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<b>let </b><font color="Maroon" title="c1">X</font> be   <a href="rltopsp1.html#NM1" title="RLTOPSP1:NM.1">LinearTopSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">X</font> is <a href="rltopsp1.html#V8" title="RLTOPSP1:attr.8">locally-convex</a> implies  ex <font color="Olive" title="b1">P</font> being  <a href="rltopsp1.html#NM2" title="RLTOPSP1:NM.2">local_base</a> of <font color="Maroon" title="c1">X</font> st <font color="Olive" title="b1">P</font> is <a href="rltopsp1.html#V1" title="RLTOPSP1:attr.1">convex-membered</a> )</span><br/>

<b>assume </b><a NAME="E1:129"/><i><font color="Green" title="E80">A1</font></i>: 
<font color="Maroon" title="c1">X</font> is <a href="rltopsp1.html#V8" title="RLTOPSP1:attr.8">locally-convex</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">P</font> being  <a href="rltopsp1.html#NM2" title="RLTOPSP1:NM.2">local_base</a> of <font color="Maroon" title="c1">X</font> st <font color="Olive" title="b1">P</font> is <a href="rltopsp1.html#V1" title="RLTOPSP1:attr.1">convex-membered</a></span><br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>] <b>means </b>( $1 is <a href="rltopsp1.html#V3" title="RLTOPSP1:attr.3">circled</a> &amp; $1 is <a href="convex1.html#V1" title="CONVEX1:attr.1">convex</a> );<br/>
<b>consider </b><font color="Maroon" title="c2">P</font> being   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E3:129"/><i><font color="Green" title="E81">A2</font></i>: 
for <font color="Olive" title="b1">V</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> holds <br/> ( <font color="Olive" title="b1">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">P</font> iff <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">V</font>] )
 <b>from </b><i><a class="ref" href="subset_1.html#S3" title="SUBSET_1:sch.3">SUBSET_1:sch 3</a>();<br/></i>
<b>reconsider </b><font color="Maroon" title="c3">P</font> = <font color="Maroon" title="c2">P</font> as   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">X</font> ;<br/>
<b>take </b>
<font color="Maroon" title="c3">P</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">P</font> is   <a href="rltopsp1.html#NM2" title="RLTOPSP1:NM.2">local_base</a> of <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c3">P</font> is <a href="rltopsp1.html#V1" title="RLTOPSP1:attr.1">convex-membered</a> )</span><br/>

<b>thus </b><a NAME="E5:129"/>
<font color="Maroon" title="c3">P</font> is   <a href="rltopsp1.html#NM2" title="RLTOPSP1:NM.2">local_base</a> of <font color="Maroon" title="c1">X</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">P</font> is <a href="rltopsp1.html#V1" title="RLTOPSP1:attr.1">convex-membered</a></span><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" title="c4">V</font> be    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow13.html#D2" title="YELLOW13:def.2">YELLOW13:def 2</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive">b<sub>1</sub></font> being   <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">V</font> )</span><br/>

<b>consider </b><font color="Maroon" title="c5">Q</font> being   <a href="rltopsp1.html#NM2" title="RLTOPSP1:NM.2">local_base</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E2:129_1"/><i><font color="Green" title="E82">A3</font></i>: 
<font color="Maroon" title="c5">Q</font> is <a href="rltopsp1.html#V1" title="RLTOPSP1:attr.1">convex-membered</a>
 <b>by </b><i><a class="txt" href="rltopsp1.html#E1:129"><i><font color="Green" title="E80">A1</font></i></a>, <a class="ref" href="rltopsp1.html#D11" target="_self" title="RLTOPSP1:def.11">Def11</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c6">U</font> being    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E4:129_1"/><i><font color="Green" title="E83">A4</font></i>: 
<font color="Maroon" title="c6">U</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Q</font>
 <b>and </b><br/><a NAME="E5:129_1"/><i><font color="Green" title="E84">A5</font></i>: 
<font color="Maroon" title="c6">U</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">V</font>
 <b>by </b><i><a class="ref" href="yellow13.html#D2" title="YELLOW13:def.2">YELLOW13:def 2</a></i>;<br/>
<a NAME="E6:129_1"/>
<font color="Maroon" title="c6">U</font> is <a href="convex1.html#V1" title="CONVEX1:attr.1">convex</a>
 
<b>by </b><i><a class="txt" href="rltopsp1.html#E2:129_1"><i><font color="Green" title="E82">A3</font></i></a>, <a class="txt" href="rltopsp1.html#E4:129_1"><i><font color="Green" title="E83">A4</font></i></a>, <a class="ref" href="rltopsp1.html#D3" target="_self" title="RLTOPSP1:def.3">Def3</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c7">W</font> being    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E8:129_1"/><i><font color="Green" title="E85">A6</font></i>: 
( <font color="Maroon" title="c7">W</font> is <a href="rltopsp1.html#V3" title="RLTOPSP1:attr.3">circled</a> &amp; <font color="Maroon" title="c7">W</font> is <a href="convex1.html#V1" title="CONVEX1:attr.1">convex</a> )
 <b>and </b><br/><a NAME="E9:129_1"/><i><font color="Green" title="E86">A7</font></i>: 
<font color="Maroon" title="c7">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">U</font>
 <b>by </b><i><a class="ref" href="rltopsp1.html#T66" target="_self" title="RLTOPSP1:th.66">Th66</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c7">W</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c7">W</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> &amp; <font color="Maroon" title="c7">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">V</font> )</span><br/>

<b>thus </b><a NAME="E10:129_1"/>
<font color="Maroon" title="c7">W</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font>
 <b>by </b><i><a class="txt" href="rltopsp1.html#E3:129"><i><font color="Green" title="E81">A2</font></i></a>, <a class="txt" href="rltopsp1.html#E8:129_1"><i><font color="Green" title="E85">A6</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">V</font></span><br/>

<b>thus </b><a NAME="E11:129_1"/>
<font color="Maroon" title="c7">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">V</font>
 <b>by </b><i><a class="txt" href="rltopsp1.html#E5:129_1"><i><font color="Green" title="E84">A5</font></i></a>, <a class="txt" href="rltopsp1.html#E9:129_1"><i><font color="Green" title="E86">A7</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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

<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="c1">X</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="rltopsp1.html#D3" target="_self" title="RLTOPSP1:def.3">RLTOPSP1:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> implies <font color="Maroon" title="c4">V</font> is <a href="convex1.html#V1" title="CONVEX1:attr.1">convex</a> )</span><br/>

<b>assume </b><a NAME="E6:129"/>
<font color="Maroon" title="c4">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">V</font> is <a href="convex1.html#V1" title="CONVEX1:attr.1">convex</a></span><br/>

<b>hence </b><a NAME="E7:129"/>
<font color="Maroon" title="c4">V</font> is <a href="convex1.html#V1" title="CONVEX1:attr.1">convex</a>
 <b>by </b><i><a class="txt" href="rltopsp1.html#E3:129"><i><font color="Green" title="E81">A2</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
