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

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</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>5</sub></font> be    <a href="yellow_9.html#M3">Refinement</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>,<span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>;<br/><b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="yellow_9.html#M3">Refinement</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>7</sub></font> be    <a href="yellow_9.html#M3">Refinement</a> of <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>;<br/>













<b>assume </b><b>that </b><br/><a NAME="E1:60"/><i><font color="Green">E27</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E2:60"/><i><font color="Green">E28</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<a NAME="E3:60"/><i><font color="Green">E29</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>1</sub></font>,<font color="Maroon">c<sub>3</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>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</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/>
<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> =  <a href="setfam_1.html#K3">INTERSECTION</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>2</sub></font> as    <a href="cantor_1.html#M1">Basis</a> of <font color="Maroon">c<sub>6</sub></font> <b>by </b><i><a class="txt" href="yellow12.html#E1:60"><i><font color="Green">E27</font></i></a>, <a class="ref" href="yellow_9.html#T60">YELLOW_9:60</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>9</sub></font> =  <a href="setfam_1.html#K3">INTERSECTION</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>3</sub></font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>4</sub></font> as    <a href="cantor_1.html#M1">Basis</a> of <font color="Maroon">c<sub>7</sub></font> <b>by </b><i><a class="txt" href="yellow12.html#E2:60"><i><font color="Green">E28</font></i></a>, <a class="ref" href="yellow_9.html#T60">YELLOW_9:60</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <span class="p1">{<span class="default"> <span class="p2"><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> where B is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>6</sub></font>, B is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>7</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> ) </span>}</span>  as    <a href="cantor_1.html#M1">Basis</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>6</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="yellow_9.html#T40">YELLOW_9:40</a></i>;<br/>
<i><font color="Green">E30</font></i>: the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>5</sub></font> = 
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>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="borsuk_1.html#K5">:]</a></span> <a href="xboole_0.html#K2">\/</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>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>
<b>by </b><i><a class="ref" href="yellow_9.html#D6">YELLOW_9:def 6</a></i>
<br/>.= 
<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>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="xboole_0.html#K2">\/</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>2</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>
<b>by </b><i><a class="txt" href="yellow12.html#E3:60"><i><font color="Green">E29</font></i></a>, <a class="ref" href="borsuk_1.html#D5">BORSUK_1:def 5</a></i>
<br/>.= 
<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>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>
<b>by </b><i><a class="txt" href="yellow12.html#E1:60"><i><font color="Green">E27</font></i></a>, <a class="txt" href="yellow12.html#E2:60"><i><font color="Green">E28</font></i></a></i>
;<br/>
<i><font color="Green">E31</font></i>: the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>6</sub></font> = 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K2">\/</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="yellow_9.html#D6">YELLOW_9:def 6</a></i>
<br/>.= 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="txt" href="yellow12.html#E1:60"><i><font color="Green">E27</font></i></a></i>
;<br/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>7</sub></font> = 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K2">\/</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="ref" href="yellow_9.html#D6">YELLOW_9:def 6</a></i>
<br/>.= 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="yellow12.html#E2:60"><i><font color="Green">E28</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E10:60"/><i><font color="Green">E32</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>6</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> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="yellow12.html#E7:60"><i><font color="Green">E30</font></i></a>, <a class="txt" href="yellow12.html#E8:60"><i><font color="Green">E31</font></i></a>, <a class="ref" href="borsuk_1.html#D5">BORSUK_1:def 5</a></i>;<br/>

<b>set </b><font color="Maroon">c<sub>11</sub></font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>3</sub></font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="subset_1.html#K9">/\</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>4</sub></font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> where B is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>, B is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>2</sub></font>, B is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>3</sub></font>, B is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>4</sub></font> : ( <font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V3">open</a> &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> &amp; <font color="Olive">b<sub>4</sub></font> is <a href="pre_topc.html#V3">open</a> ) </span>}</span> ;<br/>
<a NAME="E11:60"/><i><font color="Green">E33</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>3</sub></font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="subset_1.html#K9">/\</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>4</sub></font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> where B is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>, B is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>2</sub></font>, B is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>3</sub></font>, B is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>4</sub></font> : ( <font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V3">open</a> &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> &amp; <font color="Olive">b<sub>4</sub></font> is <a href="pre_topc.html#V3">open</a> ) </span>}</span>  is    <a href="cantor_1.html#M1">Basis</a> of <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="yellow12.html#E1:60"><i><font color="Green">E27</font></i></a>, <a class="txt" href="yellow12.html#E2:60"><i><font color="Green">E28</font></i></a>, <a class="ref" href="yellow12.html#T48" target="_self">Th48</a></i>;<br/>
<a NAME="E12:60"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>3</sub></font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="subset_1.html#K9">/\</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>4</sub></font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> where B is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>, B is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>2</sub></font>, B is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>3</sub></font>, B is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>4</sub></font> : ( <font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V3">open</a> &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> &amp; <font color="Olive">b<sub>4</sub></font> is <a href="pre_topc.html#V3">open</a> ) </span>}</span>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_4"><b>proof </b></a><div class="add">

<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a>,<a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a>
<div class="add"><b>let </b><font color="Maroon">c<sub>12</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:60_4_1"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>3</sub></font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="subset_1.html#K9">/\</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>4</sub></font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> where B is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>, B is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>2</sub></font>, B is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>3</sub></font>, B is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>4</sub></font> : ( <font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V3">open</a> &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> &amp; <font color="Olive">b<sub>4</sub></font> is <a href="pre_topc.html#V3">open</a> ) </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">c<sub>13</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>14</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>15</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>16</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>4</sub></font><b> such that </b><br/><a NAME="E3:60_4_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>12</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>13</sub></font>,<font color="Maroon">c<sub>15</sub></font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="subset_1.html#K9">/\</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>16</sub></font></span><a href="borsuk_1.html#K6">:]</a></span>
 <b>and </b><br/><a NAME="E4:60_4_1"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">c<sub>14</sub></font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">c<sub>15</sub></font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">c<sub>16</sub></font> is <a href="pre_topc.html#V3">open</a> )
 ;<br/><a NAME="E5:60_4_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <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"><font color="Maroon">c<sub>13</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>16</sub></font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span>
 <b>by </b><i><a class="txt" href="yellow12.html#E3:60_4_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="zfmisc_1.html#T123">ZFMISC_1:123</a></i>;<br/><a NAME="E6:60_4_1"/>
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="yellow12.html#E4:60_4_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/><a NAME="E7:60_4_1"/><b>then </b>
( <font color="Maroon">c<sub>13</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> )
 <b>by </b><i><a class="ref" href="setfam_1.html#D5">SETFAM_1:def 5</a></i>;<br/><b>hence </b><a NAME="E8:60_4_1"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="yellow12.html#E5:60_4_1"><i><font color="Green">E36</font></i></a></i>;<br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon">c<sub>12</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="E2:60_4"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>13</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>14</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>7</sub></font><b> such that </b><br/><a NAME="E4:60_4"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">c<sub>12</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>13</sub></font>,<font color="Maroon">c<sub>14</sub></font></span><a href="borsuk_1.html#K6">:]</a></span> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> )
 ;<br/>
<b>consider </b><font color="Maroon">c<sub>15</sub></font>, <font color="Maroon">c<sub>16</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:60_4"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">c<sub>16</sub></font> )
 <b>by </b><i><a class="txt" href="yellow12.html#E4:60_4"><i><font color="Green">E34</font></i></a>, <a class="ref" href="setfam_1.html#D5">SETFAM_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>17</sub></font>, <font color="Maroon">c<sub>18</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E8:60_4"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">c<sub>18</sub></font> )
 <b>by </b><i><a class="txt" href="yellow12.html#E4:60_4"><i><font color="Green">E34</font></i></a>, <a class="ref" href="setfam_1.html#D5">SETFAM_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>19</sub></font> = <font color="Maroon">c<sub>15</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="yellow12.html#E6:60_4"><i><font color="Green">E35</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>20</sub></font> = <font color="Maroon">c<sub>16</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="yellow12.html#E6:60_4"><i><font color="Green">E35</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>21</sub></font> = <font color="Maroon">c<sub>17</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="txt" href="yellow12.html#E8:60_4"><i><font color="Green">E36</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>22</sub></font> = <font color="Maroon">c<sub>18</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>4</sub></font> <b>by </b><i><a class="txt" href="yellow12.html#E8:60_4"><i><font color="Green">E36</font></i></a></i>;<br/>
<a NAME="E13:60_4"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">c<sub>19</sub></font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">c<sub>20</sub></font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">c<sub>21</sub></font> is <a href="pre_topc.html#V3">open</a> &amp; <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="yellow12.html#E6:60_4"><i><font color="Green">E35</font></i></a>, <a class="txt" href="yellow12.html#E8:60_4"><i><font color="Green">E36</font></i></a>, <a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>
<a NAME="E14:60_4"/>
<font color="Maroon">c<sub>12</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>19</sub></font>,<font color="Maroon">c<sub>21</sub></font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="subset_1.html#K9">/\</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>22</sub></font></span><a href="borsuk_1.html#K6">:]</a></span>
 
<b>by </b><i><a class="txt" href="yellow12.html#E4:60_4"><i><font color="Green">E34</font></i></a>, <a class="txt" href="yellow12.html#E6:60_4"><i><font color="Green">E35</font></i></a>, <a class="txt" href="yellow12.html#E8:60_4"><i><font color="Green">E36</font></i></a>, <a class="ref" href="zfmisc_1.html#T123">ZFMISC_1:123</a></i>;<br/>
<b>hence </b><a NAME="E15:60_4"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>3</sub></font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="subset_1.html#K9">/\</a> <span class="p3"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>4</sub></font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> where B is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>, B is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>2</sub></font>, B is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>3</sub></font>, B is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>4</sub></font> : ( <font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V3">open</a> &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> &amp; <font color="Olive">b<sub>4</sub></font> is <a href="pre_topc.html#V3">open</a> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="yellow12.html#E13:60_4"><i><font color="Green">E37</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E13:60"/>
the <a href="pre_topc.html#U1">topology</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>6</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> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="yellow12.html#E10:60"><i><font color="Green">E32</font></i></a>, <a class="txt" href="yellow12.html#E11:60"><i><font color="Green">E33</font></i></a>, <a class="ref" href="yellow_9.html#T25">YELLOW_9:25</a></i>;<br/>


</div>
