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

<b>let </b><font color="Maroon" title="c1">S</font>, <font color="Maroon" title="c2">T</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>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:15"/><i><font color="Green" title="E8">A1</font></i>: 
<font color="Maroon" title="c2">T</font> is <a href="waybel18.html#V2" title="WAYBEL18:attr.2">injective</a>
 <b>and </b><br/><a NAME="E2:15"/><i><font color="Green" title="E9">A2</font></i>: 
<font color="Maroon" title="c1">S</font> <a href="waybel18.html#R1" title="WAYBEL18:pred.1">is_Retract_of</a> <font color="Maroon" title="c2">T</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c3">h</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c2">T</font><b> such that </b><br/><a NAME="E4:15"/><i><font color="Green" title="E10">A3</font></i>: 
<font color="Maroon" title="c3">h</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>and </b><br/><a NAME="E5:15"/><i><font color="Green" title="E11">A4</font></i>: 
<font color="Maroon" title="c3">h</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">h</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c3">h</font>
 <b>and </b><br/><a NAME="E6:15"/><i><font color="Green" title="E12">A5</font></i>: 
 <a href="waybel18.html#K7" title="WAYBEL18:func.7">Image</a> <font color="Maroon" title="c3">h</font>,<font color="Maroon" title="c1">S</font> <a href="borsuk_3.html#R2" title="BORSUK_3:pred.2">are_homeomorphic</a> 
 <b>by </b><i><a class="txt" href="waybel25.html#E2:15"><i><font color="Green" title="E9">A2</font></i></a>, <a class="ref" href="waybel18.html#D8" title="WAYBEL18:def.8">WAYBEL18:def 8</a></i>;<br/>
<a NAME="E7:15"/><i><font color="Green" title="E13">A6</font></i>: 
 <a href="waybel18.html#K8" title="WAYBEL18:func.8">corestr</a> <font color="Maroon" title="c3">h</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 
<b>by </b><i><a class="txt" href="waybel25.html#E4:15"><i><font color="Green" title="E10">A3</font></i></a>, <a class="ref" href="waybel18.html#T11" title="WAYBEL18:th.11">WAYBEL18:11</a></i>;<br/>
<b>set </b><font color="Maroon" title="c4">F</font> =  <a href="waybel18.html#K8" title="WAYBEL18:func.8">corestr</a> <font color="Maroon" title="c3">h</font>;<br/>
<a NAME="E8:15"/>
for <font color="Olive" title="b1">W</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">T</font>  st <font color="Olive" title="b1">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="waybel18.html#K7" title="WAYBEL18:func.7">Image</a> <font color="Maroon" title="c3">h</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="waybel18.html#K8" title="WAYBEL18:func.8">corestr</a> <font color="Maroon" title="c3">h</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">W</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">W</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">T</font>;<br/>

<b>assume </b><a NAME="E1:15_1"/>
<font color="Maroon" title="c5">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="waybel18.html#K7" title="WAYBEL18:func.7">Image</a> <font color="Maroon" title="c3">h</font></span>)</span>
 ;<br/>

<a NAME="E2:15_1"/><b>then </b>
<font color="Maroon" title="c5">W</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">h</font>
 
<b>by </b><i><a class="ref" href="waybel18.html#T10" title="WAYBEL18:th.10">WAYBEL18:10</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c6">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:15_1"/><i><font color="Green" title="E14">A7</font></i>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">h</font>
 <b>and </b><br/><a NAME="E5:15_1"/><i><font color="Green" title="E15">A8</font></i>: 
<font color="Maroon" title="c5">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>thus </b><a NAME="E6:15_1"/>
<span class="p1">(<span class="default"><a href="waybel18.html#K8" title="WAYBEL18:func.8">corestr</a> <font color="Maroon" title="c3">h</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">W</font>
 <b>by </b><i><a class="txt" href="waybel25.html#E5:15_1"><i><font color="Green" title="E15">A8</font></i></a>, <a class="txt" href="waybel25.html#E5:15"><i><font color="Green" title="E11">A4</font></i></a>, <a class="txt" href="waybel25.html#E4:15_1"><i><font color="Green" title="E14">A7</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:15"/><b>then </b>
 <a href="waybel18.html#K8" title="WAYBEL18:func.8">corestr</a> <font color="Maroon" title="c3">h</font> <a href="borsuk_1.html#NR1" title="BORSUK_1:NR.1">is_a_retraction</a> 
 
<b>by </b><i><a class="ref" href="borsuk_1.html#D19" title="BORSUK_1:def.19">BORSUK_1:def 19</a></i>;<br/>
<a NAME="E10:15"/><b>then </b>
 <a href="waybel18.html#K7" title="WAYBEL18:func.7">Image</a> <font color="Maroon" title="c3">h</font> <a href="borsuk_1.html#R1" title="BORSUK_1:pred.1">is_a_retract_of</a> <font color="Maroon" title="c2">T</font>
 
<b>by </b><i><a class="txt" href="waybel25.html#E7:15"><i><font color="Green" title="E13">A6</font></i></a>, <a class="ref" href="borsuk_1.html#D20" title="BORSUK_1:def.20">BORSUK_1:def 20</a></i>;<br/>
<a NAME="E11:15"/><b>then </b>
 <a href="waybel18.html#K7" title="WAYBEL18:func.7">Image</a> <font color="Maroon" title="c3">h</font> is <a href="waybel18.html#V2" title="WAYBEL18:attr.2">injective</a>
 
<b>by </b><i><a class="txt" href="waybel25.html#E1:15"><i><font color="Green" title="E8">A1</font></i></a>, <a class="ref" href="waybel18.html#T9" title="WAYBEL18:th.9">WAYBEL18:9</a></i>;<br/>
<b>hence </b><a NAME="E12:15"/>
<font color="Maroon" title="c1">S</font> is <a href="waybel18.html#V2" title="WAYBEL18:attr.2">injective</a>
 <b>by </b><i><a class="txt" href="waybel25.html#E6:15"><i><font color="Green" title="E12">A5</font></i></a>, <a class="ref" href="waybel25.html#T6" target="_self" title="WAYBEL25:th.6">Th6</a></i>;<br/>


</div>
