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

<b>let </b><font color="Maroon">a</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">P</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 1</span>)</span>;<br/>



<b>assume </b><a NAME="E1:68"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">q</font> where <font color="Olive">q</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 1</span>)</span> :  ex <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Olive">r</font></span><a href="binarith.html#K11">*&gt;</a></span> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="binop_2.html#K7">-</a> <font color="Maroon">a</font> ) </span>}</span> 
 ;<br/>

<a NAME="E2:68"/>
for <font color="Olive">w1</font>, <font color="Olive">w2</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 1</span>)</span>  st <font color="Olive">w1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Olive">w2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> holds <br/> <a href="topreal1.html#K3">LSeg</a> <font color="Olive">w1</font>,<font color="Olive">w2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">w1</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 1</span>)</span>, <font color="Maroon">w2</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 1</span>)</span>;<br/>

<b>assume </b><a NAME="E1:68_1"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">w1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Maroon">w2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> )
 ;<br/>

<b>then consider </b><font color="Maroon">q1</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 1</span>)</span><b> such that </b><br/><a NAME="E3:68_1"/><i><font color="Green">E57</font></i>: 
( <font color="Maroon">q1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w1</font> &amp;  ex <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Maroon">q1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Olive">r</font></span><a href="binarith.html#K11">*&gt;</a></span> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="binop_2.html#K7">-</a> <font color="Maroon">a</font> ) )
 <b>by </b><i><a class="txt" href="jordan2c.html#E9:68_1"><i><font color="Green">E60</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">r1</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E5:68_1"/><i><font color="Green">E58</font></i>: 
( <font color="Maroon">q1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">r1</font></span><a href="binarith.html#K11">*&gt;</a></span> &amp; <font color="Maroon">r1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="binop_2.html#K7">-</a> <font color="Maroon">a</font> )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">q2</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 1</span>)</span><b> such that </b><br/><a NAME="E7:68_1"/><i><font color="Green">E59</font></i>: 
( <font color="Maroon">q2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w2</font> &amp;  ex <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Maroon">q2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Olive">r</font></span><a href="binarith.html#K11">*&gt;</a></span> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="binop_2.html#K7">-</a> <font color="Maroon">a</font> ) )
 <b>by </b><i><a class="txt" href="jordan2c.html#E9:68_1"><i><font color="Green">E60</font></i></a>, </i>;<br/>
<b>consider </b><font color="Maroon">r2</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E9:68_1"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">q2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">r2</font></span><a href="binarith.html#K11">*&gt;</a></span> &amp; <font color="Maroon">r2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="binop_2.html#K7">-</a> <font color="Maroon">a</font> )
 <b>by </b><i/>;<br/>
<b>thus </b><a NAME="E10:68_1"/>
 <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">w1</font>,<font color="Maroon">w2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1_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/>

<b>assume </b><a NAME="E1:68_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">w1</font>,<font color="Maroon">w2</font>
 ;<br/>

<a NAME="E2:68_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"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Olive">r</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">w1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><font color="Olive">r</font> <a href="euclid.html#K18">*</a> <font color="Maroon">w2</font></span>)</span></span>)</span> where <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r</font> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="topreal1.html#D3">TOPREAL1:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">r3</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E4:68_1_1"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">r3</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">w1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Maroon">r3</font> <a href="euclid.html#K18">*</a> <font color="Maroon">w2</font></span>)</span> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r3</font> &amp; <font color="Maroon">r3</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 ;<br/>
<a NAME="E5:68_1_1"/><i><font color="Green">E63</font></i>: 
1 <a href="binop_2.html#K10">-</a> <font color="Maroon">r3</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:68_1_1_1"/>
( <font color="Maroon">r3</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 or <font color="Maroon">r3</font> <a href="xxreal_0.html#R1">&lt;=</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_1_1_1_1"><b>suppose </b></a><a NAME="E1:68_1_1_1_1"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">r3</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:68_1_1_1_1"/><i><font color="Green">E65</font></i>: 
<span class="p1">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">r3</font></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">r1</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">r3</font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><a href="binop_2.html#K7">-</a> <font color="Maroon">a</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan2c.html#E5:68_1_1"><i><font color="Green">E63</font></i></a>, <a class="txt" href="jordan2c.html#E5:68_1_1_1_1"><i><font color="Green">E67</font></i></a>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><a NAME="E3:68_1_1_1_1"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">r3</font> <a href="binop_2.html#K11">*</a> <font color="Maroon">r2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r3</font> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><a href="binop_2.html#K7">-</a> <font color="Maroon">a</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan2c.html#E2:68_1_1_1_1"><i><font color="Green">E65</font></i></a>, <a class="ref" href="jordan2c.html#T8" target="_self">Th8</a>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><a NAME="E4:68_1_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">r3</font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><a href="binop_2.html#K7">-</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><font color="Maroon">r3</font> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><a href="binop_2.html#K7">-</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="binop_2.html#K7">-</a> <font color="Maroon">a</font>
 ;<br/><a NAME="E5:68_1_1_1_1"/><b>then </b><i><font color="Green">E67</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">r3</font></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">r1</font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><font color="Maroon">r3</font> <a href="binop_2.html#K11">*</a> <font color="Maroon">r2</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="binop_2.html#K7">-</a> <font color="Maroon">a</font>
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><i><font color="Green">E130</font></i>: <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">r3</font></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">r1</font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p3">(<span class="default"><font color="Maroon">r3</font> <a href="binop_2.html#K11">*</a> <font color="Maroon">r2</font></span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span> = 
<span class="p1"><a href="jordan2b.html#K2">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">r3</font></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">r1</font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p3">(<span class="default"><font color="Maroon">r3</font> <a href="binop_2.html#K11">*</a> <font color="Maroon">r2</font></span>)</span></span>)</span></span><a href="jordan2b.html#K2">]|</a></span>
<b>by </b><i><a class="ref" href="jordan2b.html#D2">JORDAN2B:def 2</a></i>
<br/>.= 
<span class="p1"><a href="jordan2b.html#K2">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">r3</font></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">r1</font></span>)</span></span><a href="jordan2b.html#K2">]|</a></span> <a href="euclid.html#K17">+</a> <span class="p1"><a href="jordan2b.html#K2">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">r3</font> <a href="binop_2.html#K11">*</a> <font color="Maroon">r2</font></span>)</span></span><a href="jordan2b.html#K2">]|</a></span>
<b>by </b><i><a class="ref" href="jordan2b.html#T27">JORDAN2B:27</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">r3</font></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="jordan2b.html#K2">|[</a><span class="default"><font color="Maroon">r1</font></span><a href="jordan2b.html#K2">]|</a></span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1"><a href="jordan2b.html#K2">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">r3</font> <a href="binop_2.html#K11">*</a> <font color="Maroon">r2</font></span>)</span></span><a href="jordan2b.html#K2">]|</a></span>
<b>by </b><i><a class="ref" href="jordan2b.html#T26">JORDAN2B:26</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">r3</font></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="jordan2b.html#K2">|[</a><span class="default"><font color="Maroon">r1</font></span><a href="jordan2b.html#K2">]|</a></span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Maroon">r3</font> <a href="euclid.html#K18">*</a> <span class="p2"><a href="jordan2b.html#K2">|[</a><span class="default"><font color="Maroon">r2</font></span><a href="jordan2b.html#K2">]|</a></span></span>)</span>
<b>by </b><i><a class="ref" href="jordan2b.html#T26">JORDAN2B:26</a></i>
;<br/><a NAME="E7:68_1_1_1_1"/><i><font color="Green">E131</font></i>: 
<span class="p1"><a href="jordan2b.html#K2">|[</a><span class="default"><font color="Maroon">r1</font></span><a href="jordan2b.html#K2">]|</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">w1</font>
 <b>by </b><i>, <a class="txt" href="jordan2c.html#E5:68_1_1"><i><font color="Green">E63</font></i></a>, <a class="ref" href="jordan2b.html#D2">JORDAN2B:def 2</a></i>;<br/><a NAME="E8:68_1_1_1_1"/>
<span class="p1"><a href="jordan2b.html#K2">|[</a><span class="default"><font color="Maroon">r2</font></span><a href="jordan2b.html#K2">]|</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">w2</font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E1:68_1_1_1_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jordan2c.html#E2:68_1_1_1_1"><i><font color="Green">E65</font></i></a>, <a class="ref" href="jordan2b.html#D2">JORDAN2B:def 2</a></i>;<br/><b>hence </b><a NAME="E9:68_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E9:68_1"><i><font color="Green">E60</font></i></a>, <a class="txt" href="jordan2c.html#E3:68_1_1_1_1"><i><font color="Green">E66</font></i></a>, , , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_1_1_1_2"><b>suppose </b></a><a NAME="E1:68_1_1_1_2"/>
<font color="Maroon">r3</font> <a href="xxreal_0.html#R1">&lt;=</a> 0
 ;<br/><div class="add"><a NAME="E2:68_1_1_1_2"/><b>then </b>
<font color="Maroon">r3</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i/>;<br/><b>then </b><font color="Maroon">x</font> = 
<font color="Maroon">w1</font> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default">0 <a href="euclid.html#K18">*</a> <font color="Maroon">w2</font></span>)</span>
<b>by </b><i>, <a class="ref" href="euclid.html#T33">EUCLID:33</a></i>
<br/>.= 
<font color="Maroon">w1</font> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><a href="euclid.html#K16">0.REAL</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T33">EUCLID:33</a></i>
<br/>.= 
<font color="Maroon">w1</font>
<b>by </b><i><a class="ref" href="euclid.html#T31">EUCLID:31</a></i>
;<br/><b>hence </b><a NAME="E4:68_1_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E4:68_1_1"><i><font color="Green">E61</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:68"/>
<font color="Maroon">P</font> is <a href="jordan1.html#V1">convex</a>
 <b>by </b><i><a class="ref" href="jordan1.html#D1">JORDAN1:def 1</a></i>;<br/>


</div>
