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<b>let </b><font color="Maroon">a</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">b</font> be   <a href="real_1.html#NM1">Real</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/><b>let </b><font color="Maroon">g</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">X</font> be    <a href="hidden.html#M1">set</a> ;<br/>









<b>assume </b><a NAME="E1:18"/><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> &amp; <font color="Maroon">g</font> <a href="rfunct_4.html#R1" target="_self">is_strictly_convex_on</a> <font color="Maroon">X</font> &amp; ( ( <font color="Maroon">a</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">b</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 ) or ( <font color="Maroon">a</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <font color="Maroon">b</font> <a href="xxreal_0.html#NR4" target="_self">&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">a</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">b</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 ) or ( <font color="Maroon">a</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> 0 ) or ( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">b</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 ) )
 </span><b>by </b><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">a</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">b</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 ;<br/><div class="add"><a NAME="E2:18_1_1_1"/><b>then </b>
( <font color="Maroon">a</font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">f</font> <a href="rfunct_4.html#R1" target="_self">is_strictly_convex_on</a> <font color="Maroon">X</font> &amp; <font color="Maroon">b</font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">g</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/><b>hence </b><a NAME="E3:18_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">f</font></span>)</span> <a href="seq_1.html#K6">+</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">g</font></span>)</span> <a href="rfunct_4.html#R1" target="_self">is_strictly_convex_on</a> <font color="Maroon">X</font>
 <b>by </b><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">E39</font></i>: 
( <font color="Maroon">a</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> 0 )
 ;<br/><div class="add"><a NAME="E2:18_1_1_2"/><b>then </b><i><font color="Green">E40</font></i>: 
<font color="Maroon">a</font> <a href="seq_1.html#K13">(#)</a> <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/><a NAME="E3:18_1_1_2"/>
( <font color="Maroon">g</font> <a href="rfunct_3.html#R2">is_convex_on</a> <font color="Maroon">X</font> &amp; <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> )
 <b>by </b><i>, , </i>;<br/><a NAME="E4:18_1_1_2"/><b>then </b>
<font color="Maroon">b</font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">g</font> <a href="rfunct_3.html#R2">is_convex_on</a> <font color="Maroon">X</font>
 <b>by </b><i>, <a class="ref" href="rfunct_3.html#T60">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">a</font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">f</font></span>)</span> <a href="seq_1.html#K6">+</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">g</font></span>)</span> <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><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">E41</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">b</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 ;<br/><div class="add"><a NAME="E2:18_1_1_3"/><b>then </b><i><font color="Green">E42</font></i>: 
<font color="Maroon">b</font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">g</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/><a NAME="E3:18_1_1_3"/>
( <font color="Maroon">f</font> <a href="rfunct_3.html#R2">is_convex_on</a> <font color="Maroon">X</font> &amp; <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>by </b><i>, , </i>;<br/><a NAME="E4:18_1_1_3"/><b>then </b>
<font color="Maroon">a</font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">f</font> <a href="rfunct_3.html#R2">is_convex_on</a> <font color="Maroon">X</font>
 <b>by </b><i>, <a class="ref" href="rfunct_3.html#T60">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">a</font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">f</font></span>)</span> <a href="seq_1.html#K6">+</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">g</font></span>)</span> <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></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">a</font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">f</font></span>)</span> <a href="seq_1.html#K6">+</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="seq_1.html#K13">(#)</a> <font color="Maroon">g</font></span>)</span> <a href="rfunct_4.html#R1" target="_self">is_strictly_convex_on</a> <font color="Maroon">X</font>
 ;<br/>


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