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

<b>let </b><font color="Maroon">XX</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">X</font> be  non <a href="struct_0.html#V3">empty</a> <a href="borsuk_1.html#V1" target="_self">closed</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">XX</font>;<br/><b>let </b><font color="Maroon">D</font> be   <a href="borsuk_1.html#NM2">DECOMPOSITION</a> of <font color="Maroon">X</font>;<br/>





<b>thus </b><a NAME="E1:109"/>
( <font color="Maroon">X</font> <a href="borsuk_1.html#R1" target="_self">is_a_retract_of</a> <font color="Maroon">XX</font> implies  <a href="borsuk_1.html#K19" target="_self">space</a> <font color="Maroon">D</font> <a href="borsuk_1.html#R1" target="_self">is_a_retract_of</a>  <a href="borsuk_1.html#K14" target="_self">space</a> <span class="p1">(<span class="default"><a href="borsuk_1.html#K18" target="_self">TrivExt</a> <font color="Maroon">D</font></span>)</span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="109_1"><b>proof </b></a><div class="add">

<b>given </b><font color="Maroon">R</font> being  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">XX</font>,<font color="Maroon">X</font><b> such that </b><a NAME="E1:109_1"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">R</font> <a href="borsuk_1.html#NR1" target="_self">is_a_retraction</a> 
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="borsuk_1.html#D20" target="_self">BORSUK_1:def 20</a><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="109_1_1"><b>now </b></a><div class="add"><b>given </b><font color="Maroon">xx</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">XX</font>, <font color="Maroon">xx'</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">XX</font><b> such that </b><a NAME="E1:109_1_1"/><i><font color="Green">E29</font></i>: 
<span class="p1">(<span class="default"><a href="borsuk_1.html#K15" target="_self">Proj</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K18" target="_self">TrivExt</a> <font color="Maroon">D</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">xx</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="borsuk_1.html#K15" target="_self">Proj</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K18" target="_self">TrivExt</a> <font color="Maroon">D</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">xx'</font>
 <b>and </b><br/><a NAME="E2:109_1_1"/><i><font color="Green">E31</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="borsuk_1.html#K15" target="_self">Proj</a> <font color="Maroon">D</font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">R</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">xx</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="borsuk_1.html#K15" target="_self">Proj</a> <font color="Maroon">D</font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">R</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">xx'</font>
 ;<br/><a NAME="E3:109_1_1"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">xx</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> &amp; <font color="Maroon">xx'</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> )
 <b>by </b><i><a class="ref" href="borsuk_1.html#T29" target="_self">Th29</a>, <a class="ref" href="borsuk_1.html#T30" target="_self">Th30</a>, </i>;<br/><a NAME="E4:109_1_1"/><b>then </b>
( <font color="Maroon">R</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">xx</font> <a href="hidden.html#R1">=</a> <font color="Maroon">xx</font> &amp; <font color="Maroon">R</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">xx'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">xx'</font> )
 <b>by </b><i>, <a class="ref" href="borsuk_1.html#D1" target="_self">Def1</a></i>;<br/><a NAME="E5:109_1_1"/><b>then </b><i><font color="Green">E33</font></i>: 
( <span class="p1">(<span class="default"><a href="borsuk_1.html#K15" target="_self">Proj</a> <font color="Maroon">D</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">xx</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="borsuk_1.html#K15" target="_self">Proj</a> <font color="Maroon">D</font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">R</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">xx</font> &amp; <span class="p1">(<span class="default"><a href="borsuk_1.html#K15" target="_self">Proj</a> <font color="Maroon">D</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">xx'</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="borsuk_1.html#K15" target="_self">Proj</a> <font color="Maroon">D</font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">R</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">xx'</font> )
 <b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>;<br/><a NAME="E6:109_1_1"/>
( <span class="p1">(<span class="default"><a href="borsuk_1.html#K15" target="_self">Proj</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K18" target="_self">TrivExt</a> <font color="Maroon">D</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">xx</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="borsuk_1.html#K15" target="_self">Proj</a> <font color="Maroon">D</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">xx</font> &amp; <span class="p1">(<span class="default"><a href="borsuk_1.html#K15" target="_self">Proj</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K18" target="_self">TrivExt</a> <font color="Maroon">D</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">xx'</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="borsuk_1.html#K15" target="_self">Proj</a> <font color="Maroon">D</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">xx'</font> )
 <b>by </b><i><a class="ref" href="borsuk_1.html#T31" target="_self">Th31</a>, <a class="ref" href="borsuk_1.html#T19" target="_self">Th19</a></i>;<br/><b>hence </b><a NAME="E7:109_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="borsuk_1.html#T29" target="_self">Th29</a>, <a class="ref" href="borsuk_1.html#T30" target="_self">Th30</a>, <a class="ref" href="borsuk_1.html#T32" target="_self">Th32</a></i>;<br/></div><b>end;</b></div>
<b>then consider </b><font color="Maroon">F</font> being   <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="borsuk_1.html#K14" target="_self">space</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K18" target="_self">TrivExt</a> <font color="Maroon">D</font></span>)</span></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="borsuk_1.html#K19" target="_self">space</a> <font color="Maroon">D</font></span>)</span><b> such that </b><br/><a NAME="E4:109_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">F</font> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><a href="borsuk_1.html#K15" target="_self">Proj</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K18" target="_self">TrivExt</a> <font color="Maroon">D</font></span>)</span></span>)</span> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><a href="borsuk_1.html#K15" target="_self">Proj</a> <font color="Maroon">D</font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">R</font>
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">F</font> = <font color="Maroon">F</font> as   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="borsuk_1.html#K14" target="_self">space</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K18" target="_self">TrivExt</a> <font color="Maroon">D</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="borsuk_1.html#K19" target="_self">space</a> <font color="Maroon">D</font></span>)</span> ;<br/>
<a NAME="E6:109_1"/>
<font color="Maroon">F</font> is <a href="pre_topc.html#V5">continuous</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="109_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">W</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="borsuk_1.html#K14" target="_self">space</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K18" target="_self">TrivExt</a> <font color="Maroon">D</font></span>)</span></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="borsuk_1.html#D3" target="_self">BORSUK_1:def 3</a><br/><b>let </b><font color="Maroon">G</font> be    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">W</font>;<br/>



<b>reconsider </b><font color="Maroon">FF</font> = <span class="p1">(<span class="default"><a href="borsuk_1.html#K15" target="_self">Proj</a> <font color="Maroon">D</font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">R</font> as  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">XX</font>,<span class="p1">(<span class="default"><a href="borsuk_1.html#K19" target="_self">space</a> <font color="Maroon">D</font></span>)</span> <b>by </b><i><a class="ref" href="tops_2.html#T58">TOPS_2:58</a></i>;<br/>
<a NAME="E2:109_1_2"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">FF</font> <a href="borsuk_1.html#K4" target="_self">"</a> <font color="Maroon">G</font> is    <a href="connsp_2.html#M2">a_neighborhood</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="borsuk_1.html#K15" target="_self">Proj</a> <font color="Maroon">D</font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">R</font></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">W</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 
;<br/>
<a NAME="E3:109_1_2"/>
<span class="p1">(<span class="default"><a href="borsuk_1.html#K15" target="_self">Proj</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K18" target="_self">TrivExt</a> <font color="Maroon">D</font></span>)</span></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">W</font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="borsuk_1.html#K15" target="_self">Proj</a> <font color="Maroon">D</font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">R</font></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">W</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#T29" target="_self">Th29</a>, </i>;<br/>
<b>then reconsider </b><font color="Maroon">GG</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="borsuk_1.html#K15" target="_self">Proj</a> <font color="Maroon">D</font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">R</font></span>)</span> <a href="relset_1.html#K11">"</a> <font color="Maroon">G</font> as    <a href="connsp_2.html#M2">a_neighborhood</a> of <span class="p1">(<span class="default"><a href="borsuk_1.html#K15" target="_self">Proj</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K18" target="_self">TrivExt</a> <font color="Maroon">D</font></span>)</span></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">W</font></span><a href="domain_1.html#K6">}</a></span> <b>by </b><i>, <a class="ref" href="borsuk_1.html#T30" target="_self">Th30</a></i>;<br/>
<b>reconsider </b><font color="Maroon">V'</font> = <span class="p1">(<span class="default"><a href="borsuk_1.html#K15" target="_self">Proj</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K18" target="_self">TrivExt</a> <font color="Maroon">D</font></span>)</span></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">GG</font> as    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">W</font> <b>by </b><i/>;<br/>
<b>take </b><font color="Maroon">V</font> = <font color="Maroon">V'</font>;<br/>

<a NAME="E6:109_1_2"/>
<font color="Maroon">F</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">V</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="borsuk_1.html#K15" target="_self">Proj</a> <font color="Maroon">D</font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">R</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">GG</font>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#T29" target="_self">Th29</a>, <a class="ref" href="relat_1.html#T159">RELAT_1:159</a></i>;<br/>
<b>hence </b><a NAME="E7:109_1_2"/>
<font color="Maroon">F</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">V</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="funct_1.html#T145">FUNCT_1:145</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">F'</font> = <font color="Maroon">F</font> as  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="borsuk_1.html#K14" target="_self">space</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K18" target="_self">TrivExt</a> <font color="Maroon">D</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="borsuk_1.html#K19" target="_self">space</a> <font color="Maroon">D</font></span>)</span> ;<br/>
<b>take </b><font color="Maroon">F''</font> = <font color="Maroon">F'</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="borsuk_1.html#D20" target="_self">BORSUK_1:def 20</a><br/>

<b>let </b><font color="Maroon">W</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="borsuk_1.html#K14" target="_self">space</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K18" target="_self">TrivExt</a> <font color="Maroon">D</font></span>)</span></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="borsuk_1.html#D19" target="_self">BORSUK_1:def 19</a><br/>

<b>consider </b><font color="Maroon">W'</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">XX</font><b> such that </b><br/><a NAME="E9:109_1"/><i><font color="Green">E37</font></i>: 
<span class="p1">(<span class="default"><a href="borsuk_1.html#K15" target="_self">Proj</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K18" target="_self">TrivExt</a> <font color="Maroon">D</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">W'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font>
 <b>by </b><i/>;<br/>
<b>assume </b><a NAME="E10:109_1"/>
<font color="Maroon">W</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="borsuk_1.html#K19" target="_self">space</a> <font color="Maroon">D</font></span>)</span>
 ;<br/>

<a NAME="E11:109_1"/><b>then </b>
<font color="Maroon">W'</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#T30" target="_self">Th30</a>, </i>;<br/>
<a NAME="E12:109_1"/><b>then </b>
( <font color="Maroon">W'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">R</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">W'</font> &amp; <span class="p1">(<span class="default"><a href="borsuk_1.html#K15" target="_self">Proj</a> <font color="Maroon">D</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">W'</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="borsuk_1.html#K15" target="_self">Proj</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K18" target="_self">TrivExt</a> <font color="Maroon">D</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">W'</font> )
 
<b>by </b><i>, <a class="ref" href="borsuk_1.html#D1" target="_self">Def1</a>, <a class="ref" href="borsuk_1.html#T19" target="_self">Th19</a></i>;<br/>
<a NAME="E13:109_1"/><b>then </b>
( <font color="Maroon">F'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">W</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K15" target="_self">Proj</a> <span class="p3">(<span class="default"><a href="borsuk_1.html#K18" target="_self">TrivExt</a> <font color="Maroon">D</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">W'</font> &amp; <font color="Maroon">W</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="borsuk_1.html#K15" target="_self">Proj</a> <font color="Maroon">D</font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">R</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">W'</font> )
 
<b>by </b><i><a class="ref" href="borsuk_1.html#T30" target="_self">Th30</a>, <a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>;<br/>
<b>hence </b><a NAME="E14:109_1"/>
<font color="Maroon">F''</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">W</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T29" target="_self">Th29</a></i>;<br/>


</div><b>end;</b></div>


</div>
