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<b>let </b><font color="Maroon">X</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">X0</font> be  non <a href="struct_0.html#V3">empty</a> <a href="tsp_2.html#NM1">maximal_Kolmogorov_subspace</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">r</font> be  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X</font>,<font color="Maroon">X0</font>;<br/>





<b>assume </b><a NAME="E1:37"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">r</font> <a href="borsuk_1.html#NR1" target="_self">is_a_retraction</a> 
 ;<br/>

<b>let </b><font color="Maroon">a</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">b</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X0</font>;<br/>



<b>assume </b><a NAME="E2:37"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">A</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X0</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> <b>by </b><i><a class="ref" href="tsep_1.html#T1">TSEP_1:1</a></i>;<br/>
<a NAME="E4:37"/>
<font color="Maroon">A</font> is <a href="tsp_2.html#V1" target="_self">maximal_T_0</a>
 
<b>by </b><i><a class="ref" href="tsp_2.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E5:37"/><b>then </b><i><font color="Green">E19</font></i>: 
<font color="Maroon">A</font> is <a href="tsp_1.html#V1">T_0</a>
 
<b>by </b><i/>;<br/>
<a NAME="E6:37"/>
<font color="Maroon">r</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 
<b>by </b><i><a class="txt" href="tsp_2.html#E16:37"><i><font color="Green">E26</font></i></a>, <a class="txt" href="tsp_2.html#E19:37"><i><font color="Green">E27</font></i></a>, <a class="ref" href="borsuk_1.html#D19">BORSUK_1:def 19</a></i>;<br/>
<a NAME="E7:37"/><b>then </b><i><font color="Green">E21</font></i>: 
<font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="tex_4.html#K4">MaxADSet</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">b</font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="txt" href="tsp_2.html#E16:37"><i><font color="Green">E26</font></i></a>, </i>;<br/>
<b>assume </b><a NAME="E8:37"/>
not  <a href="tex_4.html#K4">MaxADSet</a> <font color="Maroon">a</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">r</font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">b</font></span><a href="domain_1.html#K6">}</a></span>
 ;<br/>

<b>then consider </b><font color="Maroon">s</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E10:37"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="tex_4.html#K4">MaxADSet</a> <font color="Maroon">a</font>
 <b>and </b><br/><a NAME="E11:37"/><i><font color="Green">E23</font></i>: 
not <font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">r</font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">b</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">s</font> = <font color="Maroon">s</font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font> <b>by </b><i><a class="txt" href="tsp_2.html#E24:37"><i><font color="Green">E30</font></i></a></i>;<br/>
<a NAME="E13:37"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">r</font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">b</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="txt" href="tsp_2.html#E16:37"><i><font color="Green">E26</font></i></a>, <a class="txt" href="tsp_2.html#E19:37"><i><font color="Green">E27</font></i></a>, <a class="txt" href="tsp_2.html#E11:37"><i><font color="Green">E23</font></i></a></i>;<br/>
<a NAME="E14:37"/>
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>
 
;<br/>
<b>then reconsider </b><font color="Maroon">d</font> = <font color="Maroon">r</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font> <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E16:37"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">r</font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">d</font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="txt" href="tsp_2.html#E16:37"><i><font color="Green">E26</font></i></a>, <a class="txt" href="tsp_2.html#E11:37"><i><font color="Green">E23</font></i></a></i>;<br/>
<a NAME="E17:37"/>
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font></span>)</span></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T48">PRE_TOPC:48</a></i>;<br/>
<a NAME="E18:37"/><b>then </b>
<font color="Maroon">r</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E19:37"/><b>then </b><i><font color="Green">E27</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">d</font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="txt" href="tsp_2.html#E32:37"><i><font color="Green">E33</font></i></a>, <a class="ref" href="funct_2.html#T46">FUNCT_2:46</a></i>;<br/>
<a NAME="E20:37"/><i><font color="Green">E28</font></i>: 
 <a href="tex_4.html#K4">MaxADSet</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="tex_4.html#K4">MaxADSet</a> <font color="Maroon">s</font>
 
<b>by </b><i><a class="txt" href="tsp_2.html#E24:37"><i><font color="Green">E30</font></i></a>, <a class="ref" href="tex_4.html#T23">TEX_4:23</a></i>;<br/>
<a NAME="E21:37"/><b>then </b><i><font color="Green">E29</font></i>: 
 <a href="tex_4.html#K4">MaxADSet</a> <font color="Maroon">a</font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">d</font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i>, <a class="ref" href="tex_4.html#T25">TEX_4:25</a></i>;<br/>
<a NAME="E22:37"/>
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="tex_4.html#K4">MaxADSet</a> <font color="Maroon">a</font>
 
<b>by </b><i><a class="ref" href="tex_4.html#T20">TEX_4:20</a></i>;<br/>
<a NAME="E23:37"/><b>then </b>
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">d</font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E24:37"/><b>then </b><i><font color="Green">E30</font></i>: 
 <a href="pre_topc.html#K4">Cl</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">d</font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="tops_1.html#T31">TOPS_1:31</a></i>;<br/>
<a NAME="E25:37"/><i><font color="Green">E31</font></i>: 
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">s</font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="tex_4.html#K4">MaxADSet</a> <font color="Maroon">s</font>
 
<b>by </b><i><a class="ref" href="tex_4.html#T20">TEX_4:20</a></i>;<br/>
<a NAME="E26:37"/>
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">r</font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">b</font></span><a href="domain_1.html#K6">}</a></span></span>)</span>
 
<b>by </b><i><a class="txt" href="tsp_2.html#E27:37"><i><font color="Green">E32</font></i></a>, <a class="ref" href="pre_topc.html#T48">PRE_TOPC:48</a></i>;<br/>
<a NAME="E27:37"/><b>then </b><i><font color="Green">E32</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">r</font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">b</font></span><a href="domain_1.html#K6">}</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E28:37"/>
<font color="Maroon">r</font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">b</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> is <a href="pre_topc.html#V4">closed</a>
 
<b>by </b><i><a class="txt" href="tsp_2.html#E27:37"><i><font color="Green">E32</font></i></a></i>;<br/>
<a NAME="E29:37"/><b>then </b>
 <a href="tex_4.html#K4">MaxADSet</a> <font color="Maroon">s</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">r</font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">b</font></span><a href="domain_1.html#K6">}</a></span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="tsp_2.html#D2" target="_self">Def2</a>, <a class="ref" href="tex_4.html#T25">TEX_4:25</a></i>;<br/>
<a NAME="E30:37"/><b>then </b>
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">s</font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">r</font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">b</font></span><a href="domain_1.html#K6">}</a></span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E31:37"/><b>then </b>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">r</font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">b</font></span><a href="domain_1.html#K6">}</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E32:37"/><b>then </b><i><font color="Green">E33</font></i>: 
<font color="Maroon">r</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">b</font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#T46">FUNCT_2:46</a></i>;<br/>
<a NAME="E33:37"/>
 <a href="pre_topc.html#K4">Cl</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">b</font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="txt" href="tsp_2.html#E19:37"><i><font color="Green">E27</font></i></a>, <a class="ref" href="tops_3.html#T53">TOPS_3:53</a></i>;<br/>
<a NAME="E34:37"/><b>then </b>
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">d</font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E35:37"/><b>then </b>
 <a href="pre_topc.html#K4">Cl</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">d</font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="tops_1.html#T31">TOPS_1:31</a></i>;<br/>
<a NAME="E36:37"/><b>then </b>
 <a href="pre_topc.html#K4">Cl</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">d</font></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<a NAME="E37:37"/><b>then </b>
 <a href="tex_4.html#K4">MaxADSet</a> <font color="Maroon">d</font> <a href="hidden.html#R1">=</a>  <a href="tex_4.html#K4">MaxADSet</a> <font color="Maroon">a</font>
 
<b>by </b><i><a class="ref" href="tex_4.html#T51">TEX_4:51</a></i>;<br/>
<a NAME="E38:37"/><b>then </b>
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">b</font></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="txt" href="tsp_2.html#E20:37"><i><font color="Green">E28</font></i></a>, <a class="txt" href="tsp_2.html#E21:37"><i><font color="Green">E29</font></i></a>, </i>;<br/>
<a NAME="E39:37"/><b>then </b>
<font color="Maroon">r</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">b</font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<b>hence </b><a NAME="E40:37"/>
contradiction
 <b>by </b><i><a class="txt" href="tsp_2.html#E25:37"><i><font color="Green">E31</font></i></a>, <a class="ref" href="funct_2.html#T46">FUNCT_2:46</a></i>;<br/>


</div>
