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

<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>;<br/><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>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/>







<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:18_1"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">f</font> is <a href="pre_topc.html#V5">continuous</a>
 ;<br/><b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font>;<br/><a NAME="E2:18_1"/>
 <a href="waybel18.html#K6" target="_self">proj</a> <font color="Maroon">J</font>,<font color="Maroon">i</font> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:18_1"/>
<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> 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/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:18"/><i><font color="Green">E28</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> holds  <span class="p1">(<span class="default"><a href="waybel18.html#K6" target="_self">proj</a> <font color="Maroon">J</font>,<font color="Olive">i</font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font> is <a href="pre_topc.html#V5">continuous</a>
 ;<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="E3:18"/><i><font color="Green">E31</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="E4:18"/>
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">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="18_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 <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="E1:18_2"/><i><font color="Green">E32</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="E4:18_2"/><i><font color="Green">E33</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="E5:18_2"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">V</font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E6:18_2"/><i><font color="Green">E36</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="E7:18_2"/><i><font color="Green">E37</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>, </i>;<br/>
<b>reconsider </b><font color="Maroon">j</font> = <font color="Maroon">i</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> <b>by </b><i/>;<br/>
<a NAME="E9:18_2"/><i><font color="Green">E43</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="partfun1.html#K1">*</a> <font color="Maroon">f</font> is <a href="pre_topc.html#V5">continuous</a>
 
<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/>;<br/>
<a NAME="E11:18_2"/><i><font color="Green">E44</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>, </i>;<br/>
<a NAME="E12:18_2"/>
<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">j</font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font></span>)</span> <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="ref" href="tops_2.html#T55">TOPS_2:55</a></i>;<br/>
<b>hence </b><a NAME="E13:18_2"/>
<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="E5:18"/>
<font color="Maroon">f</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/>


</div>
