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<b>let </b><font color="Maroon" title="c1">X</font>, <font color="Maroon" title="c2">Y</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>;<br/><b>let </b><font color="Maroon" title="c3">f</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font>;<br/>





<a NAME="E1:30"/><i><font color="Green" title="E35">A1</font></i>: 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">Y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:30_1"/><i><font color="Green" title="E36">A2</font></i>: 
<font color="Maroon" title="c3">f</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 ;<br/><b>thus </b><a NAME="E2:30_1"/>
<font color="Maroon" title="c3">f</font> is <a href="decomp_1.html#V7" title="DECOMP_1:attr.7">alpha-continuous</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">V</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">Y</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="decomp_1.html#D28" target="_self" title="DECOMP_1:def.28">DECOMP_1:def 28</a><br/>

<b>assume </b><a NAME="E1:30_1_1"/>
<font color="Maroon" title="c4">V</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 ;<br/>

<a NAME="E2:30_1_1"/><b>then </b>
<font color="Maroon" title="c3">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c4">V</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="txt" href="decomp_1.html#E1:30"><i><font color="Green" title="E35">A1</font></i></a>, <a class="txt" href="decomp_1.html#E1:30_1"><i><font color="Green" title="E36">A2</font></i></a>, <a class="ref" href="tops_2.html#T55" title="TOPS_2:th.55">TOPS_2:55</a></i>;<br/>
<a NAME="E3:30_1_1"/><b>then </b>
<font color="Maroon" title="c3">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c4">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></i>;<br/>
<a NAME="E4:30_1_1"/><b>then </b>
<font color="Maroon" title="c3">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c4">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="decomp_1.html#K6" title="DECOMP_1:func.6">^alpha</a> </span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="decomp_1.html#K11" title="DECOMP_1:func.11">D(c,alpha)</a> <font color="Maroon" title="c1">X</font></span>)</span>
 
<b>by </b><i><a class="ref" href="decomp_1.html#T7" target="_self" title="DECOMP_1:th.7">Th7</a></i>;<br/>
<b>hence </b><a NAME="E5:30_1_1"/>
<font color="Maroon" title="c3">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c4">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> <a href="decomp_1.html#K6" title="DECOMP_1:func.6">^alpha</a> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E3:30_1"/>
<font color="Maroon" title="c3">f</font> is <a href="decomp_1.html#V10" title="DECOMP_1:attr.10">(c,alpha)-continuous</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">G</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">Y</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="decomp_1.html#D31" target="_self" title="DECOMP_1:def.31">DECOMP_1:def 31</a><br/>

<b>assume </b><a NAME="E1:30_1_2"/>
<font color="Maroon" title="c4">G</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 ;<br/>

<a NAME="E2:30_1_2"/><b>then </b>
<font color="Maroon" title="c3">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c4">G</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="txt" href="decomp_1.html#E1:30"><i><font color="Green" title="E35">A1</font></i></a>, <a class="txt" href="decomp_1.html#E1:30_1"><i><font color="Green" title="E36">A2</font></i></a>, <a class="ref" href="tops_2.html#T55" title="TOPS_2:th.55">TOPS_2:55</a></i>;<br/>
<a NAME="E3:30_1_2"/><b>then </b>
<font color="Maroon" title="c3">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c4">G</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></i>;<br/>
<a NAME="E4:30_1_2"/><b>then </b>
<font color="Maroon" title="c3">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c4">G</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="decomp_1.html#K6" title="DECOMP_1:func.6">^alpha</a> </span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="decomp_1.html#K11" title="DECOMP_1:func.11">D(c,alpha)</a> <font color="Maroon" title="c1">X</font></span>)</span>
 
<b>by </b><i><a class="ref" href="decomp_1.html#T7" target="_self" title="DECOMP_1:th.7">Th7</a></i>;<br/>
<b>hence </b><a NAME="E5:30_1_2"/>
<font color="Maroon" title="c3">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c4">G</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="decomp_1.html#K11" title="DECOMP_1:func.11">D(c,alpha)</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>


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

<b>assume </b><a NAME="E3:30"/><i><font color="Green" title="E36">A3</font></i>: 
( <font color="Maroon" title="c3">f</font> is <a href="decomp_1.html#V7" title="DECOMP_1:attr.7">alpha-continuous</a> &amp; <font color="Maroon" title="c3">f</font> is <a href="decomp_1.html#V10" title="DECOMP_1:attr.10">(c,alpha)-continuous</a> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">V</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">Y</font>;<br/><b>assume </b><a NAME="E1:30_2"/><i><font color="Green" title="E37">A4</font></i>: 
<font color="Maroon" title="c4">V</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 ;<br/><a NAME="E2:30_2"/><b>then </b><i><font color="Green" title="E38">A5</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c4">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> <a href="decomp_1.html#K6" title="DECOMP_1:func.6">^alpha</a> 
 <b>by </b><i><a class="txt" href="decomp_1.html#E3:30"><i><font color="Green" title="E36">A3</font></i></a>, <a class="ref" href="decomp_1.html#D28" target="_self" title="DECOMP_1:def.28">Def28</a></i>;<br/><a NAME="E3:30_2"/>
<font color="Maroon" title="c3">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c4">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="decomp_1.html#K11" title="DECOMP_1:func.11">D(c,alpha)</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="decomp_1.html#E3:30"><i><font color="Green" title="E36">A3</font></i></a>, <a class="txt" href="decomp_1.html#E1:30_2"><i><font color="Green" title="E37">A4</font></i></a>, <a class="ref" href="decomp_1.html#D31" target="_self" title="DECOMP_1:def.31">Def31</a></i>;<br/><a NAME="E4:30_2"/><b>then </b>
<font color="Maroon" title="c3">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c4">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="decomp_1.html#K6" title="DECOMP_1:func.6">^alpha</a> </span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="decomp_1.html#K11" title="DECOMP_1:func.11">D(c,alpha)</a> <font color="Maroon" title="c1">X</font></span>)</span>
 <b>by </b><i><a class="txt" href="decomp_1.html#E2:30_2"><i><font color="Green" title="E38">A5</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E5:30_2"/><b>then </b>
<font color="Maroon" title="c3">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c4">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="ref" href="decomp_1.html#T7" target="_self" title="DECOMP_1:th.7">Th7</a></i>;<br/><b>hence </b><a NAME="E6:30_2"/>
<font color="Maroon" title="c3">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c4">V</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:30"/>
<font color="Maroon" title="c3">f</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>by </b><i><a class="txt" href="decomp_1.html#E1:30"><i><font color="Green" title="E35">A1</font></i></a>, <a class="ref" href="tops_2.html#T55" title="TOPS_2:th.55">TOPS_2:55</a></i>;<br/>


</div>
