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<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/>





<a NAME="E1:13"/><i><font color="Green">E9</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rfunct_3.html#K21">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="rfunct_3.html#K21">-</a> <span class="p2">(<span class="default"><a href="binop_2.html#K7">-</a> <font color="Maroon">c<sub>1</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>2</sub></font> <a href="rfunct_3.html#K21">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="rfunct_3.html#D12">RFUNCT_3:def 12</a></i>;<br/>
<a NAME="E2:13"/><i><font color="Green">E10</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rfunct_3.html#K21">-</a> <font color="Maroon">c<sub>1</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="rfunct_3.html#D12">RFUNCT_3:def 12</a></i>;<br/>
<a NAME="E3:13"/>
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>2</sub></font> <a href="rfunct_3.html#K21">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rfunct_3.html#K21">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="rfunct_3.html#K21">-</a> <span class="p2">(<span class="default"><a href="binop_2.html#K7">-</a> <font color="Maroon">c<sub>1</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="13_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="E1:13_1"/><i><font color="Green">E11</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>2</sub></font> <a href="rfunct_3.html#K21">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 ;<br/>

<b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rfunct_3.html#K21">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="rfunct_3.html#K21">-</a> <span class="p2">(<span class="default"><a href="binop_2.html#K7">-</a> <font color="Maroon">c<sub>1</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"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rfunct_3.html#K21">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><a href="binop_2.html#K7">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="rfunct_4.html#E1:13"><i><font color="Green">E9</font></i></a>, <a class="ref" href="rfunct_3.html#D12">RFUNCT_3:def 12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rfunct_3.html#K21">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="binop_2.html#K9">+</a> <font color="Maroon">c<sub>1</sub></font>

<br/>.= 
<span class="p1">(<span class="default"><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> <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_2.html#K9">+</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="txt" href="rfunct_4.html#E1:13_1"><i><font color="Green">E11</font></i></a>, <a class="ref" href="rfunct_3.html#D12">RFUNCT_3:def 12</a></i>
<br/>.= 
<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> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>1</sub></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">c<sub>2</sub></font> <a href="rfunct_3.html#K21">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="rfunct_3.html#K21">-</a> <span class="p2">(<span class="default"><a href="binop_2.html#K7">-</a> <font color="Maroon">c<sub>1</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="E4:13"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="rfunct_3.html#K21">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="rfunct_3.html#K21">-</a> <span class="p1">(<span class="default"><a href="binop_2.html#K7">-</a> <font color="Maroon">c<sub>1</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#E1:13"><i><font color="Green">E9</font></i></a>, <a class="txt" href="rfunct_4.html#E2:13"><i><font color="Green">E10</font></i></a>, <a class="ref" href="partfun1.html#T34">PARTFUN1:34</a></i>;<br/>
<b>hence </b><a NAME="E5:13"/>
( <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>2</sub></font> <a href="rfunct_3.html#K21">-</a> <font color="Maroon">c<sub>1</sub></font> <a href="rfunct_4.html#R1" target="_self">is_strictly_convex_on</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="rfunct_4.html#E12"><i><font color="Green">Lemma8</font></i></a></i>;<br/>


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