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

<b>let </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font>, <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="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>,<font color="Maroon">c<sub>7</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>,<font color="Maroon">c<sub>5</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>11</sub></font> be   <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>,<font color="Maroon">c<sub>8</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>,<font color="Maroon">c<sub>5</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>12</sub></font>, <font color="Maroon">c<sub>13</sub></font> be  <a href="pre_topc.html#V4">closed</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>6</sub></font>;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:24"/><i><font color="Green">E9</font></i>: 
<font color="Maroon">c<sub>7</sub></font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>6</sub></font>
 <b>and </b><br/><a NAME="E2:24"/><i><font color="Green">E10</font></i>: 
<font color="Maroon">c<sub>8</sub></font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>6</sub></font>
 <b>and </b><br/><a NAME="E3:24"/><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>7</sub></font>
 <b>and </b><br/><a NAME="E4:24"/><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>8</sub></font>
 <b>and </b><br/><a NAME="E5:24"/><i><font color="Green">E13</font></i>: 
<span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>6</sub></font>
 <b>and </b><br/><a NAME="E6:24"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>10</sub></font> is <a href="pre_topc.html#V5">continuous</a>
 <b>and </b><br/><a NAME="E7:24"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>11</sub></font> is <a href="pre_topc.html#V5">continuous</a>
 <b>and </b><br/><a NAME="E8:24"/><i><font color="Green">E16</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> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="borsuk_1.html#K5">:]</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="borsuk_1.html#K5">:]</a></span></span>)</span> holds <br/><font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>
 ;<br/>

<a NAME="E9:24"/><i><font color="Green">E17</font></i>: 
( <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="borsuk_1.html#K5">:]</a></span> is    <a href="pre_topc.html#M1">SubSpace</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="borsuk_1.html#K5">:]</a></span> &amp; <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="borsuk_1.html#K5">:]</a></span> is    <a href="pre_topc.html#M1">SubSpace</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="borsuk_1.html#K5">:]</a></span> )
 
<b>by </b><i><a class="txt" href="topalg_3.html#E1:24"><i><font color="Green">E9</font></i></a>, <a class="txt" href="topalg_3.html#E2:24"><i><font color="Green">E10</font></i></a>, <a class="ref" href="borsuk_3.html#T17">BORSUK_3:17</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>10</sub></font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">c<sub>11</sub></font>;<br/>
<a NAME="E10:24"/><i><font color="Green">E18</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="borsuk_1.html#K5">:]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>9</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>6</sub></font></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/>
<a NAME="E11:24"/><i><font color="Green">E19</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="borsuk_1.html#K5">:]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>9</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>7</sub></font></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/>
<a NAME="E12:24"/><i><font color="Green">E20</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="borsuk_1.html#K5">:]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>9</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>8</sub></font></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/>
<a NAME="E13:24"/><i><font color="Green">E21</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E14:24"/><i><font color="Green">E22</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E15:24"/><i><font color="Green">E23</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>;<br/>
<a NAME="E16:24"/><b>then </b><i><font color="Green">E24</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>9</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>6</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="txt" href="topalg_3.html#E5:24"><i><font color="Green">E13</font></i></a>, <a class="txt" href="topalg_3.html#E11:24"><i><font color="Green">E19</font></i></a>, <a class="txt" href="topalg_3.html#E12:24"><i><font color="Green">E20</font></i></a>, <a class="txt" href="topalg_3.html#E13:24"><i><font color="Green">E21</font></i></a>, <a class="txt" href="topalg_3.html#E14:24"><i><font color="Green">E22</font></i></a>, <a class="ref" href="zfmisc_1.html#T120">ZFMISC_1:120</a></i>;<br/>
<a NAME="E17:24"/>
( <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>5</sub></font> &amp;  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> )
 
<b>by </b><i><a class="ref" href="funct_4.html#T18">FUNCT_4:18</a></i>;<br/>
<a NAME="E18:24"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>10</sub></font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">c<sub>11</sub></font> as   <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>,<font color="Maroon">c<sub>6</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>,<font color="Maroon">c<sub>5</sub></font> <b>by </b><i><a class="txt" href="topalg_3.html#E10:24"><i><font color="Green">E18</font></i></a>, <a class="txt" href="topalg_3.html#E16:24"><i><font color="Green">E24</font></i></a>, <a class="ref" href="funct_2.html#T4">FUNCT_2:4</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>15</sub></font>
;<br/>

<b>thus </b><a NAME="E20:24"/>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">c<sub>11</sub></font>
 ;<br/>

<a NAME="E21:24"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>5</sub></font> st <font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V4">closed</a> holds <br/><font color="Maroon">c<sub>15</sub></font> <a href="relset_1.html#K11">"</a> <font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V4">closed</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>16</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>5</sub></font>;<br/>

<b>assume </b><a NAME="E1:24_1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>16</sub></font> is <a href="pre_topc.html#V4">closed</a>
 ;<br/>

<i><font color="Green">E26</font></i>: <font color="Maroon">c<sub>15</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>16</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="subset_1.html#K8">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p3"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="borsuk_1.html#K5">:]</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p3"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="borsuk_1.html#K5">:]</a></span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="topalg_3.html#E10:24"><i><font color="Green">E18</font></i></a>, <a class="txt" href="topalg_3.html#E13:24"><i><font color="Green">E21</font></i></a>, <a class="txt" href="topalg_3.html#E14:24"><i><font color="Green">E22</font></i></a>, <a class="txt" href="topalg_3.html#E15:24"><i><font color="Green">E23</font></i></a>, <a class="txt" href="topalg_3.html#E16:24"><i><font color="Green">E24</font></i></a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p3"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="borsuk_1.html#K5">:]</a></span></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p3"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="borsuk_1.html#K5">:]</a></span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T23">XBOOLE_1:23</a></i>
;<br/>
<a NAME="E3:24_1"/><i><font color="Green">E27</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="pre_topc.html#K2">[#]</a> <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="borsuk_1.html#K5">:]</a></span> holds <br/><font color="Maroon">c<sub>15</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>17</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:24_1_2"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:24_1_2_1_1"/>
( <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="borsuk_1.html#K5">:]</a></span> or  not <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="borsuk_1.html#K5">:]</a></span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:24_1_2_1_1_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>
 ;<br/><div class="add"><a NAME="E2:24_1_2_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="borsuk_1.html#K5">:]</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="borsuk_1.html#K5">:]</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="topalg_3.html#E1:24_1_2"><i><font color="Green">E28</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E3:24_1_2_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="txt" href="topalg_3.html#E8:24"><i><font color="Green">E16</font></i></a></i>;<br/><b>hence </b><a NAME="E4:24_1_2_1_1_1"/>
<font color="Maroon">c<sub>15</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="txt" href="topalg_3.html#E14:24"><i><font color="Green">E22</font></i></a>, <a class="txt" href="topalg_3.html#E1:24_1_2_1_1_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:24_1_2_1_1_2"/>
not <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:24_1_2_1_1_2"/>
<font color="Maroon">c<sub>15</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="txt" href="topalg_3.html#E14:24"><i><font color="Green">E22</font></i></a>, <a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:24_1_2"/>
<font color="Maroon">c<sub>15</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font>
 ;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>17</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>thus </b><a NAME="E1:24_1_3"/>
( <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="borsuk_1.html#K5">:]</a></span></span>)</span> implies <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>16</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_3_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:24_1_3_1"/>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="borsuk_1.html#K5">:]</a></span></span>)</span>
 ;<br/>

<a NAME="E2:24_1_3_1"/><b>then </b>
( <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>15</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="borsuk_1.html#K5">:]</a></span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E3:24_1_3_1"/><b>then </b>
( <font color="Maroon">c<sub>15</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font> )
 
<b>by </b><i><a class="txt" href="topalg_3.html#E13:24"><i><font color="Green">E21</font></i></a>, <a class="txt" href="topalg_3.html#E3:24_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>
<b>hence </b><a NAME="E4:24_1_3_1"/>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>


</div><b>end;</b></div><b>assume </b><a NAME="E2:24_1_3"/>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>16</sub></font>
 ;<br/><a NAME="E3:24_1_3"/><b>then </b>
( <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>16</sub></font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/><a NAME="E4:24_1_3"/><b>then </b>
( <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="borsuk_1.html#K5">:]</a></span> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>16</sub></font> )
 <b>by </b><i><a class="txt" href="topalg_3.html#E15:24"><i><font color="Green">E23</font></i></a>, <a class="txt" href="topalg_3.html#E3:24_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E5:24_1_3"/><b>then </b>
( <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>15</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="borsuk_1.html#K5">:]</a></span> )
 <b>by </b><i><a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/><b>hence </b><a NAME="E6:24_1_3"/>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="borsuk_1.html#K5">:]</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div>
<a NAME="E5:24_1"/><b>then </b><i><font color="Green">E28</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="borsuk_1.html#K5">:]</a></span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>16</sub></font>
 
<b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>17</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>thus </b><a NAME="E1:24_1_4"/>
( <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="borsuk_1.html#K5">:]</a></span></span>)</span> implies <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>16</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_4_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:24_1_4_1"/>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="borsuk_1.html#K5">:]</a></span></span>)</span>
 ;<br/>

<a NAME="E2:24_1_4_1"/><b>then </b>
( <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>15</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="borsuk_1.html#K5">:]</a></span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E3:24_1_4_1"/><b>then </b>
( <font color="Maroon">c<sub>15</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font> )
 
<b>by </b><i><a class="txt" href="topalg_3.html#E14:24"><i><font color="Green">E22</font></i></a>, <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a>, <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>;<br/>
<b>hence </b><a NAME="E4:24_1_4_1"/>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>


</div><b>end;</b></div><b>assume </b><a NAME="E2:24_1_4"/>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>16</sub></font>
 ;<br/><a NAME="E3:24_1_4"/><b>then </b>
( <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>16</sub></font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/><a NAME="E4:24_1_4"/><b>then </b>
( <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="borsuk_1.html#K5">:]</a></span> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>16</sub></font> )
 <b>by </b><i><a class="txt" href="topalg_3.html#E15:24"><i><font color="Green">E23</font></i></a>, <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E5:24_1_4"/><b>then </b>
( <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>15</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="borsuk_1.html#K5">:]</a></span> )
 <b>by </b><i><a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/><b>hence </b><a NAME="E6:24_1_4"/>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="borsuk_1.html#K5">:]</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div>
<a NAME="E7:24_1"/><b>then </b><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>16</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="topalg_3.html#E2:24_1"><i><font color="Green">E26</font></i></a>, <a class="txt" href="topalg_3.html#E5:24_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>
<a NAME="E8:24_1"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>7</sub></font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="topalg_3.html#E1:24"><i><font color="Green">E9</font></i></a>, <a class="ref" href="tsep_1.html#T1">TSEP_1:1</a></i>;<br/>
<a NAME="E9:24_1"/><b>then </b>
( <font color="Maroon">c<sub>10</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>16</sub></font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="borsuk_1.html#K5">:]</a></span> &amp;  <a href="pre_topc.html#K2">[#]</a> <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="borsuk_1.html#K5">:]</a></span> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="borsuk_1.html#K5">:]</a></span> )
 
<b>by </b><i><a class="txt" href="topalg_3.html#E10:24"><i><font color="Green">E18</font></i></a>, <a class="txt" href="topalg_3.html#E11:24"><i><font color="Green">E19</font></i></a>, <a class="ref" href="zfmisc_1.html#T118">ZFMISC_1:118</a></i>;<br/>
<a NAME="E10:24_1"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>16</sub></font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>10</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>16</sub></font> as   <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>,<font color="Maroon">c<sub>6</sub></font></span><a href="borsuk_1.html#K5">:]</a></span> ;<br/>
<a NAME="E12:24_1"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>8</sub></font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="topalg_3.html#E2:24"><i><font color="Green">E10</font></i></a>, <a class="ref" href="tsep_1.html#T1">TSEP_1:1</a></i>;<br/>
<a NAME="E13:24_1"/><b>then </b>
( <font color="Maroon">c<sub>11</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>16</sub></font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="borsuk_1.html#K5">:]</a></span> &amp;  <a href="pre_topc.html#K2">[#]</a> <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="borsuk_1.html#K5">:]</a></span> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="borsuk_1.html#K5">:]</a></span> )
 
<b>by </b><i><a class="txt" href="topalg_3.html#E10:24"><i><font color="Green">E18</font></i></a>, <a class="txt" href="topalg_3.html#E12:24"><i><font color="Green">E20</font></i></a>, <a class="ref" href="zfmisc_1.html#T118">ZFMISC_1:118</a></i>;<br/>
<a NAME="E14:24_1"/><b>then </b>
<font color="Maroon">c<sub>11</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>16</sub></font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>18</sub></font> = <font color="Maroon">c<sub>11</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>16</sub></font> as   <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>,<font color="Maroon">c<sub>6</sub></font></span><a href="borsuk_1.html#K5">:]</a></span> ;<br/>
<a NAME="E16:24_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>16</sub></font> is <a href="pre_topc.html#V4">closed</a>
 
<b>by </b><i><a class="txt" href="topalg_3.html#E6:24"><i><font color="Green">E14</font></i></a>, <a class="txt" href="topalg_3.html#E1:24_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="pre_topc.html#D12">PRE_TOPC:def 12</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>19</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>,<font color="Maroon">c<sub>6</sub></font></span><a href="borsuk_1.html#K5">:]</a></span><b> such that </b><br/><a NAME="E18:24_1"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">c<sub>19</sub></font> is <a href="pre_topc.html#V4">closed</a> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="borsuk_1.html#K5">:]</a></span></span>)</span> )
 <b>by </b><i><a class="txt" href="topalg_3.html#E9:24"><i><font color="Green">E17</font></i></a>, <a class="ref" href="pre_topc.html#T43">PRE_TOPC:43</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>20</sub></font> = <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>,<font color="Maroon">c<sub>12</sub></font></span><a href="borsuk_1.html#K6">:]</a></span> as   <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>,<font color="Maroon">c<sub>6</sub></font></span><a href="borsuk_1.html#K5">:]</a></span> ;<br/>
<a NAME="E20:24_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>20</sub></font> is <a href="pre_topc.html#V4">closed</a>
 
<b>by </b><i><a class="ref" href="topalg_3.html#T16" target="_self">Th16</a></i>;<br/>
<a NAME="E21:24_1"/>
 <a href="pre_topc.html#K2">[#]</a> <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="borsuk_1.html#K5">:]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>,<font color="Maroon">c<sub>12</sub></font></span><a href="borsuk_1.html#K6">:]</a></span>
 
<b>by </b><i><a class="txt" href="topalg_3.html#E3:24"><i><font color="Green">E11</font></i></a>, <a class="ref" href="borsuk_3.html#T1">BORSUK_3:1</a></i>;<br/>
<a NAME="E22:24_1"/><b>then </b><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>17</sub></font> is <a href="pre_topc.html#V4">closed</a>
 
<b>by </b><i><a class="txt" href="topalg_3.html#E18:24_1"><i><font color="Green">E30</font></i></a>, <a class="txt" href="topalg_3.html#E20:24_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="tops_1.html#T35">TOPS_1:35</a></i>;<br/>
<a NAME="E23:24_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>16</sub></font> is <a href="pre_topc.html#V4">closed</a>
 
<b>by </b><i><a class="txt" href="topalg_3.html#E7:24"><i><font color="Green">E15</font></i></a>, <a class="txt" href="topalg_3.html#E1:24_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="pre_topc.html#D12">PRE_TOPC:def 12</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>21</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>,<font color="Maroon">c<sub>6</sub></font></span><a href="borsuk_1.html#K5">:]</a></span><b> such that </b><br/><a NAME="E25:24_1"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">c<sub>21</sub></font> is <a href="pre_topc.html#V4">closed</a> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="borsuk_1.html#K5">:]</a></span></span>)</span> )
 <b>by </b><i><a class="txt" href="topalg_3.html#E9:24"><i><font color="Green">E17</font></i></a>, <a class="ref" href="pre_topc.html#T43">PRE_TOPC:43</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>22</sub></font> = <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>,<font color="Maroon">c<sub>13</sub></font></span><a href="borsuk_1.html#K6">:]</a></span> as   <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>,<font color="Maroon">c<sub>6</sub></font></span><a href="borsuk_1.html#K5">:]</a></span> ;<br/>
<a NAME="E27:24_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>22</sub></font> is <a href="pre_topc.html#V4">closed</a>
 
<b>by </b><i><a class="ref" href="topalg_3.html#T16" target="_self">Th16</a></i>;<br/>
<a NAME="E28:24_1"/>
 <a href="pre_topc.html#K2">[#]</a> <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="borsuk_1.html#K5">:]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>,<font color="Maroon">c<sub>13</sub></font></span><a href="borsuk_1.html#K6">:]</a></span>
 
<b>by </b><i><a class="txt" href="topalg_3.html#E4:24"><i><font color="Green">E12</font></i></a>, <a class="ref" href="borsuk_3.html#T1">BORSUK_3:1</a></i>;<br/>
<a NAME="E29:24_1"/><b>then </b>
<font color="Maroon">c<sub>18</sub></font> is <a href="pre_topc.html#V4">closed</a>
 
<b>by </b><i><a class="txt" href="topalg_3.html#E25:24_1"><i><font color="Green">E33</font></i></a>, <a class="txt" href="topalg_3.html#E27:24_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="tops_1.html#T35">TOPS_1:35</a></i>;<br/>
<b>hence </b><a NAME="E30:24_1"/>
<font color="Maroon">c<sub>15</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>16</sub></font> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i><a class="txt" href="topalg_3.html#E7:24_1"><i><font color="Green">E29</font></i></a>, <a class="txt" href="topalg_3.html#E22:24_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="tops_1.html#T36">TOPS_1:36</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E22:24"/>
<font color="Maroon">c<sub>15</sub></font> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i><a class="ref" href="pre_topc.html#D12">PRE_TOPC:def 12</a></i>;<br/>


</div>
