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<b>let </b><font color="Maroon">X</font> be  non <a href="struct_0.html#V3">empty</a> <a href="tdlat_3.html#V3">almost_discrete</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="tdlat_3.html#V1">discrete</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X</font>;<br/>



<b>consider </b><font color="Maroon">Z0</font> being  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#V1">strict</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E2:128"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">X0</font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">Z0</font>
 <b>and </b><br/><a NAME="E3:128"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">Z0</font> is <a href="tex_2.html#V5" target="_self">maximal_discrete</a>
 <b>by </b><i><a class="ref" href="tex_2.html#T17" target="_self">Th17</a></i>;<br/>
<b>reconsider </b><font color="Maroon">Z0</font> = <font color="Maroon">Z0</font> as  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#V1">strict</a> <a href="tex_2.html#V5" target="_self">maximal_discrete</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X</font> <b>by </b><i><a class="ref" href="tex_2.html#T24" target="_self">Th24</a></i>;<br/>
<b>reconsider </b><font color="Maroon">Z1</font> = <font color="Maroon">Z0</font> as  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a> ;<br/>
<b>reconsider </b><font color="Maroon">Z1</font> = <font color="Maroon">Z1</font> as  non <a href="struct_0.html#V3">empty</a> <a href="tdlat_3.html#V1">discrete</a> <a href="pre_topc.html#NM1">TopSpace</a> ;<br/>
<b>reconsider </b><font color="Maroon">X1</font> = <font color="Maroon">X0</font> as  non <a href="struct_0.html#V3">empty</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">Z1</font> <b>by </b><i><a class="ref" href="tex_2.html#T23" target="_self">Th23</a></i>;<br/>
<b>consider </b><font color="Maroon">g</font> being  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">Z1</font>,<font color="Maroon">X1</font><b> such that </b><br/><a NAME="E9:128"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">g</font> <a href="borsuk_1.html#NR1" target="_self">is_a_retraction</a> 
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">g</font> = <font color="Maroon">g</font> as  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">Z0</font>,<font color="Maroon">X0</font> ;<br/>
<b>consider </b><font color="Maroon">h</font> being  <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">Z0</font><b> such that </b><br/><a NAME="E12:128"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">h</font> <a href="borsuk_1.html#NR1" target="_self">is_a_retraction</a> 
 <b>by </b><i><a class="ref" href="tex_2.html#D4" target="_self">Def4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">r</font> = <font color="Maroon">g</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">h</font> as  <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> <b>by </b><i><a class="ref" href="tops_2.html#T58">TOPS_2:58</a></i>;<br/>
<b>take </b>
<font color="Maroon">r</font>
;<br/>

<a NAME="E14:128"/>
for <font color="Olive">x</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X0</font> holds <br/><font color="Maroon">r</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="128_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:128_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X0</font>
 ;<br/>

<a NAME="E2:128_1"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X1</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Z1</font>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#T35">BORSUK_1:35</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">y</font> = <font color="Maroon">x</font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">Z1</font> <b>by </b><i/>;<br/>
<a NAME="E4:128_1"/>
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 
<b>by </b><i>, , <a class="ref" href="borsuk_1.html#D19">BORSUK_1:def 19</a></i>;<br/>
<a NAME="E5:128_1"/><b>then </b>
<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 
<b>by </b><i>, <a class="ref" href="borsuk_1.html#D19">BORSUK_1:def 19</a></i>;<br/>
<b>hence </b><a NAME="E6:128_1"/>
<font color="Maroon">r</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E15:128"/>
<font color="Maroon">r</font> <a href="borsuk_1.html#NR1" target="_self">is_a_retraction</a> 
 <b>by </b><i><a class="ref" href="borsuk_1.html#D19">BORSUK_1:def 19</a></i>;<br/>


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