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<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">Y</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">f</font> be   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X</font>,<font color="Maroon">Y</font>;<br/>





<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:24_1"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">f</font> is <a href="decomp_1.html#V7" target="_self">alpha-continuous</a>
 ;<br/><b>thus </b><a NAME="E2:24_1"/>
<font color="Maroon">f</font> is <a href="decomp_1.html#V5" target="_self">s-continuous</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">V</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="decomp_1.html#D26" target="_self">DECOMP_1:def 26</a><br/>

<b>assume </b><a NAME="E1:24_1_1"/>
<font color="Maroon">V</font> is <a href="pre_topc.html#V3">open</a>
 ;<br/>

<a NAME="E2:24_1_1"/><b>then </b>
<font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">V</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="decomp_1.html#K6" target="_self">^alpha</a> 
 
<b>by </b><i>, <a class="txt" href="decomp_1.html#E3:24"><i><font color="Green">E25</font></i></a></i>;<br/>
<a NAME="E3:24_1_1"/><b>then </b>
<font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">V</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="decomp_1.html#K7" target="_self">SO</a> <font color="Maroon">X</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="decomp_1.html#K16" target="_self">D(alpha,s)</a> <font color="Maroon">X</font></span>)</span>
 
<b>by </b><i><a class="ref" href="decomp_1.html#D3" target="_self">Def3</a></i>;<br/>
<b>hence </b><a NAME="E4:24_1_1"/>
<font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">V</font> <a href="hidden.html#R2">in</a>  <a href="decomp_1.html#K7" target="_self">SO</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E3:24_1"/>
<font color="Maroon">f</font> is <a href="decomp_1.html#V15" target="_self">(alpha,s)-continuous</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">G</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="decomp_1.html#D36" target="_self">DECOMP_1:def 36</a><br/>

<b>assume </b><a NAME="E1:24_1_2"/>
<font color="Maroon">G</font> is <a href="pre_topc.html#V3">open</a>
 ;<br/>

<a NAME="E2:24_1_2"/><b>then </b>
<font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">G</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="decomp_1.html#K6" target="_self">^alpha</a> 
 
<b>by </b><i>, <a class="txt" href="decomp_1.html#E3:24"><i><font color="Green">E25</font></i></a></i>;<br/>
<a NAME="E3:24_1_2"/><b>then </b>
<font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">G</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="decomp_1.html#K7" target="_self">SO</a> <font color="Maroon">X</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="decomp_1.html#K16" target="_self">D(alpha,s)</a> <font color="Maroon">X</font></span>)</span>
 
<b>by </b><i><a class="ref" href="decomp_1.html#D3" target="_self">Def3</a></i>;<br/>
<b>hence </b><a NAME="E4:24_1_2"/>
<font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">G</font> <a href="hidden.html#R2">in</a>  <a href="decomp_1.html#K16" target="_self">D(alpha,s)</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:24"/><i><font color="Green">E24</font></i>: 
( <font color="Maroon">f</font> is <a href="decomp_1.html#V5" target="_self">s-continuous</a> &amp; <font color="Maroon">f</font> is <a href="decomp_1.html#V15" target="_self">(alpha,s)-continuous</a> )
 ;<br/>

<b>let </b><font color="Maroon">V</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">Y</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="decomp_1.html#D28" target="_self">DECOMP_1:def 28</a><br/>

<b>assume </b><a NAME="E3:24"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">V</font> is <a href="pre_topc.html#V3">open</a>
 ;<br/>

<a NAME="E4:24"/><b>then </b><i><font color="Green">E26</font></i>: 
<font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">V</font> <a href="hidden.html#R2">in</a>  <a href="decomp_1.html#K7" target="_self">SO</a> <font color="Maroon">X</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:24"/>
<font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">V</font> <a href="hidden.html#R2">in</a>  <a href="decomp_1.html#K16" target="_self">D(alpha,s)</a> <font color="Maroon">X</font>
 
<b>by </b><i>, , <a class="ref" href="decomp_1.html#T4" target="_self">Th4</a></i>;<br/>
<a NAME="E6:24"/><b>then </b>
<font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">V</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="decomp_1.html#K7" target="_self">SO</a> <font color="Maroon">X</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="decomp_1.html#K16" target="_self">D(alpha,s)</a> <font color="Maroon">X</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E7:24"/>
<font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">V</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="decomp_1.html#K6" target="_self">^alpha</a> 
 <b>by </b><i><a class="ref" href="decomp_1.html#D3" target="_self">Def3</a></i>;<br/>


</div>
