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<span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>

<span class="kw">assume </span><a NAME="E1:128"/><span class="lab"><font color="Green" title="E79">A1</font></span>: 
<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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>

<span class="kw">defpred </span><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>] <span class="kw">means </span>( $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/>
<span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E3:128"/><span class="lab"><font color="Green" title="E80">A2</font></span>: 
 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>] )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="subset_1.html#S3" title="SUBSET_1:sch.3">SUBSET_1:sch 3</a>();<br/></span>
<span class="kw">reconsider </span><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/>
<span class="kw">take </span>
<font color="Maroon" title="c3">P</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>

<span class="kw">thus </span><a NAME="E5:128"/>
<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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="128_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="yellow13.html#D2" title="YELLOW13:def.2">YELLOW13:def 2</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>

<span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E2:128_1"/><span class="lab"><font color="Green" title="E81">A3</font></span>: 
<font color="Maroon" title="c5">Q</font> is  <a href="rltopsp1.html#V1" title="RLTOPSP1:attr.1">convex-membered</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:128"><span class="lab"><font color="Green" title="E79">A1</font></span></a>, <a class="ref" href="rltopsp1.html#D11" target="_self" title="RLTOPSP1:def.11">Def11</a></span>;<br/>
<span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E4:128_1"/><span class="lab"><font color="Green" title="E82">A4</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E5:128_1"/><span class="lab"><font color="Green" title="E83">A5</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow13.html#D2" title="YELLOW13:def.2">YELLOW13:def 2</a></span>;<br/>
<a NAME="E6:128_1"/>
<font color="Maroon" title="c6">U</font> is  <a href="convex1.html#V1" title="CONVEX1:attr.1">convex</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:128_1"><span class="lab"><font color="Green" title="E81">A3</font></span></a>, <a class="txt" href="#E4:128_1"><span class="lab"><font color="Green" title="E82">A4</font></span></a>, <a class="ref" href="rltopsp1.html#D3" target="_self" title="RLTOPSP1:def.3">Def3</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E8:128_1"/><span class="lab"><font color="Green" title="E84">A6</font></span>: 
( <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>  )
 <span class="kw">and </span><br/><a NAME="E9:128_1"/><span class="lab"><font color="Green" title="E85">A7</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rltopsp1.html#T65" target="_self" title="RLTOPSP1:th.65">Th66</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c7">W</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>

<span class="kw">thus </span><a NAME="E10:128_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:128"><span class="lab"><font color="Green" title="E80">A2</font></span></a>, <a class="txt" href="#E8:128_1"><span class="lab"><font color="Green" title="E84">A6</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>

<span class="kw">thus </span><a NAME="E11:128_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:128_1"><span class="lab"><font color="Green" title="E83">A5</font></span></a>, <a class="txt" href="#E9:128_1"><span class="lab"><font color="Green" title="E85">A7</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">let </span><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>; <span class="comment"><font color="firebrick">:: according to </font></span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>

<span class="kw">assume </span><a NAME="E6:128"/>
<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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>

<span class="kw">hence </span><a NAME="E7:128"/>
<font color="Maroon" title="c4">V</font> is  <a href="convex1.html#V1" title="CONVEX1:attr.1">convex</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:128"><span class="lab"><font color="Green" title="E80">A2</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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