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<div class="add">

<b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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#K18" title="BORSUK_1:func.18">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#K13" title="BORSUK_1:func.13">space</a> <span class="p1">(<span class="default"><a href="borsuk_1.html#K17" title="BORSUK_1:func.17">TrivExt</a> <font color="Olive" title="b2">D</font></span>)</span></span><br/><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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#K18" title="BORSUK_1:func.18">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#K13" title="BORSUK_1:func.13">space</a> <span class="p1">(<span class="default"><a href="borsuk_1.html#K17" title="BORSUK_1:func.17">TrivExt</a> <font color="Olive" title="b1">D</font></span>)</span></span><br/><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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#K18" title="BORSUK_1:func.18">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#K13" title="BORSUK_1:func.13">space</a> <span class="p1">(<span class="default"><a href="borsuk_1.html#K17" title="BORSUK_1:func.17">TrivExt</a> <font color="Maroon" title="c3">D</font></span>)</span> )</span><br/>





<b>thus </b><a NAME="E1:106"/>
( <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#K18" title="BORSUK_1:func.18">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#K13" title="BORSUK_1:func.13">space</a> <span class="p1">(<span class="default"><a href="borsuk_1.html#K17" title="BORSUK_1:func.17">TrivExt</a> <font color="Maroon" title="c3">D</font></span>)</span> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="106_1"><b>proof </b></a><div class="add">

<b>given </b><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><b> such that </b><a NAME="E2:106_1"/><i><font color="Green" title="E76">A1</font></i>: 
<font color="Maroon" title="c4">R</font> <a href="borsuk_1.html#NR1" title="BORSUK_1:NR.1">is_a_retraction</a> 
 ; <i><font color="Red">:: according to </font></i><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="borsuk_1.html#K18" title="BORSUK_1:func.18">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#K13" title="BORSUK_1:func.13">space</a> <span class="p1">(<span class="default"><a href="borsuk_1.html#K17" title="BORSUK_1:func.17">TrivExt</a> <font color="Maroon" title="c3">D</font></span>)</span></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="106_1_1"><b>now </b></a><div class="add"><b>given </b><font color="Maroon" title="c5">xx</font>, <font color="Maroon" title="c6">xx'</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">XX</font><b> such that </b><a NAME="E2:106_1_1"/><i><font color="Green" title="E77">A2</font></i>: 
<span class="p1">(<span class="default"><a href="borsuk_1.html#K14" title="BORSUK_1:func.14">Proj</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K17" title="BORSUK_1:func.17">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#K14" title="BORSUK_1:func.14">Proj</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K17" title="BORSUK_1:func.17">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">xx'</font>
 <b>and </b><br/><a NAME="E3:106_1_1"/><i><font color="Green" title="E78">A3</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="borsuk_1.html#K14" title="BORSUK_1:func.14">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#K14" title="BORSUK_1:func.14">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">xx'</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E4:106_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> &amp; <font color="Maroon" title="c6">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> )
 <b>by </b><i><a class="txt" href="borsuk_1.html#E2:106_1_1"><i><font color="Green" title="E77">A2</font></i></a>, <a class="txt" href="borsuk_1.html#E3:106_1_1"><i><font color="Green" title="E78">A3</font></i></a>, <a class="ref" href="borsuk_1.html#T78" target="_self" title="BORSUK_1:th.78">Th78</a></i>;<br/><a NAME="E5:106_1_1"/><b>then </b>
( <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> &amp; <font color="Maroon" title="c4">R</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">xx'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">xx'</font> )
 <b>by </b><i><a class="txt" href="borsuk_1.html#E2:106_1"><i><font color="Green" title="E76">A1</font></i></a>, <a class="ref" href="borsuk_1.html#D19" target="_self" title="BORSUK_1:def.19">Def19</a></i>;<br/><a NAME="E6:106_1_1"/><b>then </b><i><font color="Green" title="E79">A4</font></i>: 
( <span class="p1">(<span class="default"><a href="borsuk_1.html#K14" title="BORSUK_1:func.14">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#K14" title="BORSUK_1:func.14">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> &amp; <span class="p1">(<span class="default"><a href="borsuk_1.html#K14" title="BORSUK_1:func.14">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">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#K14" title="BORSUK_1:func.14">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">xx'</font> )
 <b>by </b><i><a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></i>;<br/><a NAME="E7:106_1_1"/>
( <span class="p1">(<span class="default"><a href="borsuk_1.html#K14" title="BORSUK_1:func.14">Proj</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K17" title="BORSUK_1:func.17">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#K14" title="BORSUK_1:func.14">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#K14" title="BORSUK_1:func.14">Proj</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K17" title="BORSUK_1:func.17">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">xx'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="borsuk_1.html#K14" title="BORSUK_1:func.14">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">xx'</font> )
 <b>by </b><i><a class="txt" href="borsuk_1.html#E2:106_1_1"><i><font color="Green" title="E77">A2</font></i></a>, <a class="txt" href="borsuk_1.html#E3:106_1_1"><i><font color="Green" title="E78">A3</font></i></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></i>;<br/><b>hence </b><a NAME="E8:106_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="borsuk_1.html#E2:106_1_1"><i><font color="Green" title="E77">A2</font></i></a>, <a class="txt" href="borsuk_1.html#E3:106_1_1"><i><font color="Green" title="E78">A3</font></i></a>, <a class="txt" href="borsuk_1.html#E6:106_1_1"><i><font color="Green" title="E79">A4</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>then consider </b><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#K13" title="BORSUK_1:func.13">space</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K17" title="BORSUK_1:func.17">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#K18" title="BORSUK_1:func.18">space</a> <font color="Maroon" title="c3">D</font></span>)</span><b> such that </b><br/><a NAME="E5:106_1"/><i><font color="Green" title="E77">A5</font></i>: 
<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#K14" title="BORSUK_1:func.14">Proj</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K17" title="BORSUK_1:func.17">TrivExt</a> <font color="Maroon" title="c3">D</font></span>)</span></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="borsuk_1.html#K14" title="BORSUK_1:func.14">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>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T19" target="_self" title="BORSUK_1:th.19">Th19</a></i>;<br/>
<b>reconsider </b><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#K13" title="BORSUK_1:func.13">space</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K17" title="BORSUK_1:func.17">TrivExt</a> <font color="Maroon" title="c3">D</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="borsuk_1.html#K18" title="BORSUK_1:func.18">space</a> <font color="Maroon" title="c3">D</font></span>)</span> ;<br/>
<a NAME="E7:106_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="106_1_2"><b>proof </b></a><div class="add">

<b>let </b><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#K13" title="BORSUK_1:func.13">space</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K17" title="BORSUK_1:func.17">TrivExt</a> <font color="Maroon" title="c3">D</font></span>)</span></span>)</span>; <i><font color="Red">:: according to </font></i><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:()"><i><font color="Red">::  thesis: </font></i></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#K10" title="RELSET_1:func.10">.:</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/><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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#K10" title="RELSET_1:func.10">.:</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/>



<b>reconsider </b><font color="Maroon" title="c9">FF</font> = <span class="p1">(<span class="default"><a href="borsuk_1.html#K14" title="BORSUK_1:func.14">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#K18" title="BORSUK_1:func.18">space</a> <font color="Maroon" title="c3">D</font></span>)</span> <b>by </b><i><a class="ref" href="tops_2.html#T58" title="TOPS_2:th.58">TOPS_2:58</a></i>;<br/>
<a NAME="E2:106_1_2"/>
<font color="Maroon" title="c9">FF</font> <a href="borsuk_1.html#K3" title="BORSUK_1:func.3">"</a> <font color="Maroon" title="c8">G</font> is    <a href="connsp_2.html#M2" title="CONNSP_2:mode.2">a_neighborhood</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="borsuk_1.html#K14" title="BORSUK_1:func.14">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#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><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></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c10">GG</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="borsuk_1.html#K14" title="BORSUK_1:func.14">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#K11" title="RELSET_1:func.11">"</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#K14" title="BORSUK_1:func.14">Proj</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K17" title="BORSUK_1:func.17">TrivExt</a> <font color="Maroon" title="c3">D</font></span>)</span></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</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> <b>by </b><i><a class="txt" href="borsuk_1.html#E5:106_1"><i><font color="Green" title="E77">A5</font></i></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></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">V'</font> = <span class="p1">(<span class="default"><a href="borsuk_1.html#K14" title="BORSUK_1:func.14">Proj</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K17" title="BORSUK_1:func.17">TrivExt</a> <font color="Maroon" title="c3">D</font></span>)</span></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c10">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> <b>by </b><i><a class="ref" href="borsuk_1.html#T81" target="_self" title="BORSUK_1:th.81">Th81</a></i>;<br/>
<b>take </b><font color="Maroon" title="c12">V</font> = <font color="Maroon" title="c11">V'</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">F</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</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="E5:106_1_2"/>
<font color="Maroon" title="c6">F</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</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#K14" title="BORSUK_1:func.14">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#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c10">GG</font>
 
<b>by </b><i><a class="txt" href="borsuk_1.html#E5:106_1"><i><font color="Green" title="E77">A5</font></i></a>, <a class="ref" href="relat_1.html#T159" title="RELAT_1:th.159">RELAT_1:159</a></i>;<br/>
<b>hence </b><a NAME="E6:106_1_2"/>
<font color="Maroon" title="c6">F</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</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>
 <b>by </b><i><a class="ref" href="funct_1.html#T145" title="FUNCT_1:th.145">FUNCT_1:145</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c7">F'</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#K13" title="BORSUK_1:func.13">space</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K17" title="BORSUK_1:func.17">TrivExt</a> <font color="Maroon" title="c3">D</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="borsuk_1.html#K18" title="BORSUK_1:func.18">space</a> <font color="Maroon" title="c3">D</font></span>)</span> ;<br/>
<b>take </b><font color="Maroon" title="c8">F''</font> = <font color="Maroon" title="c7">F'</font>; <i><font color="Red">:: according to </font></i><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c8">F''</font> <a href="borsuk_1.html#NR1" title="BORSUK_1:NR.1">is_a_retraction</a> </span><br/>

<b>let </b><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#K13" title="BORSUK_1:func.13">space</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K17" title="BORSUK_1:func.17">TrivExt</a> <font color="Maroon" title="c3">D</font></span>)</span></span>)</span>; <i><font color="Red">:: according to </font></i><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:()"><i><font color="Red">::  thesis: </font></i></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#K18" title="BORSUK_1:func.18">space</a> <font color="Maroon" title="c3">D</font></span>)</span> implies <font color="Maroon" title="c8">F''</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/>

<b>consider </b><font color="Maroon" title="c10">W'</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">XX</font><b> such that </b><br/><a NAME="E10:106_1"/><i><font color="Green" title="E78">A6</font></i>: 
<span class="p1">(<span class="default"><a href="borsuk_1.html#K14" title="BORSUK_1:func.14">Proj</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K17" title="BORSUK_1:func.17">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">W'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">W</font>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T71" target="_self" title="BORSUK_1:th.71">Th71</a></i>;<br/>
<b>assume </b><a NAME="E11:106_1"/>
<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#K18" title="BORSUK_1:func.18">space</a> <font color="Maroon" title="c3">D</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c8">F''</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="E12:106_1"/><b>then </b>
<font color="Maroon" title="c10">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 <font color="Maroon" title="c2">X</font>
 
<b>by </b><i><a class="txt" href="borsuk_1.html#E10:106_1"><i><font color="Green" title="E78">A6</font></i></a>, <a class="ref" href="borsuk_1.html#T79" target="_self" title="BORSUK_1:th.79">Th79</a></i>;<br/>
<a NAME="E13:106_1"/><b>then </b>
( <font color="Maroon" title="c10">W'</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">W'</font> &amp; <span class="p1">(<span class="default"><a href="borsuk_1.html#K14" title="BORSUK_1:func.14">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">W'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="borsuk_1.html#K14" title="BORSUK_1:func.14">Proj</a> <span class="p2">(<span class="default"><a href="borsuk_1.html#K17" title="BORSUK_1:func.17">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">W'</font> )
 
<b>by </b><i><a class="txt" href="borsuk_1.html#E2:106_1"><i><font color="Green" title="E76">A1</font></i></a>, <a class="ref" href="borsuk_1.html#D19" target="_self" title="BORSUK_1:def.19">Def19</a>, <a class="ref" href="borsuk_1.html#T76" target="_self" title="BORSUK_1:th.76">Th76</a></i>;<br/>
<a NAME="E14:106_1"/><b>then </b>
( <font color="Maroon" title="c7">F'</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#K14" title="BORSUK_1:func.14">Proj</a> <span class="p3">(<span class="default"><a href="borsuk_1.html#K17" title="BORSUK_1:func.17">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">W'</font> &amp; <font color="Maroon" title="c9">W</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#K14" title="BORSUK_1:func.14">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="c10">W'</font> )
 
<b>by </b><i><a class="txt" href="borsuk_1.html#E10:106_1"><i><font color="Green" title="E78">A6</font></i></a>, <a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></i>;<br/>
<b>hence </b><a NAME="E15:106_1"/>
<font color="Maroon" title="c8">F''</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>
 <b>by </b><i><a class="txt" href="borsuk_1.html#E5:106_1"><i><font color="Green" title="E77">A5</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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


</div>
