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

<b>let </b><font color="Maroon" title="c1">V</font> be   <a href="clvect_1.html#NM2" title="CLVECT_1:NM.2">ComplexLinearSpace</a>;<br/><b>let </b><font color="Maroon" title="c2">v1</font>, <font color="Maroon" title="c3">v2</font>, <font color="Maroon" title="c4">v3</font> be   <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font>;<br/><b>let </b><font color="Maroon" title="c5">L</font> be    <a href="convex4.html#M2" title="CONVEX4:mode.2">C_Linear_Combination</a> of <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c2">v1</font>,<font color="Maroon" title="c3">v2</font>,<font color="Maroon" title="c4">v3</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:124"/><i><font color="Green" title="E67">A1</font></i>: 
( <font color="Maroon" title="c2">v1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">v2</font> &amp; <font color="Maroon" title="c3">v2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">v3</font> &amp; <font color="Maroon" title="c4">v3</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">v1</font> )
 <b>and </b><br/><a NAME="E2:124"/><i><font color="Green" title="E68">A2</font></i>: 
<font color="Maroon" title="c5">L</font> is <a href="convex4.html#V3" title="CONVEX4:attr.3">convex</a>
 ;<br/>

<a NAME="E3:124"/><i><font color="Green" title="E69">A3</font></i>: 
 <a href="convex4.html#K1" title="CONVEX4:func.1">Carrier</a> <font color="Maroon" title="c5">L</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c2">v1</font>,<font color="Maroon" title="c3">v2</font>,<font color="Maroon" title="c4">v3</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span>
 
<b>by </b><i><a class="ref" href="convex4.html#D4" target="_self" title="CONVEX4:def.4">Def3</a></i>;<br/>
<a NAME="E4:124"/><i><font color="Green" title="E70">A4</font></i>: 
 <a href="convex4.html#K1" title="CONVEX4:func.1">Carrier</a> <font color="Maroon" title="c5">L</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="convex4.html#E2:124"><i><font color="Green" title="E68">A2</font></i></a>, <a class="ref" href="convex4.html#T80" target="_self" title="CONVEX4:th.80">Th221</a></i>;<br/>
<a NAME="E5:124"/>
 ex <font color="Olive" title="b1">r1</font>, <font color="Olive" title="b2">r2</font>, <font color="Olive" title="b3">r3</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <br/>( <font color="Olive" title="b1">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">v1</font> &amp; <font color="Olive" title="b2">r2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">v2</font> &amp; <font color="Olive" title="b3">r3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">v3</font> &amp; <span class="p1">(<span class="default"><font color="Olive" title="b1">r1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b2">r2</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b3">r3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Olive" title="b1">r1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  &amp; <font color="Olive" title="b2">r2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  &amp; <font color="Olive" title="b3">r3</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="124_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="124_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:124_1_1"/>
(  <a href="convex4.html#K1" title="CONVEX4:func.1">Carrier</a> <font color="Maroon" title="c5">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">v1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> or  <a href="convex4.html#K1" title="CONVEX4:func.1">Carrier</a> <font color="Maroon" title="c5">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">v2</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> or  <a href="convex4.html#K1" title="CONVEX4:func.1">Carrier</a> <font color="Maroon" title="c5">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">v3</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> or  <a href="convex4.html#K1" title="CONVEX4:func.1">Carrier</a> <font color="Maroon" title="c5">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c2">v1</font>,<font color="Maroon" title="c3">v2</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> or  <a href="convex4.html#K1" title="CONVEX4:func.1">Carrier</a> <font color="Maroon" title="c5">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c3">v2</font>,<font color="Maroon" title="c4">v3</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> or  <a href="convex4.html#K1" title="CONVEX4:func.1">Carrier</a> <font color="Maroon" title="c5">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c2">v1</font>,<font color="Maroon" title="c4">v3</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> or  <a href="convex4.html#K1" title="CONVEX4:func.1">Carrier</a> <font color="Maroon" title="c5">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c2">v1</font>,<font color="Maroon" title="c3">v2</font>,<font color="Maroon" title="c4">v3</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> )
 </span><b>by </b><i><a class="txt" href="convex4.html#E3:124"><i><font color="Green" title="E69">A3</font></i></a>, <a class="txt" href="convex4.html#E4:124"><i><font color="Green" title="E70">A4</font></i></a>, <a class="ref" href="zfmisc_1.html#T142" title="ZFMISC_1:th.142">ZFMISC_1:142</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="124_1_1_1"><b>suppose </b></a><a NAME="E1:124_1_1_1"/><i><font color="Green" title="E71">A5</font></i>: 
 <a href="convex4.html#K1" title="CONVEX4:func.1">Carrier</a> <font color="Maroon" title="c5">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">v1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 ;<br/><div class="add"><a NAME="E2:124_1_1_1"/><b>then </b><i><font color="Green" title="E72">A6</font></i>: 
 ex <font color="Olive" title="b1">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">v1</font> &amp; <font color="Olive" title="b1">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <b>by </b><i><a class="txt" href="convex4.html#E2:124"><i><font color="Green" title="E68">A2</font></i></a>, <a class="ref" href="convex4.html#T83" target="_self" title="CONVEX4:th.83">Lm211</a></i>;<br/><a NAME="E3:124_1_1_1"/>
( not <font color="Maroon" title="c3">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="convex4.html#K1" title="CONVEX4:func.1">Carrier</a> <font color="Maroon" title="c5">L</font> &amp; not <font color="Maroon" title="c4">v3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="convex4.html#K1" title="CONVEX4:func.1">Carrier</a> <font color="Maroon" title="c5">L</font> )
 <b>by </b><i><a class="txt" href="convex4.html#E1:124"><i><font color="Green" title="E67">A1</font></i></a>, <a class="txt" href="convex4.html#E1:124_1_1_1"><i><font color="Green" title="E71">A5</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E4:124_1_1_1"/><b>then </b>
( 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">v1</font> &amp;  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">v2</font> &amp;  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">v3</font> &amp; <span class="p1">(<span class="default">1 <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  &amp;  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  &amp;  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  )
 <b>by </b><i><a class="txt" href="convex4.html#E2:124_1_1_1"><i><font color="Green" title="E72">A6</font></i></a></i>;<br/><b>hence </b><a NAME="E5:124_1_1_1"/>
 ex <font color="Olive" title="b1">r1</font>, <font color="Olive" title="b2">r2</font>, <font color="Olive" title="b3">r3</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <br/>( <font color="Olive" title="b1">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">v1</font> &amp; <font color="Olive" title="b2">r2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">v2</font> &amp; <font color="Olive" title="b3">r3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">v3</font> &amp; <span class="p1">(<span class="default"><font color="Olive" title="b1">r1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b2">r2</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b3">r3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Olive" title="b1">r1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  &amp; <font color="Olive" title="b2">r2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  &amp; <font color="Olive" title="b3">r3</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="124_1_1_2"><b>suppose </b></a><a NAME="E1:124_1_1_2"/><i><font color="Green" title="E71">A7</font></i>: 
 <a href="convex4.html#K1" title="CONVEX4:func.1">Carrier</a> <font color="Maroon" title="c5">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">v2</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 ;<br/><div class="add"><a NAME="E2:124_1_1_2"/><b>then </b><i><font color="Green" title="E72">A8</font></i>: 
 ex <font color="Olive" title="b1">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">v2</font> &amp; <font color="Olive" title="b1">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <b>by </b><i><a class="txt" href="convex4.html#E2:124"><i><font color="Green" title="E68">A2</font></i></a>, <a class="ref" href="convex4.html#T83" target="_self" title="CONVEX4:th.83">Lm211</a></i>;<br/><a NAME="E3:124_1_1_2"/>
( not <font color="Maroon" title="c2">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="convex4.html#K1" title="CONVEX4:func.1">Carrier</a> <font color="Maroon" title="c5">L</font> &amp; not <font color="Maroon" title="c4">v3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="convex4.html#K1" title="CONVEX4:func.1">Carrier</a> <font color="Maroon" title="c5">L</font> )
 <b>by </b><i><a class="txt" href="convex4.html#E1:124"><i><font color="Green" title="E67">A1</font></i></a>, <a class="txt" href="convex4.html#E1:124_1_1_2"><i><font color="Green" title="E71">A7</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E4:124_1_1_2"/><b>then </b>
(  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">v1</font> &amp; 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">v2</font> &amp;  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">v3</font> &amp; <span class="p1">(<span class="default"><a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  &amp; 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  &amp;  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  )
 <b>by </b><i><a class="txt" href="convex4.html#E2:124_1_1_2"><i><font color="Green" title="E72">A8</font></i></a></i>;<br/><b>hence </b><a NAME="E5:124_1_1_2"/>
 ex <font color="Olive" title="b1">r1</font>, <font color="Olive" title="b2">r2</font>, <font color="Olive" title="b3">r3</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <br/>( <font color="Olive" title="b1">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">v1</font> &amp; <font color="Olive" title="b2">r2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">v2</font> &amp; <font color="Olive" title="b3">r3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">v3</font> &amp; <span class="p1">(<span class="default"><font color="Olive" title="b1">r1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b2">r2</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b3">r3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Olive" title="b1">r1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  &amp; <font color="Olive" title="b2">r2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  &amp; <font color="Olive" title="b3">r3</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="124_1_1_3"><b>suppose </b></a><a NAME="E1:124_1_1_3"/><i><font color="Green" title="E71">A9</font></i>: 
 <a href="convex4.html#K1" title="CONVEX4:func.1">Carrier</a> <font color="Maroon" title="c5">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">v3</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 ;<br/><div class="add"><a NAME="E2:124_1_1_3"/><b>then </b><i><font color="Green" title="E72">A10</font></i>: 
 ex <font color="Olive" title="b1">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">v3</font> &amp; <font color="Olive" title="b1">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <b>by </b><i><a class="txt" href="convex4.html#E2:124"><i><font color="Green" title="E68">A2</font></i></a>, <a class="ref" href="convex4.html#T83" target="_self" title="CONVEX4:th.83">Lm211</a></i>;<br/><a NAME="E3:124_1_1_3"/>
( not <font color="Maroon" title="c2">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="convex4.html#K1" title="CONVEX4:func.1">Carrier</a> <font color="Maroon" title="c5">L</font> &amp; not <font color="Maroon" title="c3">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="convex4.html#K1" title="CONVEX4:func.1">Carrier</a> <font color="Maroon" title="c5">L</font> )
 <b>by </b><i><a class="txt" href="convex4.html#E1:124"><i><font color="Green" title="E67">A1</font></i></a>, <a class="txt" href="convex4.html#E1:124_1_1_3"><i><font color="Green" title="E71">A9</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E4:124_1_1_3"/><b>then </b>
(  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">v1</font> &amp;  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">v2</font> &amp; 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">v3</font> &amp; <span class="p1">(<span class="default"><a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> </span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  &amp;  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  &amp; 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  )
 <b>by </b><i><a class="txt" href="convex4.html#E2:124_1_1_3"><i><font color="Green" title="E72">A10</font></i></a></i>;<br/><b>hence </b><a NAME="E5:124_1_1_3"/>
 ex <font color="Olive" title="b1">r1</font>, <font color="Olive" title="b2">r2</font>, <font color="Olive" title="b3">r3</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <br/>( <font color="Olive" title="b1">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">v1</font> &amp; <font color="Olive" title="b2">r2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">v2</font> &amp; <font color="Olive" title="b3">r3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">v3</font> &amp; <span class="p1">(<span class="default"><font color="Olive" title="b1">r1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b2">r2</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b3">r3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Olive" title="b1">r1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  &amp; <font color="Olive" title="b2">r2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  &amp; <font color="Olive" title="b3">r3</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="124_1_1_4"><b>suppose </b></a><a NAME="E1:124_1_1_4"/><i><font color="Green" title="E71">A11</font></i>: 
 <a href="convex4.html#K1" title="CONVEX4:func.1">Carrier</a> <font color="Maroon" title="c5">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c2">v1</font>,<font color="Maroon" title="c3">v2</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 ;<br/><div class="add"><a NAME="E2:124_1_1_4"/><b>then </b><i><font color="Green" title="E72">B1</font></i>: 
not <font color="Maroon" title="c4">v3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">v</font> where <font color="Olive" title="b1">v</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font> : <font color="Maroon" title="c5">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  </span>}</span> 
 <b>by </b><i><a class="txt" href="convex4.html#E1:124"><i><font color="Green" title="E67">A1</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/><b>set </b><font color="Maroon" title="c6">r3</font> =  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> ;<br/><b>consider </b><font color="Maroon" title="c7">r1</font>, <font color="Maroon" title="c8">r2</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E4:124_1_1_4"/><i><font color="Green" title="E73">A13</font></i>: 
( <font color="Maroon" title="c7">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">v1</font> &amp; <font color="Maroon" title="c8">r2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">v2</font> &amp; <font color="Maroon" title="c7">r1</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c8">r2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c7">r1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  &amp; <font color="Maroon" title="c8">r2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  )
 <b>by </b><i><a class="txt" href="convex4.html#E1:124"><i><font color="Green" title="E67">A1</font></i></a>, <a class="txt" href="convex4.html#E2:124"><i><font color="Green" title="E68">A2</font></i></a>, <a class="txt" href="convex4.html#E1:124_1_1_4"><i><font color="Green" title="E71">A11</font></i></a>, <a class="ref" href="convex4.html#T84" target="_self" title="CONVEX4:th.84">Lm212</a></i>;<br/><a NAME="E5:124_1_1_4"/>
( <font color="Maroon" title="c7">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">v1</font> &amp; <font color="Maroon" title="c8">r2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">v2</font> &amp;  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">v3</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c7">r1</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c8">r2</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c7">r1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  &amp; <font color="Maroon" title="c8">r2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  &amp;  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  )
 <b>by </b><i><a class="txt" href="convex4.html#E2:124_1_1_4"><i><font color="Green" title="E72">B1</font></i></a>, <a class="txt" href="convex4.html#E4:124_1_1_4"><i><font color="Green" title="E73">A13</font></i></a></i>;<br/><b>hence </b><a NAME="E6:124_1_1_4"/>
 ex <font color="Olive" title="b1">r1</font>, <font color="Olive" title="b2">r2</font>, <font color="Olive" title="b3">r3</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <br/>( <font color="Olive" title="b1">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">v1</font> &amp; <font color="Olive" title="b2">r2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">v2</font> &amp; <font color="Olive" title="b3">r3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">v3</font> &amp; <span class="p1">(<span class="default"><font color="Olive" title="b1">r1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b2">r2</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b3">r3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Olive" title="b1">r1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  &amp; <font color="Olive" title="b2">r2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  &amp; <font color="Olive" title="b3">r3</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="124_1_1_5"><b>suppose </b></a><a NAME="E1:124_1_1_5"/><i><font color="Green" title="E71">A14</font></i>: 
 <a href="convex4.html#K1" title="CONVEX4:func.1">Carrier</a> <font color="Maroon" title="c5">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c3">v2</font>,<font color="Maroon" title="c4">v3</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 ;<br/><div class="add"><a NAME="E2:124_1_1_5"/><b>then </b><i><font color="Green" title="E72">B2</font></i>: 
not <font color="Maroon" title="c2">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="convex4.html#K1" title="CONVEX4:func.1">Carrier</a> <font color="Maroon" title="c5">L</font>
 <b>by </b><i><a class="txt" href="convex4.html#E1:124"><i><font color="Green" title="E67">A1</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/><b>set </b><font color="Maroon" title="c6">r1</font> =  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> ;<br/><b>consider </b><font color="Maroon" title="c7">r2</font>, <font color="Maroon" title="c8">r3</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E4:124_1_1_5"/><i><font color="Green" title="E73">A16</font></i>: 
( <font color="Maroon" title="c7">r2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">v2</font> &amp; <font color="Maroon" title="c8">r3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">v3</font> &amp; <font color="Maroon" title="c7">r2</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c8">r3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c7">r2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  &amp; <font color="Maroon" title="c8">r3</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  )
 <b>by </b><i><a class="txt" href="convex4.html#E1:124"><i><font color="Green" title="E67">A1</font></i></a>, <a class="txt" href="convex4.html#E2:124"><i><font color="Green" title="E68">A2</font></i></a>, <a class="txt" href="convex4.html#E1:124_1_1_5"><i><font color="Green" title="E71">A14</font></i></a>, <a class="ref" href="convex4.html#T84" target="_self" title="CONVEX4:th.84">Lm212</a></i>;<br/><a NAME="E5:124_1_1_5"/>
(  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">v1</font> &amp; <font color="Maroon" title="c7">r2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">v2</font> &amp; <font color="Maroon" title="c8">r3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">v3</font> &amp; <span class="p1">(<span class="default"><a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c7">r2</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c8">r3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  &amp; <font color="Maroon" title="c7">r2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  &amp; <font color="Maroon" title="c8">r3</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  )
 <b>by </b><i><a class="txt" href="convex4.html#E2:124_1_1_5"><i><font color="Green" title="E72">B2</font></i></a>, <a class="txt" href="convex4.html#E4:124_1_1_5"><i><font color="Green" title="E73">A16</font></i></a></i>;<br/><b>hence </b><a NAME="E6:124_1_1_5"/>
 ex <font color="Olive" title="b1">r1</font>, <font color="Olive" title="b2">r2</font>, <font color="Olive" title="b3">r3</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <br/>( <font color="Olive" title="b1">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">v1</font> &amp; <font color="Olive" title="b2">r2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">v2</font> &amp; <font color="Olive" title="b3">r3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">v3</font> &amp; <span class="p1">(<span class="default"><font color="Olive" title="b1">r1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b2">r2</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b3">r3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Olive" title="b1">r1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  &amp; <font color="Olive" title="b2">r2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  &amp; <font color="Olive" title="b3">r3</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="124_1_1_6"><b>suppose </b></a><a NAME="E1:124_1_1_6"/><i><font color="Green" title="E71">A17</font></i>: 
 <a href="convex4.html#K1" title="CONVEX4:func.1">Carrier</a> <font color="Maroon" title="c5">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c2">v1</font>,<font color="Maroon" title="c4">v3</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 ;<br/><div class="add"><a NAME="E2:124_1_1_6"/><b>then </b><i><font color="Green" title="E72">B3</font></i>: 
not <font color="Maroon" title="c3">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="convex4.html#K1" title="CONVEX4:func.1">Carrier</a> <font color="Maroon" title="c5">L</font>
 <b>by </b><i><a class="txt" href="convex4.html#E1:124"><i><font color="Green" title="E67">A1</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/><b>set </b><font color="Maroon" title="c6">r2</font> =  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> ;<br/><b>consider </b><font color="Maroon" title="c7">r1</font>, <font color="Maroon" title="c8">r3</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E4:124_1_1_6"/><i><font color="Green" title="E73">A19</font></i>: 
( <font color="Maroon" title="c7">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">v1</font> &amp; <font color="Maroon" title="c8">r3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">v3</font> &amp; <font color="Maroon" title="c7">r1</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c8">r3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c7">r1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  &amp; <font color="Maroon" title="c8">r3</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  )
 <b>by </b><i><a class="txt" href="convex4.html#E1:124"><i><font color="Green" title="E67">A1</font></i></a>, <a class="txt" href="convex4.html#E2:124"><i><font color="Green" title="E68">A2</font></i></a>, <a class="txt" href="convex4.html#E1:124_1_1_6"><i><font color="Green" title="E71">A17</font></i></a>, <a class="ref" href="convex4.html#T84" target="_self" title="CONVEX4:th.84">Lm212</a></i>;<br/><a NAME="E5:124_1_1_6"/>
( <font color="Maroon" title="c7">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">v1</font> &amp;  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">v2</font> &amp; <font color="Maroon" title="c8">r3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">v3</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c7">r1</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c8">r3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c7">r1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  &amp;  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  &amp; <font color="Maroon" title="c8">r3</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  )
 <b>by </b><i><a class="txt" href="convex4.html#E2:124_1_1_6"><i><font color="Green" title="E72">B3</font></i></a>, <a class="txt" href="convex4.html#E4:124_1_1_6"><i><font color="Green" title="E73">A19</font></i></a></i>;<br/><b>hence </b><a NAME="E6:124_1_1_6"/>
 ex <font color="Olive" title="b1">r1</font>, <font color="Olive" title="b2">r2</font>, <font color="Olive" title="b3">r3</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <br/>( <font color="Olive" title="b1">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">v1</font> &amp; <font color="Olive" title="b2">r2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">v2</font> &amp; <font color="Olive" title="b3">r3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">v3</font> &amp; <span class="p1">(<span class="default"><font color="Olive" title="b1">r1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b2">r2</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b3">r3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Olive" title="b1">r1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  &amp; <font color="Olive" title="b2">r2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  &amp; <font color="Olive" title="b3">r3</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="124_1_1_7"><b>suppose </b></a><a NAME="E1:124_1_1_7"/>
 <a href="convex4.html#K1" title="CONVEX4:func.1">Carrier</a> <font color="Maroon" title="c5">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c2">v1</font>,<font color="Maroon" title="c3">v2</font>,<font color="Maroon" title="c4">v3</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:124_1_1_7"/>
 ex <font color="Olive" title="b1">r1</font>, <font color="Olive" title="b2">r2</font>, <font color="Olive" title="b3">r3</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <br/>( <font color="Olive" title="b1">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">v1</font> &amp; <font color="Olive" title="b2">r2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">v2</font> &amp; <font color="Olive" title="b3">r3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">v3</font> &amp; <span class="p1">(<span class="default"><font color="Olive" title="b1">r1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b2">r2</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b3">r3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Olive" title="b1">r1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  &amp; <font color="Olive" title="b2">r2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  &amp; <font color="Olive" title="b3">r3</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  )
 <b>by </b><i><a class="txt" href="convex4.html#E1:124"><i><font color="Green" title="E67">A1</font></i></a>, <a class="txt" href="convex4.html#E2:124"><i><font color="Green" title="E68">A2</font></i></a>, <a class="ref" href="convex4.html#T85" target="_self" title="CONVEX4:th.85">Lm215</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E6:124"/>
(  ex <font color="Olive" title="b1">r1</font>, <font color="Olive" title="b2">r2</font>, <font color="Olive" title="b3">r3</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <br/>( <font color="Olive" title="b1">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">v1</font> &amp; <font color="Olive" title="b2">r2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">v2</font> &amp; <font color="Olive" title="b3">r3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">v3</font> &amp; <span class="p1">(<span class="default"><font color="Olive" title="b1">r1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b2">r2</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b3">r3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Olive" title="b1">r1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  &amp; <font color="Olive" title="b2">r2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  &amp; <font color="Olive" title="b3">r3</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  ) &amp;  <a href="convex4.html#K4" title="CONVEX4:func.4">Sum</a> <font color="Maroon" title="c5">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">v1</font></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c2">v1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">v2</font></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c3">v2</font></span>)</span></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"><font color="Maroon" title="c5">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">v3</font></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c4">v3</font></span>)</span> )
 <b>by </b><i><a class="txt" href="convex4.html#E1:124"><i><font color="Green" title="E67">A1</font></i></a>, <a class="txt" href="convex4.html#E111"><i><font color="Green" title="E65">Lm214</font></i></a></i>;<br/>


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