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

<b>let </b><font color="Maroon">X</font> be   <a href="rlvect_1.html#NM2">RealLinearSpace</a>;<br/><b>let </b><font color="Maroon">M</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">r</font> be   <a href="real_1.html#NM1">Real</a>;<br/>





<b>thus </b><a NAME="E1:24"/>
<font color="Maroon">r</font> <a href="convex1.html#K1">*</a> <span class="p1">(<span class="default"><a href="convex1.html#K3">conv</a> <font color="Maroon">M</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="convex1.html#K3">conv</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="convex1.html#K1">*</a> <font color="Maroon">M</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="24_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</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="24_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:24_1_1"/>
( <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">r</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="24_1_1_1"><b>suppose </b></a><a NAME="E1:24_1_1_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_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="E1:24_1_1_1_1"/>
( <font color="Maroon">M</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">M</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="24_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:24_1_1_1_1_1"/>
<font color="Maroon">M</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:24_1_1_1_1_1"/><b>then </b>
<font color="Maroon">M</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">X</font>
 ;<br/><a NAME="E3:24_1_1_1_1_1"/><b>then </b>
 <a href="convex1.html#K3">conv</a> <font color="Maroon">M</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:24_1_1_1_1_1"/>
(  not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">r</font> <a href="convex1.html#K1">*</a> <span class="p1">(<span class="default"><a href="convex1.html#K3">conv</a> <font color="Maroon">M</font></span>)</span> or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="convex1.html#K3">conv</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="convex1.html#K1">*</a> <font color="Maroon">M</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="24_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:24_1_1_1_1_2"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">M</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:24_1_1_1_1_2"/>
<font color="Maroon">M</font> <a href="tarski.html#R1">c=</a>  <a href="convex1.html#K3">conv</a> <font color="Maroon">M</font>
 <b>by </b><i><a class="ref" href="convex2.html#T8">CONVEX2:8</a></i>;<br/><a NAME="E3:24_1_1_1_1_2"/><b>then </b>
 <a href="convex1.html#K3">conv</a> <font color="Maroon">M</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/><a NAME="E4:24_1_1_1_1_2"/><b>then </b><i><font color="Green">E38</font></i>: 
<font color="Maroon">r</font> <a href="convex1.html#K1">*</a> <span class="p1">(<span class="default"><a href="convex1.html#K3">conv</a> <font color="Maroon">M</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">X</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i>, <a class="ref" href="convex1.html#T34">CONVEX1:34</a></i>;<br/><a NAME="E5:24_1_1_1_1_2"/>
<font color="Maroon">r</font> <a href="convex1.html#K1">*</a> <font color="Maroon">M</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">X</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i>, , <a class="ref" href="convex1.html#T34">CONVEX1:34</a></i>;<br/><a NAME="E6:24_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">X</font></span>)</span></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="convex1.html#K3">conv</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="convex1.html#K1">*</a> <font color="Maroon">M</font></span>)</span>
 <b>by </b><i><a class="ref" href="convex2.html#T8">CONVEX2:8</a></i>;<br/><b>hence </b><a NAME="E7:24_1_1_1_1_2"/>
(  not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">r</font> <a href="convex1.html#K1">*</a> <span class="p1">(<span class="default"><a href="convex1.html#K3">conv</a> <font color="Maroon">M</font></span>)</span> or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="convex1.html#K3">conv</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="convex1.html#K1">*</a> <font color="Maroon">M</font></span>)</span> )
 <b>by </b><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="24_1_1_2"><b>suppose </b></a><a NAME="E1:24_1_1_2"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><b>assume </b><a NAME="E2:24_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">r</font> <a href="convex1.html#K1">*</a> <span class="p1">(<span class="default"><a href="convex1.html#K3">conv</a> <font color="Maroon">M</font></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon">v</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E4:24_1_1_2"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font>
 <b>and </b><br/><a NAME="E5:24_1_1_2"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="convex1.html#K3">conv</a> <font color="Maroon">M</font>
 ;<br/><a NAME="E6:24_1_1_2"/>
 <a href="convex1.html#K2">Convex-Family</a> <font color="Maroon">M</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="convex2.html#T7">CONVEX2:7</a></i>;<br/><b>then consider </b><font color="Maroon">Y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E8:24_1_1_2"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a>  <a href="convex1.html#K2">Convex-Family</a> <font color="Maroon">M</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">Y</font> = <font color="Maroon">Y</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> <b>by </b><i/>;<br/><a NAME="E10:24_1_1_2"/><i><font color="Green">E54</font></i>: 
for <font color="Olive">V</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">V</font> <a href="hidden.html#R2">in</a>  <a href="convex1.html#K2">Convex-Family</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="convex1.html#K1">*</a> <font color="Maroon">M</font></span>)</span> holds <br/><font color="Maroon">r</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Olive">V</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1_2_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:24_1_1_2_1"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">V</font> <a href="hidden.html#R2">in</a>  <a href="convex1.html#K2">Convex-Family</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="convex1.html#K1">*</a> <font color="Maroon">M</font></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">V</font> = <font color="Maroon">V</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> ;<br/>
<a NAME="E3:24_1_1_2_1"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">V</font> is <a href="convex1.html#V1">convex</a>
 
<b>by </b><i>, <a class="ref" href="convex1.html#D4">CONVEX1:def 4</a></i>;<br/>
<a NAME="E4:24_1_1_2_1"/>
<font color="Maroon">r</font> <a href="convex1.html#K1">*</a> <font color="Maroon">M</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font>
 
<b>by </b><i>, <a class="ref" href="convex1.html#D4">CONVEX1:def 4</a></i>;<br/>
<a NAME="E5:24_1_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">r</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">r</font> <a href="convex1.html#K1">*</a> <font color="Maroon">M</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="convex1.html#K1">*</a> <font color="Maroon">V</font>
 
<b>by </b><i><a class="ref" href="convex1.html#T39">CONVEX1:39</a></i>;<br/>
<a NAME="E6:24_1_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">r</font></span>)</span> <a href="convex1.html#K1">*</a> <font color="Maroon">M</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="convex1.html#K1">*</a> <font color="Maroon">V</font>
 
<b>by </b><i><a class="ref" href="convex1.html#T37">CONVEX1:37</a></i>;<br/>
<a NAME="E7:24_1_1_2_1"/><b>then </b>
1 <a href="convex1.html#K1">*</a> <font color="Maroon">M</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="convex1.html#K1">*</a> <font color="Maroon">V</font>
 
<b>by </b><i>, <a class="ref" href="xcmplx_0.html#D7">XCMPLX_0:def 7</a></i>;<br/>
<a NAME="E8:24_1_1_2_1"/><b>then </b><i><font color="Green">E62</font></i>: 
<font color="Maroon">M</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="convex1.html#K1">*</a> <font color="Maroon">V</font>
 
<b>by </b><i><a class="ref" href="convex1.html#T32">CONVEX1:32</a></i>;<br/>
<a NAME="E9:24_1_1_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="convex1.html#K1">*</a> <font color="Maroon">V</font> is <a href="convex1.html#V1">convex</a>
 
<b>by </b><i>, <a class="ref" href="convex1.html#T1">CONVEX1:1</a></i>;<br/>
<a NAME="E10:24_1_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="convex1.html#K1">*</a> <font color="Maroon">V</font> <a href="hidden.html#R2">in</a>  <a href="convex1.html#K2">Convex-Family</a> <font color="Maroon">M</font>
 
<b>by </b><i>, <a class="ref" href="convex1.html#D4">CONVEX1:def 4</a></i>;<br/>
<a NAME="E11:24_1_1_2_1"/><b>then </b>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="convex1.html#K1">*</a> <font color="Maroon">V</font>
 
<b>by </b><i>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">w</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E13:24_1_1_2_1"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K2">"</a> </span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w</font>
 <b>and </b><br/><a NAME="E14:24_1_1_2_1"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">w</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font>
 ;<br/>
<font color="Maroon">r</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font> = 
<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K2">"</a> </span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w</font>
<b>by </b><i>, <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">w</font>
<b>by </b><i>, <a class="ref" href="xcmplx_0.html#D7">XCMPLX_0:def 7</a></i>
<br/>.= 
<font color="Maroon">w</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="E16:24_1_1_2_1"/>
<font color="Maroon">r</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><a NAME="E11:24_1_1_2"/>
 <a href="convex1.html#K2">Convex-Family</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="convex1.html#K1">*</a> <font color="Maroon">M</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="convex2.html#T7">CONVEX2:7</a></i>;<br/><b>hence </b><a NAME="E12:24_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="convex1.html#K3">conv</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="convex1.html#K1">*</a> <font color="Maroon">M</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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

<a NAME="E2:24"/>
<font color="Maroon">M</font> <a href="tarski.html#R1">c=</a>  <a href="convex1.html#K3">conv</a> <font color="Maroon">M</font>
 
<b>by </b><i><a class="ref" href="convex2.html#T8">CONVEX2:8</a></i>;<br/>
<a NAME="E3:24"/><b>then </b><i><font color="Green">E66</font></i>: 
<font color="Maroon">r</font> <a href="convex1.html#K1">*</a> <font color="Maroon">M</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">r</font> <a href="convex1.html#K1">*</a> <span class="p1">(<span class="default"><a href="convex1.html#K3">conv</a> <font color="Maroon">M</font></span>)</span>
 
<b>by </b><i><a class="ref" href="convex1.html#T39">CONVEX1:39</a></i>;<br/>
<a NAME="E4:24"/>
<font color="Maroon">r</font> <a href="convex1.html#K1">*</a> <span class="p1">(<span class="default"><a href="convex1.html#K3">conv</a> <font color="Maroon">M</font></span>)</span> is <a href="convex1.html#V1">convex</a>
 
<b>by </b><i><a class="ref" href="convex1.html#T1">CONVEX1:1</a></i>;<br/>
<b>hence </b><a NAME="E5:24"/>
 <a href="convex1.html#K3">conv</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="convex1.html#K1">*</a> <font color="Maroon">M</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">r</font> <a href="convex1.html#K1">*</a> <span class="p1">(<span class="default"><a href="convex1.html#K3">conv</a> <font color="Maroon">M</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="convex1.html#T30">CONVEX1:30</a></i>;<br/>


</div>
