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

<b>let </b><font color="Maroon" title="c1">X</font> be   <a href="rltopsp1.html#NM1" title="RLTOPSP1:NM.1">LinearTopSpace</a>;<br/><b>let </b><font color="Maroon" title="c2">U</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>;<br/>



<b>assume </b><a NAME="E1:127"/><i><font color="Green" title="E80">A1</font></i>: 
<font color="Maroon" title="c2">U</font> is <a href="convex1.html#V1" title="CONVEX1:attr.1">convex</a>
 ;<br/>

<b>set </b><font color="Maroon" title="c3">V</font> = <font color="Maroon" title="c2">U</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="rltopsp1.html#K2" title="RLTOPSP1:func.2">-</a> <font color="Maroon" title="c2">U</font></span>)</span>;<br/>
<a NAME="E2:127"/>
 <a href="rltopsp1.html#K2" title="RLTOPSP1:func.2">-</a> <font color="Maroon" title="c2">U</font> is    <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>by </b><i><a class="ref" href="rltopsp1.html#T56" target="_self" title="RLTOPSP1:th.56">Th56</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c4">V</font> = <font color="Maroon" title="c2">U</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="rltopsp1.html#K2" title="RLTOPSP1:func.2">-</a> <font color="Maroon" title="c2">U</font></span>)</span> as    <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>by </b><i><a class="ref" href="connsp_2.html#T4" title="CONNSP_2:th.4">CONNSP_2:4</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c4">V</font>
;<br/>

<a NAME="E4:127"/><i><font color="Green" title="E81">A2</font></i>: 
 <a href="rltopsp1.html#K2" title="RLTOPSP1:func.2">-</a> <font color="Maroon" title="c2">U</font> is <a href="convex1.html#V1" title="CONVEX1:attr.1">convex</a>
 
<b>by </b><i><a class="txt" href="rltopsp1.html#E1:127"><i><font color="Green" title="E80">A1</font></i></a>, <a class="ref" href="convex1.html#T1" title="CONVEX1:th.1">CONVEX1:1</a></i>;<br/>
<b>thus </b><a NAME="E5:127"/>
<font color="Maroon" title="c4">V</font> is <a href="rltopsp1.html#V3" title="RLTOPSP1:attr.3">circled</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="127_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">r</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="rltopsp1.html#D6" target="_self" title="RLTOPSP1:def.6">RLTOPSP1:def 6</a><br/>

<b>assume </b><a NAME="E1:127_1"/><i><font color="Green" title="E82">A3</font></i>: 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c5">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 ;<br/>

<b>let </b><font color="Maroon" title="c6">u</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:127_1"/>
<font color="Maroon" title="c6">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">r</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c4">V</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c7">v</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E4:127_1"/><i><font color="Green" title="E83">A4</font></i>: 
<font color="Maroon" title="c6">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">r</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c7">v</font>
 <b>and </b><br/><a NAME="E5:127_1"/><i><font color="Green" title="E84">A5</font></i>: 
<font color="Maroon" title="c7">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">V</font>
 ;<br/>
<a NAME="E6:127_1"/><i><font color="Green" title="E85">A6</font></i>: 
<font color="Maroon" title="c7">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rltopsp1.html#K2" title="RLTOPSP1:func.2">-</a> <font color="Maroon" title="c2">U</font>
 
<b>by </b><i><a class="txt" href="rltopsp1.html#E5:127_1"><i><font color="Green" title="E84">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="E7:127_1"/><i><font color="Green" title="E86">A7</font></i>: 
<font color="Maroon" title="c7">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">U</font>
 
<b>by </b><i><a class="txt" href="rltopsp1.html#E5:127_1"><i><font color="Green" title="E84">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="E8:127_1"/>
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">V</font>
 
<b>by </b><i><a class="ref" href="connsp_2.html#T6" title="CONNSP_2:th.6">CONNSP_2:6</a></i>;<br/>
<a NAME="E9:127_1"/><b>then </b><i><font color="Green" title="E87">A8</font></i>: 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rltopsp1.html#K2" title="RLTOPSP1:func.2">-</a> <font color="Maroon" title="c2">U</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/>
<a NAME="E10:127_1"/><i><font color="Green" title="E88">A9</font></i>: 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">U</font>
 
<b>by </b><i><a class="ref" href="connsp_2.html#T6" title="CONNSP_2:th.6">CONNSP_2:6</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="127_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:127_1_1"/>
( <font color="Maroon" title="c5">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 0 or <font color="Maroon" title="c5">r</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="127_1_1_1"><b>suppose </b></a><a NAME="E1:127_1_1_1"/><i><font color="Green" title="E89">A10</font></i>: 
<font color="Maroon" title="c5">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 0
 ;<br/><div class="add"><a NAME="E2:127_1_1_1"/><b>then </b><i><font color="Green" title="E90">A11</font></i>: 
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c5">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <b>by </b><i><a class="txt" href="rltopsp1.html#E1:127_1"><i><font color="Green" title="E82">A3</font></i></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>;<br/><a NAME="E3:127_1_1_1"/><i><font color="Green" title="E91">A12</font></i>: 
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c5">r</font>
 <b>by </b><i><a class="txt" href="rltopsp1.html#E1:127_1_1_1"><i><font color="Green" title="E89">A10</font></i></a>, <a class="ref" href="xreal_1.html#T26" title="XREAL_1:th.26">XREAL_1:26</a></i>;<br/><a NAME="E4:127_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c5">r</font></span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c7">v</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c5">r</font></span>)</span></span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rltopsp1.html#K2" title="RLTOPSP1:func.2">-</a> <font color="Maroon" title="c2">U</font>
 <b>by </b><i><a class="txt" href="rltopsp1.html#E4:127"><i><font color="Green" title="E81">A2</font></i></a>, <a class="txt" href="rltopsp1.html#E6:127_1"><i><font color="Green" title="E85">A6</font></i></a>, <a class="txt" href="rltopsp1.html#E9:127_1"><i><font color="Green" title="E87">A8</font></i></a>, <a class="txt" href="rltopsp1.html#E2:127_1_1_1"><i><font color="Green" title="E90">A11</font></i></a>, <a class="ref" href="rltopsp1.html#D1" target="_self" title="RLTOPSP1:def.1">Def1</a></i>;<br/><a NAME="E5:127_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c5">r</font></span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c7">v</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rltopsp1.html#K2" title="RLTOPSP1:func.2">-</a> <font color="Maroon" title="c2">U</font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#T23" title="RLVECT_1:th.23">RLVECT_1:23</a></i>;<br/><a NAME="E6:127_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c5">r</font></span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c7">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rltopsp1.html#K2" title="RLTOPSP1:func.2">-</a> <font color="Maroon" title="c2">U</font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D7" title="RLVECT_1:def.7">RLVECT_1:def 7</a></i>;<br/><b>then consider </b><font color="Maroon" title="c8">u'</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E8:127_1_1_1"/><i><font color="Green" title="E92">A13</font></i>: 
<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c5">r</font></span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c7">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c8">u'</font>
 <b>and </b><br/><a NAME="E9:127_1_1_1"/><i><font color="Green" title="E93">A14</font></i>: 
<font color="Maroon" title="c8">u'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">U</font>
 ;<br/><i><font color="Green" title="E94">A15</font></i>: <font color="Maroon" title="c8">u'</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c8">u'</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#D9" title="RLVECT_1:def.9">RLVECT_1:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c8">u'</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D9" title="RLVECT_1:def.9">RLVECT_1:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c5">r</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c7">v</font>
<b>by </b><i><a class="txt" href="rltopsp1.html#E8:127_1_1_1"><i><font color="Green" title="E92">A13</font></i></a>, <a class="ref" href="rlvect_1.html#D9" title="RLVECT_1:def.9">RLVECT_1:def 9</a></i>
<br/>.= 
<font color="Maroon" title="c5">r</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c7">v</font>

;<br/><a NAME="E11:127_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c5">r</font></span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c7">v</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c5">r</font></span>)</span></span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">U</font>
 <b>by </b><i><a class="txt" href="rltopsp1.html#E1:127"><i><font color="Green" title="E80">A1</font></i></a>, <a class="txt" href="rltopsp1.html#E7:127_1"><i><font color="Green" title="E86">A7</font></i></a>, <a class="txt" href="rltopsp1.html#E10:127_1"><i><font color="Green" title="E88">A9</font></i></a>, <a class="txt" href="rltopsp1.html#E2:127_1_1_1"><i><font color="Green" title="E90">A11</font></i></a>, <a class="txt" href="rltopsp1.html#E3:127_1_1_1"><i><font color="Green" title="E91">A12</font></i></a>, <a class="ref" href="rltopsp1.html#D1" target="_self" title="RLTOPSP1:def.1">Def1</a></i>;<br/><a NAME="E12:127_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c5">r</font></span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c7">v</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">U</font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#T23" title="RLVECT_1:th.23">RLVECT_1:23</a></i>;<br/><a NAME="E13:127_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c5">r</font></span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c7">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">U</font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D7" title="RLVECT_1:def.7">RLVECT_1:def 7</a></i>;<br/><a NAME="E14:127_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c5">r</font></span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c7">v</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c2">U</font>
 ;<br/><a NAME="E15:127_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c5">r</font></span>)</span></span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c7">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c2">U</font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D9" title="RLVECT_1:def.9">RLVECT_1:def 9</a></i>;<br/><b>hence </b><a NAME="E16:127_1_1_1"/>
<font color="Maroon" title="c6">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">V</font>
 <b>by </b><i><a class="txt" href="rltopsp1.html#E4:127_1"><i><font color="Green" title="E83">A4</font></i></a>, <a class="txt" href="rltopsp1.html#E9:127_1_1_1"><i><font color="Green" title="E93">A14</font></i></a>, <a class="txt" href="rltopsp1.html#E10:127_1_1_1"><i><font color="Green" title="E94">A15</font></i></a>, <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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="127_1_1_2"><b>suppose </b></a><a NAME="E1:127_1_1_2"/><i><font color="Green" title="E89">A16</font></i>: 
<font color="Maroon" title="c5">r</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0
 ;<br/><div class="add"><a NAME="E2:127_1_1_2"/><i><font color="Green" title="E90">A17</font></i>: 
<font color="Maroon" title="c5">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <b>by </b><i><a class="txt" href="rltopsp1.html#E1:127_1"><i><font color="Green" title="E82">A3</font></i></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>;<br/><a NAME="E3:127_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c7">v</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c5">r</font></span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">U</font>
 <b>by </b><i><a class="txt" href="rltopsp1.html#E1:127"><i><font color="Green" title="E80">A1</font></i></a>, <a class="txt" href="rltopsp1.html#E7:127_1"><i><font color="Green" title="E86">A7</font></i></a>, <a class="txt" href="rltopsp1.html#E10:127_1"><i><font color="Green" title="E88">A9</font></i></a>, <a class="txt" href="rltopsp1.html#E1:127_1_1_2"><i><font color="Green" title="E89">A16</font></i></a>, <a class="ref" href="rltopsp1.html#D1" target="_self" title="RLTOPSP1:def.1">Def1</a></i>;<br/><a NAME="E4:127_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c7">v</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">U</font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#T23" title="RLVECT_1:th.23">RLVECT_1:23</a></i>;<br/><a NAME="E5:127_1_1_2"/><b>then </b><i><font color="Green" title="E91">A18</font></i>: 
<font color="Maroon" title="c5">r</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c7">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">U</font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D7" title="RLVECT_1:def.7">RLVECT_1:def 7</a></i>;<br/><a NAME="E6:127_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c7">v</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c5">r</font></span>)</span> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rltopsp1.html#K2" title="RLTOPSP1:func.2">-</a> <font color="Maroon" title="c2">U</font>
 <b>by </b><i><a class="txt" href="rltopsp1.html#E4:127"><i><font color="Green" title="E81">A2</font></i></a>, <a class="txt" href="rltopsp1.html#E6:127_1"><i><font color="Green" title="E85">A6</font></i></a>, <a class="txt" href="rltopsp1.html#E9:127_1"><i><font color="Green" title="E87">A8</font></i></a>, <a class="txt" href="rltopsp1.html#E1:127_1_1_2"><i><font color="Green" title="E89">A16</font></i></a>, <a class="txt" href="rltopsp1.html#E2:127_1_1_2"><i><font color="Green" title="E90">A17</font></i></a>, <a class="ref" href="rltopsp1.html#D1" target="_self" title="RLTOPSP1:def.1">Def1</a></i>;<br/><a NAME="E7:127_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c7">v</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rltopsp1.html#K2" title="RLTOPSP1:func.2">-</a> <font color="Maroon" title="c2">U</font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#T23" title="RLVECT_1:th.23">RLVECT_1:23</a></i>;<br/><a NAME="E8:127_1_1_2"/><b>then </b>
<font color="Maroon" title="c5">r</font> <a href="rlvect_1.html#K2" title="RLVECT_1:func.2">*</a> <font color="Maroon" title="c7">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rltopsp1.html#K2" title="RLTOPSP1:func.2">-</a> <font color="Maroon" title="c2">U</font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D7" title="RLVECT_1:def.7">RLVECT_1:def 7</a></i>;<br/><b>hence </b><a NAME="E9:127_1_1_2"/>
<font color="Maroon" title="c6">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">V</font>
 <b>by </b><i><a class="txt" href="rltopsp1.html#E4:127_1"><i><font color="Green" title="E83">A4</font></i></a>, <a class="txt" href="rltopsp1.html#E5:127_1_1_2"><i><font color="Green" title="E91">A18</font></i></a>, <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>

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

<b>thus </b><a NAME="E6:127"/>
<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#E1:127"><i><font color="Green" title="E80">A1</font></i></a>, <a class="txt" href="rltopsp1.html#E4:127"><i><font color="Green" title="E81">A2</font></i></a>, <a class="ref" href="convex2.html#T1" title="CONVEX2:th.1">CONVEX2:1</a></i>;<br/>

<b>thus </b><a NAME="E7:127"/>
<font color="Maroon" title="c4">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">U</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/>


</div>
