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

<b>let </b><font color="Maroon">V</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="rlvect_1.html#L2">RLSStruct</a> ;<br/><b>let </b><font color="Maroon">M</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">V</font>;<br/>



<a NAME="E1:6"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">M</font> is <a href="convex1.html#V1" target="_self">convex</a> implies for <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a>  st 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">r</font> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 holds <br/><span class="p1">(<span class="default"><font color="Olive">r</font> <a href="convex1.html#K1" target="_self">*</a> <font color="Maroon">M</font></span>)</span> <a href="rusub_4.html#K6">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Olive">r</font></span>)</span> <a href="convex1.html#K1" target="_self">*</a> <font color="Maroon">M</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">M</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:6_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">M</font> is <a href="convex1.html#V1" target="_self">convex</a>
 ;<br/>

<b>let </b><font color="Maroon">r</font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<b>assume </b><a NAME="E2:6_1"/><i><font color="Green">E35</font></i>: 
( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font> &amp; <font color="Maroon">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 )
 ;<br/>

<a NAME="E3:6_1"/>
for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="convex1.html#K1" target="_self">*</a> <font color="Maroon">M</font></span>)</span> <a href="rusub_4.html#K6">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="convex1.html#K1" target="_self">*</a> <font color="Maroon">M</font></span>)</span> holds <br/><font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>;<br/>

<b>assume </b><a NAME="E1:6_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="convex1.html#K1" target="_self">*</a> <font color="Maroon">M</font></span>)</span> <a href="rusub_4.html#K6">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="convex1.html#K1" target="_self">*</a> <font color="Maroon">M</font></span>)</span>
 ;<br/>

<a NAME="E2:6_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">u</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">v</font></span>)</span> where <font color="Olive">u</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>, <font color="Olive">v</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font> : ( <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">r</font> <a href="convex1.html#K1" target="_self">*</a> <font color="Maroon">M</font> &amp; <font color="Olive">v</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="convex1.html#K1" target="_self">*</a> <font color="Maroon">M</font> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="rusub_4.html#D10">RUSUB_4:def 10</a></i>;<br/>
<b>then consider </b><font color="Maroon">u</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>, <font color="Maroon">v</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E4:6_1_1"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">v</font> &amp; <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">r</font> <a href="convex1.html#K1" target="_self">*</a> <font color="Maroon">M</font> &amp; <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="convex1.html#K1" target="_self">*</a> <font color="Maroon">M</font> )
 ;<br/>
<a NAME="E5:6_1_1"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_1.html#K3">*</a> <font color="Olive">w</font></span>)</span> where <font color="Olive">w</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font> : <font color="Olive">w</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> </span>}</span> 
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">w1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E7:6_1_1"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w1</font> &amp; <font color="Maroon">w1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> )
 ;<br/>
<a NAME="E8:6_1_1"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Olive">w</font></span>)</span> where <font color="Olive">w</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font> : <font color="Olive">w</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> </span>}</span> 
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">w2</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E10:6_1_1"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w2</font> &amp; <font color="Maroon">w2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> )
 ;<br/>
<b>thus </b><a NAME="E11:6_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 <b>by </b><i>, , , , , </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:6_1"/>
<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="convex1.html#K1" target="_self">*</a> <font color="Maroon">M</font></span>)</span> <a href="rusub_4.html#K6">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="convex1.html#K1" target="_self">*</a> <font color="Maroon">M</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">M</font>
 <b>by </b><i><a class="ref" href="subset_1.html#T7">SUBSET_1:7</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:6"/>
( ( for <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a>  st 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">r</font> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 holds <br/><span class="p1">(<span class="default"><font color="Olive">r</font> <a href="convex1.html#K1" target="_self">*</a> <font color="Maroon">M</font></span>)</span> <a href="rusub_4.html#K6">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Olive">r</font></span>)</span> <a href="convex1.html#K1" target="_self">*</a> <font color="Maroon">M</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">M</font> ) implies <font color="Maroon">M</font> is <a href="convex1.html#V1" target="_self">convex</a> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:6_2"/><i><font color="Green">E47</font></i>: 
for <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a>  st 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">r</font> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 holds <br/><span class="p1">(<span class="default"><font color="Olive">r</font> <a href="convex1.html#K1" target="_self">*</a> <font color="Maroon">M</font></span>)</span> <a href="rusub_4.html#K6">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Olive">r</font></span>)</span> <a href="convex1.html#K1" target="_self">*</a> <font color="Maroon">M</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">M</font>
 ;<br/>

<b>let </b><font color="Maroon">u</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>, <font color="Maroon">v</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="convex1.html#D2" target="_self">CONVEX1:def 2</a><br/><b>let </b><font color="Maroon">r</font> be   <a href="real_1.html#NM1">Real</a>;<br/>



<b>assume </b><a NAME="E2:6_2"/><i><font color="Green">E51</font></i>: 
( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font> &amp; <font color="Maroon">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 )
 ;<br/>

<b>assume </b><a NAME="E3:6_2"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> &amp; <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> )
 ;<br/>

<a NAME="E4:6_2"/><b>then </b>
<font color="Maroon">r</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_1.html#K3">*</a> <font color="Olive">w</font></span>)</span> where <font color="Olive">w</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font> : <font color="Olive">w</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> </span>}</span> 
 
;<br/>
<a NAME="E5:6_2"/><b>then </b><i><font color="Green">E53</font></i>: 
<font color="Maroon">r</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">r</font> <a href="convex1.html#K1" target="_self">*</a> <font color="Maroon">M</font>
 
;<br/>
<a NAME="E6:6_2"/>
<span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Olive">w</font></span>)</span> where <font color="Olive">w</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font> : <font color="Olive">w</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> </span>}</span> 
 
<b>by </b><i/>;<br/>
<a NAME="E7:6_2"/><b>then </b>
<span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="convex1.html#K1" target="_self">*</a> <font color="Maroon">M</font>
 
;<br/>
<a NAME="E8:6_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">p</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">q</font></span>)</span> where <font color="Olive">p</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font>, <font color="Olive">q</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font> : ( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">r</font> <a href="convex1.html#K1" target="_self">*</a> <font color="Maroon">M</font> &amp; <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="convex1.html#K1" target="_self">*</a> <font color="Maroon">M</font> ) </span>}</span> 
 
<b>by </b><i/>;<br/>
<a NAME="E9:6_2"/><b>then </b><i><font color="Green">E55</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="convex1.html#K1" target="_self">*</a> <font color="Maroon">M</font></span>)</span> <a href="rusub_4.html#K6">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="convex1.html#K1" target="_self">*</a> <font color="Maroon">M</font></span>)</span>
 
<b>by </b><i><a class="ref" href="rusub_4.html#D10">RUSUB_4:def 10</a></i>;<br/>
<a NAME="E10:6_2"/>
<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="convex1.html#K1" target="_self">*</a> <font color="Maroon">M</font></span>)</span> <a href="rusub_4.html#K6">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="convex1.html#K1" target="_self">*</a> <font color="Maroon">M</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">M</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E11:6_2"/>
<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:6"/>
( <font color="Maroon">M</font> is <a href="convex1.html#V1" target="_self">convex</a> iff for <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a>  st 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">r</font> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 holds <br/><span class="p1">(<span class="default"><font color="Olive">r</font> <a href="convex1.html#K1" target="_self">*</a> <font color="Maroon">M</font></span>)</span> <a href="rusub_4.html#K6">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Olive">r</font></span>)</span> <a href="convex1.html#K1" target="_self">*</a> <font color="Maroon">M</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">M</font> )
 <b>by </b><i/>;<br/>


</div>
