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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>reconsider </b><font color="Maroon">M</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="E2:32"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">M</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>
 
;<br/>
<a NAME="E3:32"/>
<font color="Maroon">M</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="E4:32"/><b>then </b><i><font color="Green">E18</font></i>: 
<font color="Maroon">M</font> is <a href="tsp_1.html#V1">T_0</a>
 
<b>by </b><i/>;<br/>
<b>assume </b><a NAME="E5:32"/><i><font color="Green">E19</font></i>: 
for <font color="Olive">a</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font> holds  <font color="Maroon">r</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a</font> <a href="hidden.html#R2">in</a>  <a href="tex_4.html#K4">MaxADSet</a> <font color="Olive">a</font>
 ;<br/>

<a NAME="E6:32"/>
for <font color="Olive">a</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font> holds  <font color="Maroon">M</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="tex_4.html#K4">MaxADSet</a> <font color="Olive">a</font></span>)</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="Olive">a</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1"><b>proof </b></a><div class="add">

<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>reconsider </b><font color="Maroon">s</font> = <font color="Maroon">r</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</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="E2:32_1"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">M</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">s</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">s</font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E3:32_1"/>
<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>by </b><i/>;<br/>
<b>hence </b><a NAME="E4:32_1"/>
<font color="Maroon">M</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"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i>, <a class="ref" href="tex_4.html#T23">TEX_4:23</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:32"/>
<font color="Maroon">r</font> <a href="borsuk_1.html#NR1" target="_self">is_a_retraction</a> 
 <b>by </b><i/>;<br/>


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