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<b>set </b><font color="Maroon">c<sub>7</sub></font> =  <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>5</sub></font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>5</sub></font> #);<br/>
<b>set </b><font color="Maroon">c<sub>8</sub></font> =  <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>6</sub></font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>6</sub></font> #);<br/>
<a NAME="E1:20_1_1"/><i><font color="Green">E9</font></i>: 
 <a href="pre_topc.html#G1">TopStruct</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>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>,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>5</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="borsuk_1.html#K5">[:</a><span class="default"><a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>5</sub></font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>5</sub></font> #),<a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>6</sub></font>,the <a href="pre_topc.html#U1">topology</a> of <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="topalg_3.html#T14" target="_self">Th14</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:20_1_1_1"/>
(  not <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>5</sub></font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>5</sub></font> #),<a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>6</sub></font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>6</sub></font> #)</span><a href="borsuk_1.html#K5">:]</a></span> is <a href="struct_0.html#V3">empty</a> or <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>5</sub></font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>5</sub></font> #),<a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>6</sub></font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>6</sub></font> #)</span><a href="borsuk_1.html#K5">:]</a></span> is <a href="struct_0.html#V3">empty</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_1_1_1"><b>suppose </b></a><a NAME="E1:20_1_1_1_1"/><i><font color="Green">E10</font></i>: 
not <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>5</sub></font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>5</sub></font> #),<a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>6</sub></font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>6</sub></font> #)</span><a href="borsuk_1.html#K5">:]</a></span> is <a href="struct_0.html#V3">empty</a>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1_1_1_1"><b>now </b></a><div class="add"><b>set </b><font color="Maroon">c<sub>9</sub></font> =  <a href="compts_1.html#K1">1TopSp</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span>;<br/><b>given </b><font color="Maroon">c<sub>10</sub></font> being   <a href="struct_0.html#NM7">Function</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>5</sub></font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>5</sub></font> #),<a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>6</sub></font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>6</sub></font> #)</span><a href="borsuk_1.html#K5">:]</a></span>,<span class="p1">(<span class="default"><a href="compts_1.html#K1">1TopSp</a> <span class="p2"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span></span>)</span><b> such that </b><a NAME="E1:20_1_1_1_1_1"/><i><font color="Green">E11</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="E2:20_1_1_1_1_1"/><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>10</sub></font> is <a href="funct_2.html#V2">onto</a>
 ;<br/><a NAME="E3:20_1_1_1_1_1"/><i><font color="Green">E13</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="compts_1.html#K1">1TopSp</a> <span class="p2"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span>
 <b>by </b><i><a class="ref" href="pcomps_1.html#T8">PCOMPS_1:8</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>11</sub></font> = 0, <font color="Maroon">c<sub>12</sub></font> = 1 as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="compts_1.html#K1">1TopSp</a> <span class="p2"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span></span>)</span> <b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><a NAME="E5:20_1_1_1_1_1"/><i><font color="Green">E14</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"><a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>5</sub></font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>5</sub></font> #),<a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>6</sub></font>,the <a href="pre_topc.html#U1">topology</a> of <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="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E6:20_1_1_1_1_1"/><i><font color="Green">E15</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"><a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>5</sub></font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>5</sub></font> #),<a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>6</sub></font>,the <a href="pre_topc.html#U1">topology</a> of <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 <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>5</sub></font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>5</sub></font> #),the <a href="struct_0.html#U1">carrier</a> of <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>6</sub></font>,the <a href="pre_topc.html#U1">topology</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="E7:20_1_1_1_1_1"/><i><font color="Green">E16</font></i>: 
 <a href="relset_1.html#K5">rng</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">(<span class="default"><a href="compts_1.html#K1">1TopSp</a> <span class="p2"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="topalg_3.html#E2:20_1_1_1_1_1"><i><font color="Green">E12</font></i></a>, <a class="ref" href="funct_2.html#D3">FUNCT_2:def 3</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>13</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E9:20_1_1_1_1_1"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>13</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>
 <b>and </b><br/><a NAME="E10:20_1_1_1_1_1"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>14</sub></font>, <font color="Maroon">c<sub>15</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E12:20_1_1_1_1_1"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>5</sub></font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>5</sub></font> #)
 <b>and </b><br/><a NAME="E13:20_1_1_1_1_1"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>6</sub></font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>6</sub></font> #)
 <b>and </b><br/><a NAME="E14:20_1_1_1_1_1"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>13</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>14</sub></font>,<font color="Maroon">c<sub>15</sub></font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="txt" href="topalg_3.html#E6:20_1_1_1_1_1"><i><font color="Green">E15</font></i></a>, <a class="txt" href="topalg_3.html#E9:20_1_1_1_1_1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>16</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E16:20_1_1_1_1_1"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>16</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>
 <b>and </b><br/><a NAME="E17:20_1_1_1_1_1"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="topalg_3.html#E7:20_1_1_1_1_1"><i><font color="Green">E16</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>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="E19:20_1_1_1_1_1"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>5</sub></font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>5</sub></font> #)
 <b>and </b><br/><a NAME="E20:20_1_1_1_1_1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>6</sub></font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>6</sub></font> #)
 <b>and </b><br/><a NAME="E21:20_1_1_1_1_1"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>16</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>17</sub></font>,<font color="Maroon">c<sub>18</sub></font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="txt" href="topalg_3.html#E6:20_1_1_1_1_1"><i><font color="Green">E15</font></i></a>, <a class="txt" href="topalg_3.html#E16:20_1_1_1_1_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>19</sub></font> = <font color="Maroon">c<sub>14</sub></font>, <font color="Maroon">c<sub>20</sub></font> = <font color="Maroon">c<sub>17</sub></font> as   <a href="pre_topc.html#NM2">Point</a> of <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>5</sub></font>,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="topalg_3.html#E12:20_1_1_1_1_1"><i><font color="Green">E19</font></i></a>, <a class="txt" href="topalg_3.html#E19:20_1_1_1_1_1"><i><font color="Green">E24</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>21</sub></font> = <font color="Maroon">c<sub>15</sub></font>, <font color="Maroon">c<sub>22</sub></font> = <font color="Maroon">c<sub>18</sub></font> as   <a href="pre_topc.html#NM2">Point</a> of <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>6</sub></font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>6</sub></font> #) <b>by </b><i><a class="txt" href="topalg_3.html#E13:20_1_1_1_1_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="topalg_3.html#E20:20_1_1_1_1_1"><i><font color="Green">E25</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>23</sub></font> =  <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>5</sub></font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>5</sub></font> #), <font color="Maroon">c<sub>24</sub></font> =  <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>6</sub></font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>6</sub></font> #) as  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a> <b>by </b><i><a class="txt" href="topalg_3.html#E1:20_1_1_1_1"><i><font color="Green">E10</font></i></a>, <a class="ref" href="borsuk_3.html#T4">BORSUK_3:4</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>25</sub></font> = <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>19</sub></font></span><a href="tarski.html#K1">}</a></span> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>23</sub></font> <b>by </b><i><a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>26</sub></font> = <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>22</sub></font></span><a href="tarski.html#K1">}</a></span> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>24</sub></font> <b>by </b><i><a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/><a NAME="E27:20_1_1_1_1_1"/>
<font color="Maroon">c<sub>24</sub></font>,<span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>25</sub></font></span>)</span>,<font color="Maroon">c<sub>24</sub></font></span><a href="borsuk_1.html#K5">:]</a></span> <a href="borsuk_3.html#R1">are_homeomorphic</a> 
 <b>by </b><i><a class="ref" href="borsuk_3.html#T15">BORSUK_3:15</a></i>;<br/><a NAME="E28:20_1_1_1_1_1"/><b>then </b><i><font color="Green">E27</font></i>: 
<span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>25</sub></font></span>)</span>,<font color="Maroon">c<sub>24</sub></font></span><a href="borsuk_1.html#K5">:]</a></span> is <a href="connsp_1.html#V1">connected</a>
 <b>by </b><i><a class="ref" href="topreal6.html#T25">TOPREAL6:25</a></i>;<br/><a NAME="E29:20_1_1_1_1_1"/>
<font color="Maroon">c<sub>23</sub></font>,<span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>24</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>26</sub></font></span>)</span>,<font color="Maroon">c<sub>23</sub></font></span><a href="borsuk_1.html#K5">:]</a></span> <a href="borsuk_3.html#R1">are_homeomorphic</a> 
 <b>by </b><i><a class="ref" href="borsuk_3.html#T15">BORSUK_3:15</a></i>;<br/><a NAME="E30:20_1_1_1_1_1"/><b>then </b><i><font color="Green">E28</font></i>: 
<span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>24</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>26</sub></font></span>)</span>,<font color="Maroon">c<sub>23</sub></font></span><a href="borsuk_1.html#K5">:]</a></span> is <a href="connsp_1.html#V1">connected</a>
 <b>by </b><i><a class="ref" href="topreal6.html#T25">TOPREAL6:25</a></i>;<br/><a NAME="E31:20_1_1_1_1_1"/>
<span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>23</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>24</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>26</sub></font></span>)</span></span><a href="borsuk_1.html#K5">:]</a></span>,<span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>24</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>26</sub></font></span>)</span>,<font color="Maroon">c<sub>23</sub></font></span><a href="borsuk_1.html#K5">:]</a></span> <a href="borsuk_3.html#R1">are_homeomorphic</a> 
 <b>by </b><i><a class="ref" href="yellow12.html#T44">YELLOW12:44</a></i>;<br/><a NAME="E32:20_1_1_1_1_1"/><b>then </b><i><font color="Green">E29</font></i>: 
<span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>23</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>24</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>26</sub></font></span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> is <a href="connsp_1.html#V1">connected</a>
 <b>by </b><i><a class="txt" href="topalg_3.html#E30:20_1_1_1_1_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="topreal6.html#T25">TOPREAL6:25</a></i>;<br/><b>set </b><font color="Maroon">c<sub>27</sub></font> = <font color="Maroon">c<sub>10</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>25</sub></font>,<span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>24</sub></font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span>;<br/><a NAME="E33:20_1_1_1_1_1"/><i><font color="Green">E30</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>25</sub></font>,<span class="p3">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>24</sub></font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <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>,<span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>24</sub></font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span>
 <b>by </b><i><a class="txt" href="topalg_3.html#E5:20_1_1_1_1_1"><i><font color="Green">E14</font></i></a>, <a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/><a NAME="E34:20_1_1_1_1_1"/><i><font color="Green">E31</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"><span class="p2">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>25</sub></font></span>)</span>,<font color="Maroon">c<sub>24</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 <span class="p2">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>25</sub></font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>24</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="E35:20_1_1_1_1_1"/><i><font color="Green">E32</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>25</sub></font>,<span class="p3">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>24</sub></font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <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"><span class="p2">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>25</sub></font></span>)</span>,<font color="Maroon">c<sub>24</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>
 <b>by </b><i><a class="txt" href="topalg_3.html#E33:20_1_1_1_1_1"><i><font color="Green">E30</font></i></a>, <a class="txt" href="topalg_3.html#E34:20_1_1_1_1_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/><a NAME="E36:20_1_1_1_1_1"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>25</sub></font>,<span class="p3">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>24</sub></font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="compts_1.html#K1">1TopSp</a> <span class="p2"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span></span>)</span>
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>28</sub></font> = <font color="Maroon">c<sub>10</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>25</sub></font>,<span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>24</sub></font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span> as   <a href="struct_0.html#NM7">Function</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>25</sub></font></span>)</span>,<font color="Maroon">c<sub>24</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>,<span class="p1">(<span class="default"><a href="compts_1.html#K1">1TopSp</a> <span class="p2"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span></span>)</span> <b>by </b><i><a class="txt" href="topalg_3.html#E35:20_1_1_1_1_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="funct_2.html#T4">FUNCT_2:4</a></i>;<br/><span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>25</sub></font></span>)</span>,<font color="Maroon">c<sub>24</sub></font></span><a href="borsuk_1.html#K5">:]</a></span> = 
<span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>25</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>24</sub></font> <a href="pre_topc.html#K3">|</a> <span class="p3">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>24</sub></font></span>)</span></span>)</span></span><a href="borsuk_1.html#K5">:]</a></span>
<b>by </b><i><a class="ref" href="tsep_1.html#T3">TSEP_1:3</a></i>
<br/>.= 
<span class="p1"><a href="borsuk_1.html#K5">[:</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#K5">:]</a></span> <a href="pre_topc.html#K3">|</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>25</sub></font>,<span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>24</sub></font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span>
<b>by </b><i><a class="ref" href="borsuk_3.html#T26">BORSUK_3:26</a></i>
;<br/><a NAME="E39:20_1_1_1_1_1"/><b>then </b><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>28</sub></font> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i><a class="txt" href="topalg_3.html#E1:20_1_1_1_1_1"><i><font color="Green">E11</font></i></a>, <a class="ref" href="topmetr.html#T10">TOPMETR:10</a></i>;<br/><b>set </b><font color="Maroon">c<sub>29</sub></font> = <font color="Maroon">c<sub>10</sub></font> <a href="partfun1.html#K2">|</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>23</sub></font></span>)</span>,<font color="Maroon">c<sub>26</sub></font></span><a href="borsuk_1.html#K6">:]</a></span>;<br/><a NAME="E40:20_1_1_1_1_1"/><i><font color="Green">E34</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>23</sub></font></span>)</span>,<font color="Maroon">c<sub>26</sub></font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</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>23</sub></font></span>)</span>,<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_3.html#E5:20_1_1_1_1_1"><i><font color="Green">E14</font></i></a>, <a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/><a NAME="E41:20_1_1_1_1_1"/><i><font color="Green">E35</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>23</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>24</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>26</sub></font></span>)</span></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>23</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon">c<sub>24</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>26</sub></font></span>)</span></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="E42:20_1_1_1_1_1"/><i><font color="Green">E36</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>23</sub></font></span>)</span>,<font color="Maroon">c<sub>26</sub></font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <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>23</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>24</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>26</sub></font></span>)</span></span><a href="borsuk_1.html#K5">:]</a></span>
 <b>by </b><i><a class="txt" href="topalg_3.html#E40:20_1_1_1_1_1"><i><font color="Green">E34</font></i></a>, <a class="txt" href="topalg_3.html#E41:20_1_1_1_1_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/><a NAME="E43:20_1_1_1_1_1"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="partfun1.html#K2">|</a> <span class="p2"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p3">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>23</sub></font></span>)</span>,<font color="Maroon">c<sub>26</sub></font></span><a href="borsuk_1.html#K6">:]</a></span></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="compts_1.html#K1">1TopSp</a> <span class="p2"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span></span>)</span>
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>30</sub></font> = <font color="Maroon">c<sub>10</sub></font> <a href="partfun1.html#K2">|</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>23</sub></font></span>)</span>,<font color="Maroon">c<sub>26</sub></font></span><a href="borsuk_1.html#K6">:]</a></span> 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>23</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>24</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>26</sub></font></span>)</span></span><a href="borsuk_1.html#K5">:]</a></span>,<span class="p1">(<span class="default"><a href="compts_1.html#K1">1TopSp</a> <span class="p2"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span></span>)</span> <b>by </b><i><a class="txt" href="topalg_3.html#E42:20_1_1_1_1_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="funct_2.html#T4">FUNCT_2:4</a></i>;<br/><span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>23</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>24</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>26</sub></font></span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> = 
<span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="pre_topc.html#K3">|</a> <span class="p3">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>23</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>24</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>26</sub></font></span>)</span></span><a href="borsuk_1.html#K5">:]</a></span>
<b>by </b><i><a class="ref" href="tsep_1.html#T3">TSEP_1:3</a></i>
<br/>.= 
<span class="p1"><a href="borsuk_1.html#K5">[:</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#K5">:]</a></span> <a href="pre_topc.html#K3">|</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>23</sub></font></span>)</span>,<font color="Maroon">c<sub>26</sub></font></span><a href="borsuk_1.html#K6">:]</a></span>
<b>by </b><i><a class="ref" href="borsuk_3.html#T26">BORSUK_3:26</a></i>
;<br/><a NAME="E46:20_1_1_1_1_1"/><b>then </b><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>30</sub></font> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i><a class="txt" href="topalg_3.html#E1:20_1_1_1_1_1"><i><font color="Green">E11</font></i></a>, <a class="ref" href="topmetr.html#T10">TOPMETR:10</a></i>;<br/><div><i><font color="Green">E38</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1_1_1_1_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:20_1_1_1_1_1_3"/><i><font color="Green">E39</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>19</sub></font>,<font color="Maroon">c<sub>22</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><a NAME="E2:20_1_1_1_1_1_3"/>
<font color="Maroon">c<sub>28</sub></font> is <a href="funct_2.html#V2">onto</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1_1_1_1_3_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:20_1_1_1_1_1_3_1"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>28</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="compts_1.html#K1">1TopSp</a> <span class="p2"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span></span>)</span>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a>,<a class="ref" href="funct_2.html#D3">FUNCT_2:def 3</a><br/>

<b>let </b><font color="Maroon">c<sub>31</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="E2:20_1_1_1_1_1_3_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="compts_1.html#K1">1TopSp</a> <span class="p2"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span></span>)</span>
 ;<br/>

<a NAME="E3:20_1_1_1_1_1_3_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>25</sub></font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_1_1_1_1_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:20_1_1_1_1_1_3_1_1"/>
( <font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R1">=</a> 1 )
 </span><b>by </b><i><a class="txt" href="topalg_3.html#E3:20_1_1_1_1_1"><i><font color="Green">E13</font></i></a>, <a class="txt" href="topalg_3.html#E2:20_1_1_1_1_1_3_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_1_1_1_1_3_1_1_1"><b>suppose </b></a><a NAME="E1:20_1_1_1_1_1_3_1_1_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:20_1_1_1_1_1_3_1_1_1"/><i><font color="Green">E43</font></i>: 
<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>21</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>28</sub></font>
 <b>by </b><i><a class="txt" href="topalg_3.html#E33:20_1_1_1_1_1"><i><font color="Green">E30</font></i></a>, <a class="txt" href="topalg_3.html#E3:20_1_1_1_1_1_3_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E3:20_1_1_1_1_1_3_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>28</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>19</sub></font>,<font color="Maroon">c<sub>21</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>31</sub></font>
 <b>by </b><i><a class="txt" href="topalg_3.html#E10:20_1_1_1_1_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="topalg_3.html#E14:20_1_1_1_1_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="topalg_3.html#E33:20_1_1_1_1_1"><i><font color="Green">E30</font></i></a>, <a class="txt" href="topalg_3.html#E1:20_1_1_1_1_1_3_1_1_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/><b>hence </b><a NAME="E4:20_1_1_1_1_1_3_1_1_1"/>
<font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>28</sub></font>
 <b>by </b><i><a class="txt" href="topalg_3.html#E2:20_1_1_1_1_1_3_1_1_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_1_1_1_1_3_1_1_2"><b>suppose </b></a><a NAME="E1:20_1_1_1_1_1_3_1_1_2"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:20_1_1_1_1_1_3_1_1_2"/>
<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>22</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="topalg_3.html#E5:20_1_1_1_1_1"><i><font color="Green">E14</font></i></a>, <a class="txt" href="topalg_3.html#E6:20_1_1_1_1_1"><i><font color="Green">E15</font></i></a>, <a class="txt" href="topalg_3.html#E12:20_1_1_1_1_1"><i><font color="Green">E19</font></i></a>, <a class="txt" href="topalg_3.html#E20:20_1_1_1_1_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E3:20_1_1_1_1_1_3_1_1_2"/><b>then </b><i><font color="Green">E43</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>19</sub></font>,<font color="Maroon">c<sub>22</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E4:20_1_1_1_1_1_3_1_1_2"/><i><font color="Green">E44</font></i>: 
<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>22</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>28</sub></font>
 <b>by </b><i><a class="txt" href="topalg_3.html#E33:20_1_1_1_1_1"><i><font color="Green">E30</font></i></a>, <a class="txt" href="topalg_3.html#E3:20_1_1_1_1_1_3_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><b>then </b><font color="Maroon">c<sub>28</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>19</sub></font>,<font color="Maroon">c<sub>22</sub></font></span><a href="tarski.html#K4">]</a></span> = 
<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>19</sub></font>,<font color="Maroon">c<sub>22</sub></font></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="topalg_3.html#E33:20_1_1_1_1_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon">c<sub>31</sub></font>
<b>by </b><i><a class="txt" href="topalg_3.html#E3:20_1_1_1_1_1"><i><font color="Green">E13</font></i></a>, <a class="txt" href="topalg_3.html#E1:20_1_1_1_1_1_3"><i><font color="Green">E39</font></i></a>, <a class="txt" href="topalg_3.html#E1:20_1_1_1_1_1_3_1_1_2"><i><font color="Green">E42</font></i></a>, <a class="txt" href="topalg_3.html#E3:20_1_1_1_1_1_3_1_1_2"><i><font color="Green">E43</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>
;<br/><b>hence </b><a NAME="E6:20_1_1_1_1_1_3_1_1_2"/>
<font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>28</sub></font>
 <b>by </b><i><a class="txt" href="topalg_3.html#E4:20_1_1_1_1_1_3_1_1_2"><i><font color="Green">E44</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>hence </b><a NAME="E3:20_1_1_1_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="topalg_3.html#E28:20_1_1_1_1_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="topalg_3.html#E39:20_1_1_1_1_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="topalg_3.html#T8" target="_self">Th8</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1_1_1_1_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:20_1_1_1_1_1_4"/><i><font color="Green">E39</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>19</sub></font>,<font color="Maroon">c<sub>22</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 ;<br/><a NAME="E2:20_1_1_1_1_1_4"/>
<font color="Maroon">c<sub>30</sub></font> is <a href="funct_2.html#V2">onto</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1_1_1_1_4_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:20_1_1_1_1_1_4_1"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>30</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="compts_1.html#K1">1TopSp</a> <span class="p2"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span></span>)</span>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a>,<a class="ref" href="funct_2.html#D3">FUNCT_2:def 3</a><br/>

<b>let </b><font color="Maroon">c<sub>31</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="E2:20_1_1_1_1_1_4_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="compts_1.html#K1">1TopSp</a> <span class="p2"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span></span>)</span>
 ;<br/>

<a NAME="E3:20_1_1_1_1_1_4_1"/><i><font color="Green">E41</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="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_1_1_1_1_4_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:20_1_1_1_1_1_4_1_1"/>
( <font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R1">=</a> 0 )
 </span><b>by </b><i><a class="txt" href="topalg_3.html#E3:20_1_1_1_1_1"><i><font color="Green">E13</font></i></a>, <a class="txt" href="topalg_3.html#E2:20_1_1_1_1_1_4_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_1_1_1_1_4_1_1_1"><b>suppose </b></a><a NAME="E1:20_1_1_1_1_1_4_1_1_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:20_1_1_1_1_1_4_1_1_1"/><i><font color="Green">E43</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>22</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>30</sub></font>
 <b>by </b><i><a class="txt" href="topalg_3.html#E40:20_1_1_1_1_1"><i><font color="Green">E34</font></i></a>, <a class="txt" href="topalg_3.html#E3:20_1_1_1_1_1_4_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E3:20_1_1_1_1_1_4_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>30</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>20</sub></font>,<font color="Maroon">c<sub>22</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>31</sub></font>
 <b>by </b><i><a class="txt" href="topalg_3.html#E17:20_1_1_1_1_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="topalg_3.html#E21:20_1_1_1_1_1"><i><font color="Green">E26</font></i></a>, <a class="txt" href="topalg_3.html#E40:20_1_1_1_1_1"><i><font color="Green">E34</font></i></a>, <a class="txt" href="topalg_3.html#E1:20_1_1_1_1_1_4_1_1_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/><b>hence </b><a NAME="E4:20_1_1_1_1_1_4_1_1_1"/>
<font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>30</sub></font>
 <b>by </b><i><a class="txt" href="topalg_3.html#E2:20_1_1_1_1_1_4_1_1_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_1_1_1_1_4_1_1_2"><b>suppose </b></a><a NAME="E1:20_1_1_1_1_1_4_1_1_2"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:20_1_1_1_1_1_4_1_1_2"/>
<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>22</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="topalg_3.html#E5:20_1_1_1_1_1"><i><font color="Green">E14</font></i></a>, <a class="txt" href="topalg_3.html#E6:20_1_1_1_1_1"><i><font color="Green">E15</font></i></a>, <a class="txt" href="topalg_3.html#E12:20_1_1_1_1_1"><i><font color="Green">E19</font></i></a>, <a class="txt" href="topalg_3.html#E20:20_1_1_1_1_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E3:20_1_1_1_1_1_4_1_1_2"/><b>then </b><i><font color="Green">E43</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>19</sub></font>,<font color="Maroon">c<sub>22</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E4:20_1_1_1_1_1_4_1_1_2"/><i><font color="Green">E44</font></i>: 
<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>22</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>30</sub></font>
 <b>by </b><i><a class="txt" href="topalg_3.html#E40:20_1_1_1_1_1"><i><font color="Green">E34</font></i></a>, <a class="txt" href="topalg_3.html#E3:20_1_1_1_1_1_4_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><b>then </b><font color="Maroon">c<sub>30</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>19</sub></font>,<font color="Maroon">c<sub>22</sub></font></span><a href="tarski.html#K4">]</a></span> = 
<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>19</sub></font>,<font color="Maroon">c<sub>22</sub></font></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="topalg_3.html#E40:20_1_1_1_1_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon">c<sub>31</sub></font>
<b>by </b><i><a class="txt" href="topalg_3.html#E3:20_1_1_1_1_1"><i><font color="Green">E13</font></i></a>, <a class="txt" href="topalg_3.html#E1:20_1_1_1_1_1_4"><i><font color="Green">E39</font></i></a>, <a class="txt" href="topalg_3.html#E1:20_1_1_1_1_1_4_1_1_2"><i><font color="Green">E42</font></i></a>, <a class="txt" href="topalg_3.html#E3:20_1_1_1_1_1_4_1_1_2"><i><font color="Green">E43</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>
;<br/><b>hence </b><a NAME="E6:20_1_1_1_1_1_4_1_1_2"/>
<font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>30</sub></font>
 <b>by </b><i><a class="txt" href="topalg_3.html#E4:20_1_1_1_1_1_4_1_1_2"><i><font color="Green">E44</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>hence </b><a NAME="E3:20_1_1_1_1_1_4"/>
contradiction
 <b>by </b><i><a class="txt" href="topalg_3.html#E32:20_1_1_1_1_1"><i><font color="Green">E29</font></i></a>, <a class="txt" href="topalg_3.html#E46:20_1_1_1_1_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="topalg_3.html#T8" target="_self">Th8</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E49:20_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="topalg_3.html#E47:20_1_1_1_1_1"><i><font color="Green">E38</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:20_1_1_1_1"/>
<span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="borsuk_1.html#K5">:]</a></span> is <a href="connsp_1.html#V1">connected</a>
 <b>by </b><i><a class="txt" href="topalg_3.html#E1:20_1_1"><i><font color="Green">E9</font></i></a>, <a class="txt" href="topalg_3.html#E1:20_1_1_1_1"><i><font color="Green">E10</font></i></a>, <a class="ref" href="topalg_3.html#T8" target="_self">Th8</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_1_1_2"><b>suppose </b></a><a NAME="E1:20_1_1_1_2"/><i><font color="Green">E10</font></i>: 
<span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>5</sub></font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>5</sub></font> #),<a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>6</sub></font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>6</sub></font> #)</span><a href="borsuk_1.html#K5">:]</a></span> is <a href="struct_0.html#V3">empty</a>
 ;<br/><div class="add"><a NAME="E2:20_1_1_1_2"/>
for <font color="Olive">b<sub>1</sub></font> being <a href="struct_0.html#V3">empty</a>  <a href="pre_topc.html#L1">TopStruct</a>  holds <font color="Olive">b<sub>1</sub></font> is <a href="connsp_1.html#V1">connected</a>
 ;<br/><b>hence </b><a NAME="E3:20_1_1_1_2"/>
<span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="borsuk_1.html#K5">:]</a></span> is <a href="connsp_1.html#V1">connected</a>
 <b>by </b><i><a class="txt" href="topalg_3.html#E1:20_1_1"><i><font color="Green">E9</font></i></a>, <a class="txt" href="topalg_3.html#E1:20_1_1_1_2"><i><font color="Green">E10</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
