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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="rlvect_1.html#NM2">RealLinearSpace</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/>





<b>thus </b><a NAME="E1:21"/>
<font color="Maroon">c<sub>3</sub></font> <a href="convex1.html#K1">*</a> <span class="p1">(<span class="default"><a href="circled1.html#K2" target="_self">Cir</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="circled1.html#K2" target="_self">Cir</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="convex1.html#K1">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:21_1_1"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_1_1"><b>suppose </b></a><a NAME="E1:21_1_1_1"/><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:21_1_1_1_1"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:21_1_1_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:21_1_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><a NAME="E3:21_1_1_1_1_1"/><b>then </b>
 <a href="circled1.html#K2" target="_self">Cir</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="circled1.html#T14" target="_self">Th14</a></i>;<br/><b>hence </b><a NAME="E4:21_1_1_1_1_1"/>
(  not <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="convex1.html#K1">*</a> <span class="p1">(<span class="default"><a href="circled1.html#K2" target="_self">Cir</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="circled1.html#K2" target="_self">Cir</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="convex1.html#K1">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> )
 <b>by </b><i><a class="ref" href="convex1.html#T33">CONVEX1:33</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:21_1_1_1_1_2"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:21_1_1_1_1_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a>  <a href="circled1.html#K2" target="_self">Cir</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="circled1.html#T11" target="_self">Th11</a></i>;<br/><a NAME="E3:21_1_1_1_1_2"/><b>then </b>
 <a href="circled1.html#K2" target="_self">Cir</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="circled1.html#E1:21_1_1_1_1_2"><i><font color="Green">E13</font></i></a>, <a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/><a NAME="E4:21_1_1_1_1_2"/><b>then </b><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="convex1.html#K1">*</a> <span class="p1">(<span class="default"><a href="circled1.html#K2" target="_self">Cir</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="txt" href="circled1.html#E1:21_1_1_1"><i><font color="Green">E12</font></i></a>, <a class="ref" href="convex1.html#T34">CONVEX1:34</a></i>;<br/><a NAME="E5:21_1_1_1_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="convex1.html#K1">*</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="txt" href="circled1.html#E1:21_1_1_1"><i><font color="Green">E12</font></i></a>, <a class="txt" href="circled1.html#E1:21_1_1_1_1_2"><i><font color="Green">E13</font></i></a>, <a class="ref" href="convex1.html#T34">CONVEX1:34</a></i>;<br/><a NAME="E6:21_1_1_1_1_2"/><b>then </b>
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="circled1.html#K2" target="_self">Cir</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="convex1.html#K1">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="circled1.html#T11" target="_self">Th11</a></i>;<br/><b>hence </b><a NAME="E7:21_1_1_1_1_2"/>
(  not <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="convex1.html#K1">*</a> <span class="p1">(<span class="default"><a href="circled1.html#K2" target="_self">Cir</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="circled1.html#K2" target="_self">Cir</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="convex1.html#K1">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="circled1.html#E4:21_1_1_1_1_2"><i><font color="Green">E14</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_1_2"><b>suppose </b></a><a NAME="E1:21_1_1_2"/><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:21_1_1_2"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="convex1.html#K1">*</a> <span class="p1">(<span class="default"><a href="circled1.html#K2" target="_self">Cir</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="circled1.html#K2" target="_self">Cir</a> <font color="Maroon">c<sub>2</sub></font> </span>}</span> 
 <b>by </b><i><a class="ref" href="convex1.html#D1">CONVEX1:def 1</a></i>;<br/><b>assume </b><a NAME="E3:21_1_1_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="convex1.html#K1">*</a> <span class="p1">(<span class="default"><a href="circled1.html#K2" target="_self">Cir</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E5:21_1_1_2"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>5</sub></font>
 <b>and </b><br/><a NAME="E6:21_1_1_2"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="circled1.html#K2" target="_self">Cir</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="circled1.html#E2:21_1_1_2"><i><font color="Green">E13</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>6</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E8:21_1_1_2"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="circled1.html#K1" target="_self">Circled-Family</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="circled1.html#E8:21_1_1_2"><i><font color="Green">E16</font></i></a></i>;<br/><a NAME="E10:21_1_1_2"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K6">meet</a> <span class="p1">(<span class="default"><a href="circled1.html#K1" target="_self">Circled-Family</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="circled1.html#E6:21_1_1_2"><i><font color="Green">E15</font></i></a></i>;<br/><a NAME="E11:21_1_1_2"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="circled1.html#K1" target="_self">Circled-Family</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="convex1.html#K1">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> holds <br/><font color="Maroon">c<sub>3</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:21_1_1_2_1"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="circled1.html#K1" target="_self">Circled-Family</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="convex1.html#K1">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>8</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E3:21_1_1_2_1"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>9</sub></font> is <a href="rltopsp1.html#V3">circled</a>
 
<b>by </b><i><a class="txt" href="circled1.html#E1:21_1_1_2_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="circled1.html#D2" target="_self">Def2</a></i>;<br/>
<a NAME="E4:21_1_1_2_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="convex1.html#K1">*</a> <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="circled1.html#E1:21_1_1_2_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="circled1.html#D2" target="_self">Def2</a></i>;<br/>
<a NAME="E5:21_1_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="convex1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="convex1.html#K1">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="convex1.html#K1">*</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="ref" href="convex1.html#T39">CONVEX1:39</a></i>;<br/>
<a NAME="E6:21_1_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="convex1.html#K1">*</a> <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="convex1.html#K1">*</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="ref" href="convex1.html#T37">CONVEX1:37</a></i>;<br/>
<a NAME="E7:21_1_1_2_1"/><b>then </b>
1 <a href="convex1.html#K1">*</a> <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="convex1.html#K1">*</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="circled1.html#E1:21_1_1_2"><i><font color="Green">E12</font></i></a>, <a class="ref" href="xcmplx_0.html#D7">XCMPLX_0:def 7</a></i>;<br/>
<a NAME="E8:21_1_1_2_1"/><b>then </b><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="convex1.html#K1">*</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="ref" href="convex1.html#T32">CONVEX1:32</a></i>;<br/>
<a NAME="E9:21_1_1_2_1"/><i><font color="Green">E21</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="convex1.html#K1">*</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="convex1.html#D1">CONVEX1:def 1</a></i>;<br/>
<a NAME="E10:21_1_1_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="convex1.html#K1">*</a> <font color="Maroon">c<sub>9</sub></font> is <a href="rltopsp1.html#V3">circled</a>
 
<b>by </b><i><a class="txt" href="circled1.html#E3:21_1_1_2_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="circled1.html#T2" target="_self">Th2</a></i>;<br/>
<a NAME="E11:21_1_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="convex1.html#K1">*</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="circled1.html#K1" target="_self">Circled-Family</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="circled1.html#E8:21_1_1_2_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="circled1.html#D2" target="_self">Def2</a></i>;<br/>
<a NAME="E12:21_1_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="convex1.html#K1">*</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="circled1.html#E10:21_1_1_2"><i><font color="Green">E17</font></i></a>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>10</sub></font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E14:21_1_1_2_1"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>10</sub></font>
 <b>and </b><br/><a NAME="E15:21_1_1_2_1"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="circled1.html#E9:21_1_1_2_1"><i><font color="Green">E21</font></i></a></i>;<br/>
<font color="Maroon">c<sub>3</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>5</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="real_1.html#K2">"</a> </span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>10</sub></font>
<b>by </b><i><a class="txt" href="circled1.html#E14:21_1_1_2_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="rlvect_1.html#D9">RLVECT_1:def 9</a></i>
<br/>.= 
1 <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>10</sub></font>
<b>by </b><i><a class="txt" href="circled1.html#E1:21_1_1_2"><i><font color="Green">E12</font></i></a>, <a class="ref" href="xcmplx_0.html#D7">XCMPLX_0:def 7</a></i>
<br/>.= 
<font color="Maroon">c<sub>10</sub></font>
<b>by </b><i><a class="ref" href="rlvect_1.html#D9">RLVECT_1:def 9</a></i>
;<br/>
<b>hence </b><a NAME="E17:21_1_1_2_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="circled1.html#E15:21_1_1_2_1"><i><font color="Green">E23</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E12:21_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K6">meet</a> <span class="p1">(<span class="default"><a href="circled1.html#K1" target="_self">Circled-Family</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="convex1.html#K1">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/><b>hence </b><a NAME="E13:21_1_1_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="circled1.html#K2" target="_self">Cir</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="convex1.html#K1">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="circled1.html#E5:21_1_1_2"><i><font color="Green">E14</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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

<a NAME="E2:21"/>
<font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a>  <a href="circled1.html#K2" target="_self">Cir</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="circled1.html#T11" target="_self">Th11</a></i>;<br/>
<a NAME="E3:21"/><b>then </b><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="convex1.html#K1">*</a> <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font> <a href="convex1.html#K1">*</a> <span class="p1">(<span class="default"><a href="circled1.html#K2" target="_self">Cir</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="convex1.html#T39">CONVEX1:39</a></i>;<br/>
<a NAME="E4:21"/>
<font color="Maroon">c<sub>3</sub></font> <a href="convex1.html#K1">*</a> <span class="p1">(<span class="default"><a href="circled1.html#K2" target="_self">Cir</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> is <a href="rltopsp1.html#V3">circled</a>
 
<b>by </b><i><a class="ref" href="circled1.html#T2" target="_self">Th2</a></i>;<br/>
<b>hence </b><a NAME="E5:21"/>
 <a href="circled1.html#K2" target="_self">Cir</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="convex1.html#K1">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font> <a href="convex1.html#K1">*</a> <span class="p1">(<span class="default"><a href="circled1.html#K2" target="_self">Cir</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="circled1.html#E3:21"><i><font color="Green">E12</font></i></a>, <a class="ref" href="circled1.html#T12" target="_self">Th12</a></i>;<br/>


</div>
