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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">c<sub>2</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>3</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>





<b>assume </b><a NAME="E1:15"/><i><font color="Green">E11</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<a NAME="E2:15"/><b>then </b><i><font color="Green">E12</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="binop_2.html#K12">/</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="real_2.html#T127">REAL_2:127</a></i>;<br/>
<a NAME="E3:15"/><i><font color="Green">E13</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K12">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K13">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="ref" href="seq_1.html#D6">SEQ_1:def 6</a></i>;<br/>
<a NAME="E4:15"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="supinf_1.html#K3">REAL</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K12">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K13">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="supinf_1.html#K3">REAL</a> ;<br/>

<b>assume </b><a NAME="E1:15_1"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/>

<b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K12">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K13">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font> = 
<span class="p1">(<span class="default">1 <a href="binop_2.html#K12">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="rfunct_4.html#E3:15"><i><font color="Green">E13</font></i></a>, <a class="ref" href="seq_1.html#D6">SEQ_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="binop_2.html#K12">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="rfunct_4.html#E1:15_1"><i><font color="Green">E14</font></i></a>, <a class="ref" href="seq_1.html#D6">SEQ_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K12">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>

<br/>.= 
1 <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="rfunct_4.html#E1:15"><i><font color="Green">E11</font></i></a>, <a class="ref" href="xcmplx_1.html#T107">XCMPLX_1:107</a></i>
;<br/>
<b>hence </b><a NAME="E3:15_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K12">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K13">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E5:15"/><b>then </b>
<span class="p1">(<span class="default">1 <a href="binop_2.html#K12">/</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K13">(#)</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="rfunct_4.html#E3:15"><i><font color="Green">E13</font></i></a>, <a class="ref" href="partfun1.html#T34">PARTFUN1:34</a></i>;<br/>
<b>hence </b><a NAME="E6:15"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="rfunct_4.html#R1" target="_self">is_strictly_convex_on</a> <font color="Maroon">c<sub>3</sub></font> iff <font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">c<sub>2</sub></font> <a href="rfunct_4.html#R1" target="_self">is_strictly_convex_on</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="rfunct_4.html#E1:15"><i><font color="Green">E11</font></i></a>, <a class="txt" href="rfunct_4.html#E2:15"><i><font color="Green">E12</font></i></a>, <a class="txt" href="rfunct_4.html#E14"><i><font color="Green">Lemma9</font></i></a></i>;<br/>


</div>
