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

<b>let </b><font color="Maroon">T</font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>, <font color="Maroon">S</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="E1:29"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">T</font> is <a href="waybel18.html#V2" target="_self">injective</a>
 ;<br/>

<b>let </b><font color="Maroon">f</font> be   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">T</font>,<font color="Maroon">S</font>;<br/>

<b>assume </b><a NAME="E2:29"/><i><font color="Green">E28</font></i>: 
 <a href="waybel18.html#K8" target="_self">corestr</a> <font color="Maroon">f</font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a> 
 ;<br/>

<a NAME="E3:29"/><b>then </b><i><font color="Green">E31</font></i>: 
 <a href="waybel18.html#K8" target="_self">corestr</a> <font color="Maroon">f</font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i><a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon">g</font> being   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="waybel18.html#K7" target="_self">Image</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">T</font><b> such that </b><br/><a NAME="E5:29"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">g</font> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><a href="waybel18.html#K8" target="_self">corestr</a> <font color="Maroon">f</font></span>)</span> <a href="tops_2.html#NK3" target="_self">"</a> 
 ;<br/>
<a NAME="E6:29"/>
<font color="Maroon">g</font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i>, , <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon">h</font> being   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">S</font>,<font color="Maroon">T</font><b> such that </b><br/><a NAME="E8:29"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">h</font> is <a href="pre_topc.html#V5">continuous</a>
 <b>and </b><br/><a NAME="E9:29"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">h</font> <a href="partfun1.html#K2">|</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel18.html#K7" target="_self">Image</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font>
 <b>by </b><i>, </i>;<br/>
<a NAME="E10:29"/>
<font color="Maroon">g</font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a> 
 
<b>by </b><i>, , <a class="ref" href="tops_2.html#T70">TOPS_2:70</a></i>;<br/>
<a NAME="E11:29"/><b>then </b><i><font color="Green">E36</font></i>: 
<font color="Maroon">g</font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i><a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">p</font> =  <a href="tmap_1.html#K4">incl</a> <span class="p1">(<span class="default"><a href="waybel18.html#K7" target="_self">Image</a> <font color="Maroon">f</font></span>)</span> as   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="waybel18.html#K7" target="_self">Image</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">S</font> ;<br/>
<a NAME="E13:29"/>
<font color="Maroon">p</font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i><a class="ref" href="tmap_1.html#T98">TMAP_1:98</a></i>;<br/>
<a NAME="E14:29"/><b>then </b><i><font color="Green">E37</font></i>: 
<font color="Maroon">p</font> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><a href="waybel18.html#K8" target="_self">corestr</a> <font color="Maroon">f</font></span>)</span> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i>, <a class="ref" href="tops_2.html#T58">TOPS_2:58</a></i>;<br/>
<i><font color="Green">E43</font></i>:  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="waybel18.html#K8" target="_self">corestr</a> <font color="Maroon">f</font></span>)</span></span>)</span> = 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
<br/>.= 
 <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/>
<b>take </b><font color="Maroon">F</font> = <font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">h</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel18.html#D8" target="_self">WAYBEL18:def 8</a><br/>

<a NAME="E16:29"/><i><font color="Green">E44</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel18.html#K7" target="_self">Image</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font>
 
<b>by </b><i/>;<br/>
<a NAME="E17:29"/>
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">h</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E18:29"/><b>then </b><i><font color="Green">E66</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">h</font> <a href="tarski.html#R1">c=</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/>
<a NAME="E19:29"/><i><font color="Green">E73</font></i>: 
for <font color="Olive">P</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font> holds  <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Olive">P</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="waybel18.html#K8" target="_self">corestr</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="relset_1.html#K11">"</a> <font color="Olive">P</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">P</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font>;<br/>

<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a>,<a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</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="E1:29_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">P</font>
 ;<br/><a NAME="E2:29_2_1"/><b>then </b><i><font color="Green">E74</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/><a NAME="E3:29_2_1"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</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/><a NAME="E4:29_2_1"/><b>then </b><i><font color="Green">E75</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 <span class="p1">(<span class="default"><a href="waybel18.html#K7" target="_self">Image</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">S</font> <b>by </b><i><a class="txt" href="waybel18.html#E1:29"><i><font color="Green">E27</font></i></a></i>;<br/><span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="waybel18.html#K8" target="_self">corestr</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
<font color="Maroon">p</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="txt" href="waybel18.html#E1:29"><i><font color="Green">E27</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon">y</font>
<b>by </b><i><a class="txt" href="waybel18.html#E2:29"><i><font color="Green">E28</font></i></a>, <a class="ref" href="tmap_1.html#T95">TMAP_1:95</a></i>
;<br/><b>hence </b><a NAME="E7:29_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="waybel18.html#K8" target="_self">corestr</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="relset_1.html#K11">"</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="waybel18.html#T2" target="_self">Th2</a>, <a class="txt" href="waybel18.html#E1:29"><i><font color="Green">E27</font></i></a>, <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon">x</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="E2:29_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="waybel18.html#K8" target="_self">corestr</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="relset_1.html#K11">"</a> <font color="Maroon">P</font>
 ;<br/>

<a NAME="E3:29_2"/><b>then </b><i><font color="Green">E76</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="waybel18.html#K8" target="_self">corestr</a> <font color="Maroon">f</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="waybel18.html#K8" target="_self">corestr</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> )
 
<b>by </b><i><a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>
<a NAME="E4:29_2"/><b>then </b><i><font color="Green">E77</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>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="waybel18.html#T2" target="_self">Th2</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E5:29_2"/><b>then </b><i><font color="Green">E78</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 <span class="p1">(<span class="default"><a href="waybel18.html#K7" target="_self">Image</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">S</font> <b>by </b><i><a class="txt" href="waybel18.html#E2:29"><i><font color="Green">E28</font></i></a></i>;<br/>
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="waybel18.html#K8" target="_self">corestr</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
<font color="Maroon">p</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="waybel18.html#T2" target="_self">Th2</a>, <a class="txt" href="waybel18.html#E1:29"><i><font color="Green">E27</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon">y</font>
<b>by </b><i>, <a class="ref" href="tmap_1.html#T95">TMAP_1:95</a></i>
;<br/>
<b>hence </b><a NAME="E8:29_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="waybel18.html#T2" target="_self">Th2</a>, <a class="txt" href="waybel18.html#E1:29"><i><font color="Green">E27</font></i></a>, <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E20:29"/>
for <font color="Olive">P</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font>  st <font color="Olive">P</font> is <a href="pre_topc.html#V3">open</a> holds <br/><font color="Maroon">F</font> <a href="relset_1.html#K11">"</a> <font color="Olive">P</font> is <a href="pre_topc.html#V3">open</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">P</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font>;<br/>

<b>assume </b><a NAME="E1:29_3"/>
<font color="Maroon">P</font> is <a href="pre_topc.html#V3">open</a>
 ;<br/>

<a NAME="E2:29_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="waybel18.html#K8" target="_self">corestr</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="relset_1.html#K11">"</a> <font color="Maroon">P</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i>, <a class="ref" href="tops_2.html#T55">TOPS_2:55</a></i>;<br/>
<a NAME="E3:29_3"/><b>then </b>
<font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">P</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i/>;<br/>
<a NAME="E4:29_3"/><b>then </b>
<font color="Maroon">h</font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">P</font></span>)</span> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i>, <a class="ref" href="tops_2.html#T55">TOPS_2:55</a></i>;<br/>
<b>hence </b><a NAME="E5:29_3"/>
<font color="Maroon">F</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">P</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="ref" href="relat_1.html#T181">RELAT_1:181</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E21:29"/>
<font color="Maroon">F</font> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i><a class="ref" href="tops_2.html#T55">TOPS_2:55</a></i>;<br/>

<a NAME="E22:29"/><i><font color="Green">E79</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E23:29"/>
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">T</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">h</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="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_4"><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="E1:29_4"/><i><font color="Green">E80</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">T</font>
 ;<br/>

<a NAME="E2:29_4"/><b>then </b><i><font color="Green">E81</font></i>: 
<font color="Maroon">x</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/>
<a NAME="E3:29_4"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="waybel18.html#K8" target="_self">corestr</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel18.html#E2:29"><i><font color="Green">E28</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E4:29_4"/><i><font color="Green">E83</font></i>: 
 <a href="waybel18.html#K8" target="_self">corestr</a> <font color="Maroon">f</font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/>
<a NAME="E5:29_4"/><i><font color="Green">E95</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="waybel18.html#K8" target="_self">corestr</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="waybel18.html#K7" target="_self">Image</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D3">FUNCT_2:def 3</a></i>;<br/>
<a NAME="E6:29_4"/><i><font color="Green">E96</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>  <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/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">h</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">h</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"><span class="p2">(<span class="default"><a href="waybel18.html#K8" target="_self">corestr</a> <font color="Maroon">f</font></span>)</span> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel18.html#K8" target="_self">corestr</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i>, , , <a class="txt" href="waybel18.html#E8:29"><i><font color="Green">E33</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel18.html#K8" target="_self">corestr</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel18.html#K8" target="_self">corestr</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="txt" href="waybel18.html#E3:29"><i><font color="Green">E31</font></i></a>, <a class="txt" href="waybel18.html#E5:29"><i><font color="Green">E32</font></i></a>, <a class="ref" href="tops_2.html#D4">TOPS_2:def 4</a></i>
<br/>.= 
<font color="Maroon">x</font>
<b>by </b><i>, <a class="txt" href="waybel18.html#E3:29"><i><font color="Green">E31</font></i></a>, <a class="ref" href="funct_1.html#T56">FUNCT_1:56</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E24:29"/><b>then </b><i><font color="Green">E97</font></i>: 
<font color="Maroon">h</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="pralg_3.html#K1">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>
 
<b>by </b><i><a class="txt" href="waybel18.html#E1:29"><i><font color="Green">E27</font></i></a>, <a class="ref" href="funct_1.html#T34">FUNCT_1:34</a></i>;<br/>
<b>thus </b><font color="Maroon">F</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">h</font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">h</font>
<b>by </b><i><a class="ref" href="relat_1.html#T55">RELAT_1:55</a></i>
<br/>.= 
<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="grcat_1.html#K7">id</a> <font color="Maroon">T</font></span>)</span></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">h</font>
<b>by </b><i><a class="ref" href="relat_1.html#T55">RELAT_1:55</a>, <a class="txt" href="waybel18.html#E2:29"><i><font color="Green">E28</font></i></a></i>
<br/>.= 
<font color="Maroon">F</font>
<b>by </b><i><a class="ref" href="grcat_1.html#T12">GRCAT_1:12</a></i>
;<br/>

<a NAME="E26:29"/><i><font color="Green">E98</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_6"><b>proof </b></a><div class="add">

<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="E1:29_6"/>
<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>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:29_6"/><i><font color="Green">E99</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font>
 <b>and </b><br/><a NAME="E4:29_6"/><i><font color="Green">E100</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <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#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E5:29_6"/>
<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">S</font>
 
<b>by </b><i/>;<br/>
<a NAME="E6:29_6"/><b>then </b><i><font color="Green">E102</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E7:29_6"/><b>then </b><i><font color="Green">E103</font></i>: 
<font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">h</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E8:29_6"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 
<b>by </b><i><a class="txt" href="waybel18.html#E3:29"><i><font color="Green">E31</font></i></a>, <a class="txt" href="waybel18.html#E5:29"><i><font color="Green">E32</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/>
<b>hence </b><a NAME="E9:29_6"/>
<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="txt" href="waybel18.html#E8:29"><i><font color="Green">E33</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<b>set </b><font color="Maroon">H</font> = <font color="Maroon">h</font> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><a href="tmap_1.html#K4">incl</a> <span class="p2">(<span class="default"><a href="waybel18.html#K7" target="_self">Image</a> <font color="Maroon">F</font></span>)</span></span>)</span>;<br/>
<a NAME="E27:29"/><i><font color="Green">E104</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="tmap_1.html#K4">incl</a> <span class="p2">(<span class="default"><a href="waybel18.html#K7" target="_self">Image</a> <font color="Maroon">F</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel18.html#K7" target="_self">Image</a> <font color="Maroon">F</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E28:29"/><i><font color="Green">E105</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="tmap_1.html#K4">incl</a> <span class="p3">(<span class="default"><a href="waybel18.html#K7" target="_self">Image</a> <font color="Maroon">F</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="waybel18.html#K7" target="_self">Image</a> <font color="Maroon">F</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E29:29"/><i><font color="Green">E106</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="tmap_1.html#K4">incl</a> <span class="p3">(<span class="default"><a href="waybel18.html#K7" target="_self">Image</a> <font color="Maroon">F</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_7"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:29_7"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="tmap_1.html#K4">incl</a> <span class="p3">(<span class="default"><a href="waybel18.html#K7" target="_self">Image</a> <font color="Maroon">F</font></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</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="E2:29_7"/><i><font color="Green">E107</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font>
 ;<br/>

<a NAME="E3:29_7"/><b>then </b><i><font color="Green">E108</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">T</font>
 
;<br/>
<a NAME="E4:29_7"/><i><font color="Green">E109</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="tmap_1.html#K4">incl</a> <span class="p3">(<span class="default"><a href="waybel18.html#K7" target="_self">Image</a> <font color="Maroon">F</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">F</font>
 
<b>by </b><i><a class="txt" href="waybel18.html#E3:29"><i><font color="Green">E31</font></i></a>, </i>;<br/>
<a NAME="E5:29_7"/><i><font color="Green">E110</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/>
<a NAME="E6:29_7"/><b>then </b><i><font color="Green">E111</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <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/>
<a NAME="E7:29_7"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <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">S</font>
 
;<br/>
<a NAME="E8:29_7"/><b>then </b><i><font color="Green">E112</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <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/>
<i><font color="Green">E113</font></i>: <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">y</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">h</font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
<b>by </b><i><a class="txt" href="waybel18.html#E11:29"><i><font color="Green">E36</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<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="grcat_1.html#K7">id</a> <font color="Maroon">T</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
<b>by </b><i><a class="ref" href="relat_1.html#T55">RELAT_1:55</a>, <a class="txt" href="waybel18.html#E2:29"><i><font color="Green">E28</font></i></a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
<b>by </b><i><a class="ref" href="grcat_1.html#T12">GRCAT_1:12</a></i>
;<br/>
<a NAME="E10:29_7"/><b>then </b><i><font color="Green">E114</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="tmap_1.html#K4">incl</a> <span class="p3">(<span class="default"><a href="waybel18.html#K7" target="_self">Image</a> <font color="Maroon">F</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel18.html#E9:29"><i><font color="Green">E35</font></i></a>, , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">pp</font> = <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">S</font> <b>by </b><i><a class="txt" href="waybel18.html#E14:29"><i><font color="Green">E37</font></i></a></i>;<br/>
<a NAME="E12:29_7"/>
<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">y</font></span>)</span> <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/>
<a NAME="E13:29_7"/><b>then </b><i><font color="Green">E115</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</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="waybel18.html#K7" target="_self">Image</a> <font color="Maroon">F</font></span>)</span>
 
<b>by </b><i><a class="ref" href="waybel18.html#T3" target="_self">Th3</a>, </i>;<br/>
<a NAME="E14:29_7"/><b>then </b><i><font color="Green">E116</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tmap_1.html#K4">incl</a> <span class="p3">(<span class="default"><a href="waybel18.html#K7" target="_self">Image</a> <font color="Maroon">F</font></span>)</span></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i>, <a class="txt" href="waybel18.html#E11:29"><i><font color="Green">E36</font></i></a>, <a class="ref" href="funct_1.html#T21">FUNCT_1:21</a></i>;<br/>
<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="tmap_1.html#K4">incl</a> <span class="p3">(<span class="default"><a href="waybel18.html#K7" target="_self">Image</a> <font color="Maroon">F</font></span>)</span></span>)</span></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">y</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="partfun1.html#K1">*</a> <span class="p3">(<span class="default"><a href="tmap_1.html#K4">incl</a> <span class="p4">(<span class="default"><a href="waybel18.html#K7" target="_self">Image</a> <font color="Maroon">F</font></span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
<b>by </b><i><a class="txt" href="waybel18.html#E11:29"><i><font color="Green">E36</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="tmap_1.html#K4">incl</a> <span class="p4">(<span class="default"><a href="waybel18.html#K7" target="_self">Image</a> <font color="Maroon">F</font></span>)</span></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
<b>by </b><i><a class="ref" href="relat_1.html#T55">RELAT_1:55</a></i>
<br/>.= 
<font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="tmap_1.html#K4">incl</a> <span class="p4">(<span class="default"><a href="waybel18.html#K7" target="_self">Image</a> <font color="Maroon">F</font></span>)</span></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tmap_1.html#K4">incl</a> <span class="p3">(<span class="default"><a href="waybel18.html#K7" target="_self">Image</a> <font color="Maroon">F</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">pp</font></span>)</span>
<b>by </b><i><a class="txt" href="waybel18.html#E11:29"><i><font color="Green">E36</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon">h</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">y</font></span>)</span>
<b>by </b><i>, <a class="ref" href="tmap_1.html#T95">TMAP_1:95</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="grcat_1.html#K7">id</a> <font color="Maroon">T</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
<b>by </b><i><a class="txt" href="waybel18.html#E11:29"><i><font color="Green">E36</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a>, <a class="txt" href="waybel18.html#E2:29"><i><font color="Green">E28</font></i></a></i>
<br/>.= 
<font color="Maroon">y</font>
<b>by </b><i><a class="txt" href="waybel18.html#E8:29"><i><font color="Green">E33</font></i></a>, <a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>
;<br/>
<b>hence </b><a NAME="E16:29_7"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="tmap_1.html#K4">incl</a> <span class="p3">(<span class="default"><a href="waybel18.html#K7" target="_self">Image</a> <font color="Maroon">F</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E30:29"/><i><font color="Green">E117</font></i>: 
<font color="Maroon">h</font> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><a href="tmap_1.html#K4">incl</a> <span class="p2">(<span class="default"><a href="waybel18.html#K7" target="_self">Image</a> <font color="Maroon">F</font></span>)</span></span>)</span> is <a href="funct_1.html#V2">one-to-one</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="waybel18.html#K7" target="_self">Image</a> <font color="Maroon">F</font></span>)</span>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="waybel18.html#K7" target="_self">Image</a> <font color="Maroon">F</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_1.html#D1">WAYBEL_1:def 1</a><br/>

<b>assume </b><a NAME="E1:29_8"/><i><font color="Green">E118</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="tmap_1.html#K4">incl</a> <span class="p3">(<span class="default"><a href="waybel18.html#K7" target="_self">Image</a> <font color="Maroon">F</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="tmap_1.html#K4">incl</a> <span class="p3">(<span class="default"><a href="waybel18.html#K7" target="_self">Image</a> <font color="Maroon">F</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font>
 ;<br/>

<a NAME="E2:29_8"/><i><font color="Green">E119</font></i>: 
<font color="Maroon">x</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="waybel18.html#K7" target="_self">Image</a> <font color="Maroon">F</font></span>)</span>
 
;<br/>
<a NAME="E3:29_8"/><b>then </b><i><font color="Green">E120</font></i>: 
<font color="Maroon">x</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/>;<br/>
<a NAME="E4:29_8"/><i><font color="Green">E121</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="tmap_1.html#K4">incl</a> <span class="p2">(<span class="default"><a href="waybel18.html#K7" target="_self">Image</a> <font color="Maroon">F</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel18.html#E9:29"><i><font color="Green">E35</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E5:29_8"/><i><font color="Green">E122</font></i>: 
<font color="Maroon">y</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="waybel18.html#K7" target="_self">Image</a> <font color="Maroon">F</font></span>)</span>
 
;<br/>
<a NAME="E6:29_8"/><b>then </b><i><font color="Green">E123</font></i>: 
<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/>;<br/>
<a NAME="E7:29_8"/><i><font color="Green">E124</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="tmap_1.html#K4">incl</a> <span class="p2">(<span class="default"><a href="waybel18.html#K7" target="_self">Image</a> <font color="Maroon">F</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">a</font> = <font color="Maroon">x</font>, <font color="Maroon">b</font> = <font color="Maroon">y</font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">S</font> <b>by </b><i><a class="txt" href="waybel18.html#E11:29"><i><font color="Green">E36</font></i></a>, <a class="ref" href="waybel18.html#T3" target="_self">Th3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font>, <font color="Maroon">y'</font> = <font color="Maroon">y</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="waybel18.html#K7" target="_self">Image</a> <font color="Maroon">f</font></span>)</span> <b>by </b><i>, <a class="txt" href="waybel18.html#E11:29"><i><font color="Green">E36</font></i></a>, <a class="ref" href="waybel18.html#T3" target="_self">Th3</a>, </i>;<br/>
<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x'</font> = 
<font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tmap_1.html#K4">incl</a> <span class="p3">(<span class="default"><a href="waybel18.html#K7" target="_self">Image</a> <font color="Maroon">F</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i><a class="ref" href="tmap_1.html#T95">TMAP_1:95</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="tmap_1.html#K4">incl</a> <span class="p3">(<span class="default"><a href="waybel18.html#K7" target="_self">Image</a> <font color="Maroon">F</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font>
<b>by </b><i>, <a class="txt" href="waybel18.html#E14:29"><i><font color="Green">E37</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tmap_1.html#K4">incl</a> <span class="p3">(<span class="default"><a href="waybel18.html#K7" target="_self">Image</a> <font color="Maroon">F</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font></span>)</span>
<b>by </b><i>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
<b>by </b><i><a class="ref" href="tmap_1.html#T95">TMAP_1:95</a></i>
<br/>.= 
<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y'</font>
<b>by </b><i>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
;<br/>
<b>hence </b><a NAME="E11:29_8"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 <b>by </b><i>, <a class="ref" href="waybel_1.html#D1">WAYBEL_1:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E31:29"/>
 <a href="tmap_1.html#K4">incl</a> <span class="p1">(<span class="default"><a href="waybel18.html#K7" target="_self">Image</a> <font color="Maroon">F</font></span>)</span> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i><a class="ref" href="tmap_1.html#T98">TMAP_1:98</a></i>;<br/>
<a NAME="E32:29"/><b>then </b><i><font color="Green">E125</font></i>: 
<font color="Maroon">h</font> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><a href="tmap_1.html#K4">incl</a> <span class="p2">(<span class="default"><a href="waybel18.html#K7" target="_self">Image</a> <font color="Maroon">F</font></span>)</span></span>)</span> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i>, <a class="ref" href="tops_2.html#T58">TOPS_2:58</a></i>;<br/>
<a NAME="E33:29"/>
for <font color="Olive">P</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="waybel18.html#K7" target="_self">Image</a> <font color="Maroon">F</font></span>)</span>  st <font color="Olive">P</font> is <a href="pre_topc.html#V3">open</a> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="partfun1.html#K1">*</a> <span class="p3">(<span class="default"><a href="tmap_1.html#K4">incl</a> <span class="p4">(<span class="default"><a href="waybel18.html#K7" target="_self">Image</a> <font color="Maroon">F</font></span>)</span></span>)</span></span>)</span> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="relset_1.html#K11">"</a> <font color="Olive">P</font> is <a href="pre_topc.html#V3">open</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_9"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">P</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="waybel18.html#K7" target="_self">Image</a> <font color="Maroon">F</font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:29_9"/>
<font color="Maroon">P</font> is <a href="pre_topc.html#V3">open</a>
 ;<br/>

<a NAME="E2:29_9"/><b>then </b>
<font color="Maroon">P</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="waybel18.html#K7" target="_self">Image</a> <font color="Maroon">F</font></span>)</span>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon">Q</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E4:29_9"/><i><font color="Green">E127</font></i>: 
<font color="Maroon">Q</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">S</font>
 <b>and </b><br/><a NAME="E5:29_9"/><i><font color="Green">E128</font></i>: 
<font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Q</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><a href="waybel18.html#K7" target="_self">Image</a> <font color="Maroon">F</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="pre_topc.html#D9">PRE_TOPC:def 9</a></i>;<br/>
<b>reconsider </b><font color="Maroon">Q</font> = <font color="Maroon">Q</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font> ;<br/>
<a NAME="E7:29_9"/><i><font color="Green">E129</font></i>: 
<font color="Maroon">Q</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="txt" href="waybel18.html#E9:29"><i><font color="Green">E35</font></i></a>, <a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>
<a NAME="E8:29_9"/>
<font color="Maroon">p</font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i><a class="ref" href="tmap_1.html#T98">TMAP_1:98</a></i>;<br/>
<a NAME="E9:29_9"/><b>then </b>
<font color="Maroon">p</font> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><a href="waybel18.html#K8" target="_self">corestr</a> <font color="Maroon">f</font></span>)</span> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i>, <a class="ref" href="tops_2.html#T58">TOPS_2:58</a></i>;<br/>
<a NAME="E10:29_9"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="waybel18.html#K8" target="_self">corestr</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="relset_1.html#K11">"</a> <font color="Maroon">Q</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="txt" href="waybel18.html#E14:29"><i><font color="Green">E37</font></i></a>, <a class="ref" href="tops_2.html#T55">TOPS_2:55</a></i>;<br/>
<a NAME="E11:29_9"/><b>then </b><i><font color="Green">E130</font></i>: 
<font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">Q</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i/>;<br/>
<a NAME="E12:29_9"/><i><font color="Green">E131</font></i>: 
<span class="p1">(<span class="default"><a href="tmap_1.html#K4">incl</a> <span class="p2">(<span class="default"><a href="waybel18.html#K7" target="_self">Image</a> <font color="Maroon">F</font></span>)</span></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_9_1"><b>proof </b></a><div class="add">

<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a>,<a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a>
<div class="add"><b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:29_9_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="tmap_1.html#K4">incl</a> <span class="p2">(<span class="default"><a href="waybel18.html#K7" target="_self">Image</a> <font color="Maroon">F</font></span>)</span></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">P</font>
 ;<br/><b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:29_9_1_1"/><i><font color="Green">E132</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="tmap_1.html#K4">incl</a> <span class="p2">(<span class="default"><a href="waybel18.html#K7" target="_self">Image</a> <font color="Maroon">F</font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E4:29_9_1_1"/><i><font color="Green">E133</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 <b>and </b><br/><a NAME="E5:29_9_1_1"/><i><font color="Green">E134</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="tmap_1.html#K4">incl</a> <span class="p2">(<span class="default"><a href="waybel18.html#K7" target="_self">Image</a> <font color="Maroon">F</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/><a NAME="E6:29_9_1_1"/><i><font color="Green">E135</font></i>: 
<font color="Maroon">x</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="waybel18.html#K7" target="_self">Image</a> <font color="Maroon">F</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E7:29_9_1_1"/><b>then </b>
<font color="Maroon">x</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/>;<br/><b>then reconsider </b><font color="Maroon">xx</font> = <font color="Maroon">x</font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">S</font> ;<br/><a NAME="E9:29_9_1_1"/>
<span class="p1">(<span class="default"><a href="tmap_1.html#K4">incl</a> <span class="p2">(<span class="default"><a href="waybel18.html#K7" target="_self">Image</a> <font color="Maroon">F</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">xx</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="txt" href="waybel18.html#E15:29"><i><font color="Green">E43</font></i></a>, <a class="ref" href="tmap_1.html#T95">TMAP_1:95</a></i>;<br/><b>hence </b><a NAME="E10:29_9_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 <b>by </b><i>, </i>;<br/></div>
<b>end;</b></div>

<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="E2:29_9_1"/><i><font color="Green">E136</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 ;<br/>

<a NAME="E3:29_9_1"/><b>then </b><i><font color="Green">E137</font></i>: 
<font color="Maroon">y</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="waybel18.html#K7" target="_self">Image</a> <font color="Maroon">F</font></span>)</span>
 
;<br/>
<a NAME="E4:29_9_1"/><b>then </b><i><font color="Green">E138</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="tmap_1.html#K4">incl</a> <span class="p2">(<span class="default"><a href="waybel18.html#K7" target="_self">Image</a> <font color="Maroon">F</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E5:29_9_1"/>
<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>, </i>;<br/>
<b>then reconsider </b><font color="Maroon">yy</font> = <font color="Maroon">y</font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">S</font> ;<br/>
<a NAME="E7:29_9_1"/>
<font color="Maroon">yy</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="tmap_1.html#K4">incl</a> <span class="p2">(<span class="default"><a href="waybel18.html#K7" target="_self">Image</a> <font color="Maroon">F</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
 
<b>by </b><i>, <a class="ref" href="tmap_1.html#T95">TMAP_1:95</a></i>;<br/>
<b>hence </b><a NAME="E8:29_9_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="tmap_1.html#K4">incl</a> <span class="p2">(<span class="default"><a href="waybel18.html#K7" target="_self">Image</a> <font color="Maroon">F</font></span>)</span></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">P</font>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E13:29_9"/><i><font color="Green">E139</font></i>: 
<font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">Q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">P</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_9_2"><b>proof </b></a><div class="add">

<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a>,<a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</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="E1:29_9_2_1"/><i><font color="Green">E140</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">Q</font>
 ;<br/><a NAME="E2:29_9_2_1"/><b>then </b><i><font color="Green">E141</font></i>: 
<font color="Maroon">x</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_1.html#D13">FUNCT_1:def 13</a></i>;<br/><a NAME="E3:29_9_2_1"/><i><font color="Green">E142</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> <font color="Maroon">Q</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/><a NAME="E4:29_9_2_1"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</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/><a NAME="E5:29_9_2_1"/><b>then </b><i><font color="Green">E143</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">S</font>
 ;<br/><a NAME="E6:29_9_2_1"/><b>then </b><i><font color="Green">E144</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>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E7:29_9_2_1"/><i><font color="Green">E145</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>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="txt" href="waybel18.html#E16:29"><i><font color="Green">E44</font></i></a>, <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> <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> = 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="grcat_1.html#K7">id</a> <font color="Maroon">T</font></span>)</span> <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>, <a class="txt" href="waybel18.html#E2:29"><i><font color="Green">E28</font></i></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/><a NAME="E9:29_9_2_1"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</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="waybel18.html#D1" target="_self">Def1</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E10:29_9_2_1"/><b>then </b>
<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 <span class="p1">(<span class="default"><a href="waybel18.html#K7" target="_self">Image</a> <font color="Maroon">F</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E11:29_9_2_1"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="waybel18.html#K7" target="_self">Image</a> <font color="Maroon">F</font></span>)</span>
 ;<br/><a NAME="E12:29_9_2_1"/><b>then </b><i><font color="Green">E146</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> <font color="Maroon">P</font>
 <b>by </b><i><a class="txt" href="waybel18.html#E11:29"><i><font color="Green">E36</font></i></a>, <a class="txt" href="waybel18.html#E15:29"><i><font color="Green">E43</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><font color="Maroon">h</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> = 
<span class="p1">(<span class="default"><a href="grcat_1.html#K7">id</a> <font color="Maroon">T</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a>, <a class="txt" href="waybel18.html#E2:29"><i><font color="Green">E28</font></i></a></i>
<br/>.= 
<font color="Maroon">x</font>
<b>by </b><i>, <a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>
;<br/><b>hence </b><a NAME="E14:29_9_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">h</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">P</font>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon">x</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="E2:29_9_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">h</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">P</font>
 ;<br/>

<b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:29_9_2"/><i><font color="Green">E147</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">h</font>
 <b>and </b><br/><a NAME="E5:29_9_2"/><i><font color="Green">E148</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 <b>and </b><br/><a NAME="E6:29_9_2"/><i><font color="Green">E149</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E7:29_9_2"/><i><font color="Green">E150</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">h</font>
 
<b>by </b><i>, <a class="txt" href="waybel18.html#E15:29"><i><font color="Green">E43</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E8:29_9_2"/><i><font color="Green">E151</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font>
 
<b>by </b><i><a class="txt" href="waybel18.html#E11:29"><i><font color="Green">E36</font></i></a>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E9:29_9_2"/>
<font color="Maroon">y</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="waybel18.html#K7" target="_self">Image</a> <font color="Maroon">F</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E10:29_9_2"/><b>then </b><i><font color="Green">E152</font></i>: 
<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/>;<br/>
<a NAME="E11:29_9_2"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</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="txt" href="waybel18.html#E16:29"><i><font color="Green">E44</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">a</font> = <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>, <font color="Maroon">b</font> = <font color="Maroon">y</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="waybel18.html#K7" target="_self">Image</a> <font color="Maroon">f</font></span>)</span> <b>by </b><i>, <a class="ref" href="waybel18.html#D1" target="_self">Def1</a>, </i>;<br/>
<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font> = 
<font color="Maroon">h</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#T72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="grcat_1.html#K7">id</a> <font color="Maroon">T</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i>, <a class="txt" href="waybel18.html#E16:29"><i><font color="Green">E44</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a>, <a class="txt" href="waybel18.html#E2:29"><i><font color="Green">E28</font></i></a></i>
<br/>.= 
<font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
<b>by </b><i><a class="txt" href="waybel18.html#E15:29"><i><font color="Green">E43</font></i></a>, <a class="txt" href="waybel18.html#E16:29"><i><font color="Green">E44</font></i></a>, <a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>
<br/>.= 
<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font>
<b>by </b><i>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
;<br/>
<a NAME="E14:29_9_2"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font>
 
<b>by </b><i>, , <a class="ref" href="waybel_1.html#D1">WAYBEL_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E15:29_9_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">Q</font>
 <b>by </b><i>, <a class="txt" href="waybel18.html#E16:29"><i><font color="Green">E44</font></i></a>, <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E14:29_9"/>
<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="tmap_1.html#K4">incl</a> <span class="p3">(<span class="default"><a href="waybel18.html#K7" target="_self">Image</a> <font color="Maroon">F</font></span>)</span></span>)</span></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">P</font>
 
<b>by </b><i><a class="ref" href="waybel18.html#T3" target="_self">Th3</a>, <a class="ref" href="relat_1.html#T159">RELAT_1:159</a></i>;<br/>
<b>hence </b><a NAME="E15:29_9"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="partfun1.html#K1">*</a> <span class="p3">(<span class="default"><a href="tmap_1.html#K4">incl</a> <span class="p4">(<span class="default"><a href="waybel18.html#K7" target="_self">Image</a> <font color="Maroon">F</font></span>)</span></span>)</span></span>)</span> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="relset_1.html#K11">"</a> <font color="Maroon">P</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="txt" href="waybel18.html#E5:29"><i><font color="Green">E32</font></i></a>, <a class="txt" href="waybel18.html#E8:29"><i><font color="Green">E33</font></i></a>, , , <a class="ref" href="tops_2.html#T67">TOPS_2:67</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E34:29"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="tmap_1.html#K4">incl</a> <span class="p3">(<span class="default"><a href="waybel18.html#K7" target="_self">Image</a> <font color="Maroon">F</font></span>)</span></span>)</span></span>)</span> <a href="tops_2.html#NK3" target="_self">"</a>  is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i><a class="ref" href="tops_2.html#T55">TOPS_2:55</a></i>;<br/>
<a NAME="E35:29"/><b>then </b>
<font color="Maroon">h</font> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><a href="tmap_1.html#K4">incl</a> <span class="p2">(<span class="default"><a href="waybel18.html#K7" target="_self">Image</a> <font color="Maroon">F</font></span>)</span></span>)</span> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a> 
 
<b>by </b><i><a class="txt" href="waybel18.html#E3:29"><i><font color="Green">E31</font></i></a>, <a class="txt" href="waybel18.html#E5:29"><i><font color="Green">E32</font></i></a>, <a class="txt" href="waybel18.html#E8:29"><i><font color="Green">E33</font></i></a>, , <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/>
<b>hence </b><a NAME="E36:29"/>
 <a href="waybel18.html#K7" target="_self">Image</a> <font color="Maroon">F</font>,<font color="Maroon">T</font> <a href="t_0topsp.html#R1">are_homeomorphic</a> 
 <b>by </b><i><a class="ref" href="t_0topsp.html#D1">T_0TOPSP:def 1</a></i>;<br/>


</div>
