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

<b>let </b><font color="Maroon" title="c1">r</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/><b>let </b><font color="Maroon" title="c2">f</font> be   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of  <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a> , <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a> ;<br/><b>let </b><font color="Maroon" title="c3">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>





<a NAME="E1:13"/><i><font color="Green" title="E9">A1</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">-</a> <font color="Maroon" title="c1">r</font></span>)</span> <a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">-</a> <span class="p2">(<span class="default"><a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <font color="Maroon" title="c1">r</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">-</a> <font color="Maroon" title="c1">r</font></span>)</span>
 
<b>by </b><i><a class="ref" href="rfunct_3.html#D12" title="RFUNCT_3:def.12">RFUNCT_3:def 12</a></i>;<br/>
<a NAME="E2:13"/><i><font color="Green" title="E10">A2</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">-</a> <font color="Maroon" title="c1">r</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">f</font>
 
<b>by </b><i><a class="ref" href="rfunct_3.html#D12" title="RFUNCT_3:def.12">RFUNCT_3:def 12</a></i>;<br/>
<a NAME="E3:13"/>
for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">-</a> <font color="Maroon" title="c1">r</font></span>)</span> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">-</a> <font color="Maroon" title="c1">r</font></span>)</span> <a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">-</a> <span class="p2">(<span class="default"><a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <font color="Maroon" title="c1">r</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a> ;<br/>

<b>assume </b><a NAME="E1:13_1"/><i><font color="Green" title="E11">A3</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">-</a> <font color="Maroon" title="c1">r</font></span>)</span>
 ;<br/>

<b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">-</a> <font color="Maroon" title="c1">r</font></span>)</span> <a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">-</a> <span class="p2">(<span class="default"><a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <font color="Maroon" title="c1">r</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">x</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">-</a> <font color="Maroon" title="c1">r</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p1">(<span class="default"><a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <font color="Maroon" title="c1">r</font></span>)</span>
<b>by </b><i><a class="txt" href="rfunct_4.html#E1:13"><i><font color="Green" title="E9">A1</font></i></a>, <a class="ref" href="rfunct_3.html#D12" title="RFUNCT_3:def.12">RFUNCT_3:def 12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">-</a> <font color="Maroon" title="c1">r</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <font color="Maroon" title="c1">r</font>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c1">r</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <font color="Maroon" title="c1">r</font>
<b>by </b><i><a class="txt" href="rfunct_4.html#E1:13_1"><i><font color="Green" title="E11">A3</font></i></a>, <a class="ref" href="rfunct_3.html#D12" title="RFUNCT_3:def.12">RFUNCT_3:def 12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">r</font> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c1">r</font></span>)</span>

;<br/>
<b>hence </b><a NAME="E3:13_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">-</a> <font color="Maroon" title="c1">r</font></span>)</span> <a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">-</a> <span class="p2">(<span class="default"><a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <font color="Maroon" title="c1">r</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c4">x</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E4:13"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">-</a> <font color="Maroon" title="c1">r</font></span>)</span> <a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">-</a> <span class="p1">(<span class="default"><a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <font color="Maroon" title="c1">r</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font>
 
<b>by </b><i><a class="txt" href="rfunct_4.html#E1:13"><i><font color="Green" title="E9">A1</font></i></a>, <a class="txt" href="rfunct_4.html#E2:13"><i><font color="Green" title="E10">A2</font></i></a>, <a class="ref" href="partfun1.html#T34" title="PARTFUN1:th.34">PARTFUN1:34</a></i>;<br/>
<b>hence </b><a NAME="E5:13"/>
( <font color="Maroon" title="c2">f</font> <a href="rfunct_4.html#R1" title="RFUNCT_4:pred.1">is_strictly_convex_on</a> <font color="Maroon" title="c3">X</font> iff <font color="Maroon" title="c2">f</font> <a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">-</a> <font color="Maroon" title="c1">r</font> <a href="rfunct_4.html#R1" title="RFUNCT_4:pred.1">is_strictly_convex_on</a> <font color="Maroon" title="c3">X</font> )
 <b>by </b><i><a class="txt" href="rfunct_4.html#E12"><i><font color="Green" title="E8">Lm1</font></i></a></i>;<br/>


</div>
