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<b>let </b><font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c2">b</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/><b>let </b><font color="Maroon" title="c3">f</font>, <font color="Maroon" title="c4">g</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="c5">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>









<b>assume </b><a NAME="E1:18"/><i><font color="Green" title="E13">A1</font></i>: 
( <font color="Maroon" title="c3">f</font> <a href="rfunct_4.html#R1" title="RFUNCT_4:pred.1">is_strictly_convex_on</a> <font color="Maroon" title="c5">X</font> &amp; <font color="Maroon" title="c4">g</font> <a href="rfunct_4.html#R1" title="RFUNCT_4:pred.1">is_strictly_convex_on</a> <font color="Maroon" title="c5">X</font> &amp; ( ( <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 &amp; <font color="Maroon" title="c2">b</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 ) or ( <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 &amp; <font color="Maroon" title="c2">b</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 ) ) )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:18_1_1"/>
( ( <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 &amp; <font color="Maroon" title="c2">b</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 ) or ( <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 &amp; <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 ) or ( <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 &amp; <font color="Maroon" title="c2">b</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 ) )
 </span><b>by </b><i><a class="txt" href="rfunct_4.html#E1:18"><i><font color="Green" title="E13">A1</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_1_1"><b>suppose </b></a><a NAME="E1:18_1_1_1"/>
( <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 &amp; <font color="Maroon" title="c2">b</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 )
 ;<br/><div class="add"><a NAME="E2:18_1_1_1"/><b>then </b>
( <font color="Maroon" title="c1">a</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c3">f</font> <a href="rfunct_4.html#R1" title="RFUNCT_4:pred.1">is_strictly_convex_on</a> <font color="Maroon" title="c5">X</font> &amp; <font color="Maroon" title="c2">b</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c4">g</font> <a href="rfunct_4.html#R1" title="RFUNCT_4:pred.1">is_strictly_convex_on</a> <font color="Maroon" title="c5">X</font> )
 <b>by </b><i><a class="txt" href="rfunct_4.html#E1:18"><i><font color="Green" title="E13">A1</font></i></a>, <a class="ref" href="rfunct_4.html#T12" target="_self" title="RFUNCT_4:th.12">Th12</a></i>;<br/><b>hence </b><a NAME="E3:18_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="seq_1.html#K6" title="SEQ_1:func.6">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="rfunct_4.html#R1" title="RFUNCT_4:pred.1">is_strictly_convex_on</a> <font color="Maroon" title="c5">X</font>
 <b>by </b><i><a class="ref" href="rfunct_4.html#T13" target="_self" title="RFUNCT_4:th.13">Th13</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_1_2"><b>suppose </b></a><a NAME="E1:18_1_1_2"/><i><font color="Green" title="E14">A2</font></i>: 
( <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 &amp; <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 )
 ;<br/><div class="add"><a NAME="E2:18_1_1_2"/><b>then </b><i><font color="Green" title="E15">A3</font></i>: 
<font color="Maroon" title="c1">a</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c3">f</font> <a href="rfunct_4.html#R1" title="RFUNCT_4:pred.1">is_strictly_convex_on</a> <font color="Maroon" title="c5">X</font>
 <b>by </b><i><a class="txt" href="rfunct_4.html#E1:18"><i><font color="Green" title="E13">A1</font></i></a>, <a class="ref" href="rfunct_4.html#T12" target="_self" title="RFUNCT_4:th.12">Th12</a></i>;<br/><a NAME="E3:18_1_1_2"/>
( <font color="Maroon" title="c4">g</font> <a href="rfunct_3.html#R2" title="RFUNCT_3:pred.2">is_convex_on</a> <font color="Maroon" title="c5">X</font> &amp; <font color="Maroon" title="c5">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">g</font> )
 <b>by </b><i><a class="txt" href="rfunct_4.html#E1:18"><i><font color="Green" title="E13">A1</font></i></a>, <a class="ref" href="rfunct_4.html#D1" target="_self" title="RFUNCT_4:def.1">Def1</a>, <a class="ref" href="rfunct_4.html#T6" target="_self" title="RFUNCT_4:th.6">Th6</a></i>;<br/><a NAME="E4:18_1_1_2"/><b>then </b>
<font color="Maroon" title="c2">b</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c4">g</font> <a href="rfunct_3.html#R2" title="RFUNCT_3:pred.2">is_convex_on</a> <font color="Maroon" title="c5">X</font>
 <b>by </b><i><a class="txt" href="rfunct_4.html#E1:18_1_1_2"><i><font color="Green" title="E14">A2</font></i></a>, <a class="ref" href="rfunct_3.html#T60" title="RFUNCT_3:th.60">RFUNCT_3:60</a></i>;<br/><b>hence </b><a NAME="E5:18_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="seq_1.html#K6" title="SEQ_1:func.6">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="rfunct_4.html#R1" title="RFUNCT_4:pred.1">is_strictly_convex_on</a> <font color="Maroon" title="c5">X</font>
 <b>by </b><i><a class="txt" href="rfunct_4.html#E2:18_1_1_2"><i><font color="Green" title="E15">A3</font></i></a>, <a class="ref" href="rfunct_4.html#T14" target="_self" title="RFUNCT_4:th.14">Th14</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_1_3"><b>suppose </b></a><a NAME="E1:18_1_1_3"/><i><font color="Green" title="E14">A4</font></i>: 
( <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 &amp; <font color="Maroon" title="c2">b</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 )
 ;<br/><div class="add"><a NAME="E2:18_1_1_3"/><b>then </b><i><font color="Green" title="E15">A5</font></i>: 
<font color="Maroon" title="c2">b</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c4">g</font> <a href="rfunct_4.html#R1" title="RFUNCT_4:pred.1">is_strictly_convex_on</a> <font color="Maroon" title="c5">X</font>
 <b>by </b><i><a class="txt" href="rfunct_4.html#E1:18"><i><font color="Green" title="E13">A1</font></i></a>, <a class="ref" href="rfunct_4.html#T12" target="_self" title="RFUNCT_4:th.12">Th12</a></i>;<br/><a NAME="E3:18_1_1_3"/>
( <font color="Maroon" title="c3">f</font> <a href="rfunct_3.html#R2" title="RFUNCT_3:pred.2">is_convex_on</a> <font color="Maroon" title="c5">X</font> &amp; <font color="Maroon" title="c5">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">f</font> )
 <b>by </b><i><a class="txt" href="rfunct_4.html#E1:18"><i><font color="Green" title="E13">A1</font></i></a>, <a class="ref" href="rfunct_4.html#D1" target="_self" title="RFUNCT_4:def.1">Def1</a>, <a class="ref" href="rfunct_4.html#T6" target="_self" title="RFUNCT_4:th.6">Th6</a></i>;<br/><a NAME="E4:18_1_1_3"/><b>then </b>
<font color="Maroon" title="c1">a</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c3">f</font> <a href="rfunct_3.html#R2" title="RFUNCT_3:pred.2">is_convex_on</a> <font color="Maroon" title="c5">X</font>
 <b>by </b><i><a class="txt" href="rfunct_4.html#E1:18_1_1_3"><i><font color="Green" title="E14">A4</font></i></a>, <a class="ref" href="rfunct_3.html#T60" title="RFUNCT_3:th.60">RFUNCT_3:60</a></i>;<br/><b>hence </b><a NAME="E5:18_1_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="seq_1.html#K6" title="SEQ_1:func.6">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="rfunct_4.html#R1" title="RFUNCT_4:pred.1">is_strictly_convex_on</a> <font color="Maroon" title="c5">X</font>
 <b>by </b><i><a class="txt" href="rfunct_4.html#E2:18_1_1_3"><i><font color="Green" title="E15">A5</font></i></a>, <a class="ref" href="rfunct_4.html#T14" target="_self" title="RFUNCT_4:th.14">Th14</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:18"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="seq_1.html#K6" title="SEQ_1:func.6">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="rfunct_4.html#R1" title="RFUNCT_4:pred.1">is_strictly_convex_on</a> <font color="Maroon" title="c5">X</font>
 ;<br/>


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