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

<b>let </b><font color="Maroon">I</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">J</font> be  <a href="waybel_3.html#V4">non-Empty</a> <a href="waybel18.html#V1" target="_self">TopSpace-yielding</a>  <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">I</font>;<br/>



<b>assume </b><a NAME="E1:20"/><i><font color="Green">E27</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> holds  <font color="Maroon">J</font> <a href="waybel18.html#K4" target="_self">.</a> <font color="Olive">i</font> is <a href="waybel18.html#V2" target="_self">injective</a>
 ;<br/>

<b>let </b><font color="Maroon">X</font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel18.html#D5" target="_self">WAYBEL18:def 5</a><br/><b>let </b><font color="Maroon">f</font> be   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X</font>,<span class="p1">(<span class="default"><a href="waybel18.html#K3" target="_self">product</a> <font color="Maroon">J</font></span>)</span>;<br/>



<b>assume </b><a NAME="E2:20"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">f</font> is <a href="pre_topc.html#V5">continuous</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="E3:20"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">X</font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">Y</font>
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">i1</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> st <br/>( <font color="Olive">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp;  ex <font color="Olive">g</font> being  <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">Y</font>,<span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="waybel18.html#K4" target="_self">.</a> <font color="Olive">i1</font></span>)</span> st <br/>( <font color="Olive">g</font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Olive">g</font> <a href="partfun1.html#K2">|</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel18.html#K6" target="_self">proj</a> <font color="Maroon">J</font>,<font color="Olive">i1</font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">g</font> ) );<br/>
<a NAME="E4:20"/><i><font color="Green">E32</font></i>: 
for <font color="Olive">i</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> holds <br/> ex <font color="Olive">u</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">i</font>,<font color="Olive">u</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:20_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">i1</font> = <font color="Maroon">i</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> ;<br/>
<a NAME="E3:20_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">J</font> <a href="waybel18.html#K4" target="_self">.</a> <font color="Maroon">i1</font> is <a href="waybel18.html#V2" target="_self">injective</a>
 
<b>by </b><i/>;<br/>
<a NAME="E4:20_1"/>
<span class="p1">(<span class="default"><a href="waybel18.html#K6" target="_self">proj</a> <font color="Maroon">J</font>,<font color="Maroon">i1</font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i>, </i>;<br/>
<b>then consider </b><font color="Maroon">g</font> being   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">Y</font>,<span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="waybel18.html#K4" target="_self">.</a> <font color="Maroon">i1</font></span>)</span><b> such that </b><br/><a NAME="E6:20_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">g</font> is <a href="pre_topc.html#V5">continuous</a>
 <b>and </b><br/><a NAME="E7:20_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel18.html#K6" target="_self">proj</a> <font color="Maroon">J</font>,<font color="Maroon">i1</font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font>
 <b>by </b><i>, , </i>;<br/>
<b>take </b>
<font color="Maroon">g</font>
;<br/>

<b>take </b>
<font color="Maroon">i1</font>
;<br/>

<b>thus </b><a NAME="E8:20_1"/>
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> &amp;  ex <font color="Olive">g</font> being  <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">Y</font>,<span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="waybel18.html#K4" target="_self">.</a> <font color="Maroon">i1</font></span>)</span> st <br/>( <font color="Olive">g</font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Olive">g</font> <a href="partfun1.html#K2">|</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel18.html#K6" target="_self">proj</a> <font color="Maroon">J</font>,<font color="Maroon">i1</font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font> &amp; <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Olive">g</font> ) )
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">G</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E6:20"/><i><font color="Green">E37</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font>
 <b>and </b><br/><a NAME="E7:20"/><i><font color="Green">E43</font></i>: 
for <font color="Olive">i</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">i</font>,<font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font>]
 <b>from </b><i><a class="ref" href="zfrefle1.html#S1">ZFREFLE1:sch 1</a>();<br/></i>
<a NAME="E8:20"/><i><font color="Green">E44</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1"><a href="funct_6.html#K6">&lt;:</a><span class="default"><font color="Maroon">G</font></span><a href="funct_6.html#K6">:&gt;</a></span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_2"><b>proof </b></a><div class="add">

<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="E1:20_2"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1"><a href="funct_6.html#K6">&lt;:</a><span class="default"><font color="Maroon">G</font></span><a href="funct_6.html#K6">:&gt;</a></span>
 ;<br/>

<b>consider </b><font color="Maroon">j</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:20_2"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font><b> such that </b><br/><a NAME="E5:20_2"/>
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font>
 <b>and </b><br/><a NAME="E6:20_2"/><i><font color="Green">E74</font></i>: 
 ex <font color="Olive">g</font> being  <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">Y</font>,<span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="waybel18.html#K4" target="_self">.</a> <font color="Maroon">i</font></span>)</span> st <br/>( <font color="Olive">g</font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Olive">g</font> <a href="partfun1.html#K2">|</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel18.html#K6" target="_self">proj</a> <font color="Maroon">J</font>,<font color="Maroon">i</font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font> &amp; <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Olive">g</font> )
 <b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">g</font> being   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">Y</font>,<span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="waybel18.html#K4" target="_self">.</a> <font color="Maroon">i</font></span>)</span><b> such that </b><br/><a NAME="E8:20_2"/>
( <font color="Maroon">g</font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel18.html#K6" target="_self">proj</a> <font color="Maroon">J</font>,<font color="Maroon">i</font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font> )
 <b>and </b><br/><a NAME="E9:20_2"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font>
 <b>by </b><i/>;<br/>
<a NAME="E10:20_2"/>
<font color="Maroon">g</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">G</font>
 
<b>by </b><i>, , , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E11:20_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font>
 
<b>by </b><i>, <a class="ref" href="funct_6.html#T52">FUNCT_6:52</a></i>;<br/>
<b>hence </b><a NAME="E12:20_2"/>
<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/>


</div><b>end;</b></div>
<a NAME="E9:20"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1"><a href="funct_6.html#K6">&lt;:</a><span class="default"><font color="Maroon">G</font></span><a href="funct_6.html#K6">:&gt;</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_3"><b>proof </b></a><div class="add">

<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="E1:20_3"/><i><font color="Green">E76</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/>

<b>consider </b><font color="Maroon">i</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:20_3"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon">j</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font><b> such that </b><br/><a NAME="E5:20_3"/>
<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font>
 <b>and </b><br/><a NAME="E6:20_3"/><i><font color="Green">E78</font></i>: 
 ex <font color="Olive">g</font> being  <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">Y</font>,<span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="waybel18.html#K4" target="_self">.</a> <font color="Maroon">j</font></span>)</span> st <br/>( <font color="Olive">g</font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Olive">g</font> <a href="partfun1.html#K2">|</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel18.html#K6" target="_self">proj</a> <font color="Maroon">J</font>,<font color="Maroon">j</font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font> &amp; <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Olive">g</font> )
 <b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">g</font> being   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">Y</font>,<span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="waybel18.html#K4" target="_self">.</a> <font color="Maroon">j</font></span>)</span><b> such that </b><br/><a NAME="E8:20_3"/>
<font color="Maroon">g</font> is <a href="pre_topc.html#V5">continuous</a>
 <b>and </b><br/><a NAME="E9:20_3"/>
<font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel18.html#K6" target="_self">proj</a> <font color="Maroon">J</font>,<font color="Maroon">j</font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E10:20_3"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font>
 <b>by </b><i/>;<br/>
<a NAME="E11:20_3"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">g</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">G</font>
 
<b>by </b><i>, , , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E12:20_3"/>
for <font color="Olive">f'</font> being  <a href="funct_1.html#NM1">Function</a>  st <font color="Olive">f'</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">G</font> holds <br/><font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Olive">f'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">f'</font> be   <a href="funct_1.html#NM1">Function</a>;<br/>

<b>assume </b><a NAME="E1:20_3_1"/>
<font color="Maroon">f'</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">G</font>
 ;<br/>

<b>then consider </b><font color="Maroon">k</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:20_3_1"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">G</font>
 <b>and </b><br/><a NAME="E4:20_3_1"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">f'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon">i1</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font><b> such that </b><br/><a NAME="E6:20_3_1"/>
<font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 <b>and </b><br/><a NAME="E7:20_3_1"/><i><font color="Green">E83</font></i>: 
 ex <font color="Olive">ff</font> being  <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">Y</font>,<span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="waybel18.html#K4" target="_self">.</a> <font color="Maroon">i1</font></span>)</span> 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#K2">|</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel18.html#K6" target="_self">proj</a> <font color="Maroon">J</font>,<font color="Maroon">i1</font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font> &amp; <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Olive">ff</font> )
 <b>by </b><i>, , </i>;<br/>
<b>consider </b><font color="Maroon">ff</font> being   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">Y</font>,<span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="waybel18.html#K4" target="_self">.</a> <font color="Maroon">i1</font></span>)</span><b> such that </b><br/><a NAME="E9:20_3_1"/>
<font color="Maroon">ff</font> is <a href="pre_topc.html#V5">continuous</a>
 <b>and </b><br/><a NAME="E10:20_3_1"/>
<font color="Maroon">ff</font> <a href="partfun1.html#K2">|</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel18.html#K6" target="_self">proj</a> <font color="Maroon">J</font>,<font color="Maroon">i1</font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E11:20_3_1"/><i><font color="Green">E95</font></i>: 
<font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">ff</font>
 <b>by </b><i/>;<br/>
<b>thus </b><a NAME="E12:20_3_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f'</font>
 <b>by </b><i>, , <a class="txt" href="waybel18.html#E1:20"><i><font color="Green">E27</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E13:20_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1"><a href="funct_6.html#K6">&lt;:</a><span class="default"><font color="Maroon">G</font></span><a href="funct_6.html#K6">:&gt;</a></span>
 <b>by </b><i><a class="ref" href="waybel18.html#T2" target="_self">Th2</a>, <a class="ref" href="funct_6.html#T53">FUNCT_6:53</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:20"/><b>then </b><i><font color="Green">E96</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1"><a href="funct_6.html#K6">&lt;:</a><span class="default"><font color="Maroon">G</font></span><a href="funct_6.html#K6">:&gt;</a></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="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<a NAME="E11:20"/><i><font color="Green">E97</font></i>: 
<font color="Maroon">G</font> is <a href="funcop_1.html#V1">Function-yielding</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">j</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="funcop_1.html#D6">FUNCOP_1:def 6</a><br/>

<b>assume </b><a NAME="E1:20_4"/>
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">G</font>
 ;<br/>

<b>then consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font><b> such that </b><br/><a NAME="E3:20_4"/>
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font>
 <b>and </b><br/><a NAME="E4:20_4"/><i><font color="Green">E98</font></i>: 
 ex <font color="Olive">g</font> being  <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">Y</font>,<span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="waybel18.html#K4" target="_self">.</a> <font color="Maroon">i</font></span>)</span> st <br/>( <font color="Olive">g</font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Olive">g</font> <a href="partfun1.html#K2">|</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel18.html#K6" target="_self">proj</a> <font color="Maroon">J</font>,<font color="Maroon">i</font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font> &amp; <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Olive">g</font> )
 <b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">g</font> being   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">Y</font>,<span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="waybel18.html#K4" target="_self">.</a> <font color="Maroon">i</font></span>)</span><b> such that </b><br/><a NAME="E6:20_4"/>
( <font color="Maroon">g</font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel18.html#K6" target="_self">proj</a> <font color="Maroon">J</font>,<font color="Maroon">i</font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font> )
 <b>and </b><br/><a NAME="E7:20_4"/><i><font color="Green">E99</font></i>: 
<font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font>
 <b>by </b><i/>;<br/>
<b>thus </b><a NAME="E8:20_4"/>
<font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> is   <a href="funct_1.html#NM1">Function</a>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E12:20"/><i><font color="Green">E100</font></i>: 
 <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K3">rngs</a> <font color="Maroon">G</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">J</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_5"><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="E1:20_5"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K3">rngs</a> <font color="Maroon">G</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">h</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E3:20_5"/><i><font color="Green">E102</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font>
 <b>and </b><br/><a NAME="E4:20_5"/><i><font color="Green">E103</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="funct_6.html#K3">rngs</a> <font color="Maroon">G</font></span>)</span>
 <b>and </b><br/><a NAME="E5:20_5"/><i><font color="Green">E104</font></i>: 
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>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="funct_6.html#K3">rngs</a> <font color="Maroon">G</font></span>)</span> holds <br/><font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="funct_6.html#K3">rngs</a> <font color="Maroon">G</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>
 <b>by </b><i><a class="ref" href="card_3.html#D5">CARD_3:def 5</a></i>;<br/>
<i><font color="Green">E105</font></i>:  <a href="relat_1.html#K1">dom</a> <font color="Maroon">h</font> = 
<font color="Maroon">I</font>
<b>by </b><i>, , <a class="ref" href="waybel18.html#T2" target="_self">Th2</a>, <a class="ref" href="funct_6.html#T90">FUNCT_6:90</a></i>
<br/>.= 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">J</font></span>)</span>
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>
;<br/>
<a NAME="E7:20_5"/>
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>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">J</font></span>)</span> holds <br/><font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">J</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_5_2"><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="E1:20_5_2"/><i><font color="Green">E106</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">J</font></span>)</span>
 ;<br/>

<a NAME="E2:20_5_2"/><b>then </b><i><font color="Green">E107</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> <span class="p1">(<span class="default"><a href="funct_6.html#K3">rngs</a> <font color="Maroon">G</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="waybel18.html#T2" target="_self">Th2</a>, , </i>;<br/>
<a NAME="E3:20_5_2"/><i><font color="Green">E108</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 
<b>by </b><i>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font><b> such that </b><br/><a NAME="E5:20_5_2"/><i><font color="Green">E109</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>and </b><br/><a NAME="E6:20_5_2"/><i><font color="Green">E110</font></i>: 
 ex <font color="Olive">g</font> being  <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">Y</font>,<span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="waybel18.html#K4" target="_self">.</a> <font color="Maroon">i</font></span>)</span> st <br/>( <font color="Olive">g</font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Olive">g</font> <a href="partfun1.html#K2">|</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel18.html#K6" target="_self">proj</a> <font color="Maroon">J</font>,<font color="Maroon">i</font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font> &amp; <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">g</font> )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">g</font> being   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">Y</font>,<span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="waybel18.html#K4" target="_self">.</a> <font color="Maroon">i</font></span>)</span><b> such that </b><br/><a NAME="E8:20_5_2"/>
( <font color="Maroon">g</font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel18.html#K6" target="_self">proj</a> <font color="Maroon">J</font>,<font color="Maroon">i</font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font> )
 <b>and </b><br/><a NAME="E9:20_5_2"/><i><font color="Green">E111</font></i>: 
<font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font>
 <b>by </b><i/>;<br/>
<a NAME="E10:20_5_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">G</font>
 
<b>by </b><i>, , <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E11:20_5_2"/><b>then </b><i><font color="Green">E112</font></i>: 
<span class="p1">(<span class="default"><a href="funct_6.html#K3">rngs</a> <font color="Maroon">G</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">g</font>
 
<b>by </b><i><a class="txt" href="waybel18.html#E3:20"><i><font color="Green">E31</font></i></a>, <a class="ref" href="funct_6.html#T31">FUNCT_6:31</a></i>;<br/>
<b>consider </b><font color="Maroon">R</font> being    <a href="struct_0.html#L1">1-sorted</a> <b> such that </b><br/><a NAME="E13:20_5_2"/><i><font color="Green">E113</font></i>: 
<font color="Maroon">R</font> <a href="hidden.html#R1">=</a> <font color="Maroon">J</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>and </b><br/><a NAME="E14:20_5_2"/><i><font color="Green">E114</font></i>: 
<span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">J</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>
 <b>by </b><i><a class="txt" href="waybel18.html#E2:20"><i><font color="Green">E28</font></i></a>, <a class="ref" href="pralg_1.html#D13">PRALG_1:def 13</a></i>;<br/>
<b>thus </b><a NAME="E15:20_5_2"/>
<font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">J</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#E1:20"><i><font color="Green">E27</font></i></a>, , <a class="txt" href="waybel18.html#E4:20"><i><font color="Green">E32</font></i></a>, , </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:20_5"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">J</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="card_3.html#D5">CARD_3:def 5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E13:20"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1"><a href="funct_6.html#K6">&lt;:</a><span class="default"><font color="Maroon">G</font></span><a href="funct_6.html#K6">:&gt;</a></span> <a href="tarski.html#R1">c=</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K3">rngs</a> <font color="Maroon">G</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_6.html#T49">FUNCT_6:49</a></i>;<br/>
<a NAME="E14:20"/><b>then </b><i><font color="Green">E115</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1"><a href="funct_6.html#K6">&lt;:</a><span class="default"><font color="Maroon">G</font></span><a href="funct_6.html#K6">:&gt;</a></span> <a href="tarski.html#R1">c=</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">J</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E15:20"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <span class="p1"><a href="funct_6.html#K6">&lt;:</a><span class="default"><font color="Maroon">G</font></span><a href="funct_6.html#K6">:&gt;</a></span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel18.html#K3" target="_self">product</a> <font color="Maroon">J</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">h</font> = <span class="p1"><a href="funct_6.html#K6">&lt;:</a><span class="default"><font color="Maroon">G</font></span><a href="funct_6.html#K6">:&gt;</a></span> as   <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>,the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel18.html#K3" target="_self">product</a> <font color="Maroon">J</font></span>)</span> <b>by </b><i>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/>
<a NAME="E17:20"/><i><font color="Green">E116</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</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/>
<i><font color="Green">E117</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#K2">|</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">h</font></span>)</span> <a href="subset_1.html#K8">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>
<b>by </b><i><a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>
<br/>.= 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font> <a href="xboole_0.html#K3">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
<br/>.= 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>
<b>by </b><i><a class="ref" href="waybel18.html#T2" target="_self">Th2</a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</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/>
<a NAME="E19:20"/><i><font color="Green">E118</font></i>: 
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>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="partfun1.html#K2">|</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="partfun1.html#K2">|</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</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="20_7"><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="E1:20_7"/><i><font color="Green">E119</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">h</font> <a href="partfun1.html#K2">|</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font></span>)</span>
 ;<br/>

<a NAME="E2:20_7"/><b>then </b><i><font color="Green">E120</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">X</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:20_7"/><b>then </b><i><font color="Green">E121</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>
 
<b>by </b><i><a class="ref" href="waybel18.html#T2" target="_self">Th2</a></i>;<br/>
<a NAME="E4:20_7"/>
<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="E5:20_7"/><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#K3" target="_self">product</a> <font color="Maroon">J</font></span>)</span>
 
;<br/>
<a NAME="E6:20_7"/><b>then </b><i><font color="Green">E122</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="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">J</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">y</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E8:20_7"/><i><font color="Green">E123</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 <b>and </b><br/><a NAME="E9:20_7"/><i><font color="Green">E124</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">J</font></span>)</span>
 <b>and </b><br/><a NAME="E10:20_7"/>
for <font color="Olive">i</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">J</font></span>)</span> holds <br/><font color="Maroon">y</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">J</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font>
 <b>by </b><i><a class="ref" href="card_3.html#D5">CARD_3:def 5</a></i>;<br/>
<a NAME="E11:20_7"/><i><font color="Green">E125</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#K2">|</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">h</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T99">RELAT_1:99</a></i>;<br/>
<a NAME="E12:20_7"/>
<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="partfun1.html#K2">|</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font></span>)</span> <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> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="partfun1.html#K2">|</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E13:20_7"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="partfun1.html#K2">|</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font></span>)</span> <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="txt" href="waybel18.html#E4:20"><i><font color="Green">E32</font></i></a></i>;<br/>
<a NAME="E14:20_7"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="partfun1.html#K2">|</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font></span>)</span> <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#K3" target="_self">product</a> <font color="Maroon">J</font></span>)</span>
 
;<br/>
<a NAME="E15:20_7"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="partfun1.html#K2">|</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">J</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">z</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E17:20_7"/><i><font color="Green">E127</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="partfun1.html#K2">|</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font>
 <b>and </b><br/><a NAME="E18:20_7"/><i><font color="Green">E128</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">J</font></span>)</span>
 <b>and </b><br/><a NAME="E19:20_7"/>
for <font color="Olive">i</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">J</font></span>)</span> holds <br/><font color="Maroon">z</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">J</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font>
 <b>by </b><i><a class="ref" href="card_3.html#D5">CARD_3:def 5</a></i>;<br/>
<a NAME="E20:20_7"/>
for <font color="Olive">j</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">y</font> holds <br/><font color="Maroon">y</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_7_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:20_7_1"/>
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">y</font>
 ;<br/>

<a NAME="E2:20_7_1"/><b>then </b><i><font color="Green">E129</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 
<b>by </b><i><a class="txt" href="waybel18.html#E3:20"><i><font color="Green">E31</font></i></a>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font><b> such that </b><br/><a NAME="E4:20_7_1"/><i><font color="Green">E130</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font>
 <b>and </b><br/><a NAME="E5:20_7_1"/><i><font color="Green">E131</font></i>: 
 ex <font color="Olive">g</font> being  <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">Y</font>,<span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="waybel18.html#K4" target="_self">.</a> <font color="Maroon">i</font></span>)</span> st <br/>( <font color="Olive">g</font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Olive">g</font> <a href="partfun1.html#K2">|</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel18.html#K6" target="_self">proj</a> <font color="Maroon">J</font>,<font color="Maroon">i</font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font> &amp; <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Olive">g</font> )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">g</font> being   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">Y</font>,<span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="waybel18.html#K4" target="_self">.</a> <font color="Maroon">i</font></span>)</span><b> such that </b><br/><a NAME="E7:20_7_1"/>
<font color="Maroon">g</font> is <a href="pre_topc.html#V5">continuous</a>
 <b>and </b><br/><a NAME="E8:20_7_1"/><i><font color="Green">E132</font></i>: 
<font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel18.html#K6" target="_self">proj</a> <font color="Maroon">J</font>,<font color="Maroon">i</font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E9:20_7_1"/><i><font color="Green">E133</font></i>: 
<font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font>
 <b>by </b><i><a class="txt" href="waybel18.html#E6:20"><i><font color="Green">E37</font></i></a></i>;<br/>
<a NAME="E10:20_7_1"/><i><font color="Green">E134</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="txt" href="waybel18.html#E2:20"><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="E11:20_7_1"/><i><font color="Green">E135</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_3.html#K3">proj</a> <span class="p2">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">J</font></span>)</span>,<font color="Maroon">i</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="pralg_3.html#D2">PRALG_3:def 2</a></i>;<br/>
<a NAME="E12:20_7_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="funct_6.html#K6">&lt;:</a><span class="default"><font color="Maroon">G</font></span><a href="funct_6.html#K6">:&gt;</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="txt" href="waybel18.html#E1:20"><i><font color="Green">E27</font></i></a>, , <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/>
<b>hence </b><font color="Maroon">z</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> = 
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i>, , <a class="ref" href="waybel18.html#T3" target="_self">Th3</a>, , <a class="ref" href="funct_6.html#T54">FUNCT_6:54</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel18.html#K6" target="_self">proj</a> <font color="Maroon">J</font>,<font color="Maroon">i</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">x</font>
<b>by </b><i><a class="txt" href="waybel18.html#E1:20"><i><font color="Green">E27</font></i></a>, , <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="pralg_3.html#K3">proj</a> <span class="p2">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">J</font></span>)</span>,<font color="Maroon">i</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#E1:20"><i><font color="Green">E27</font></i></a>, , <a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
<br/>.= 
<font color="Maroon">y</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
<b>by </b><i>, , <a class="ref" href="pralg_3.html#D2">PRALG_3:def 2</a></i>
;<br/><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E21:20_7"/>
<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="partfun1.html#K2">|</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</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="txt" href="waybel18.html#E3:20"><i><font color="Green">E31</font></i></a>, , , <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon">h</font> = <font color="Maroon">h</font> as   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">Y</font>,<span class="p1">(<span class="default"><a href="waybel18.html#K3" target="_self">product</a> <font color="Maroon">J</font></span>)</span> ;<br/>
<b>take </b>
<font color="Maroon">h</font>
;<br/>

<b>set </b><font color="Maroon">B</font> =  <a href="waybel18.html#K2" target="_self">product_prebasis</a> <font color="Maroon">J</font>;<br/>
<a NAME="E21:20"/><i><font color="Green">E136</font></i>: 
 <a href="waybel18.html#K2" target="_self">product_prebasis</a> <font color="Maroon">J</font> is    <a href="cantor_1.html#M2">prebasis</a> of  <a href="waybel18.html#K3" target="_self">product</a> <font color="Maroon">J</font>
 
<b>by </b><i/>;<br/>
<a NAME="E22:20"/>
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#K3" target="_self">product</a> <font color="Maroon">J</font></span>)</span>  st <font color="Olive">P</font> <a href="hidden.html#R2">in</a>  <a href="waybel18.html#K2" target="_self">product_prebasis</a> <font color="Maroon">J</font> holds <br/><font color="Maroon">h</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="20_8"><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#K3" target="_self">product</a> <font color="Maroon">J</font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:20_8"/><i><font color="Green">E137</font></i>: 
<font color="Maroon">P</font> <a href="hidden.html#R2">in</a>  <a href="waybel18.html#K2" target="_self">product_prebasis</a> <font color="Maroon">J</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">p</font> = <font color="Maroon">P</font> as   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="card_3.html#K4">product</a> <span class="p2">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">J</font></span>)</span></span>)</span> <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">i</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">T</font> being    <a href="pre_topc.html#L1">TopStruct</a> , <font color="Maroon">V</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E4:20_8"/><i><font color="Green">E138</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 <b>and </b><br/><a NAME="E5:20_8"/><i><font color="Green">E139</font></i>: 
<font color="Maroon">V</font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E6:20_8"/><i><font color="Green">E140</font></i>: 
<font color="Maroon">T</font> <a href="hidden.html#R1">=</a> <font color="Maroon">J</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>and </b><br/><a NAME="E7:20_8"/><i><font color="Green">E141</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">J</font></span>)</span> <a href="funct_7.html#K2">+*</a> <font color="Maroon">i</font>,<font color="Maroon">V</font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel18.html#E1:20"><i><font color="Green">E27</font></i></a>, </i>;<br/>
<b>consider </b><font color="Maroon">j</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font><b> such that </b><br/><a NAME="E9:20_8"/><i><font color="Green">E142</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font>
 <b>and </b><br/><a NAME="E10:20_8"/><i><font color="Green">E143</font></i>: 
 ex <font color="Olive">g</font> being  <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">Y</font>,<span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="waybel18.html#K4" target="_self">.</a> <font color="Maroon">j</font></span>)</span> st <br/>( <font color="Olive">g</font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Olive">g</font> <a href="partfun1.html#K2">|</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel18.html#K6" target="_self">proj</a> <font color="Maroon">J</font>,<font color="Maroon">j</font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font> &amp; <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Olive">g</font> )
 <b>by </b><i>, <a class="txt" href="waybel18.html#E2:20"><i><font color="Green">E28</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">g</font> being   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">Y</font>,<span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="waybel18.html#K4" target="_self">.</a> <font color="Maroon">j</font></span>)</span><b> such that </b><br/><a NAME="E12:20_8"/><i><font color="Green">E144</font></i>: 
<font color="Maroon">g</font> is <a href="pre_topc.html#V5">continuous</a>
 <b>and </b><br/><a NAME="E13:20_8"/>
<font color="Maroon">g</font> <a href="partfun1.html#K2">|</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel18.html#K6" target="_self">proj</a> <font color="Maroon">J</font>,<font color="Maroon">j</font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E14:20_8"/><i><font color="Green">E145</font></i>: 
<font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font>
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">V</font> = <font color="Maroon">V</font> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="waybel18.html#K4" target="_self">.</a> <font color="Maroon">j</font></span>)</span> <b>by </b><i>, <a class="txt" href="waybel18.html#E4:20"><i><font color="Green">E32</font></i></a></i>;<br/>
<a NAME="E16:20_8"/><i><font color="Green">E146</font></i>: 
<span class="p1">(<span class="default"><a href="waybel18.html#K6" target="_self">proj</a> <font color="Maroon">J</font>,<font color="Maroon">j</font></span>)</span> <a href="relset_1.html#K11">"</a> <font color="Maroon">V</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font>
 
<b>by </b><i><a class="txt" href="waybel18.html#E3:20"><i><font color="Green">E31</font></i></a>, <a class="txt" href="waybel18.html#E4:20"><i><font color="Green">E32</font></i></a>, </i>;<br/>
<a NAME="E17:20_8"/><i><font color="Green">E147</font></i>: 
<font color="Maroon">g</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">V</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i>, , <a class="txt" href="waybel18.html#E4:20"><i><font color="Green">E32</font></i></a>, , <a class="ref" href="tops_2.html#T55">TOPS_2:55</a></i>;<br/>
<i><font color="Green">E148</font></i>:  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> = 
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 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="E19:20_8"/><i><font color="Green">E149</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="waybel18.html#K6" target="_self">proj</a> <font color="Maroon">J</font>,<font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">J</font></span>)</span>
 
<b>by </b><i><a class="ref" href="pralg_3.html#D2">PRALG_3:def 2</a></i>;<br/>
<a NAME="E20:20_8"/><i><font color="Green">E150</font></i>: 
<font color="Maroon">h</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">P</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">g</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">V</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_8_2"><b>proof </b></a><div class="add">

<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="E1:20_8_2"/><i><font color="Green">E151</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">h</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">P</font>
 ;<br/>

<a NAME="E2:20_8_2"/><b>then </b><i><font color="Green">E152</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> &amp; <font color="Maroon">h</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="E3:20_8_2"/><i><font color="Green">E153</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">g</font>
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>
<a NAME="E4:20_8_2"/><i><font color="Green">E154</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> <span class="p1">(<span class="default"><a href="waybel18.html#K6" target="_self">proj</a> <font color="Maroon">J</font>,<font color="Maroon">j</font></span>)</span> <a href="relset_1.html#K11">"</a> <font color="Maroon">V</font>
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>
<a NAME="E5:20_8_2"/><b>then </b><i><font color="Green">E155</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#K4">dom</a> <span class="p1">(<span class="default"><a href="waybel18.html#K6" target="_self">proj</a> <font color="Maroon">J</font>,<font color="Maroon">j</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>
<a NAME="E6:20_8_2"/>
<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="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">J</font></span>)</span>
 
<b>by </b><i><a class="ref" href="waybel18.html#T3" target="_self">Th3</a>, <a class="txt" href="waybel18.html#E8:20"><i><font color="Green">E44</font></i></a>, <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>
<b>then consider </b><font color="Maroon">y</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E8:20_8_2"/><i><font color="Green">E156</font></i>: 
<font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 <b>and </b><br/><a NAME="E9:20_8_2"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">J</font></span>)</span>
 <b>and </b><br/><a NAME="E10:20_8_2"/>
for <font color="Olive">i</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">J</font></span>)</span> holds <br/><font color="Maroon">y</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">J</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font>
 <b>by </b><i><a class="ref" href="card_3.html#D5">CARD_3:def 5</a></i>;<br/>
<a NAME="E11:20_8_2"/><i><font color="Green">E157</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_3.html#K3">proj</a> <span class="p2">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">J</font></span>)</span>,<font color="Maroon">j</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="waybel18.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E12:20_8_2"/>
<span class="p1">(<span class="default"><a href="waybel18.html#K6" target="_self">proj</a> <font color="Maroon">J</font>,<font color="Maroon">j</font></span>)</span> <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> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 
<b>by </b><i><a class="ref" href="waybel18.html#D1" target="_self">Def1</a>, , <a class="ref" href="pralg_3.html#D2">PRALG_3:def 2</a></i>;<br/>
<a NAME="E13:20_8_2"/><b>then </b><i><font color="Green">E158</font></i>: 
<font color="Maroon">y</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font>
 
<b>by </b><i><a class="txt" href="waybel18.html#E8:20"><i><font color="Green">E44</font></i></a>, <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>
<a NAME="E14:20_8_2"/>
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 
<b>by </b><i>, <a class="txt" href="waybel18.html#E4:20"><i><font color="Green">E32</font></i></a>, , , <a class="ref" href="waybel18.html#D1" target="_self">Def1</a>, <a class="ref" href="funct_6.html#T54">FUNCT_6:54</a></i>;<br/>
<b>hence </b><a NAME="E15:20_8_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">g</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">V</font>
 <b>by </b><i><a class="txt" href="waybel18.html#E7:20"><i><font color="Green">E43</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="E21:20_8"/>
<font color="Maroon">g</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">V</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">h</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">P</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_8_3"><b>proof </b></a><div class="add">

<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="E1:20_8_3"/><i><font color="Green">E159</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">g</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">V</font>
 ;<br/>

<a NAME="E2:20_8_3"/><b>then </b><i><font color="Green">E160</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_1.html#D13">FUNCT_1:def 13</a></i>;<br/>
<a NAME="E3:20_8_3"/><b>then </b><i><font color="Green">E161</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="E4:20_8_3"/><b>then </b><i><font color="Green">E162</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="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">J</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">y</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E6:20_8_3"/><i><font color="Green">E163</font></i>: 
<font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 <b>and </b><br/><a NAME="E7:20_8_3"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">J</font></span>)</span>
 <b>and </b><br/><a NAME="E8:20_8_3"/>
for <font color="Olive">i</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">J</font></span>)</span> holds <br/><font color="Maroon">y</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">J</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font>
 <b>by </b><i>, <a class="txt" href="waybel18.html#E7:20"><i><font color="Green">E43</font></i></a>, <a class="ref" href="card_3.html#D5">CARD_3:def 5</a></i>;<br/>
<a NAME="E9:20_8_3"/><i><font color="Green">E164</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_3.html#K3">proj</a> <span class="p2">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">J</font></span>)</span>,<font color="Maroon">j</font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel18.html#E8:20"><i><font color="Green">E44</font></i></a>, , <a class="ref" href="pralg_3.html#D2">PRALG_3:def 2</a></i>;<br/>
<a NAME="E10:20_8_3"/><i><font color="Green">E165</font></i>: 
<span class="p1">(<span class="default"><a href="waybel18.html#K6" target="_self">proj</a> <font color="Maroon">J</font>,<font color="Maroon">j</font></span>)</span> <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> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 
<b>by </b><i>, <a class="ref" href="waybel18.html#D1" target="_self">Def1</a>, <a class="ref" href="pralg_3.html#D2">PRALG_3:def 2</a></i>;<br/>
<a NAME="E11:20_8_3"/>
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 
<b>by </b><i>, <a class="txt" href="waybel18.html#E4:20"><i><font color="Green">E32</font></i></a>, , , , <a class="ref" href="funct_6.html#T54">FUNCT_6:54</a></i>;<br/>
<a NAME="E12:20_8_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="waybel18.html#K6" target="_self">proj</a> <font color="Maroon">J</font>,<font color="Maroon">j</font></span>)</span> <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#R2">in</a> <font color="Maroon">V</font>
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>
<a NAME="E13:20_8_3"/><b>then </b>
<font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="waybel18.html#K6" target="_self">proj</a> <font color="Maroon">J</font>,<font color="Maroon">j</font></span>)</span> <a href="relset_1.html#K11">"</a> <font color="Maroon">V</font>
 
<b>by </b><i>, <a class="ref" href="waybel18.html#T3" target="_self">Th3</a>, <a class="txt" href="waybel18.html#E7:20"><i><font color="Green">E43</font></i></a>, <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>
<b>hence </b><a NAME="E14:20_8_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">h</font> <a href="relset_1.html#K11">"</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/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E22:20_8"/>
<font color="Maroon">h</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="txt" href="waybel18.html#E6:20"><i><font color="Green">E37</font></i></a>, , <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E23:20"/>
<font color="Maroon">h</font> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i>, <a class="ref" href="yellow_9.html#T36">YELLOW_9:36</a></i>;<br/>

<b>thus </b><a NAME="E24:20"/>
<font color="Maroon">h</font> <a href="partfun1.html#K2">|</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


</div>
