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<div class="add">

<b>let </b><font color="Maroon">X</font> be   <a href="rltopsp1.html#NM1">LinearTopSpace</a>;<br/>

<b>assume </b><a NAME="E1:129"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">X</font> is <a href="rltopsp1.html#V8" target="_self">locally-convex</a>
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>] <b>means </b>( <font color="Maroon">a<sub>1</sub></font> is <a href="rltopsp1.html#V3" target="_self">circled</a> &amp; <font color="Maroon">a<sub>1</sub></font> is <a href="convex1.html#V1">convex</a> );<br/>
<b>consider </b><font color="Maroon">P</font> being   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E3:129"/><i><font color="Green">E37</font></i>: 
for <font color="Olive">V</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> holds <br/> ( <font color="Olive">V</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> iff <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">V</font>] )
 <b>from </b><i><a class="ref" href="subset_1.html#S3">SUBSET_1:sch 3</a>();<br/></i>
<b>reconsider </b><font color="Maroon">P</font> = <font color="Maroon">P</font> as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">X</font> ;<br/>
<b>take </b>
<font color="Maroon">P</font>
;<br/>

<b>thus </b><a NAME="E5:129"/>
<font color="Maroon">P</font> is   <a href="rltopsp1.html#NM2">local_base</a> of <font color="Maroon">X</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="129_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">V</font> be    <a href="connsp_2.html#M1">a_neighborhood</a> of  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow13.html#D2">YELLOW13:def 2</a><br/>

<b>consider </b><font color="Maroon">Q</font> being   <a href="rltopsp1.html#NM2">local_base</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E2:129_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">Q</font> is <a href="rltopsp1.html#V1" target="_self">convex-membered</a>
 <b>by </b><i><a class="ref" href="rltopsp1.html#T13" target="_self">Th13</a>, </i>;<br/>
<b>consider </b><font color="Maroon">U</font> being    <a href="connsp_2.html#M1">a_neighborhood</a> of  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font><b> such that </b><br/><a NAME="E4:129_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">U</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font>
 <b>and </b><br/><a NAME="E5:129_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">U</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font>
 <b>by </b><i><a class="ref" href="yellow13.html#D2">YELLOW13:def 2</a></i>;<br/>
<a NAME="E6:129_1"/>
<font color="Maroon">U</font> is <a href="convex1.html#V1">convex</a>
 
<b>by </b><i>, , </i>;<br/>
<b>then consider </b><font color="Maroon">W</font> being    <a href="connsp_2.html#M1">a_neighborhood</a> of  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font><b> such that </b><br/><a NAME="E8:129_1"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">W</font> is <a href="rltopsp1.html#V3" target="_self">circled</a> &amp; <font color="Maroon">W</font> is <a href="convex1.html#V1">convex</a> )
 <b>and </b><br/><a NAME="E9:129_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">W</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">U</font>
 <b>by </b><i><a class="ref" href="rltopsp1.html#T12" target="_self">Th12</a></i>;<br/>
<b>take </b>
<font color="Maroon">W</font>
;<br/>

<b>thus </b><a NAME="E10:129_1"/>
<font color="Maroon">W</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="rltopsp1.html#T15" target="_self">Th15</a>, </i>;<br/>

<b>thus </b><a NAME="E11:129_1"/>
<font color="Maroon">W</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font>
 <b>by </b><i>, <a class="ref" href="rltopsp1.html#D1" target="_self">Def1</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>


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

<b>let </b><font color="Maroon">V</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="rltopsp1.html#D3" target="_self">RLTOPSP1:def 3</a><br/>

<b>assume </b><a NAME="E6:129"/>
<font color="Maroon">V</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 ;<br/>

<b>hence </b><a NAME="E7:129"/>
<font color="Maroon">V</font> is <a href="convex1.html#V1">convex</a>
 <b>by </b><i><a class="ref" href="rltopsp1.html#T15" target="_self">Th15</a></i>;<br/>


</div>
