color="Maroon">c<sub>10</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S8">FUNCT_2:sch 8</a>(<a class="txt" href="waybel_9.html#E11:86"><i><font color="Green">E42</font></i></a>);<br/></i>
<b>set </b><font color="Maroon">c<sub>11</sub></font> =  <a href="waybel_0.html#G1">NetStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>9</sub></font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font> #);<br/>
<a NAME="E14:86"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <a href="waybel_0.html#G1">NetStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>9</sub></font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font> #)ex <font color="Olive">b<sub>3</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <a href="waybel_0.html#G1">NetStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>9</sub></font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font> #) st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">b<sub>3</sub></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">c<sub>12</sub></font>, <font color="Maroon">c<sub>13</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <a href="waybel_0.html#G1">NetStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>9</sub></font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font> #);<br/>

<a NAME="E1:86_2"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="waybel_9.html#E9:86"><i><font color="Green">E40</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>14</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>15</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K10">OpenNeighborhoods</a> <font color="Maroon">c<sub>3</sub></font></span>)</span><b> such that </b><br/><a NAME="E3:86_2"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></font></span><a href="domain_1.html#K1">]</a></span>
 <b>and </b><br/><a NAME="E4:86_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>15</sub></font>
 ;<br/>
<a NAME="E5:86_2"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="waybel_9.html#E9:86"><i><font color="Green">E40</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>16</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>17</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K10">OpenNeighborhoods</a> <font color="Maroon">c<sub>3</sub></font></span>)</span><b> such that </b><br/><a NAME="E7:86_2"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></font></span><a href="domain_1.html#K1">]</a></span>
 <b>and </b><br/><a NAME="E8:86_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>17</sub></font>
 ;<br/>
<b>consider </b><font color="Maroon">c<sub>18</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E10:86_2"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">c<sub>14</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>18</sub></font> )
 <b>by </b><i><a class="ref" href="yellow_6.html#D5">YELLOW_6:def 5</a></i>;<br/>
<a NAME="E11:86_2"/>
( <font color="Maroon">c<sub>15</sub></font> is    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> is    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="ref" href="waybel_9.html#T21" target="_self">Th21</a></i>;<br/>
<a NAME="E12:86_2"/><b>then </b>
<font color="Maroon">c<sub>15</sub></font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">c<sub>17</sub></font> is    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="connsp_2.html#T4">CONNSP_2:4</a></i>;<br/>
<a NAME="E13:86_2"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="waybel_0.html#R2">is_often_in</a> <font color="Maroon">c<sub>15</sub></font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">c<sub>17</sub></font>
 
<b>by </b><i><a class="txt" href="waybel_9.html#E1:86"><i><font color="Green">E39</font></i></a>, <a class="ref" href="waybel_9.html#D9" target="_self">Def9</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>19</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E15:86_2"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>19</sub></font>
 <b>and </b><br/><a NAME="E16:86_2"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>15</sub></font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="ref" href="waybel_0.html#D12">WAYBEL_0:def 12</a></i>;<br/>
<a NAME="E17:86_2"/>
<font color="Maroon">c<sub>15</sub></font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">c<sub>17</sub></font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K10">OpenNeighborhoods</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="waybel_9.html#T22" target="_self">Th22</a></i>;<br/>
<a NAME="E18:86_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>19</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">c<sub>17</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="waybel_9.html#E16:86_2"><i><font color="Green">E48</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>20</sub></font> = <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>19</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">c<sub>17</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span> as   <a href="struct_0.html#NM1">Element</a> of <a href="waybel_0.html#G1">NetStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>9</sub></font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font> #) <b>by </b><i><a class="txt" href="waybel_9.html#E9:86"><i><font color="Green">E40</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>20</sub></font>
;<br/>

<b>reconsider </b><font color="Maroon">c<sub>21</sub></font> = <font color="Maroon">c<sub>12</sub></font>, <font color="Maroon">c<sub>22</sub></font> = <font color="Maroon">c<sub>13</sub></font>, <font color="Maroon">c<sub>23</sub></font> = <font color="Maroon">c<sub>20</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>9</sub></font> ;<br/>
<a NAME="E21:86_2"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">c<sub>15</sub></font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">c<sub>17</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">c<sub>17</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>17</sub></font> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E22:86_2"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">c<sub>14</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>19</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>19</sub></font> )
 
<b>by </b><i><a class="txt" href="waybel_9.html#E10:86_2"><i><font color="Green">E46</font></i></a>, <a class="txt" href="waybel_9.html#E15:86_2"><i><font color="Green">E47</font></i></a>, <a class="ref" href="yellow_0.html#D2">YELLOW_0:def 2</a></i>;<br/>
<a NAME="E23:86_2"/>
( <font color="Maroon">c<sub>21</sub></font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>22</sub></font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>23</sub></font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>22</sub></font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> &amp; <font color="Maroon">c<sub>23</sub></font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">c<sub>17</sub></font> )
 
<b>by </b><i><a class="txt" href="waybel_9.html#E3:86_2"><i><font color="Green">E44</font></i></a>, <a class="txt" href="waybel_9.html#E7:86_2"><i><font color="Green">E45</font></i></a>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/>
<a NAME="E24:86_2"/><b>then </b>
( <font color="Maroon">c<sub>21</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>23</sub></font> &amp; <font color="Maroon">c<sub>22</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>23</sub></font> )
 
<b>by </b><i><a class="txt" href="waybel_9.html#E10:86"><i><font color="Green">E41</font></i></a>, <a class="txt" href="waybel_9.html#E21:86_2"><i><font color="Green">E49</font></i></a>, <a class="txt" href="waybel_9.html#E22:86_2"><i><font color="Green">E50</font></i></a></i>;<br/>
<b>hence </b><a NAME="E25:86_2"/>
( <font color="Maroon">c<sub>12</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>20</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>20</sub></font> )
 <b>by </b><i><a class="ref" href="yellow_0.html#T1">YELLOW_0:1</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>12</sub></font> =  <a href="waybel_0.html#G1">NetStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>9</sub></font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font> #) as   <a href="waybel_0.html#NM1">prenet</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="yellow_6.html#D5">YELLOW_6:def 5</a></i>;<br/>
<a NAME="E16:86"/>
<font color="Maroon">c<sub>12</sub></font> is <a href="orders_2.html#V3">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">c<sub>13</sub></font>, <font color="Maroon">c<sub>14</sub></font>, <font color="Maroon">c<sub>15</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>12</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_0.html#D2">YELLOW_0:def 2</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:86_3"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>14</sub></font>
 <b>and </b><br/><a NAME="E2:86_3"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>15</sub></font>
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>13</sub></font>, <font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>14</sub></font>, <font color="Maroon">c<sub>18</sub></font> = <font color="Maroon">c<sub>15</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>9</sub></font> ;<br/>
<a NAME="E4:86_3"/>
<font color="Maroon">c<sub>16</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>17</sub></font>
 
<b>by </b><i><a class="txt" href="waybel_9.html#E1:86_3"><i><font color="Green">E44</font></i></a>, <a class="ref" href="yellow_0.html#T1">YELLOW_0:1</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>19</sub></font>, <font color="Maroon">c<sub>20</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E6:86_3"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <a href="mcart_1.html#K1">`1</a>  &amp; <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> <a href="mcart_1.html#K1">`1</a>  &amp; <font color="Maroon">c<sub>19</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>20</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="mcart_1.html#K2">`2</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>16</sub></font> <a href="mcart_1.html#K2">`2</a>  )
 <b>by </b><i><a class="txt" href="waybel_9.html#E10:86"><i><font color="Green">E41</font></i></a></i>;<br/>
<a NAME="E7:86_3"/>
<font color="Maroon">c<sub>17</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>18</sub></font>
 
<b>by </b><i><a class="txt" href="waybel_9.html#E2:86_3"><i><font color="Green">E45</font></i></a>, <a class="ref" href="yellow_0.html#T1">YELLOW_0:1</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>21</sub></font>, <font color="Maroon">c<sub>22</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E9:86_3"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> <a href="mcart_1.html#K1">`1</a>  &amp; <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> <a href="mcart_1.html#K1">`1</a>  &amp; <font color="Maroon">c<sub>21</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>18</sub></font> <a href="mcart_1.html#K2">`2</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>17</sub></font> <a href="mcart_1.html#K2">`2</a>  )
 <b>by </b><i><a class="txt" href="waybel_9.html#E10:86"><i><font color="Green">E41</font></i></a></i>;<br/>
<a NAME="E10:86_3"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>22</sub></font>
 
<b>by </b><i><a class="txt" href="waybel_9.html#E6:86_3"><i><font color="Green">E46</font></i></a>, <a class="txt" href="waybel_9.html#E9:86_3"><i><font color="Green">E47</font></i></a>, <a class="ref" href="yellow_0.html#D2">YELLOW_0:def 2</a></i>;<br/>
<a NAME="E11:86_3"/>
<font color="Maroon">c<sub>18</sub></font> <a href="mcart_1.html#K2">`2</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>16</sub></font> <a href="mcart_1.html#K2">`2</a> 
 
<b>by </b><i><a class="txt" href="waybel_9.html#E6:86_3"><i><font color="Green">E46</font></i></a>, <a class="txt" href="waybel_9.html#E9:86_3"><i><font color="Green">E47</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E12:86_3"/><b>then </b>
<font color="Maroon">c<sub>16</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>18</sub></font>
 
<b>by </b><i><a class="txt" href="waybel_9.html#E10:86"><i><font color="Green">E41</font></i></a>, <a class="txt" href="waybel_9.html#E6:86_3"><i><font color="Green">E46</font></i></a>, <a class="txt" href="waybel_9.html#E9:86_3"><i><font color="Green">E47</font></i></a>, <a class="txt" href="waybel_9.html#E10:86_3"><i><font color="Green">E48</font></i></a></i>;<br/>
<b>hence </b><a NAME="E13:86_3"/>
<font color="Maroon">c<sub>13</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T1">YELLOW_0:1</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>12</sub></font> as   <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E18:86"/>
<font color="Maroon">c<sub>13</sub></font> is    <a href="yellow_6.html#M2">subnet</a> of <font color="Maroon">c<sub>2</sub></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">Element</a> of <font color="Maroon">c<sub>9</sub></font>) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <font color="Maroon">a<sub>1</sub></font> <a href="mcart_1.html#K1">`1</a> ;<br/>
<a NAME="E1:86_4"/><i><font color="Green">E44</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>9</sub></font> holds <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b<sub>1</sub></font>) <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></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">c<sub>14</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>9</sub></font>;<br/>

<a NAME="E1:86_4_1"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="waybel_9.html#E9:86"><i><font color="Green">E40</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>15</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>16</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K10">OpenNeighborhoods</a> <font color="Maroon">c<sub>3</sub></font></span>)</span><b> such that </b><br/><a NAME="E3:86_4_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font></span><a href="domain_1.html#K1">]</a></span>
 <b>and </b><br/><a NAME="E4:86_4_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>16</sub></font>
 ;<br/>
<a NAME="E5:86_4_1"/>
<font color="Maroon">c<sub>14</sub></font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font>
 
<b>by </b><i><a class="txt" href="waybel_9.html#E3:86_4_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/>
<b>hence </b><a NAME="E6:86_4_1"/>
<font color="Maroon">c<sub>14</sub></font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>14</sub></font> being   <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>9</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E3:86_4"/><i><font color="Green">E45</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>9</sub></font> holds <font color="Maroon">c<sub>14</sub></font> <a href="yellow_6.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b<sub>1</sub></font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S8">FUNCT_2:sch 8</a>(<a class="txt" href="waybel_9.html#E1:86_4"><i><font color="Green">E44</font></i></a>);<br/></i>
<b>reconsider </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>14</sub></font> as   <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>2</sub></font> ;<br/>
<b>take </b>
<font color="Maroon">c<sub>15</sub></font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_6.html#D12">YELLOW_6:def 12</a><br/>

<a NAME="E5:86_4"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>13</sub></font> holds <br/><font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">c<sub>2</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></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">c<sub>16</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:86_4_2"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>13</sub></font>
 ;<br/>

<b>hence </b><font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font> = 
the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="mcart_1.html#K1">`1</a> </span>)</span>
<b>by </b><i><a class="txt" href="waybel_9.html#E13:86"><i><font color="Green">E43</font></i></a></i>
<br/>.= 
the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="waybel_9.html#E3:86_4"><i><font color="Green">E45</font></i></a>, <a class="txt" href="waybel_9.html#E1:86_4_2"><i><font color="Green">E46</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default">the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">c<sub>2</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font>
<b>by </b><i><a class="txt" href="waybel_9.html#E1:86_4_2"><i><font color="Green">E46</font></i></a>, <a class="ref" href="funct_2.html#T21">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">mapping</a> of <font color="Maroon">c<sub>13</sub></font> <a href="funct_2.html#R2">=</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">c<sub>2</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="ref" href="funct_2.html#T18">FUNCT_2:18</a></i>;<br/>

<b>let </b><font color="Maroon">c<sub>16</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>

<a NAME="E7:86_4"/>
<font color="Maroon">c<sub>2</sub></font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font>
 
;<br/>
<a NAME="E8:86_4"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 
;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>17</sub></font> = <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K1">]</a></span> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>13</sub></font> <b>by </b><i><a class="txt" href="waybel_9.html#E9:86"><i><font color="Green">E40</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>17</sub></font>
;<br/>

<b>let </b><font color="Maroon">c<sub>18</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>13</sub></font>;<br/>

<b>assume </b><a NAME="E10:86_4"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>18</sub></font>
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>19</sub></font> = <font color="Maroon">c<sub>17</sub></font>, <font color="Maroon">c<sub>20</sub></font> = <font color="Maroon">c<sub>18</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>9</sub></font> ;<br/>
<a NAME="E12:86_4"/>
<font color="Maroon">c<sub>19</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>20</sub></font>
 
<b>by </b><i><a class="txt" href="waybel_9.html#E10:86_4"><i><font color="Green">E46</font></i></a>, <a class="ref" href="yellow_0.html#T1">YELLOW_0:1</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>21</sub></font>, <font color="Maroon">c<sub>22</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E14:86_4"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> <a href="mcart_1.html#K1">`1</a>  &amp; <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> <a href="mcart_1.html#K1">`1</a>  &amp; <font color="Maroon">c<sub>21</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>22</sub></font> )
 <b>and </b><br/><a NAME="E15:86_4"/>
<font color="Maroon">c<sub>20</sub></font> <a href="mcart_1.html#K2">`2</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>19</sub></font> <a href="mcart_1.html#K2">`2</a> 
 <b>by </b><i><a class="txt" href="waybel_9.html#E10:86"><i><font color="Green">E41</font></i></a></i>;<br/>
<a NAME="E16:86_4"/>
<font color="Maroon">c<sub>15</sub></font> <a href="yellow_6.html#K3">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> <a href="mcart_1.html#K1">`1</a> 
 
<b>by </b><i><a class="txt" href="waybel_9.html#E3:86_4"><i><font color="Green">E45</font></i></a></i>;<br/>
<b>hence </b><a NAME="E17:86_4"/>
<font color="Maroon">c<sub>16</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>15</sub></font> <a href="yellow_6.html#K3">.</a> <font color="Maroon">c<sub>18</sub></font>
 <b>by </b><i><a class="txt" href="waybel_9.html#E14:86_4"><i><font color="Green">E47</font></i></a>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>13</sub></font> as    <a href="yellow_6.html#M2">subnet</a> of <font color="Maroon">c<sub>2</sub></font> ;<br/>
<b>take </b>
<font color="Maroon">c<sub>14</sub></font>
;<br/>

<a NAME="E20:86"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">c<sub>3</sub></font> holds <font color="Maroon">c<sub>14</sub></font> <a href="waybel_0.html#R1">is_eventually_in</a> <font color="Olive">b<sub>1</sub></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">c<sub>15</sub></font> be    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">c<sub>3</sub></font>;<br/>

<b>consider </b><font color="Maroon">c<sub>16</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>
<a NAME="E2:86_5"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>15</sub></font>
 
<b>by </b><i><a class="ref" href="connsp_2.html#D1">CONNSP_2:def 1</a></i>;<br/>
<a NAME="E3:86_5"/>
 <a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>15</sub></font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="ref" href="tops_1.html#T51">TOPS_1:51</a></i>;<br/>
<a NAME="E4:86_5"/><b>then </b><i><font color="Green">E45</font></i>: 
 <a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K10">OpenNeighborhoods</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel_9.html#E2:86_5"><i><font color="Green">E44</font></i></a>, <a class="ref" href="yellow_6.html#T39">YELLOW_6:39</a></i>;<br/>
<a NAME="E5:86_5"/><b>then </b>
 <a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>15</sub></font> is    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="waybel_9.html#T21" target="_self">Th21</a></i>;<br/>
<a NAME="E6:86_5"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="waybel_0.html#R2">is_often_in</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>15</sub></font>
 
<b>by </b><i><a class="txt" href="waybel_9.html#E1:86"><i><font color="Green">E39</font></i></a>, <a class="ref" href="waybel_9.html#D9" target="_self">Def9</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>17</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E8:86_5"/>
<font color="Maroon">c<sub>16</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>17</sub></font>
 <b>and </b><br/><a NAME="E9:86_5"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="ref" href="waybel_0.html#D12">WAYBEL_0:def 12</a></i>;<br/>
<a NAME="E10:86_5"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="waybel_0.html#R1">is_eventually_in</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>15</sub></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">[</a><span class="default"><font color="Maroon">c<sub>17</sub></font>,<span class="p2">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="waybel_9.html#E4:86_5"><i><font color="Green">E45</font></i></a>, <a class="txt" href="waybel_9.html#E9:86_5"><i><font color="Green">E46</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>18</sub></font> = <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>17</sub></font>,<span class="p2">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>14</sub></font> <b>by </b><i><a class="txt" href="waybel_9.html#E9:86"><i><font color="Green">E40</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>18</sub></font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D11">WAYBEL_0:def 11</a><br/>

<b>let </b><font color="Maroon">c<sub>19</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>14</sub></font>;<br/>

<b>assume </b><a NAME="E3:86_5_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>19</sub></font>
 ;<br/>

<a NAME="E4:86_5_1"/>
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="waybel_9.html#E9:86"><i><font color="Green">E40</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>20</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>21</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K10">OpenNeighborhoods</a> <font color="Maroon">c<sub>3</sub></font></span>)</span><b> such that </b><br/><a NAME="E6:86_5_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>21</sub></font></span><a href="domain_1.html#K1">]</a></span>
 <b>and </b><br/><a NAME="E7:86_5_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>21</sub></font>
 ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>22</sub></font> = <font color="Maroon">c<sub>18</sub></font>, <font color="Maroon">c<sub>23</sub></font> = <font color="Maroon">c<sub>19</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>9</sub></font> ;<br/>
<a NAME="E9:86_5_1"/>
<font color="Maroon">c<sub>22</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>23</sub></font>
 
<b>by </b><i><a class="txt" href="waybel_9.html#E3:86_5_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="yellow_0.html#T1">YELLOW_0:1</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>24</sub></font>, <font color="Maroon">c<sub>25</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E11:86_5_1"/>
( <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> <a href="mcart_1.html#K1">`1</a>  &amp; <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> <a href="mcart_1.html#K1">`1</a>  &amp; <font color="Maroon">c<sub>24</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>25</sub></font> )
 <b>and </b><br/><a NAME="E12:86_5_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="mcart_1.html#K2">`2</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>18</sub></font> <a href="mcart_1.html#K2">`2</a> 
 <b>by </b><i><a class="txt" href="waybel_9.html#E10:86"><i><font color="Green">E41</font></i></a></i>;<br/>
<a NAME="E13:86_5_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>15</sub></font>
 
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/>
<a NAME="E14:86_5_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font>
 
<b>by </b><i><a class="txt" href="waybel_9.html#E6:86_5_1"><i><font color="Green">E49</font></i></a>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/>
<font color="Maroon">c<sub>14</sub></font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">c<sub>19</sub></font> = 
the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">c<sub>14</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>19</sub></font>

<br/>.= 
the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="mcart_1.html#K1">`1</a> </span>)</span>
<b>by </b><i><a class="txt" href="waybel_9.html#E13:86"><i><font color="Green">E43</font></i></a></i>
<br/>.= 
the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>20</sub></font>
<b>by </b><i><a class="txt" href="waybel_9.html#E6:86_5_1"><i><font color="Green">E49</font></i></a>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font>

;<br/>
<b>hence </b><a NAME="E16:86_5_1"/>
<font color="Maroon">c<sub>14</sub></font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="txt" href="waybel_9.html#E7:86_5_1"><i><font color="Green">E50</font></i></a>, <a class="txt" href="waybel_9.html#E12:86_5_1"><i><font color="Green">E51</font></i></a>, <a class="txt" href="waybel_9.html#E13:86_5_1"><i><font color="Green">E52</font></i></a>, <a class="txt" href="waybel_9.html#E14:86_5_1"><i><font color="Green">E53</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:86_5"/>
 <a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>15</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>15</sub></font>
 
<b>by </b><i><a class="ref" href="tops_1.html#T44">TOPS_1:44</a></i>;<br/>
<b>hence </b><a NAME="E12:86_5"/>
<font color="Maroon">c<sub>14</sub></font> <a href="waybel_0.html#R1">is_eventually_in</a> <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="txt" href="waybel_9.html#E10:86_5"><i><font color="Green">E47</font></i></a>, <a class="ref" href="waybel_0.html#T8">WAYBEL_0:8</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E21:86"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K11">Lim</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="ref" href="yellow_6.html#D18">YELLOW_6:def 18</a></i>;<br/>


</div>
