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

<b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">f</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/>



<a NAME="E1:10"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">f</font> <a href="rfunct_4.html#R1" target="_self">is_strictly_convex_on</a> <font color="Maroon">X</font> implies ( <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> &amp; ( for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Olive">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Olive">a</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b</font> &amp; <font color="Olive">b</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">c</font> holds <br/><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">b</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive">c</font> <a href="binop_2.html#K10">-</a> <font color="Olive">b</font></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p3">(<span class="default"><font color="Olive">c</font> <a href="binop_2.html#K10">-</a> <font color="Olive">a</font></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">a</font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive">b</font> <a href="binop_2.html#K10">-</a> <font color="Olive">a</font></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p3">(<span class="default"><font color="Olive">c</font> <a href="binop_2.html#K10">-</a> <font color="Olive">a</font></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">c</font></span>)</span></span>)</span> ) ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:10_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">f</font> <a href="rfunct_4.html#R1" target="_self">is_strictly_convex_on</a> <font color="Maroon">X</font>
 ;<br/>

<a NAME="E2:10_1"/>
for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Olive">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Olive">a</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b</font> &amp; <font color="Olive">b</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">c</font> holds <br/><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">b</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive">c</font> <a href="binop_2.html#K10">-</a> <font color="Olive">b</font></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p3">(<span class="default"><font color="Olive">c</font> <a href="binop_2.html#K10">-</a> <font color="Olive">a</font></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">a</font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive">b</font> <a href="binop_2.html#K10">-</a> <font color="Olive">a</font></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p3">(<span class="default"><font color="Olive">c</font> <a href="binop_2.html#K10">-</a> <font color="Olive">a</font></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">c</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:10_1_1"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">a</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b</font> &amp; <font color="Maroon">b</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c</font> )
 ;<br/>

<a NAME="E2:10_1_1"/><b>then </b><i><font color="Green">E41</font></i>: 
( <font color="Maroon">c</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">b</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font> &amp; 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">b</font> )
 
<b>by </b><i><a class="ref" href="real_2.html#T105">REAL_2:105</a>, <a class="ref" href="xreal_1.html#T52">XREAL_1:52</a></i>;<br/>
<a NAME="E3:10_1_1"/><i><font color="Green">E42</font></i>: 
( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">b</font></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">b</font></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 )
 
<b>by </b><i>, <a class="ref" href="real_2.html#T127">REAL_2:127</a>, <a class="ref" href="real_2.html#T142">REAL_2:142</a></i>;<br/>
<b>set </b><font color="Maroon">p</font> = <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">b</font></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span>;<br/>
<a NAME="E4:10_1_1"/>
( 1 <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">b</font></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">b</font></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p3">(<span class="default"><font color="Maroon">c</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">a</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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">b</font></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p4">(<span class="default"><font color="Maroon">c</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">c</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1_1"><b>proof </b></a><div class="add">

<i><font color="Green">E43</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">b</font></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">b</font></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T63">XCMPLX_1:63</a></i>
<br/>.= 
1
<b>by </b><i>, <a class="ref" href="xcmplx_1.html#T60">XCMPLX_1:60</a></i>
;<br/>
<b>hence </b><a NAME="E2:10_1_1_1"/>
1 <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">b</font></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span>
 ;<br/>

<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">b</font></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p3">(<span class="default"><font color="Maroon">c</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">a</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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">b</font></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p4">(<span class="default"><font color="Maroon">c</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">c</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="binop_2.html#K11">*</a> <span class="p3">(<span class="default"><font color="Maroon">c</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p3">(<span class="default"><font color="Maroon">c</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="xcmplx_1.html#T75">XCMPLX_1:75</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="binop_2.html#K11">*</a> <span class="p3">(<span class="default"><font color="Maroon">c</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="binop_2.html#K11">*</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T75">XCMPLX_1:75</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c</font> <a href="binop_2.html#K11">*</a> <font color="Maroon">a</font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="binop_2.html#K11">*</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T63">XCMPLX_1:63</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span>

;<br/>
<b>hence </b><a NAME="E4:10_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">b</font></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p3">(<span class="default"><font color="Maroon">c</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">a</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> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">b</font></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p4">(<span class="default"><font color="Maroon">c</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">c</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 <b>by </b><i>, <a class="ref" href="xcmplx_1.html#T90">XCMPLX_1:90</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:10_1_1"/>
<font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">b</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">b</font></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p3">(<span class="default"><font color="Maroon">c</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p3">(<span class="default"><font color="Maroon">c</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c</font></span>)</span></span>)</span>
 <b>by </b><i>, , , </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:10_1"/>
( <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> &amp; ( for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Olive">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Olive">a</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b</font> &amp; <font color="Olive">b</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">c</font> holds <br/><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">b</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive">c</font> <a href="binop_2.html#K10">-</a> <font color="Olive">b</font></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p3">(<span class="default"><font color="Olive">c</font> <a href="binop_2.html#K10">-</a> <font color="Olive">a</font></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">a</font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive">b</font> <a href="binop_2.html#K10">-</a> <font color="Olive">a</font></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p3">(<span class="default"><font color="Olive">c</font> <a href="binop_2.html#K10">-</a> <font color="Olive">a</font></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">c</font></span>)</span></span>)</span> ) )
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:10"/>
( <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> &amp; ( for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Olive">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Olive">a</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b</font> &amp; <font color="Olive">b</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">c</font> holds <br/><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">b</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive">c</font> <a href="binop_2.html#K10">-</a> <font color="Olive">b</font></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p3">(<span class="default"><font color="Olive">c</font> <a href="binop_2.html#K10">-</a> <font color="Olive">a</font></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">a</font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive">b</font> <a href="binop_2.html#K10">-</a> <font color="Olive">a</font></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p3">(<span class="default"><font color="Olive">c</font> <a href="binop_2.html#K10">-</a> <font color="Olive">a</font></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">c</font></span>)</span></span>)</span> ) implies <font color="Maroon">f</font> <a href="rfunct_4.html#R1" target="_self">is_strictly_convex_on</a> <font color="Maroon">X</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_2"><b>proof </b></a><div class="add">

<b>assume </b><b>that </b><br/><a NAME="E1:10_2"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">X</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E2:10_2"/><i><font color="Green">E45</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Olive">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Olive">a</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b</font> &amp; <font color="Olive">b</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">c</font> holds <br/><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">b</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive">c</font> <a href="binop_2.html#K10">-</a> <font color="Olive">b</font></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p3">(<span class="default"><font color="Olive">c</font> <a href="binop_2.html#K10">-</a> <font color="Olive">a</font></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">a</font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive">b</font> <a href="binop_2.html#K10">-</a> <font color="Olive">a</font></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p3">(<span class="default"><font color="Olive">c</font> <a href="binop_2.html#K10">-</a> <font color="Olive">a</font></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">c</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E3:10_2"/>
for <font color="Olive">p</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">p</font> &amp; <font color="Olive">p</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 holds <br/>for <font color="Olive">r</font>, <font color="Olive">s</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="binop_2.html#K11">*</a> <font color="Olive">r</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">p</font></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Olive">s</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Olive">r</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">s</font> holds <br/><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive">p</font> <a href="binop_2.html#K11">*</a> <font color="Olive">r</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">p</font></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Olive">s</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">r</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">p</font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">s</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_2_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:10_2_1"/><i><font color="Green">E46</font></i>: 
( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">p</font> &amp; <font color="Maroon">p</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 )
 ;<br/>

<a NAME="E2:10_2_1"/>
for <font color="Olive">r</font>, <font color="Olive">s</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="binop_2.html#K11">*</a> <font color="Olive">r</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">p</font></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Olive">s</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Olive">r</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">s</font> holds <br/><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="binop_2.html#K11">*</a> <font color="Olive">r</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">p</font></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Olive">s</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">r</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">p</font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">s</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">r</font> be   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">s</font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<b>assume </b><a NAME="E1:10_2_1_1"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="binop_2.html#K11">*</a> <font color="Maroon">r</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">p</font></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">s</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">r</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">s</font> )
 ;<br/>

<a NAME="E2:10_2_1_1"/>
<font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="binop_2.html#K11">*</a> <font color="Maroon">r</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">p</font></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">r</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">p</font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">s</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_2_1_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_2_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_2_1_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="E1:10_2_1_1_1_1_1"/>
( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">p</font> &amp; <font color="Maroon">p</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_2_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:10_2_1_1_1_1_1_1"/><i><font color="Green">E48</font></i>: 
( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">p</font> &amp; <font color="Maroon">p</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 )
 ;<br/><div class="add"><a NAME="E2:10_2_1_1_1_1_1_1"/><b>then </b>
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="binop_2.html#K9">+</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/><a NAME="E3:10_2_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E49</font></i>: 
( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="binop_2.html#K10">-</a> <font color="Maroon">p</font> &amp; 1 <a href="binop_2.html#K10">-</a> <font color="Maroon">p</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 )
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T21">XREAL_1:21</a>, <a class="ref" href="xreal_1.html#T52">XREAL_1:52</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_2_1_1_1_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_2_1_1_1_1_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="E1:10_2_1_1_1_1_1_1_1_1"/>
( <font color="Maroon">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">s</font> or <font color="Maroon">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">s</font> )
 </span><b>by </b><i>, <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="10_2_1_1_1_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:10_2_1_1_1_1_1_1_1_1_1"/>
<font color="Maroon">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">s</font>
 ;<br/><div class="add"><a NAME="E2:10_2_1_1_1_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E50</font></i>: 
<font color="Maroon">r</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">s</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T52">XREAL_1:52</a></i>;<br/><b>set </b><font color="Maroon">t</font> = <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="binop_2.html#K11">*</a> <font color="Maroon">r</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">p</font></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">s</font></span>)</span>;<br/><a NAME="E3:10_2_1_1_1_1_1_1_1_1_1"/><i><font color="Green">E62</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="binop_2.html#K11">*</a> <font color="Maroon">r</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">p</font></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">s</font></span>)</span>
 ;<br/><a NAME="E4:10_2_1_1_1_1_1_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="binop_2.html#K11">*</a> <font color="Maroon">r</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">p</font></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> <font color="Maroon">s</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i>, , <a class="ref" href="real_2.html#T122">REAL_2:122</a></i>;<br/><a NAME="E5:10_2_1_1_1_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E64</font></i>: 
<font color="Maroon">s</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="binop_2.html#K11">*</a> <font color="Maroon">r</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">p</font></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">s</font></span>)</span>
 <b>by </b><i><a class="ref" href="real_2.html#T106">REAL_2:106</a></i>;<br/><a NAME="E6:10_2_1_1_1_1_1_1_1_1_1"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">r</font> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="binop_2.html#K11">*</a> <font color="Maroon">r</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">p</font></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">p</font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">s</font></span>)</span>
 ;<br/><a NAME="E7:10_2_1_1_1_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon">r</font> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="binop_2.html#K11">*</a> <font color="Maroon">r</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">p</font></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i>, , <a class="ref" href="real_2.html#T122">REAL_2:122</a></i>;<br/><a NAME="E8:10_2_1_1_1_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E66</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="binop_2.html#K11">*</a> <font color="Maroon">r</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">p</font></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">s</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="real_2.html#T106">REAL_2:106</a></i>;<br/><a NAME="E9:10_2_1_1_1_1_1_1_1_1_1"/>
( <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="binop_2.html#K11">*</a> <font color="Maroon">r</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">p</font></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">s</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">s</font></span>)</span> <a href="hidden.html#R1">=</a> 1 <a href="binop_2.html#K10">-</a> <font color="Maroon">p</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="binop_2.html#K11">*</a> <font color="Maroon">r</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">p</font></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> <font color="Maroon">s</font></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">s</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> )
 <b>by </b><i>, , , <a class="ref" href="xcmplx_1.html#T90">XCMPLX_1:90</a></i>;<br/><b>hence </b><a NAME="E10:10_2_1_1_1_1_1_1_1_1_1"/>
<font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="binop_2.html#K11">*</a> <font color="Maroon">r</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">p</font></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">r</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">p</font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">s</font></span>)</span></span>)</span>
 <b>by </b><i>, , , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_2_1_1_1_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:10_2_1_1_1_1_1_1_1_1_2"/>
<font color="Maroon">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">s</font>
 ;<br/><div class="add"><a NAME="E2:10_2_1_1_1_1_1_1_1_1_2"/><b>then </b><i><font color="Green">E67</font></i>: 
<font color="Maroon">s</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T52">XREAL_1:52</a></i>;<br/><b>set </b><font color="Maroon">t</font> = <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="binop_2.html#K11">*</a> <font color="Maroon">r</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">p</font></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">s</font></span>)</span>;<br/><a NAME="E3:10_2_1_1_1_1_1_1_1_1_2"/><i><font color="Green">E68</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="binop_2.html#K11">*</a> <font color="Maroon">r</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">p</font></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K10">-</a> <font color="Maroon">p</font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">r</font></span>)</span>
 ;<br/><a NAME="E4:10_2_1_1_1_1_1_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="binop_2.html#K11">*</a> <font color="Maroon">r</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">p</font></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> <font color="Maroon">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i>, , <a class="ref" href="real_2.html#T122">REAL_2:122</a></i>;<br/><a NAME="E5:10_2_1_1_1_1_1_1_1_1_2"/><b>then </b><i><font color="Green">E69</font></i>: 
<font color="Maroon">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="binop_2.html#K11">*</a> <font color="Maroon">r</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">p</font></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">s</font></span>)</span>
 <b>by </b><i><a class="ref" href="real_2.html#T106">REAL_2:106</a></i>;<br/><a NAME="E6:10_2_1_1_1_1_1_1_1_1_2"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">s</font> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="binop_2.html#K11">*</a> <font color="Maroon">r</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">p</font></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">r</font></span>)</span>
 ;<br/><a NAME="E7:10_2_1_1_1_1_1_1_1_1_2"/><b>then </b>
<font color="Maroon">s</font> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="binop_2.html#K11">*</a> <font color="Maroon">r</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">p</font></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i>, , <a class="ref" href="real_2.html#T122">REAL_2:122</a></i>;<br/><a NAME="E8:10_2_1_1_1_1_1_1_1_1_2"/><b>then </b><i><font color="Green">E71</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="binop_2.html#K11">*</a> <font color="Maroon">r</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">p</font></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">s</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">s</font>
 <b>by </b><i><a class="ref" href="real_2.html#T106">REAL_2:106</a></i>;<br/><a NAME="E9:10_2_1_1_1_1_1_1_1_1_2"/>
( <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="binop_2.html#K10">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="binop_2.html#K11">*</a> <font color="Maroon">r</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">p</font></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">s</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">r</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="binop_2.html#K11">*</a> <font color="Maroon">r</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">p</font></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> <font color="Maroon">r</font></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">r</font></span>)</span> <a href="hidden.html#R1">=</a> 1 <a href="binop_2.html#K10">-</a> <font color="Maroon">p</font> )
 <b>by </b><i>, , , <a class="ref" href="xcmplx_1.html#T90">XCMPLX_1:90</a></i>;<br/><b>hence </b><a NAME="E10:10_2_1_1_1_1_1_1_1_1_2"/>
<font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="binop_2.html#K11">*</a> <font color="Maroon">r</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">p</font></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">r</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">p</font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">s</font></span>)</span></span>)</span>
 <b>by </b><i>, , , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E5:10_2_1_1_1_1_1_1"/>
<font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="binop_2.html#K11">*</a> <font color="Maroon">r</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">p</font></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">r</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">p</font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">s</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="E2:10_2_1_1_1"/>
<font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="binop_2.html#K11">*</a> <font color="Maroon">r</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">p</font></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">r</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">p</font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">s</font></span>)</span></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:10_2_1_1"/>
<font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="binop_2.html#K11">*</a> <font color="Maroon">r</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">p</font></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">r</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">p</font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">s</font></span>)</span></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:10_2_1"/>
for <font color="Olive">r</font>, <font color="Olive">s</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="binop_2.html#K11">*</a> <font color="Olive">r</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">p</font></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Olive">s</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Olive">r</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">s</font> holds <br/><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="binop_2.html#K11">*</a> <font color="Olive">r</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">p</font></span>)</span> <a href="binop_2.html#K11">*</a> <font color="Olive">s</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">r</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">p</font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">s</font></span>)</span></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:10_2"/>
<font color="Maroon">f</font> <a href="rfunct_4.html#R1" target="_self">is_strictly_convex_on</a> <font color="Maroon">X</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:10"/>
( <font color="Maroon">f</font> <a href="rfunct_4.html#R1" target="_self">is_strictly_convex_on</a> <font color="Maroon">X</font> iff ( <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> &amp; ( for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Olive">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Olive">a</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b</font> &amp; <font color="Olive">b</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">c</font> holds <br/><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">b</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive">c</font> <a href="binop_2.html#K10">-</a> <font color="Olive">b</font></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p3">(<span class="default"><font color="Olive">c</font> <a href="binop_2.html#K10">-</a> <font color="Olive">a</font></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">a</font></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive">b</font> <a href="binop_2.html#K10">-</a> <font color="Olive">a</font></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p3">(<span class="default"><font color="Olive">c</font> <a href="binop_2.html#K10">-</a> <font color="Olive">a</font></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">c</font></span>)</span></span>)</span> ) ) )
 <b>by </b><i/>;<br/>


</div>
