<?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="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <a href="convfun1.html#K4" target="_self">RLS_Real</a> , <a href="numbers.html#K6">ExtREAL</a> ;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <a href="convfun1.html#K4" target="_self">RLS_Real</a> ;<br/>





<b>assume </b><a NAME="E1:43"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">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> holds <br/>( ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> implies <font color="Maroon">c<sub>2</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> ) &amp; ( not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> implies <font color="Maroon">c<sub>2</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) ) ) )
 ;<br/>

<a NAME="E2:43"/><i><font color="Green">E34</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="convfun1.html#K4" target="_self">RLS_Real</a>  holds <font color="Maroon">c<sub>2</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K2">-infty</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="convfun1.html#K4" target="_self">RLS_Real</a> ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="real_1.html#NM1">Real</a> ;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:43_1_1"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> or  not <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_1_1_1"><b>suppose </b></a><a NAME="E1:43_1_1_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><div class="add"><a NAME="E2:43_1_1_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a> 
 ;<br/><b>hence </b><a NAME="E3:43_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K2">-infty</a> 
 <b>by </b><i><a class="txt" href="convfun1.html#E1:43"><i><font color="Green">E33</font></i></a>, <a class="txt" href="convfun1.html#E1:43_1_1_1"><i><font color="Green">E35</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_1_1_2"><b>suppose </b></a><a NAME="E1:43_1_1_2"/>
not <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:43_1_1_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K2">-infty</a> 
 <b>by </b><i><a class="txt" href="convfun1.html#E1:43"><i><font color="Green">E33</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E3:43"/><i><font color="Green">E35</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="convfun1.html#K4" target="_self">RLS_Real</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> holds <br/><font color="Maroon">c<sub>2</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="convfun1.html#K4" target="_self">RLS_Real</a> ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="real_1.html#NM1">Real</a> ;<br/>
<b>assume </b><a NAME="E2:43_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<a NAME="E3:43_2"/><b>then </b>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <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="convfun1.html#E1:43"><i><font color="Green">E33</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:43_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a> 
 ;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E4:43"/>
( <font color="Maroon">c<sub>2</sub></font> is <a href="convfun1.html#V1" target="_self">convex</a> implies ( <font color="Maroon">c<sub>1</sub></font> <a href="rfunct_3.html#R2">is_convex_on</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> is <a href="convex1.html#V1">convex</a> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:43_3"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="convfun1.html#V1" target="_self">convex</a>
 ;<br/>

<b>thus </b><a NAME="E2:43_3"/>
<font color="Maroon">c<sub>1</sub></font> <a href="rfunct_3.html#R2">is_convex_on</a> <font color="Maroon">c<sub>3</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_3_1"><b>proof </b></a><div class="add">

<a NAME="E1:43_3_1"/>
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>3</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</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>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#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="43_3_1_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:43_3_1_1"/><i><font color="Green">E37</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 ;<br/>

<b>let </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<b>assume </b><a NAME="E2:43_3_1_1"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="binop_2.html#K11">*</a> <font color="Maroon">c<sub>5</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>4</sub></font></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/>

<a NAME="E3:43_3_1_1"/><b>then </b><i><font color="Green">E38</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>6</sub></font> &amp; <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>4</sub></font> <a href="binop_2.html#K11">*</a> <font color="Maroon">c<sub>5</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>4</sub></font></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="binop_2.html#K11">*</a> <font color="Maroon">c<sub>5</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>4</sub></font></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="convfun1.html#E1:43"><i><font color="Green">E33</font></i></a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_3_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:43_3_1_1_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 1 or ( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_3_1_1_1_1"><b>suppose </b></a><a NAME="E1:43_3_1_1_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:43_3_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>4</sub></font> <a href="binop_2.html#K11">*</a> <font color="Maroon">c<sub>5</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>4</sub></font></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">c<sub>6</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>4</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> <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>4</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>6</sub></font></span>)</span></span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_3_1_1_1_2"><b>suppose </b></a><a NAME="E1:43_3_1_1_1_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:43_3_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>4</sub></font> <a href="binop_2.html#K11">*</a> <font color="Maroon">c<sub>5</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>4</sub></font></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">c<sub>6</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>4</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> <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>4</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>6</sub></font></span>)</span></span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_3_1_1_1_3"><b>suppose </b></a><a NAME="E1:43_3_1_1_1_3"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 )
 ;<br/><div class="add"><a NAME="E2:43_3_1_1_1_3"/><b>then </b><i><font color="Green">E39</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;</a> 1 )
 <b>by </b><i><a class="txt" href="convfun1.html#E1:43_3_1_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>5</sub></font> as   <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="convfun1.html#K4" target="_self">RLS_Real</a>  ;<br/><b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>6</sub></font> as   <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="convfun1.html#K4" target="_self">RLS_Real</a>  ;<br/><a NAME="E5:43_3_1_1_1_3"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</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>4</sub></font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p3">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="convfun1.html#E2:43"><i><font color="Green">E34</font></i></a>, <a class="txt" href="convfun1.html#E1:43_3"><i><font color="Green">E36</font></i></a>, <a class="txt" href="convfun1.html#E2:43_3_1_1_1_3"><i><font color="Green">E39</font></i></a>, <a class="ref" href="convfun1.html#T9" target="_self">Th9</a></i>;<br/><a NAME="E6:43_3_1_1_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</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>7</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>4</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>8</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p3">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="convfun1.html#E3:43_3_1_1"><i><font color="Green">E38</font></i></a>, <a class="txt" href="convfun1.html#E34"><i><font color="Green">Lemma31</font></i></a></i>;<br/><a NAME="E7:43_3_1_1_1_3"/><b>then </b><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</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>4</sub></font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure6.html#K1">R_EAL</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</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>5</sub></font></span>)</span></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>4</sub></font></span>)</span> <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>6</sub></font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="convfun1.html#E5:43_3_1_1_1_3"><i><font color="Green">E40</font></i></a>, <a class="ref" href="measure6.html#D1">MEASURE6:def 1</a></i>;<br/><a NAME="E8:43_3_1_1_1_3"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="binop_2.html#K11">*</a> <font color="Maroon">c<sub>5</sub></font> &amp; <span class="p1">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="convfun1.html#E9"><i><font color="Green">Lemma7</font></i></a></i>;<br/><a NAME="E9:43_3_1_1_1_3"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</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>4</sub></font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <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>4</sub></font> <a href="binop_2.html#K11">*</a> <font color="Maroon">c<sub>5</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>4</sub></font></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="convfun1.html#E3:43_3_1_1"><i><font color="Green">E38</font></i></a>, <a class="txt" href="convfun1.html#E10"><i><font color="Green">Lemma8</font></i></a></i>;<br/><a NAME="E10:43_3_1_1_1_3"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</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>4</sub></font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="measure6.html#K1">R_EAL</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="binop_2.html#K11">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="measure6.html#D1">MEASURE6:def 1</a></i>;<br/><b>hence </b><a NAME="E11:43_3_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>4</sub></font> <a href="binop_2.html#K11">*</a> <font color="Maroon">c<sub>5</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>4</sub></font></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">c<sub>6</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>4</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> <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>4</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>6</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="convfun1.html#E7:43_3_1_1_1_3"><i><font color="Green">E41</font></i></a>, <a class="ref" href="measure6.html#T13">MEASURE6:13</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E2:43_3_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="rfunct_3.html#R2">is_convex_on</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="convfun1.html#E1:43"><i><font color="Green">E33</font></i></a>, <a class="ref" href="rfunct_3.html#D13">RFUNCT_3:def 13</a></i>;<br/>


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

<b>thus </b><a NAME="E3:43_3"/>
<font color="Maroon">c<sub>3</sub></font> is <a href="convex1.html#V1">convex</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_3_2"><b>proof </b></a><div class="add">

<a NAME="E1:43_3_2"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="convfun1.html#K4" target="_self">RLS_Real</a> <br/>for <font color="Olive">b<sub>3</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>3</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;</a> 1 &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> holds <br/><span class="p1">(<span class="default"><font color="Olive">b<sub>3</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</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>3</sub></font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_3_2_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="rlvect_1.html#NM1">VECTOR</a> of <a href="convfun1.html#K4" target="_self">RLS_Real</a> ;<br/><b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/>



<b>assume </b><a NAME="E1:43_3_2_1"/><i><font color="Green">E37</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;</a> 1 &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/>

<a NAME="E2:43_3_2_1"/><b>then </b><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</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>6</sub></font></span>)</span> <a href="rlvect_1.html#K3">*</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"><span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p3">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="convfun1.html#E2:43"><i><font color="Green">E34</font></i></a>, <a class="txt" href="convfun1.html#E1:43_3"><i><font color="Green">E36</font></i></a>, <a class="ref" href="convfun1.html#T9" target="_self">Th9</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>2</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>4</sub></font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="txt" href="convfun1.html#E3:43"><i><font color="Green">E35</font></i></a>, <a class="txt" href="convfun1.html#E1:43_3_2_1"><i><font color="Green">E37</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>2</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>5</sub></font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="txt" href="convfun1.html#E3:43"><i><font color="Green">E35</font></i></a>, <a class="txt" href="convfun1.html#E1:43_3_2_1"><i><font color="Green">E37</font></i></a></i>;<br/>
<a NAME="E5:43_3_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K11">*</a> <font color="Maroon">c<sub>7</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>6</sub></font></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p3">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="convfun1.html#E34"><i><font color="Green">Lemma31</font></i></a></i>;<br/>
<a NAME="E6:43_3_2_1"/><b>then </b><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</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>6</sub></font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K1">+infty</a> 
 
<b>by </b><i><a class="txt" href="convfun1.html#E2:43_3_2_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="supinf_1.html#T18">SUPINF_1:18</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</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>6</sub></font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> as   <a href="real_1.html#NM1">Real</a> ;<br/>
<a NAME="E8:43_3_2_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="convfun1.html#E1:43"><i><font color="Green">E33</font></i></a>, <a class="txt" href="convfun1.html#E6:43_3_2_1"><i><font color="Green">E39</font></i></a></i>;<br/>
<b>hence </b><a NAME="E9:43_3_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</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>6</sub></font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:43_3_2"/>
<font color="Maroon">c<sub>3</sub></font> is <a href="convex1.html#V1">convex</a>
 <b>by </b><i><a class="ref" href="convex1.html#D2">CONVEX1:def 2</a></i>;<br/>


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


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

<b>thus </b><a NAME="E5:43"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="rfunct_3.html#R2">is_convex_on</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> is <a href="convex1.html#V1">convex</a> implies <font color="Maroon">c<sub>2</sub></font> is <a href="convfun1.html#V1" target="_self">convex</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_4"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:43_4"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="rfunct_3.html#R2">is_convex_on</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> is <a href="convex1.html#V1">convex</a> )
 ;<br/>

<a NAME="E2:43_4"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="convfun1.html#K4" target="_self">RLS_Real</a> <br/>for <font color="Olive">b<sub>3</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>3</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;</a> 1 holds <br/><font color="Maroon">c<sub>2</sub></font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive">b<sub>3</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</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>3</sub></font></span>)</span> <a href="rlvect_1.html#K3">*</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"><span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p3">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Olive">b<sub>3</sub></font></span>)</span></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_4_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="rlvect_1.html#NM1">VECTOR</a> of <a href="convfun1.html#K4" target="_self">RLS_Real</a> ;<br/><b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/>



<b>assume </b><a NAME="E1:43_4_1"/><i><font color="Green">E37</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;</a> 1 )
 ;<br/>

<a NAME="E2:43_4_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="measure6.html#D1">MEASURE6:def 1</a></i>;<br/>
<a NAME="E3:43_4_1"/><b>then </b><i><font color="Green">E38</font></i>: 
 <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="convfun1.html#E1:43_4_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="extreal1.html#T18">EXTREAL1:18</a></i>;<br/>
<a NAME="E4:43_4_1"/><i><font color="Green">E39</font></i>: 
1 <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="txt" href="convfun1.html#E1:43_4_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="xreal_1.html#T52">XREAL_1:52</a></i>;<br/>
<a NAME="E5:43_4_1"/>
1 <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="measure6.html#K1">R_EAL</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="measure6.html#D1">MEASURE6:def 1</a></i>;<br/>
<a NAME="E6:43_4_1"/><b>then </b><i><font color="Green">E40</font></i>: 
 <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="measure6.html#K1">R_EAL</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="convfun1.html#E4:43_4_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="extreal1.html#T18">EXTREAL1:18</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="real_1.html#NM1">Real</a> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>5</sub></font> as   <a href="real_1.html#NM1">Real</a> ;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_4_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:43_4_1_1"/>
( ( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> ) or ( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; not <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> ) or ( not <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> ) or ( not <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; not <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_4_1_1_1"><b>suppose </b></a><a NAME="E1:43_4_1_1_1"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/><div class="add"><a NAME="E2:43_4_1_1_1"/><b>then </b><i><font color="Green">E42</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</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>6</sub></font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="convfun1.html#E1:43_4"><i><font color="Green">E36</font></i></a>, <a class="txt" href="convfun1.html#E1:43_4_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="convex1.html#D2">CONVEX1:def 2</a></i>;<br/><a NAME="E3:43_4_1_1_1"/>
( <font color="Maroon">c<sub>6</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K11">*</a> <font color="Maroon">c<sub>7</sub></font> &amp; <span class="p1">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">c<sub>8</sub></font> )
 <b>by </b><i><a class="txt" href="convfun1.html#E9"><i><font color="Green">Lemma7</font></i></a></i>;<br/><a NAME="E4:43_4_1_1_1"/><b>then </b><i><font color="Green">E43</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</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>6</sub></font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K11">*</a> <font color="Maroon">c<sub>7</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>6</sub></font></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="convfun1.html#E10"><i><font color="Green">Lemma8</font></i></a></i>;<br/><a NAME="E5:43_4_1_1_1"/><b>then </b>
<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>6</sub></font> <a href="binop_2.html#K11">*</a> <font color="Maroon">c<sub>7</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>6</sub></font></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">c<sub>8</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>6</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>7</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>6</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>8</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="convfun1.html#E1:43_4"><i><font color="Green">E36</font></i></a>, <a class="txt" href="convfun1.html#E1:43_4_1"><i><font color="Green">E37</font></i></a>, <a class="txt" href="convfun1.html#E1:43_4_1_1_1"><i><font color="Green">E41</font></i></a>, <a class="txt" href="convfun1.html#E2:43_4_1_1_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="rfunct_3.html#D13">RFUNCT_3:def 13</a></i>;<br/><a NAME="E6:43_4_1_1_1"/><b>then </b><i><font color="Green">E44</font></i>: 
 <a href="measure6.html#K1">R_EAL</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K11">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="measure6.html#K1">R_EAL</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</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>7</sub></font></span>)</span></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>6</sub></font></span>)</span> <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>8</sub></font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="measure6.html#T13">MEASURE6:13</a></i>;<br/><a NAME="E7:43_4_1_1_1"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>5</sub></font> &amp; <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>6</sub></font> <a href="binop_2.html#K11">*</a> <font color="Maroon">c<sub>7</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>6</sub></font></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</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>6</sub></font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="convfun1.html#E1:43"><i><font color="Green">E33</font></i></a>, <a class="txt" href="convfun1.html#E1:43_4_1_1_1"><i><font color="Green">E41</font></i></a>, <a class="txt" href="convfun1.html#E2:43_4_1_1_1"><i><font color="Green">E42</font></i></a>, <a class="txt" href="convfun1.html#E4:43_4_1_1_1"><i><font color="Green">E43</font></i></a></i>;<br/><a NAME="E8:43_4_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</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>7</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>6</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>8</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p3">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="convfun1.html#E34"><i><font color="Green">Lemma31</font></i></a></i>;<br/><a NAME="E9:43_4_1_1_1"/><b>then </b>
 <a href="measure6.html#K1">R_EAL</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</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>7</sub></font></span>)</span></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>6</sub></font></span>)</span> <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>8</sub></font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p3">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="measure6.html#D1">MEASURE6:def 1</a></i>;<br/><b>hence </b><a NAME="E10:43_4_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</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>6</sub></font></span>)</span> <a href="rlvect_1.html#K3">*</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"><span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p3">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="convfun1.html#E6:43_4_1_1_1"><i><font color="Green">E44</font></i></a>, <a class="txt" href="convfun1.html#E7:43_4_1_1_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="measure6.html#D1">MEASURE6:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_4_1_1_2"><b>suppose </b></a><a NAME="E1:43_4_1_1_2"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; not <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/><div class="add"><a NAME="E2:43_4_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 <b>by </b><i><a class="txt" href="convfun1.html#E1:43"><i><font color="Green">E33</font></i></a></i>;<br/><a NAME="E3:43_4_1_1_2"/><b>then </b><i><font color="Green">E42</font></i>: 
<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p2">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 <b>by </b><i><a class="txt" href="convfun1.html#E6:43_4_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="extreal1.html#D1">EXTREAL1:def 1</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>2</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>7</sub></font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="txt" href="convfun1.html#E3:43"><i><font color="Green">E35</font></i></a>, <a class="txt" href="convfun1.html#E1:43_4_1_1_2"><i><font color="Green">E41</font></i></a></i>;<br/><a NAME="E5:43_4_1_1_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="measure6.html#D1">MEASURE6:def 1</a></i>;<br/><a NAME="E6:43_4_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K11">*</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="extreal1.html#T13">EXTREAL1:13</a></i>;<br/><a NAME="E7:43_4_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a> 
 ;<br/><a NAME="E8:43_4_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p3">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 <b>by </b><i><a class="txt" href="convfun1.html#E3:43_4_1_1_2"><i><font color="Green">E42</font></i></a>, <a class="ref" href="supinf_1.html#T6">SUPINF_1:6</a>, <a class="ref" href="supinf_2.html#D2">SUPINF_2:def 2</a></i>;<br/><b>hence </b><a NAME="E9:43_4_1_1_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</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>6</sub></font></span>)</span> <a href="rlvect_1.html#K3">*</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"><span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p3">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="supinf_1.html#T20">SUPINF_1:20</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_4_1_1_3"><b>suppose </b></a><a NAME="E1:43_4_1_1_3"/><i><font color="Green">E41</font></i>: 
( not <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/><div class="add"><a NAME="E2:43_4_1_1_3"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 <b>by </b><i><a class="txt" href="convfun1.html#E1:43"><i><font color="Green">E33</font></i></a></i>;<br/><a NAME="E3:43_4_1_1_3"/><b>then </b><i><font color="Green">E42</font></i>: 
<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 <b>by </b><i><a class="txt" href="convfun1.html#E3:43_4_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="extreal1.html#D1">EXTREAL1:def 1</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>2</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>8</sub></font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="txt" href="convfun1.html#E3:43"><i><font color="Green">E35</font></i></a>, <a class="txt" href="convfun1.html#E1:43_4_1_1_3"><i><font color="Green">E41</font></i></a></i>;<br/><a NAME="E5:43_4_1_1_3"/>
1 <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="measure6.html#K1">R_EAL</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="measure6.html#D1">MEASURE6:def 1</a></i>;<br/><a NAME="E6:43_4_1_1_3"/><b>then </b>
<span class="p1">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p2">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="extreal1.html#T13">EXTREAL1:13</a></i>;<br/><a NAME="E7:43_4_1_1_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p2">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a> 
 ;<br/><a NAME="E8:43_4_1_1_3"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p3">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 <b>by </b><i><a class="txt" href="convfun1.html#E3:43_4_1_1_3"><i><font color="Green">E42</font></i></a>, <a class="ref" href="supinf_1.html#T6">SUPINF_1:6</a>, <a class="ref" href="supinf_2.html#D2">SUPINF_2:def 2</a></i>;<br/><b>hence </b><a NAME="E9:43_4_1_1_3"/>
<font color="Maroon">c<sub>2</sub></font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</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>6</sub></font></span>)</span> <a href="rlvect_1.html#K3">*</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"><span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p3">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="supinf_1.html#T20">SUPINF_1:20</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_4_1_1_4"><b>suppose </b></a><a NAME="E1:43_4_1_1_4"/>
( not <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; not <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/><div class="add"><a NAME="E2:43_4_1_1_4"/><b>then </b>
( <font color="Maroon">c<sub>2</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">c<sub>2</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  )
 <b>by </b><i><a class="txt" href="convfun1.html#E1:43"><i><font color="Green">E33</font></i></a></i>;<br/><a NAME="E3:43_4_1_1_4"/><b>then </b>
( <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <span class="p1">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p2">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  )
 <b>by </b><i><a class="txt" href="convfun1.html#E3:43_4_1"><i><font color="Green">E38</font></i></a>, <a class="txt" href="convfun1.html#E6:43_4_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="extreal1.html#D1">EXTREAL1:def 1</a></i>;<br/><a NAME="E4:43_4_1_1_4"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p3">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 <b>by </b><i><a class="ref" href="supinf_2.html#D2">SUPINF_2:def 2</a></i>;<br/><b>hence </b><a NAME="E5:43_4_1_1_4"/>
<font color="Maroon">c<sub>2</sub></font> <a href="mesfunc1.html#K4">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="rlvect_1.html#K4">+</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>6</sub></font></span>)</span> <a href="rlvect_1.html#K3">*</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"><span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="measure6.html#K1">R_EAL</a> <span class="p3">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="extreal1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="supinf_1.html#T20">SUPINF_1:20</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E3:43_4"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="convfun1.html#V1" target="_self">convex</a>
 <b>by </b><i><a class="txt" href="convfun1.html#E2:43"><i><font color="Green">E34</font></i></a>, <a class="ref" href="convfun1.html#T9" target="_self">Th9</a></i>;<br/>


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


</div>
