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

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></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">c<sub>3</sub></font> be   <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/>





<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:29_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="decomp_1.html#V9" target="_self">sp-continuous</a>
 ;<br/><b>thus </b><a NAME="E2:29_1"/>
<font color="Maroon">c<sub>3</sub></font> is <a href="decomp_1.html#V8" target="_self">ps-continuous</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>2</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="decomp_1.html#D29" target="_self">DECOMP_1:def 29</a><br/>

<b>assume </b><a NAME="E1:29_1_1"/>
<font color="Maroon">c<sub>4</sub></font> is <a href="pre_topc.html#V3">open</a>
 ;<br/>

<a NAME="E2:29_1_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="decomp_1.html#K9" target="_self">SPO</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="decomp_1.html#E1:29_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="decomp_1.html#D30" target="_self">Def30</a></i>;<br/>
<a NAME="E3:29_1_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="decomp_1.html#K10" target="_self">PSO</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="decomp_1.html#K20" target="_self">D(sp,ps)</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="decomp_1.html#T17" target="_self">Th17</a></i>;<br/>
<b>hence </b><a NAME="E4:29_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="decomp_1.html#K10" target="_self">PSO</a> <font color="Maroon">c<sub>1</sub></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="E3:29_1"/>
<font color="Maroon">c<sub>3</sub></font> is <a href="decomp_1.html#V19" target="_self">(sp,ps)-continuous</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>2</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="decomp_1.html#D40" target="_self">DECOMP_1:def 40</a><br/>

<b>assume </b><a NAME="E1:29_1_2"/>
<font color="Maroon">c<sub>4</sub></font> is <a href="pre_topc.html#V3">open</a>
 ;<br/>

<a NAME="E2:29_1_2"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="decomp_1.html#K9" target="_self">SPO</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="decomp_1.html#E1:29_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="decomp_1.html#D30" target="_self">Def30</a></i>;<br/>
<a NAME="E3:29_1_2"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="decomp_1.html#K10" target="_self">PSO</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="decomp_1.html#K20" target="_self">D(sp,ps)</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="decomp_1.html#T17" target="_self">Th17</a></i>;<br/>
<b>hence </b><a NAME="E4:29_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="decomp_1.html#K20" target="_self">D(sp,ps)</a> <font color="Maroon">c<sub>1</sub></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="E2:29"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> is <a href="decomp_1.html#V8" target="_self">ps-continuous</a> &amp; <font color="Maroon">c<sub>3</sub></font> is <a href="decomp_1.html#V19" target="_self">(sp,ps)-continuous</a> )
 ;<br/>

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>2</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="decomp_1.html#D30" target="_self">DECOMP_1:def 30</a><br/>

<b>assume </b><a NAME="E3:29"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>4</sub></font> is <a href="pre_topc.html#V3">open</a>
 ;<br/>

<a NAME="E4:29"/><b>then </b><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="decomp_1.html#K10" target="_self">PSO</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="decomp_1.html#E2:29"><i><font color="Green">E38</font></i></a>, <a class="ref" href="decomp_1.html#D29" target="_self">Def29</a></i>;<br/>
<a NAME="E5:29"/>
<font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="decomp_1.html#K20" target="_self">D(sp,ps)</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="decomp_1.html#E2:29"><i><font color="Green">E38</font></i></a>, <a class="txt" href="decomp_1.html#E3:29"><i><font color="Green">E39</font></i></a>, <a class="ref" href="decomp_1.html#D40" target="_self">Def40</a></i>;<br/>
<a NAME="E6:29"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="decomp_1.html#K10" target="_self">PSO</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="decomp_1.html#K20" target="_self">D(sp,ps)</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="decomp_1.html#E4:29"><i><font color="Green">E40</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E7:29"/>
<font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="decomp_1.html#K9" target="_self">SPO</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="decomp_1.html#T17" target="_self">Th17</a></i>;<br/>


</div>
