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

<span class="kw">let </span><font color="Maroon" title="c1">XX</font> be  non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">X</font> being non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="borsuk_1.html#V1" title="BORSUK_1:attr.1">closed</a>   <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c1">XX</font><br/> for <font color="Olive" title="b2">D</font> being  <a href="borsuk_1.html#NM2" title="BORSUK_1:NM.2">DECOMPOSITION</a> of <font color="Olive" title="b1">X</font>  st <font color="Olive" title="b1">X</font> <a href="borsuk_1.html#R1" title="BORSUK_1:pred.1">is_a_retract_of</a> <font color="Maroon" title="c1">XX</font> holds <br/> <a href="borsuk_1.html#K17" title="BORSUK_1:func.17">space</a> <font color="Olive" title="b2">D</font> <a href="borsuk_1.html#R1" title="BORSUK_1:pred.1">is_a_retract_of</a>  <a href="borsuk_1.html#K12" title="BORSUK_1:func.12">space</a> <span class="p1">(<span class="default"><a href="borsuk_1.html#K16" title="BORSUK_1:func.16">TrivExt</a> <font color="Olive" title="b2">D</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">X</font> be  non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="borsuk_1.html#V1" title="BORSUK_1:attr.1">closed</a>   <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c1">XX</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">D</font> being  <a href="borsuk_1.html#NM2" title="BORSUK_1:NM.2">DECOMPOSITION</a> of <font color="Maroon" title="c2">X</font>  st <font color="Maroon" title="c2">X</font> <a href="borsuk_1.html#R1" title="BORSUK_1:pred.1">is_a_retract_of</a> <font color="Maroon" title="c1">XX</font> holds <br/> <a href="borsuk_1.html#K17" title="BORSUK_1:func.17">space</a> <font color="Olive" title="b1">D</font> <a href="borsuk_1.html#R1" title="BORSUK_1:pred.1">is_a_retract_of</a>  <a href="borsuk_1.html#K12" title="BORSUK_1:func.12">space</a> <span class="p1">(<span class="default"><a href="borsuk_1.html#K16" title="BORSUK_1:func.16">TrivExt</a> <font color="Olive" title="b1">D</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c3">D</font> be   <a href="borsuk_1.html#NM2" title="BORSUK_1:NM.2">DECOMPOSITION</a> of <font color="Maroon" title="c2">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">X</font> <a href="borsuk_1.html#R1" title="BORSUK_1:pred.1">is_a_retract_of</a> <font color="Maroon" title="c1">XX</font> implies  <a href="borsuk_1.html#K17" title="BORSUK_1:func.17">space</a> <font color="Maroon" title="c3">D</font> <a href="borsuk_1.html#R1" title="BORSUK_1:pred.1">is_a_retract_of</a>  <a href="borsuk_1.html#K12" title="BORSUK_1:func.12">space</a> <span class="p1">(<span class="default"><a href="borsuk_1.html#K16" title="BORSUK_1:func.16">TrivExt</a> <font color="Maroon" title="c3">D</font></span>)</span> )</span><br/>





<span class="kw">thus </span><a NAME="E1:117"/>
( <font color="Maroon" title="c2">X</font> <a href="borsuk_1.html#R1" title="BORSUK_1:pred.1">is_a_retract_of</a> <font color="Maroon" title="c1">XX</font> implies  <a href="borsuk_1.html#K17" title="BORSUK_1:func.17">space</a> <font color="Maroon" title="c3">D</font> <a href="borsuk_1.html#R1" title="BORSUK_1:pred.1">is_a_retract_of</a>  <a href="borsuk_1.html#K12" title="BORSUK_1:func.12">space</a> <span class="p1">(<span class="default"><a href="borsuk_1.html#K16" title="BORSUK_1:func.16">TrivExt</a> <font color="Maroon" title="c3">D</font></span>)</span> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="117_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">given </span><font color="Maroon" title="c4">R</font> being   <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">XX</font>,<font color="Maroon" title="c2">X</font><span class="kw"> such that </span><a NAME="E2:117_1"/><span class="lab"><font color="Green" title="E73">A1</font></span>: 
<font color="Maroon" title="c4">R</font> is  <a href="borsuk_1.html#V3" title="BORSUK_1:attr.3">being_a_retraction</a> 
 ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="borsuk_1.html#D20" target="_self" title="BORSUK_1:def.20">BORSUK_1:def 20</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="borsuk_1.html#K17" title="BORSUK_1:func.17">space</a> <font color="Maroon" title="c3">D</font> <a href="borsuk_1.html#R1" title="BORSUK_1:pred.1">is_a_retract_of</a>  <a href="borsuk_1.html#K12" title="BORSUK_1:func.12">space</a> <span class="p1">(<span class="default"><a href="borsuk_1.html#K16" title="BORSUK_1:func.16">TrivExt</a> <font color="Maroon" title="c3">D</font></span>)</span></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="117_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">given </span><font color="Maroon" title="c5">xx</font>, <font color="Maroon" title="c6">xx9</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">XX</font><span class="kw"> such that </span><a NAME="E2:117_1_1"/><span class="lab"><font color="Green" title="E74">A2</font></span>: 
( <span class="p1">(<span class="default"><a href="borsuk_1.html#K13" title="BORSUK_1:func.13">Proj</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K16" title="BORSUK_1:func.16">TrivExt</a> <font color="Maroon" title="c3">D</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">xx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="borsuk_1.html#K13" title="BORSUK_1:func.13">Proj</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K16" title="BORSUK_1:func.16">TrivExt</a> <font color="Maroon" title="c3">D</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">xx9</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="borsuk_1.html#K13" title="BORSUK_1:func.13">Proj</a> <font color="Maroon" title="c3">D</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">R</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">xx</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="borsuk_1.html#K13" title="BORSUK_1:func.13">Proj</a> <font color="Maroon" title="c3">D</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">R</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">xx9</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E3:117_1_1"/>
<font color="Maroon" title="c5">xx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:117_1_1"><span class="lab"><font color="Green" title="E74">A2</font></span></a>, <a class="ref" href="borsuk_1.html#T78" target="_self" title="BORSUK_1:th.78">Th78</a></span>;<br/><a NAME="E4:117_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">R</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">xx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">xx</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:117_1"><span class="lab"><font color="Green" title="E73">A1</font></span></a>, <a class="ref" href="borsuk_1.html#D19" target="_self" title="BORSUK_1:def.19">Def19</a></span>;<br/><a NAME="E5:117_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A3</font></span>: 
<span class="p1">(<span class="default"><a href="borsuk_1.html#K13" title="BORSUK_1:func.13">Proj</a> <font color="Maroon" title="c3">D</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">xx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="borsuk_1.html#K13" title="BORSUK_1:func.13">Proj</a> <font color="Maroon" title="c3">D</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">R</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">xx</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></span>;<br/><a NAME="E6:117_1_1"/>
<font color="Maroon" title="c6">xx9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:117_1_1"><span class="lab"><font color="Green" title="E74">A2</font></span></a>, <a class="ref" href="borsuk_1.html#T78" target="_self" title="BORSUK_1:th.78">Th78</a></span>;<br/><a NAME="E7:117_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E76">A4</font></span>: 
<font color="Maroon" title="c4">R</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">xx9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">xx9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:117_1"><span class="lab"><font color="Green" title="E73">A1</font></span></a>, <a class="ref" href="borsuk_1.html#D19" target="_self" title="BORSUK_1:def.19">Def19</a></span>;<br/><a NAME="E8:117_1_1"/>
( <span class="p1">(<span class="default"><a href="borsuk_1.html#K13" title="BORSUK_1:func.13">Proj</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K16" title="BORSUK_1:func.16">TrivExt</a> <font color="Maroon" title="c3">D</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">xx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="borsuk_1.html#K13" title="BORSUK_1:func.13">Proj</a> <font color="Maroon" title="c3">D</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">xx</font> &amp; <span class="p1">(<span class="default"><a href="borsuk_1.html#K13" title="BORSUK_1:func.13">Proj</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K16" title="BORSUK_1:func.16">TrivExt</a> <font color="Maroon" title="c3">D</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">xx9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="borsuk_1.html#K13" title="BORSUK_1:func.13">Proj</a> <font color="Maroon" title="c3">D</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">xx9</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:117_1_1"><span class="lab"><font color="Green" title="E74">A2</font></span></a>, <a class="ref" href="borsuk_1.html#T76" target="_self" title="BORSUK_1:th.76">Th76</a>, <a class="ref" href="borsuk_1.html#T78" target="_self" title="BORSUK_1:th.78">Th78</a></span>;<br/><span class="kw">hence </span><a NAME="E9:117_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:117_1_1"><span class="lab"><font color="Green" title="E74">A2</font></span></a>, <a class="txt" href="#E7:117_1_1"><span class="lab"><font color="Green" title="E76">A4</font></span></a>, <a class="txt" href="#E5:117_1_1"><span class="lab"><font color="Green" title="E75">A3</font></span></a>, <a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">F</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="borsuk_1.html#K12" title="BORSUK_1:func.12">space</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K16" title="BORSUK_1:func.16">TrivExt</a> <font color="Maroon" title="c3">D</font></span>)</span></span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="borsuk_1.html#K17" title="BORSUK_1:func.17">space</a> <font color="Maroon" title="c3">D</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E5:117_1"/><span class="lab"><font color="Green" title="E74">A5</font></span>: 
<font color="Maroon" title="c5">F</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="borsuk_1.html#K13" title="BORSUK_1:func.13">Proj</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K16" title="BORSUK_1:func.16">TrivExt</a> <font color="Maroon" title="c3">D</font></span>)</span></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><a href="borsuk_1.html#K13" title="BORSUK_1:func.13">Proj</a> <font color="Maroon" title="c3">D</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">R</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_1.html#T19" target="_self" title="BORSUK_1:th.19">Th19</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c6">F</font> = <font color="Maroon" title="c5">F</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="borsuk_1.html#K12" title="BORSUK_1:func.12">space</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K16" title="BORSUK_1:func.16">TrivExt</a> <font color="Maroon" title="c3">D</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="borsuk_1.html#K17" title="BORSUK_1:func.17">space</a> <font color="Maroon" title="c3">D</font></span>)</span> ;<br/>
<a NAME="E7:117_1"/>
<font color="Maroon" title="c6">F</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="117_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">W</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="borsuk_1.html#K12" title="BORSUK_1:func.12">space</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K16" title="BORSUK_1:func.16">TrivExt</a> <font color="Maroon" title="c3">D</font></span>)</span></span>)</span>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="borsuk_1.html#D3" target="_self" title="BORSUK_1:def.3">BORSUK_1:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">G</font> being   <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c6">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">W</font>  ex <font color="Olive" title="b2">H</font> being   <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c7">W</font> st <font color="Maroon" title="c6">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b2">H</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">G</font></span><br/><span class="kw">let </span><font color="Maroon" title="c8">G</font> be    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c6">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">W</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">H</font> being   <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c7">W</font> st <font color="Maroon" title="c6">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b1">H</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">G</font></span><br/>



<span class="kw">reconsider </span><font color="Maroon" title="c9">GG</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="borsuk_1.html#K13" title="BORSUK_1:func.13">Proj</a> <font color="Maroon" title="c3">D</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">R</font></span>)</span> <a href="borsuk_1.html#K2" title="BORSUK_1:func.2">"</a> <font color="Maroon" title="c8">G</font> as    <a href="connsp_2.html#M2" title="CONNSP_2:mode.2">a_neighborhood</a> of <span class="p1">(<span class="default"><a href="borsuk_1.html#K13" title="BORSUK_1:func.13">Proj</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K16" title="BORSUK_1:func.16">TrivExt</a> <font color="Maroon" title="c3">D</font></span>)</span></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c7">W</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:117_1"><span class="lab"><font color="Green" title="E74">A5</font></span></a>, <a class="ref" href="borsuk_1.html#T20" target="_self" title="BORSUK_1:th.20">Th20</a>, <a class="ref" href="borsuk_1.html#T39" target="_self" title="BORSUK_1:th.39">Th39</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">FF</font> = <span class="p1">(<span class="default"><a href="borsuk_1.html#K13" title="BORSUK_1:func.13">Proj</a> <font color="Maroon" title="c3">D</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">R</font> as   <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">XX</font>,<span class="p1">(<span class="default"><a href="borsuk_1.html#K17" title="BORSUK_1:func.17">space</a> <font color="Maroon" title="c3">D</font></span>)</span> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">V9</font> = <span class="p1">(<span class="default"><a href="borsuk_1.html#K13" title="BORSUK_1:func.13">Proj</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K16" title="BORSUK_1:func.16">TrivExt</a> <font color="Maroon" title="c3">D</font></span>)</span></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c9">GG</font> as    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c7">W</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_1.html#T81" target="_self" title="BORSUK_1:th.81">Th81</a></span>;<br/>
<span class="kw">take </span><font color="Maroon" title="c12">V</font> = <font color="Maroon" title="c11">V9</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c12">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">G</font></span><br/>

<a NAME="E4:117_1_2"/>
<font color="Maroon" title="c6">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c12">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="borsuk_1.html#K13" title="BORSUK_1:func.13">Proj</a> <font color="Maroon" title="c3">D</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">R</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c9">GG</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:117_1"><span class="lab"><font color="Green" title="E74">A5</font></span></a>, <a class="ref" href="relat_1.html#T159" title="RELAT_1:th.159">RELAT_1:159</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:117_1_2"/>
<font color="Maroon" title="c6">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c12">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">G</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T145" title="FUNCT_1:th.145">FUNCT_1:145</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">F9</font> = <font color="Maroon" title="c6">F</font> as   <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="borsuk_1.html#K12" title="BORSUK_1:func.12">space</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K16" title="BORSUK_1:func.16">TrivExt</a> <font color="Maroon" title="c3">D</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="borsuk_1.html#K17" title="BORSUK_1:func.17">space</a> <font color="Maroon" title="c3">D</font></span>)</span> ;<br/>
<span class="kw">take </span><font color="Maroon" title="c8">F99</font> = <font color="Maroon" title="c7">F9</font>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="borsuk_1.html#D20" target="_self" title="BORSUK_1:def.20">BORSUK_1:def 20</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">F99</font> is  <a href="borsuk_1.html#V3" title="BORSUK_1:attr.3">being_a_retraction</a> </span><br/>

<span class="kw">let </span><font color="Maroon" title="c9">W</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="borsuk_1.html#K12" title="BORSUK_1:func.12">space</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K16" title="BORSUK_1:func.16">TrivExt</a> <font color="Maroon" title="c3">D</font></span>)</span></span>)</span>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="borsuk_1.html#D19" target="_self" title="BORSUK_1:def.19">BORSUK_1:def 19</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">W</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="borsuk_1.html#K17" title="BORSUK_1:func.17">space</a> <font color="Maroon" title="c3">D</font></span>)</span> implies <font color="Maroon" title="c8">F99</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">W</font> )</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c10">W9</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">XX</font><span class="kw"> such that </span><br/><a NAME="E10:117_1"/><span class="lab"><font color="Green" title="E75">A6</font></span>: 
<span class="p1">(<span class="default"><a href="borsuk_1.html#K13" title="BORSUK_1:func.13">Proj</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K16" title="BORSUK_1:func.16">TrivExt</a> <font color="Maroon" title="c3">D</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">W9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">W</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_1.html#T71" target="_self" title="BORSUK_1:th.71">Th71</a></span>;<br/>
<span class="kw">assume </span><a NAME="E11:117_1"/><span class="lab"><font color="Green" title="E76">A7</font></span>: 
<font color="Maroon" title="c9">W</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="borsuk_1.html#K17" title="BORSUK_1:func.17">space</a> <font color="Maroon" title="c3">D</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">F99</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">W</font></span><br/>

<a NAME="E12:117_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">W9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:117_1"><span class="lab"><font color="Green" title="E75">A6</font></span></a>, <a class="ref" href="borsuk_1.html#T79" target="_self" title="BORSUK_1:th.79">Th79</a></span>;<br/>
<a NAME="E13:117_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E77">A8</font></span>: 
<font color="Maroon" title="c10">W9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">R</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">W9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:117_1"><span class="lab"><font color="Green" title="E73">A1</font></span></a>, <a class="ref" href="borsuk_1.html#D19" target="_self" title="BORSUK_1:def.19">Def19</a></span>;<br/>
<a NAME="E14:117_1"/><span class="lab"><font color="Green" title="E78">A9</font></span>: 
<font color="Maroon" title="c7">F9</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K13" title="BORSUK_1:func.13">Proj</a> <span class="p3">(<span class="default"><a href="borsuk_1.html#K16" title="BORSUK_1:func.16">TrivExt</a> <font color="Maroon" title="c3">D</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">W9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:117_1"><span class="lab"><font color="Green" title="E75">A6</font></span></a>, <a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></span>;<br/>
<a NAME="E15:117_1"/>
<span class="p1">(<span class="default"><a href="borsuk_1.html#K13" title="BORSUK_1:func.13">Proj</a> <font color="Maroon" title="c3">D</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">W9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="borsuk_1.html#K13" title="BORSUK_1:func.13">Proj</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K16" title="BORSUK_1:func.16">TrivExt</a> <font color="Maroon" title="c3">D</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">W9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:117_1"><span class="lab"><font color="Green" title="E75">A6</font></span></a>, <a class="txt" href="#E11:117_1"><span class="lab"><font color="Green" title="E76">A7</font></span></a>, <a class="ref" href="borsuk_1.html#T76" target="_self" title="BORSUK_1:th.76">Th76</a>, <a class="ref" href="borsuk_1.html#T79" target="_self" title="BORSUK_1:th.79">Th79</a></span>;<br/>
<span class="kw">hence </span><a NAME="E16:117_1"/>
<font color="Maroon" title="c8">F99</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">W</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:117_1"><span class="lab"><font color="Green" title="E74">A5</font></span></a>, <a class="txt" href="#E10:117_1"><span class="lab"><font color="Green" title="E75">A6</font></span></a>, <a class="txt" href="#E13:117_1"><span class="lab"><font color="Green" title="E77">A8</font></span></a>, <a class="txt" href="#E14:117_1"><span class="lab"><font color="Green" title="E78">A9</font></span></a>, <a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>


</div>
