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<b>let </b><font color="Maroon">J</font> be  non <a href="struct_0.html#V3">empty</a> <a href="waybel18.html#V2">injective</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">Y</font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/>



<b>assume </b><a NAME="E1:16"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">J</font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">Y</font>
 ;<br/>

<b>then consider </b><font color="Maroon">f</font> being   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">Y</font>,<font color="Maroon">J</font><b> such that </b><br/><a NAME="E3:16"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">f</font> is <a href="pre_topc.html#V5">continuous</a>
 <b>and </b><br/><a NAME="E4:16"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">J</font> <a href="hidden.html#R1">=</a>  <a href="grcat_1.html#K7">id</a> <font color="Maroon">J</font>
 <b>by </b><i><a class="ref" href="waybel18.html#D5">WAYBEL18:def 5</a></i>;<br/>
<a NAME="E5:16"/><i><font color="Green">E47</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">J</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</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">ff</font> = <font color="Maroon">f</font> as   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">Y</font>,<font color="Maroon">Y</font> <b>by </b><i><a class="ref" href="funct_2.html#T9">FUNCT_2:9</a></i>;<br/>
<a NAME="E7:16"/>
 ex <font color="Olive">ff</font> being  <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">Y</font>,<font color="Maroon">Y</font> st <br/>( <font color="Olive">ff</font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Olive">ff</font> <a href="partfun1.html#K1">*</a> <font color="Olive">ff</font> <a href="funct_2.html#R2">=</a> <font color="Olive">ff</font> &amp;  <a href="waybel18.html#K7">Image</a> <font color="Olive">ff</font>,<font color="Maroon">J</font> <a href="borsuk_3.html#R2">are_homeomorphic</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon">ff</font>
;<br/>

<b>thus </b><a NAME="E1:16_1"/>
<font color="Maroon">ff</font> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i>, , <a class="ref" href="topmetr.html#T7">TOPMETR:7</a></i>;<br/>

<a NAME="E2:16_1"/><i><font color="Green">E54</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">ff</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">ff</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E3:16_1"/><i><font color="Green">E55</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E4:16_1"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   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">Y</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:16_1_1"/><i><font color="Green">E64</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">Y</font>
 ;<br/>

<a NAME="E2:16_1_1"/><b>then </b><i><font color="Green">E65</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <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">J</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="grcat_1.html#K7">id</a> <font color="Maroon">J</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i>, <a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>
;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:16_1"/>
<font color="Maroon">ff</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">ff</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">ff</font>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>

<a NAME="E6:16_1"/><i><font color="Green">E68</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">J</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:16_1_2"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">J</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><br/>

<b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:16_1_2"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">J</font>
 ;<br/>

<a NAME="E3:16_1_2"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>
 
<b>by </b><i/>;<br/>
<a NAME="E4:16_1_2"/><b>then </b><i><font color="Green">E71</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> = 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">J</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
<b>by </b><i>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon">y</font>
<b>by </b><i>, , <a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>
;<br/>
<b>hence </b><a NAME="E6:16_1_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon">M</font> =  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">J</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font> <b>by </b><i/>;<br/>
<i><font color="Green">E72</font></i>: the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">Y</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">M</font></span>)</span> = 
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon">Y</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">M</font></span>)</span>

<br/>.= 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">J</font>
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>
;<br/>
<a NAME="E9:16_1"/>
 <a href="waybel18.html#K7">Image</a> <font color="Maroon">ff</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">J</font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">J</font> #)
 
<b>by </b><i>, , , <a class="ref" href="tsep_1.html#T5">TSEP_1:5</a></i>;<br/>
<b>hence </b><a NAME="E10:16_1"/>
 <a href="waybel18.html#K7">Image</a> <font color="Maroon">ff</font>,<font color="Maroon">J</font> <a href="borsuk_3.html#R2">are_homeomorphic</a> 
 <b>by </b><i><a class="ref" href="yellow14.html#T20">YELLOW14:20</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:16"/>
<font color="Maroon">J</font> <a href="waybel18.html#R1">is_Retract_of</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="waybel18.html#D8">WAYBEL18:def 8</a></i>;<br/>


</div>
