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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="partfun1.html#NM1">PartFunc</a> of  <a href="supinf_1.html#K3">REAL</a> , <a href="supinf_1.html#K3">REAL</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><a NAME="E1:7"/><i><font color="Green">E8</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="rfunct_4.html#R1" target="_self">is_strictly_convex_on</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E2:7"/><b>then </b><i><font color="Green">E9</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></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">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 holds <br/>for <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being  <a href="real_1.html#NM1">Real</a> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="binop_2.html#K11">*</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>3</sub></font> holds <br/><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="binop_2.html#K11">*</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Olive">b<sub>3</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>3</sub></font></span>)</span></span>)</span> ) )
 
<b>by </b><i><a class="ref" href="rfunct_4.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E3:7"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> st 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 holds <br/>for <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being  <a href="real_1.html#NM1">Real</a> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="binop_2.html#K11">*</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> holds <br/><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="binop_2.html#K11">*</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Olive">b<sub>3</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>3</sub></font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<b>assume </b><a NAME="E1:7_1"/><i><font color="Green">E10</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 ;<br/>

<a NAME="E2:7_1"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="real_1.html#NM1">Real</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="binop_2.html#K11">*</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> holds <br/><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="binop_2.html#K11">*</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Olive">b<sub>2</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<b>assume </b><a NAME="E1:7_1_1"/><i><font color="Green">E11</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="binop_2.html#K11">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_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="E1:7_1_1_1_1"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 1 or ( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 ) )
 </span><b>by </b><i><a class="txt" href="rfunct_4.html#E1:7_1"><i><font color="Green">E10</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:7_1_1_1_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:7_1_1_1_1_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="binop_2.html#K11">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:7_1_1_1_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:7_1_1_1_1_2"/>
<font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="binop_2.html#K11">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1_1_1_1_3"><b>suppose </b></a><a NAME="E1:7_1_1_1_1_3"/><i><font color="Green">E12</font></i>: 
( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1_1_1_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1_1_1_1_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:7_1_1_1_1_3_1_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>5</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1_1_1_1_3_1_1_1"><b>suppose </b></a><a NAME="E1:7_1_1_1_1_3_1_1_1"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><div class="add"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="binop_2.html#K11">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> = 
<font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="binop_2.html#K11">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="binop_2.html#K9">+</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="binop_2.html#K11">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>

<br/>.= 
<font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="rfunct_4.html#E1:7_1_1_1_1_3_1_1_1"><i><font color="Green">E13</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="binop_2.html#K11">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="rfunct_4.html#E1:7_1_1_1_1_3_1_1_1"><i><font color="Green">E13</font></i></a></i>
;<br/><b>hence </b><a NAME="E3:7_1_1_1_1_3_1_1_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="binop_2.html#K11">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1_1_1_1_3_1_1_2"><b>suppose </b></a><a NAME="E1:7_1_1_1_1_3_1_1_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:7_1_1_1_1_3_1_1_2"/>
<font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="binop_2.html#K11">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="rfunct_4.html#E1:7"><i><font color="Green">E8</font></i></a>, <a class="txt" href="rfunct_4.html#E1:7_1_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="rfunct_4.html#E1:7_1_1_1_1_3"><i><font color="Green">E12</font></i></a>, <a class="ref" href="rfunct_4.html#D1" target="_self">Def1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:7_1_1_1_1_3"/>
<font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="binop_2.html#K11">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:7_1_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="binop_2.html#K11">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:7_1"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="real_1.html#NM1">Real</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="binop_2.html#K11">*</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> holds <br/><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="binop_2.html#K11">*</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Olive">b<sub>2</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:7"/>
<font color="Maroon">c<sub>1</sub></font> <a href="rfunct_3.html#R2">is_convex_on</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="rfunct_4.html#E2:7"><i><font color="Green">E9</font></i></a>, <a class="ref" href="rfunct_3.html#D13">RFUNCT_3:def 13</a></i>;<br/>


</div>
