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

<b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="struct_0.html#NM3">Subset-Family</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K8">Tunit_circle</a> 2</span>)</span>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:39"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="pre_topc.html#R1">is_a_cover_of</a>  <a href="toprealb.html#K8">Tunit_circle</a> 2
 <b>and </b><br/><a NAME="E2:39"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>8</sub></font> is <a href="tops_2.html#V1">open</a>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>9</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>10</sub></font> be  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM7">Function</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span>,<span class="p1">(<span class="default"><a href="toprealb.html#K8">Tunit_circle</a> 2</span>)</span>;<br/><b>let </b><font color="Maroon">c<sub>11</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>9</sub></font>;<br/>





<b>set </b><font color="Maroon">c<sub>12</sub></font> =  <a href="topmetr.html#K4">Closed-Interval-TSpace</a> 0,1;<br/>
<b>set </b><font color="Maroon">c<sub>13</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>9</sub></font>;<br/>
<a NAME="E3:39"/><i><font color="Green">E39</font></i>: 
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="toprealb.html#K8">Tunit_circle</a> 2</span>)</span> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E1:39"><i><font color="Green">E37</font></i></a>, <a class="ref" href="pre_topc.html#D8">PRE_TOPC:def 8</a></i>;<br/>
<a NAME="E4:39"/><i><font color="Green">E40</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a> ex <font color="Olive">b<sub>2</sub></font> being non <a href="xboole_0.html#V1">empty</a> <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K8">Tunit_circle</a> 2</span>)</span> st <br/>( <font color="Maroon">c<sub>10</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Olive">b<sub>1</sub></font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>14</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a> ;<br/>

<b>consider </b><font color="Maroon">c<sub>15</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E2:39_1"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>14</sub></font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>15</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="topalg_5.html#E3:39"><i><font color="Green">E39</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>15</sub></font> as  non <a href="xboole_0.html#V1">empty</a> <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K8">Tunit_circle</a> 2</span>)</span> <b>by </b><i><a class="txt" href="topalg_5.html#E2:39"><i><font color="Green">E38</font></i></a>, <a class="txt" href="topalg_5.html#E2:39_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="tops_2.html#D1">TOPS_2:def 1</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>16</sub></font>
;<br/>

<b>thus </b><a NAME="E4:39_1"/>
( <font color="Maroon">c<sub>10</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>14</sub></font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>16</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="topalg_5.html#E2:39_1"><i><font color="Green">E41</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:39"/>
ex <font color="Olive">b<sub>1</sub></font> being non <a href="xboole_0.html#V1">empty</a> <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K8">Tunit_circle</a> 2</span>)</span> st <br/>( <font color="Maroon">c<sub>10</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<a href="borsuk_1.html#K21">0[01]</a> </span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</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="topalg_5.html#E4:39"><i><font color="Green">E40</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>8</sub></font> 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> being <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K8">Tunit_circle</a> 2</span>)</span> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">a<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> );<br/>
<a NAME="E7:39"/><i><font color="Green">E41</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> ex <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>14</sub></font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>15</sub></font> be    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> ;<br/>

<a NAME="E1:39_2"/>
ex <font color="Olive">b<sub>1</sub></font> being non <a href="xboole_0.html#V1">empty</a> <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K8">Tunit_circle</a> 2</span>)</span> st <br/>( <font color="Maroon">c<sub>10</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>15</sub></font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</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="topalg_5.html#E4:39"><i><font color="Green">E40</font></i></a></i>;<br/>
<b>hence </b><a NAME="E2:39_2"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>14</sub></font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>15</sub></font>,<font color="Olive">b<sub>1</sub></font>]
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>15</sub></font> being   <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> ,<font color="Maroon">c<sub>14</sub></font><b> such that </b><br/><a NAME="E9:39"/><i><font color="Green">E42</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a>  holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>15</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</a>(<a class="txt" href="topalg_5.html#E7:39"><i><font color="Green">E41</font></i></a>);<br/></i>
<b>defpred </b><font color="Maroon">S<sub>2</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> being <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>9</sub></font>ex <font color="Olive">b<sub>2</sub></font> being <a href="pre_topc.html#V3">open</a> <a href="connsp_1.html#V2">connected</a> <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K5">I[01]</a>  st <br/>( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>15</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="borsuk_1.html#K6">:]</a></span> );<br/>
<a NAME="E10:39"/><i><font color="Green">E43</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> ex <font color="Olive">b<sub>2</sub></font> being  <a href="subset_1.html#NM2">Subset</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>9</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>16</sub></font> be    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> ;<br/>

<b>consider </b><font color="Maroon">c<sub>17</sub></font> being  <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K8">Tunit_circle</a> 2</span>)</span><b> such that </b><br/><a NAME="E2:39_3"/>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>14</sub></font>
 <b>and </b><br/><a NAME="E3:39_3"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>16</sub></font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>17</sub></font>
 <b>and </b><br/><a NAME="E4:39_3"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E9:39"><i><font color="Green">E42</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>18</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span><b> such that </b><br/><a NAME="E6:39_3"/><i><font color="Green">E46</font></i>: 
<span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>16</sub></font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>18</sub></font>
 <b>and </b><br/><a NAME="E7:39_3"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>18</sub></font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E8:39_3"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>18</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E3:39_3"><i><font color="Green">E44</font></i></a>, <a class="ref" href="jgraph_2.html#T20">JGRAPH_2:20</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>19</sub></font> being   <a href="struct_0.html#NM3">Subset-Family</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span><b> such that </b><br/><a NAME="E10:39_3"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>19</sub></font>
 <b>and </b><br/><a NAME="E11:39_3"/><i><font color="Green">E50</font></i>: 
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> <font color="Maroon">c<sub>19</sub></font> holds <br/>ex <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>9</sub></font>ex <font color="Olive">b<sub>3</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K5">I[01]</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font></span><a href="borsuk_1.html#K6">:]</a></span> &amp; <font color="Olive">b<sub>2</sub></font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">b<sub>3</sub></font> is <a href="pre_topc.html#V3">open</a> )
 <b>by </b><i><a class="txt" href="topalg_5.html#E7:39_3"><i><font color="Green">E47</font></i></a>, <a class="ref" href="borsuk_1.html#T45">BORSUK_1:45</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>20</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E13:39_3"/><i><font color="Green">E51</font></i>: 
<span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>16</sub></font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>20</sub></font>
 <b>and </b><br/><a NAME="E14:39_3"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>19</sub></font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E6:39_3"><i><font color="Green">E46</font></i></a>, <a class="txt" href="topalg_5.html#E10:39_3"><i><font color="Green">E49</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>21</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>22</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K5">I[01]</a> <b> such that </b><br/><a NAME="E16:39_3"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>22</sub></font></span><a href="borsuk_1.html#K6">:]</a></span>
 <b>and </b><br/><a NAME="E17:39_3"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">c<sub>21</sub></font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E18:39_3"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">c<sub>22</sub></font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="txt" href="topalg_5.html#E11:39_3"><i><font color="Green">E50</font></i></a>, <a class="txt" href="topalg_5.html#E14:39_3"><i><font color="Green">E52</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>23</sub></font> = <font color="Maroon">c<sub>21</sub></font> as  <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>9</sub></font> <b>by </b><i><a class="txt" href="topalg_5.html#E17:39_3"><i><font color="Green">E54</font></i></a></i>;<br/>
<a NAME="E20:39_3"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>22</sub></font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E13:39_3"><i><font color="Green">E51</font></i></a>, <a class="txt" href="topalg_5.html#E16:39_3"><i><font color="Green">E53</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>24</sub></font> = <font color="Maroon">c<sub>22</sub></font> as  non <a href="xboole_0.html#V1">empty</a> <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K5">I[01]</a>  <b>by </b><i><a class="txt" href="topalg_5.html#E18:39_3"><i><font color="Green">E55</font></i></a>, <a class="txt" href="topalg_5.html#E13:39_3"><i><font color="Green">E51</font></i></a>, <a class="txt" href="topalg_5.html#E16:39_3"><i><font color="Green">E53</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>25</sub></font> = <font color="Maroon">c<sub>16</sub></font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <b>by </b><i><a class="txt" href="topalg_5.html#E20:39_3"><i><font color="Green">E56</font></i></a>, <a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/>
<a NAME="E23:39_3"/><i><font color="Green">E57</font></i>: 
 <a href="connsp_3.html#K6">skl</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a>  <a href="connsp_1.html#K1">skl</a> <font color="Maroon">c<sub>25</sub></font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E20:39_3"><i><font color="Green">E56</font></i></a>, <a class="ref" href="connsp_3.html#D7">CONNSP_3:def 7</a></i>;<br/>
<a NAME="E24:39_3"/>
 <a href="connsp_1.html#K1">skl</a> <font color="Maroon">c<sub>25</sub></font> <a href="connsp_1.html#R3">is_a_component_of</a> <a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>24</sub></font>
 
<b>by </b><i><a class="ref" href="connsp_1.html#T43">CONNSP_1:43</a></i>;<br/>
<a NAME="E25:39_3"/><b>then </b>
 <a href="connsp_1.html#K1">skl</a> <font color="Maroon">c<sub>25</sub></font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="ref" href="connsp_2.html#T21">CONNSP_2:21</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>26</sub></font> =  <a href="connsp_3.html#K6">skl</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>24</sub></font> as  <a href="pre_topc.html#V3">open</a> <a href="connsp_1.html#V2">connected</a> <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K5">I[01]</a>  <b>by </b><i><a class="txt" href="topalg_5.html#E20:39_3"><i><font color="Green">E56</font></i></a>, <a class="txt" href="topalg_5.html#E23:39_3"><i><font color="Green">E57</font></i></a>, <a class="ref" href="tsep_1.html#T17">TSEP_1:17</a>, <a class="ref" href="connsp_3.html#T35">CONNSP_3:35</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>27</sub></font> = <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>26</sub></font></span><a href="borsuk_1.html#K6">:]</a></span> as   <a href="subset_1.html#NM2">Subset</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>9</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span> <b>by </b><i><a class="ref" href="borsuk_1.html#D5">BORSUK_1:def 5</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>27</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>23</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>26</sub></font>
;<br/>

<b>thus </b><a NAME="E28:39_3"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>23</sub></font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E13:39_3"><i><font color="Green">E51</font></i></a>, <a class="txt" href="topalg_5.html#E16:39_3"><i><font color="Green">E53</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>

<b>thus </b><a NAME="E29:39_3"/>
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>26</sub></font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E20:39_3"><i><font color="Green">E56</font></i></a>, <a class="ref" href="connsp_3.html#T26">CONNSP_3:26</a></i>;<br/>

<a NAME="E30:39_3"/>
<font color="Maroon">c<sub>26</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>24</sub></font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E20:39_3"><i><font color="Green">E56</font></i></a>, <a class="ref" href="topalg_5.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E31:39_3"/><b>then </b><i><font color="Green">E58</font></i>: 
<font color="Maroon">c<sub>27</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>24</sub></font></span><a href="borsuk_1.html#K6">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T118">ZFMISC_1:118</a></i>;<br/>
<a NAME="E32:39_3"/>
<span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>24</sub></font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>18</sub></font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E14:39_3"><i><font color="Green">E52</font></i></a>, <a class="txt" href="topalg_5.html#E10:39_3"><i><font color="Green">E49</font></i></a>, <a class="txt" href="topalg_5.html#E16:39_3"><i><font color="Green">E53</font></i></a>, <a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/>
<a NAME="E33:39_3"/><b>then </b>
<font color="Maroon">c<sub>27</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>18</sub></font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E31:39_3"><i><font color="Green">E58</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E34:39_3"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>27</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>10</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>18</sub></font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T156">RELAT_1:156</a></i>;<br/>
<b>hence </b><a NAME="E35:39_3"/>
<font color="Maroon">c<sub>10</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>26</sub></font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>15</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E4:39_3"><i><font color="Green">E45</font></i></a>, <a class="txt" href="topalg_5.html#E8:39_3"><i><font color="Green">E48</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>

<b>thus </b><a NAME="E36:39_3"/>
<font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>26</sub></font></span><a href="borsuk_1.html#K6">:]</a></span>
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>16</sub></font> being   <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> , <a href="pcomps_1.html#K1">bool</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>9</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span><b> such that </b><br/><a NAME="E12:39"/><i><font color="Green">E44</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a>  holds <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>16</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</a>(<a class="txt" href="topalg_5.html#E10:39"><i><font color="Green">E43</font></i></a>);<br/></i>
<a NAME="E13:39"/><i><font color="Green">E45</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>17</sub></font> =  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>16</sub></font> as   <a href="struct_0.html#NM3">Subset-Family</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span> <b>by </b><i><a class="ref" href="borsuk_1.html#D5">BORSUK_1:def 5</a></i>;<br/>
<a NAME="E15:39"/>
<span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="domain_1.html#K6">}</a></span>,<span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <a href="topmetr.html#K5">I[01]</a> </span>)</span></span><a href="borsuk_1.html#K6">:]</a></span> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>17</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>18</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:39_4"/>
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="domain_1.html#K6">}</a></span>,<span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <a href="topmetr.html#K5">I[01]</a> </span>)</span></span><a href="borsuk_1.html#K6">:]</a></span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>19</sub></font>, <font color="Maroon">c<sub>20</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:39_4"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>and </b><br/><a NAME="E4:39_4"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <a href="topmetr.html#K5">I[01]</a> 
 <b>and </b><br/><a NAME="E5:39_4"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>20</sub></font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>21</sub></font> = <font color="Maroon">c<sub>20</sub></font> as   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a>  <b>by </b><i><a class="txt" href="topalg_5.html#E4:39_4"><i><font color="Green">E47</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>22</sub></font> being  <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>23</sub></font> being  <a href="pre_topc.html#V3">open</a> <a href="connsp_1.html#V2">connected</a> <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K5">I[01]</a> <b> such that </b><br/><a NAME="E8:39_4"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>23</sub></font> )
 <b>and </b><br/><a NAME="E9:39_4"/>
<font color="Maroon">c<sub>10</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>22</sub></font>,<font color="Maroon">c<sub>23</sub></font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>15</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>21</sub></font>
 <b>and </b><br/><a NAME="E10:39_4"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>22</sub></font>,<font color="Maroon">c<sub>23</sub></font></span><a href="borsuk_1.html#K6">:]</a></span>
 <b>by </b><i><a class="txt" href="topalg_5.html#E12:39"><i><font color="Green">E44</font></i></a></i>;<br/>
<a NAME="E11:39_4"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E3:39_4"><i><font color="Green">E46</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E12:39_4"/><i><font color="Green">E52</font></i>: 
<span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>21</sub></font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>22</sub></font>,<font color="Maroon">c<sub>23</sub></font></span><a href="borsuk_1.html#K6">:]</a></span>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E8:39_4"><i><font color="Green">E49</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E13:39_4"/>
<span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>22</sub></font>,<font color="Maroon">c<sub>23</sub></font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>17</sub></font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E13:39"><i><font color="Green">E45</font></i></a>, <a class="txt" href="topalg_5.html#E10:39_4"><i><font color="Green">E50</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E14:39_4"/>
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E12:39_4"><i><font color="Green">E52</font></i></a>, <a class="txt" href="topalg_5.html#E5:39_4"><i><font color="Green">E48</font></i></a>, <a class="txt" href="topalg_5.html#E11:39_4"><i><font color="Green">E51</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E16:39"/><b>then </b><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="compts_1.html#R1">is_a_cover_of</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="domain_1.html#K6">}</a></span>,<span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <a href="topmetr.html#K5">I[01]</a> </span>)</span></span><a href="borsuk_1.html#K6">:]</a></span>
 
<b>by </b><i><a class="ref" href="compts_1.html#D1">COMPTS_1:def 1</a></i>;<br/>
<a NAME="E17:39"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>17</sub></font> is <a href="tops_2.html#V1">open</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>18</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span>; <i><font color="Red">:: according to </font></i><a class="ref" href="tops_2.html#D1">TOPS_2:def 1</a><br/>

<b>assume </b><a NAME="E1:39_5"/>
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>17</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>19</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:39_5"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>16</sub></font>
 <b>and </b><br/><a NAME="E4:39_5"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E5:39_5"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>16</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font>]
 
<b>by </b><i><a class="txt" href="topalg_5.html#E12:39"><i><font color="Green">E44</font></i></a>, <a class="txt" href="topalg_5.html#E3:39_5"><i><font color="Green">E48</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:39_5"/>
<font color="Maroon">c<sub>18</sub></font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="txt" href="topalg_5.html#E4:39_5"><i><font color="Green">E49</font></i></a>, <a class="ref" href="borsuk_1.html#T46">BORSUK_1:46</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E18:39"/>
<span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="domain_1.html#K6">}</a></span>,<span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <a href="topmetr.html#K5">I[01]</a> </span>)</span></span><a href="borsuk_1.html#K6">:]</a></span> is <a href="compts_1.html#V6">compact</a>
 
<b>by </b><i><a class="ref" href="borsuk_3.html#T27">BORSUK_3:27</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>18</sub></font> being   <a href="struct_0.html#NM3">Subset-Family</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span><b> such that </b><br/><a NAME="E20:39"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>17</sub></font>
 <b>and </b><br/><a NAME="E21:39"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="compts_1.html#R1">is_a_cover_of</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="domain_1.html#K6">}</a></span>,<span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <a href="topmetr.html#K5">I[01]</a> </span>)</span></span><a href="borsuk_1.html#K6">:]</a></span>
 <b>and </b><br/><a NAME="E22:39"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>18</sub></font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="txt" href="topalg_5.html#E16:39"><i><font color="Green">E46</font></i></a>, <a class="txt" href="topalg_5.html#E17:39"><i><font color="Green">E47</font></i></a>, <a class="ref" href="compts_1.html#D7">COMPTS_1:def 7</a></i>;<br/>
<a NAME="E23:39"/><i><font color="Green">E51</font></i>: 
<span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="domain_1.html#K6">}</a></span>,<span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <a href="topmetr.html#K5">I[01]</a> </span>)</span></span><a href="borsuk_1.html#K6">:]</a></span> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>18</sub></font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E21:39"><i><font color="Green">E49</font></i></a>, <a class="ref" href="compts_1.html#D1">COMPTS_1:def 1</a></i>;<br/>
<a NAME="E24:39"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>19</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is  <a href="pre_topc.html#V3">open</a> <a href="connsp_1.html#V2">connected</a> <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K5">I[01]</a>  : ex <font color="Olive">b<sub>1</sub></font> being <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>9</sub></font> st <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>18</sub></font> </span>}</span> ;<br/>
<a NAME="E25:39"/>
<span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is  <a href="pre_topc.html#V3">open</a> <a href="connsp_1.html#V2">connected</a> <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K5">I[01]</a>  : ex <font color="Olive">b<sub>1</sub></font> being <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>9</sub></font> st <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>18</sub></font> </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="pcomps_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>20</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:39_6"/>
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is  <a href="pre_topc.html#V3">open</a> <a href="connsp_1.html#V2">connected</a> <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K5">I[01]</a>  : ex <font color="Olive">b<sub>1</sub></font> being <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>9</sub></font> st <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>18</sub></font> </span>}</span> 
 ;<br/>

<a NAME="E2:39_6"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being <a href="pre_topc.html#V3">open</a> <a href="connsp_1.html#V2">connected</a> <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K5">I[01]</a>  st <br/>( <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; ex <font color="Olive">b<sub>2</sub></font> being <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>9</sub></font> st <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>18</sub></font> )
 
;<br/>
<b>hence </b><a NAME="E3:39_6"/>
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> 
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>20</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is  <a href="pre_topc.html#V3">open</a> <a href="connsp_1.html#V2">connected</a> <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K5">I[01]</a>  : ex <font color="Olive">b<sub>1</sub></font> being <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>9</sub></font> st <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>18</sub></font> </span>}</span>  as   <a href="struct_0.html#NM3">Subset-Family</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> 0,1</span>)</span> <b>by </b><i><a class="ref" href="topmetr.html#T27">TOPMETR:27</a></i>;<br/>
<a NAME="E27:39"/><i><font color="Green">E53</font></i>: 
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> 0,1</span>)</span> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>20</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>21</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:39_7"/>
<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> 0,1</span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>22</sub></font> = <font color="Maroon">c<sub>21</sub></font> as   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a>  <b>by </b><i><a class="ref" href="topmetr.html#T27">TOPMETR:27</a></i>;<br/>
<a NAME="E3:39_7"/>
<span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>22</sub></font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="domain_1.html#K6">}</a></span>,<span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <a href="topmetr.html#K5">I[01]</a> </span>)</span></span><a href="borsuk_1.html#K6">:]</a></span>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E24:39"><i><font color="Green">E52</font></i></a>, <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>23</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:39_7"/><i><font color="Green">E54</font></i>: 
<span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>22</sub></font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>23</sub></font>
 <b>and </b><br/><a NAME="E6:39_7"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>18</sub></font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E23:39"><i><font color="Green">E51</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>24</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E8:39_7"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>16</sub></font>
 <b>and </b><br/><a NAME="E9:39_7"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E6:39_7"><i><font color="Green">E55</font></i></a>, <a class="txt" href="topalg_5.html#E20:39"><i><font color="Green">E48</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>25</sub></font> being  <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>26</sub></font> being  <a href="pre_topc.html#V3">open</a> <a href="connsp_1.html#V2">connected</a> <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K5">I[01]</a> <b> such that </b><br/><a NAME="E11:39_7"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>25</sub></font>
 <b>and </b><br/><a NAME="E12:39_7"/>
<font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>26</sub></font>
 <b>and </b><br/><a NAME="E13:39_7"/>
<font color="Maroon">c<sub>10</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>25</sub></font>,<font color="Maroon">c<sub>26</sub></font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>15</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font>
 <b>and </b><br/><a NAME="E14:39_7"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>25</sub></font>,<font color="Maroon">c<sub>26</sub></font></span><a href="borsuk_1.html#K6">:]</a></span>
 <b>by </b><i><a class="txt" href="topalg_5.html#E8:39_7"><i><font color="Green">E56</font></i></a>, <a class="txt" href="topalg_5.html#E12:39"><i><font color="Green">E44</font></i></a></i>;<br/>
<a NAME="E15:39_7"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>26</sub></font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E5:39_7"><i><font color="Green">E54</font></i></a>, <a class="txt" href="topalg_5.html#E9:39_7"><i><font color="Green">E57</font></i></a>, <a class="txt" href="topalg_5.html#E14:39_7"><i><font color="Green">E58</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E16:39_7"/>
<font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>20</sub></font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E14:39_7"><i><font color="Green">E58</font></i></a>, <a class="txt" href="topalg_5.html#E6:39_7"><i><font color="Green">E55</font></i></a>, <a class="txt" href="topalg_5.html#E9:39_7"><i><font color="Green">E57</font></i></a></i>;<br/>
<b>hence </b><a NAME="E17:39_7"/>
<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>20</sub></font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E15:39_7"><i><font color="Green">E59</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E28:39"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="pre_topc.html#R1">is_a_cover_of</a>  <a href="topmetr.html#K4">Closed-Interval-TSpace</a> 0,1
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_8"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:39_8"/>
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> 0,1</span>)</span> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>20</sub></font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E27:39"><i><font color="Green">E53</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="pre_topc.html#D8">PRE_TOPC:def 8</a>,<a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><br/>

<b>thus </b><a NAME="E2:39_8"/>
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>20</sub></font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> 0,1</span>)</span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E29:39"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">c<sub>20</sub></font> is <a href="tops_2.html#V1">open</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_9"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>21</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> 0,1</span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="tops_2.html#D1">TOPS_2:def 1</a><br/>

<b>assume </b><a NAME="E1:39_9"/>
<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>20</sub></font>
 ;<br/>

<a NAME="E2:39_9"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being <a href="pre_topc.html#V3">open</a> <a href="connsp_1.html#V2">connected</a> <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K5">I[01]</a>  st <br/>( <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; ex <font color="Olive">b<sub>2</sub></font> being <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>9</sub></font> st <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>18</sub></font> )
 
;<br/>
<b>hence </b><a NAME="E3:39_9"/>
<font color="Maroon">c<sub>21</sub></font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="ref" href="topmetr.html#T27">TOPMETR:27</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>21</sub></font> being    <a href="rcomp_3.html#M1">IntervalCover</a> of <font color="Maroon">c<sub>20</sub></font>;<br/>
<b>consider </b><font color="Maroon">c<sub>22</sub></font> being    <a href="rcomp_3.html#M2">IntervalCoverPts</a> of <font color="Maroon">c<sub>21</sub></font>;<br/>
<a NAME="E32:39"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">c<sub>20</sub></font> is <a href="rcomp_3.html#V1">connected</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_10"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>23</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> 0,1</span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="rcomp_3.html#D1">RCOMP_3:def 1</a><br/>

<b>assume </b><a NAME="E1:39_10"/>
<font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>20</sub></font>
 ;<br/>

<a NAME="E2:39_10"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being <a href="pre_topc.html#V3">open</a> <a href="connsp_1.html#V2">connected</a> <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K5">I[01]</a>  st <br/>( <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; ex <font color="Olive">b<sub>2</sub></font> being <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>9</sub></font> st <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>18</sub></font> )
 
;<br/>
<b>hence </b><a NAME="E3:39_10"/>
<font color="Maroon">c<sub>23</sub></font> is <a href="connsp_1.html#V2">connected</a>
 <b>by </b><i><a class="ref" href="topmetr.html#T27">TOPMETR:27</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>23</sub></font> = <font color="Maroon">c<sub>22</sub></font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="txt" href="topalg_5.html#E29:39"><i><font color="Green">E55</font></i></a>, <a class="txt" href="topalg_5.html#E28:39"><i><font color="Green">E54</font></i></a>, <a class="txt" href="topalg_5.html#E41"><i><font color="Green">Lemma35</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>23</sub></font>
;<br/>

<b>thus </b><a NAME="E34:39"/>
( <font color="Maroon">c<sub>23</sub></font> <a href="seq_1.html#K2">.</a> 1 <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">c<sub>23</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i><a class="txt" href="topalg_5.html#E29:39"><i><font color="Green">E55</font></i></a>, <a class="txt" href="topalg_5.html#E28:39"><i><font color="Green">E54</font></i></a>, <a class="ref" href="rcomp_3.html#D3">RCOMP_3:def 3</a>, <a class="txt" href="topalg_5.html#E32:39"><i><font color="Green">E56</font></i></a></i>;<br/>

<b>thus </b><a NAME="E35:39"/>
<font color="Maroon">c<sub>23</sub></font> is <a href="seqm_3.html#V1">increasing</a>
 <b>by </b><i><a class="txt" href="topalg_5.html#E29:39"><i><font color="Green">E55</font></i></a>, <a class="txt" href="topalg_5.html#E28:39"><i><font color="Green">E54</font></i></a>, <a class="ref" href="rcomp_3.html#T79">RCOMP_3:79</a>, <a class="txt" href="topalg_5.html#E32:39"><i><font color="Green">E56</font></i></a></i>;<br/>

<b>set </b><font color="Maroon">c<sub>24</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is  <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>9</sub></font> : ( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> &amp; ex <font color="Olive">b<sub>1</sub></font> being <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K5">I[01]</a>  st <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>18</sub></font> ) </span>}</span> ;<br/>
<a NAME="E36:39"/>
<span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is  <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>9</sub></font> : ( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> &amp; ex <font color="Olive">b<sub>1</sub></font> being <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K5">I[01]</a>  st <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>18</sub></font> ) </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="pcomps_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>9</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_11"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>25</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:39_11"/>
<font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is  <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>9</sub></font> : ( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> &amp; ex <font color="Olive">b<sub>1</sub></font> being <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K5">I[01]</a>  st <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>18</sub></font> ) </span>}</span> 
 ;<br/>

<a NAME="E2:39_11"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>9</sub></font> st <br/>( <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> &amp; ex <font color="Olive">b<sub>2</sub></font> being <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K5">I[01]</a>  st <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>18</sub></font> )
 
;<br/>
<b>hence </b><a NAME="E3:39_11"/>
<font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>9</sub></font>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>25</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is  <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>9</sub></font> : ( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> &amp; ex <font color="Olive">b<sub>1</sub></font> being <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K5">I[01]</a>  st <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>18</sub></font> ) </span>}</span>  as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>9</sub></font> ;<br/>
<a NAME="E38:39"/>
<span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<a href="borsuk_1.html#K21">0[01]</a> </span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="domain_1.html#K6">}</a></span>,<span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <a href="topmetr.html#K5">I[01]</a> </span>)</span></span><a href="borsuk_1.html#K6">:]</a></span>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E24:39"><i><font color="Green">E52</font></i></a>, <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>26</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E40:39"/>
<span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<a href="borsuk_1.html#K21">0[01]</a> </span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>26</sub></font>
 <b>and </b><br/><a NAME="E41:39"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>18</sub></font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E23:39"><i><font color="Green">E51</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>27</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E43:39"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>16</sub></font>
 <b>and </b><br/><a NAME="E44:39"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>26</sub></font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E41:39"><i><font color="Green">E57</font></i></a>, <a class="txt" href="topalg_5.html#E20:39"><i><font color="Green">E48</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>28</sub></font> being  <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>29</sub></font> being  <a href="pre_topc.html#V3">open</a> <a href="connsp_1.html#V2">connected</a> <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K5">I[01]</a> <b> such that </b><br/><a NAME="E46:39"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>28</sub></font>
 <b>and </b><br/><a NAME="E47:39"/>
( <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>29</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>28</sub></font>,<font color="Maroon">c<sub>29</sub></font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>15</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>27</sub></font> )
 <b>and </b><br/><a NAME="E48:39"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>28</sub></font>,<font color="Maroon">c<sub>29</sub></font></span><a href="borsuk_1.html#K6">:]</a></span>
 <b>by </b><i><a class="txt" href="topalg_5.html#E12:39"><i><font color="Green">E44</font></i></a>, <a class="txt" href="topalg_5.html#E43:39"><i><font color="Green">E58</font></i></a></i>;<br/>
<a NAME="E49:39"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>25</sub></font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E46:39"><i><font color="Green">E60</font></i></a>, <a class="txt" href="topalg_5.html#E48:39"><i><font color="Green">E61</font></i></a>, <a class="txt" href="topalg_5.html#E41:39"><i><font color="Green">E57</font></i></a>, <a class="txt" href="topalg_5.html#E44:39"><i><font color="Green">E59</font></i></a></i>;<br/>
<a NAME="E50:39"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">c<sub>25</sub></font> is <a href="tops_2.html#V1">open</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_12"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>30</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>9</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="tops_2.html#D1">TOPS_2:def 1</a><br/>

<b>assume </b><a NAME="E1:39_12"/>
<font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>25</sub></font>
 ;<br/>

<a NAME="E2:39_12"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>9</sub></font> st <br/>( <font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> &amp; ex <font color="Olive">b<sub>2</sub></font> being <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K5">I[01]</a>  st <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>18</sub></font> )
 
;<br/>
<b>hence </b><a NAME="E3:39_12"/>
<font color="Maroon">c<sub>30</sub></font> is <a href="pre_topc.html#V3">open</a>
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>30</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E52:39"/><i><font color="Green">E64</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font>
 <b>and </b><br/><a NAME="E53:39"/><i><font color="Green">E65</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K5">omega</a> 
 <b>by </b><i><a class="txt" href="topalg_5.html#E22:39"><i><font color="Green">E50</font></i></a>, <a class="ref" href="finset_1.html#D1">FINSET_1:def 1</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>3</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> being <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>9</sub></font>ex <font color="Olive">b<sub>2</sub></font> being non <a href="xboole_0.html#V1">empty</a> <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K5">I[01]</a>  st <br/>( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>30</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="borsuk_1.html#K6">:]</a></span> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> );<br/>
<a NAME="E54:39"/><i><font color="Green">E66</font></i>: 
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>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>30</sub></font> holds <br/>ex <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>25</sub></font> &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_13"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>31</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:39_13"/>
<font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>30</sub></font>
 ;<br/>

<a NAME="E2:39_13"/><b>then </b><i><font color="Green">E67</font></i>: 
<font color="Maroon">c<sub>30</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>30</sub></font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>32</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:39_13"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>16</sub></font>
 <b>and </b><br/><a NAME="E5:39_13"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>30</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>31</sub></font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E20:39"><i><font color="Green">E48</font></i></a>, <a class="txt" href="topalg_5.html#E52:39"><i><font color="Green">E64</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>33</sub></font> being  <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>34</sub></font> being  <a href="pre_topc.html#V3">open</a> <a href="connsp_1.html#V2">connected</a> <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K5">I[01]</a> <b> such that </b><br/><a NAME="E7:39_13"/><i><font color="Green">E70</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>33</sub></font> &amp; <font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>34</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>33</sub></font>,<font color="Maroon">c<sub>34</sub></font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>15</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>32</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>33</sub></font>,<font color="Maroon">c<sub>34</sub></font></span><a href="borsuk_1.html#K6">:]</a></span> )
 <b>by </b><i><a class="txt" href="topalg_5.html#E12:39"><i><font color="Green">E44</font></i></a>, <a class="txt" href="topalg_5.html#E4:39_13"><i><font color="Green">E68</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>33</sub></font>
;<br/>

<b>thus </b><a NAME="E8:39_13"/>
( <font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>25</sub></font> &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">c<sub>31</sub></font>,<font color="Maroon">c<sub>33</sub></font>] )
 <b>by </b><i><a class="txt" href="topalg_5.html#E5:39_13"><i><font color="Green">E69</font></i></a>, <a class="txt" href="topalg_5.html#E7:39_13"><i><font color="Green">E70</font></i></a>, <a class="txt" href="topalg_5.html#E52:39"><i><font color="Green">E64</font></i></a>, <a class="txt" href="topalg_5.html#E2:39_13"><i><font color="Green">E67</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>31</sub></font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>30</sub></font>,<font color="Maroon">c<sub>25</sub></font><b> such that </b><br/><a NAME="E56:39"/><i><font color="Green">E67</font></i>: 
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>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>30</sub></font> holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>31</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S1">FUNCT_2:sch 1</a>(<a class="txt" href="topalg_5.html#E54:39"><i><font color="Green">E66</font></i></a>);<br/></i>
<a NAME="E57:39"/><i><font color="Green">E68</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>30</sub></font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E49:39"><i><font color="Green">E62</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E58:39"/><i><font color="Green">E69</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>25</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_14"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:39_14"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>31</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>25</sub></font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><br/>

<b>let </b><font color="Maroon">c<sub>32</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:39_14"/>
<font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>25</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>33</sub></font> being  <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>9</sub></font><b> such that </b><br/><a NAME="E4:39_14"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>33</sub></font>
 <b>and </b><br/><a NAME="E5:39_14"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>33</sub></font>
 <b>and </b><br/><a NAME="E6:39_14"/><i><font color="Green">E71</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K5">I[01]</a>  st <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>33</sub></font>,<font color="Olive">b<sub>1</sub></font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>18</sub></font>
 ;<br/>
<b>consider </b><font color="Maroon">c<sub>34</sub></font> being  <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K5">I[01]</a> <b> such that </b><br/><a NAME="E8:39_14"/><i><font color="Green">E72</font></i>: 
<span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>33</sub></font>,<font color="Maroon">c<sub>34</sub></font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>18</sub></font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E6:39_14"><i><font color="Green">E71</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>35</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E10:39_14"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">c<sub>35</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>30</sub></font>
 <b>and </b><br/><a NAME="E11:39_14"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">c<sub>30</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>35</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>33</sub></font>,<font color="Maroon">c<sub>34</sub></font></span><a href="borsuk_1.html#K6">:]</a></span>
 <b>by </b><i><a class="txt" href="topalg_5.html#E52:39"><i><font color="Green">E64</font></i></a>, <a class="txt" href="topalg_5.html#E8:39_14"><i><font color="Green">E72</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>36</sub></font> being  <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>37</sub></font> being  non <a href="xboole_0.html#V1">empty</a> <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K5">I[01]</a> <b> such that </b><br/><a NAME="E13:39_14"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>36</sub></font>
 <b>and </b><br/><a NAME="E14:39_14"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">c<sub>30</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>35</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>36</sub></font>,<font color="Maroon">c<sub>37</sub></font></span><a href="borsuk_1.html#K6">:]</a></span>
 <b>and </b><br/><a NAME="E15:39_14"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">c<sub>31</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>35</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>36</sub></font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E10:39_14"><i><font color="Green">E73</font></i></a>, <a class="txt" href="topalg_5.html#E56:39"><i><font color="Green">E67</font></i></a></i>;<br/>
<a NAME="E16:39_14"/>
<font color="Maroon">c<sub>36</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>33</sub></font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E11:39_14"><i><font color="Green">E74</font></i></a>, <a class="txt" href="topalg_5.html#E14:39_14"><i><font color="Green">E76</font></i></a>, <a class="txt" href="topalg_5.html#E13:39_14"><i><font color="Green">E75</font></i></a>, <a class="ref" href="zfmisc_1.html#T134">ZFMISC_1:134</a></i>;<br/>
<b>hence </b><a NAME="E17:39_14"/>
<font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>31</sub></font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E4:39_14"><i><font color="Green">E70</font></i></a>, <a class="txt" href="topalg_5.html#E57:39"><i><font color="Green">E68</font></i></a>, <a class="txt" href="topalg_5.html#E10:39_14"><i><font color="Green">E73</font></i></a>, <a class="txt" href="topalg_5.html#E15:39_14"><i><font color="Green">E77</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E59:39"/>
<font color="Maroon">c<sub>25</sub></font> is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E53:39"><i><font color="Green">E65</font></i></a>, <a class="txt" href="topalg_5.html#E58:39"><i><font color="Green">E69</font></i></a>, <a class="txt" href="topalg_5.html#E57:39"><i><font color="Green">E68</font></i></a>, <a class="ref" href="finset_1.html#D1">FINSET_1:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>32</sub></font> =  <a href="setfam_1.html#K6">meet</a> <font color="Maroon">c<sub>25</sub></font> as  <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>9</sub></font> <b>by </b><i><a class="txt" href="topalg_5.html#E50:39"><i><font color="Green">E63</font></i></a>, <a class="ref" href="tops_2.html#T27">TOPS_2:27</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>32</sub></font>
;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_15"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>33</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:39_15"/>
<font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>25</sub></font>
 ;<br/><a NAME="E2:39_15"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>9</sub></font> st <br/>( <font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> &amp; ex <font color="Olive">b<sub>2</sub></font> being <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K5">I[01]</a>  st <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>18</sub></font> )
 ;<br/><b>hence </b><a NAME="E3:39_15"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>33</sub></font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E62:39"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>32</sub></font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E49:39"><i><font color="Green">E62</font></i></a>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>

<b>let </b><font color="Maroon">c<sub>33</sub></font> be  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E63:39"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>23</sub></font>
 <b>and </b><br/><a NAME="E64:39"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">c<sub>33</sub></font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>23</sub></font>
 ;<br/>

<a NAME="E65:39"/><i><font color="Green">E72</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>33</sub></font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E63:39"><i><font color="Green">E70</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E66:39"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">c<sub>33</sub></font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>23</sub></font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E64:39"><i><font color="Green">E71</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E67:39"/><b>then </b><i><font color="Green">E74</font></i>: 
<font color="Maroon">c<sub>33</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>23</sub></font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E68:39"/><i><font color="Green">E75</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>21</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>20</sub></font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E29:39"><i><font color="Green">E55</font></i></a>, <a class="txt" href="topalg_5.html#E28:39"><i><font color="Green">E54</font></i></a>, <a class="ref" href="rcomp_3.html#D2">RCOMP_3:def 2</a>, <a class="txt" href="topalg_5.html#E32:39"><i><font color="Green">E56</font></i></a></i>;<br/>
<a NAME="E69:39"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="txt" href="topalg_5.html#E29:39"><i><font color="Green">E55</font></i></a>, <a class="txt" href="topalg_5.html#E28:39"><i><font color="Green">E54</font></i></a>, <a class="ref" href="rcomp_3.html#D3">RCOMP_3:def 3</a>, <a class="txt" href="topalg_5.html#E32:39"><i><font color="Green">E56</font></i></a></i>;<br/>
<a NAME="E70:39"/><b>then </b>
<font color="Maroon">c<sub>33</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>21</sub></font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E66:39"><i><font color="Green">E73</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E71:39"/><b>then </b>
<font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>21</sub></font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E65:39"><i><font color="Green">E72</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E72:39"/><b>then </b>
<font color="Maroon">c<sub>21</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>21</sub></font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E73:39"/><b>then </b>
<font color="Maroon">c<sub>21</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>20</sub></font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E68:39"><i><font color="Green">E75</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>34</sub></font> being  <a href="pre_topc.html#V3">open</a> <a href="connsp_1.html#V2">connected</a> <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K5">I[01]</a> <b> such that </b><br/><a NAME="E75:39"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">c<sub>21</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>34</sub></font>
 <b>and </b><br/><a NAME="E76:39"/><i><font color="Green">E77</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>9</sub></font> st <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>34</sub></font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>18</sub></font>
 ;<br/>
<b>consider </b><font color="Maroon">c<sub>35</sub></font> being  <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>9</sub></font><b> such that </b><br/><a NAME="E78:39"/><i><font color="Green">E78</font></i>: 
<span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>35</sub></font>,<font color="Maroon">c<sub>34</sub></font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>18</sub></font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E76:39"><i><font color="Green">E77</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>36</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E80:39"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">c<sub>36</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>16</sub></font>
 <b>and </b><br/><a NAME="E81:39"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>36</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>35</sub></font>,<font color="Maroon">c<sub>34</sub></font></span><a href="borsuk_1.html#K6">:]</a></span>
 <b>by </b><i><a class="txt" href="topalg_5.html#E78:39"><i><font color="Green">E78</font></i></a>, <a class="txt" href="topalg_5.html#E20:39"><i><font color="Green">E48</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>37</sub></font> being  <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>38</sub></font> being  <a href="pre_topc.html#V3">open</a> <a href="connsp_1.html#V2">connected</a> <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K5">I[01]</a> <b> such that </b><br/><a NAME="E83:39"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>37</sub></font>
 <b>and </b><br/><a NAME="E84:39"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">c<sub>36</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>38</sub></font>
 <b>and </b><br/><a NAME="E85:39"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>37</sub></font>,<font color="Maroon">c<sub>38</sub></font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>15</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>36</sub></font>
 <b>and </b><br/><a NAME="E86:39"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>36</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>37</sub></font>,<font color="Maroon">c<sub>38</sub></font></span><a href="borsuk_1.html#K6">:]</a></span>
 <b>by </b><i><a class="txt" href="topalg_5.html#E12:39"><i><font color="Green">E44</font></i></a>, <a class="txt" href="topalg_5.html#E80:39"><i><font color="Green">E79</font></i></a></i>;<br/>
<a NAME="E87:39"/><i><font color="Green">E85</font></i>: 
( <font color="Maroon">c<sub>35</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>37</sub></font> &amp; <font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>38</sub></font> )
 
<b>by </b><i><a class="txt" href="topalg_5.html#E86:39"><i><font color="Green">E84</font></i></a>, <a class="txt" href="topalg_5.html#E81:39"><i><font color="Green">E80</font></i></a>, <a class="txt" href="topalg_5.html#E83:39"><i><font color="Green">E81</font></i></a>, <a class="txt" href="topalg_5.html#E84:39"><i><font color="Green">E82</font></i></a>, <a class="ref" href="zfmisc_1.html#T134">ZFMISC_1:134</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>39</sub></font> being  <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K8">Tunit_circle</a> 2</span>)</span><b> such that </b><br/><a NAME="E89:39"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">c<sub>39</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>14</sub></font>
 <b>and </b><br/><a NAME="E90:39"/><i><font color="Green">E87</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>36</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>39</sub></font>
 <b>and </b><br/><a NAME="E91:39"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>36</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>39</sub></font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E9:39"><i><font color="Green">E42</font></i></a>, <a class="txt" href="topalg_5.html#E80:39"><i><font color="Green">E79</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>40</sub></font> = <font color="Maroon">c<sub>39</sub></font> as  non <a href="xboole_0.html#V1">empty</a> <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K8">Tunit_circle</a> 2</span>)</span> <b>by </b><i><a class="txt" href="topalg_5.html#E90:39"><i><font color="Green">E87</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>40</sub></font>
;<br/>

<b>thus </b><a NAME="E93:39"/>
<font color="Maroon">c<sub>40</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="topalg_5.html#E89:39"><i><font color="Green">E86</font></i></a></i>;<br/>

<a NAME="E94:39"/><i><font color="Green">E89</font></i>: 
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>33</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>33</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>34</sub></font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E67:39"><i><font color="Green">E74</font></i></a>, <a class="txt" href="topalg_5.html#E75:39"><i><font color="Green">E76</font></i></a>, <a class="txt" href="topalg_5.html#E29:39"><i><font color="Green">E55</font></i></a>, <a class="txt" href="topalg_5.html#E28:39"><i><font color="Green">E54</font></i></a>, <a class="txt" href="topalg_5.html#E65:39"><i><font color="Green">E72</font></i></a>, <a class="ref" href="rcomp_3.html#T80">RCOMP_3:80</a>, <a class="txt" href="topalg_5.html#E32:39"><i><font color="Green">E56</font></i></a></i>;<br/>
<a NAME="E95:39"/>
<font color="Maroon">c<sub>35</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>25</sub></font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E83:39"><i><font color="Green">E81</font></i></a>, <a class="txt" href="topalg_5.html#E87:39"><i><font color="Green">E85</font></i></a>, <a class="txt" href="topalg_5.html#E78:39"><i><font color="Green">E78</font></i></a></i>;<br/>
<a NAME="E96:39"/><b>then </b>
<font color="Maroon">c<sub>32</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>35</sub></font>
 
<b>by </b><i><a class="ref" href="setfam_1.html#T4">SETFAM_1:4</a></i>;<br/>
<a NAME="E97:39"/><b>then </b>
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>32</sub></font>,<span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>33</sub></font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>33</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>37</sub></font>,<font color="Maroon">c<sub>38</sub></font></span><a href="borsuk_1.html#K6">:]</a></span>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E94:39"><i><font color="Green">E89</font></i></a>, <a class="txt" href="topalg_5.html#E86:39"><i><font color="Green">E84</font></i></a>, <a class="txt" href="topalg_5.html#E81:39"><i><font color="Green">E80</font></i></a>, <a class="ref" href="zfmisc_1.html#T119">ZFMISC_1:119</a></i>;<br/>
<a NAME="E98:39"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>32</sub></font>,<span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>33</sub></font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>33</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>10</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>37</sub></font>,<font color="Maroon">c<sub>38</sub></font></span><a href="borsuk_1.html#K6">:]</a></span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T156">RELAT_1:156</a></i>;<br/>
<b>hence </b><a NAME="E99:39"/>
<font color="Maroon">c<sub>10</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>32</sub></font>,<span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>33</sub></font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>33</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>40</sub></font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E91:39"><i><font color="Green">E88</font></i></a>, <a class="txt" href="topalg_5.html#E85:39"><i><font color="Green">E83</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>


</div>
