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<b>let </b><font color="Maroon">F</font> be   <a href="partfun1.html#NM1">PartFunc</a> of  <a href="numbers.html#K1">REAL</a> , <a href="numbers.html#K1">REAL</a> ;<br/><b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">r</font> be   <a href="real_1.html#NM1">Real</a>;<br/>





<b>assume </b><a NAME="E1:82"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">F</font> <a href="rfunct_3.html#R2" target="_self">is_convex_on</a> <font color="Maroon">X</font>
 ;<br/>

<a NAME="E2:82"/><i><font color="Green">E31</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">r</font></span>)</span>
 
<b>by </b><i><a class="txt" href="rfunct_3.html#E10"><i><font color="Green">Lemma45</font></i></a></i>;<br/>
<a NAME="E3:82"/><i><font color="Green">E32</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K19" target="_self">max+</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">r</font></span>)</span>
 
<b>by </b><i><a class="txt" href="rfunct_3.html#E11:82"><i><font color="Green">E35</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:82"/>
<font color="Maroon">X</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K19" target="_self">max+</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">r</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="rfunct_3.html#T9" target="_self">Th9</a>, <a class="txt" href="rfunct_3.html#E11"><i><font color="Green">Lemma48</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="rfunct_3.html#D13" target="_self">RFUNCT_3:def 13</a><br/>

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

<b>assume </b><a NAME="E5:82"/><i><font color="Green">E33</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p</font> &amp; <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 ;<br/>

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

<b>assume </b><a NAME="E6:82"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">y</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="real_1.html#K4">*</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">y</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 ;<br/>

<a NAME="E7:82"/><b>then </b>
<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="real_1.html#K4">*</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">y</font></span>)</span></span>)</span>
 
<b>by </b><i>, , <a class="txt" href="rfunct_3.html#E11"><i><font color="Green">Lemma48</font></i></a></i>;<br/>
<a NAME="E8:82"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="real_1.html#K4">*</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">r</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<a NAME="E9:82"/><b>then </b>
 <a href="xxreal_0.html#K4">max</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="real_1.html#K4">*</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span>,0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="xxreal_0.html#K4">max</a> <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="real_1.html#K4">*</a> <span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span>,0
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T26">XXREAL_0:26</a></i>;<br/>
<a NAME="E10:82"/><b>then </b>
 <a href="rfunct_3.html#K2" target="_self">max+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="real_1.html#K4">*</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="xxreal_0.html#K4">max</a> <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="real_1.html#K4">*</a> <span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span>,0
 
;<br/>
<a NAME="E11:82"/><b>then </b><i><font color="Green">E35</font></i>: 
 <a href="rfunct_3.html#K2" target="_self">max+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="real_1.html#K4">*</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="rfunct_3.html#K2" target="_self">max+</a> <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="real_1.html#K4">*</a> <span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span>
 
;<br/>
<a NAME="E12:82"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">X</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font>
 
<b>by </b><i>, <a class="txt" href="rfunct_3.html#E11"><i><font color="Green">Lemma48</font></i></a></i>;<br/>
<i><font color="Green">E94</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">r</font> = 
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">y</font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">r</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">y</font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rfunct_3.html#T9" target="_self">Th9</a>, <a class="ref" href="rfunct_3.html#T11" target="_self">Th11</a>, <a class="ref" href="rfunct_3.html#T13" target="_self">Th13</a>, <a class="txt" href="rfunct_3.html#E10"><i><font color="Green">Lemma45</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">r</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">r</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">y</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rfunct_3.html#T9" target="_self">Th9</a>, <a class="ref" href="rfunct_3.html#T11" target="_self">Th11</a>, <a class="ref" href="rfunct_3.html#T13" target="_self">Th13</a>, <a class="txt" href="rfunct_3.html#E10"><i><font color="Green">Lemma45</font></i></a></i>
;<br/>
<a NAME="E14:82"/><i><font color="Green">E95</font></i>: 
 <a href="rfunct_3.html#K2" target="_self">max+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">r</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">r</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K2" target="_self">max+</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">r</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K2" target="_self">max+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">r</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E15:82"/>
0 <a href="real_1.html#K3">+</a> <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> 1
 
<b>by </b><i/>;<br/>
<a NAME="E16:82"/><b>then </b>
0 <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="real_1.html#K5">-</a> <font color="Maroon">p</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/>
<a NAME="E17:82"/><b>then </b>
(  <a href="rfunct_3.html#K2" target="_self">max+</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">r</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K2" target="_self">max+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">r</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> &amp;  <a href="rfunct_3.html#K2" target="_self">max+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">r</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K2" target="_self">max+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">r</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">y</font></span>)</span></span>)</span> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E18:82"/><b>then </b>
 <a href="rfunct_3.html#K2" target="_self">max+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="real_1.html#K4">*</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K2" target="_self">max+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">r</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K2" target="_self">max+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">r</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="rfunct_3.html#T12" target="_self">Th12</a>, , , <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<a NAME="E19:82"/><b>then </b>
 <a href="rfunct_3.html#K2" target="_self">max+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">r</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="real_1.html#K4">*</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K2" target="_self">max+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">r</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K2" target="_self">max+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">r</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="rfunct_3.html#T9" target="_self">Th9</a>, <a class="ref" href="rfunct_3.html#T11" target="_self">Th11</a>, <a class="ref" href="rfunct_3.html#T13" target="_self">Th13</a>, <a class="txt" href="rfunct_3.html#E10"><i><font color="Green">Lemma45</font></i></a></i>;<br/>
<a NAME="E20:82"/><b>then </b>
<span class="p1">(<span class="default"><a href="rfunct_3.html#K19" target="_self">max+</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">r</font></span>)</span></span>)</span> <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="real_1.html#K4">*</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K2" target="_self">max+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">r</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K2" target="_self">max+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">r</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="rfunct_3.html#T9" target="_self">Th9</a>, <a class="ref" href="rfunct_3.html#T10" target="_self">Th10</a>, <a class="ref" href="rfunct_3.html#T11" target="_self">Th11</a>, <a class="ref" href="rfunct_3.html#T13" target="_self">Th13</a>, <a class="txt" href="rfunct_3.html#E11:82"><i><font color="Green">E35</font></i></a></i>;<br/>
<a NAME="E21:82"/><b>then </b>
<span class="p1">(<span class="default"><a href="rfunct_3.html#K19" target="_self">max+</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">r</font></span>)</span></span>)</span> <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="real_1.html#K4">*</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rfunct_3.html#K19" target="_self">max+</a> <span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K2" target="_self">max+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">r</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="rfunct_3.html#T9" target="_self">Th9</a>, <a class="ref" href="rfunct_3.html#T10" target="_self">Th10</a>, <a class="ref" href="rfunct_3.html#T11" target="_self">Th11</a>, <a class="ref" href="rfunct_3.html#T13" target="_self">Th13</a>, <a class="txt" href="rfunct_3.html#E11:82"><i><font color="Green">E35</font></i></a></i>;<br/>
<b>hence </b><a NAME="E22:82"/>
<span class="p1">(<span class="default"><a href="rfunct_3.html#K19" target="_self">max+</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">r</font></span>)</span></span>)</span> <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="real_1.html#K4">*</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rfunct_3.html#K19" target="_self">max+</a> <span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">p</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rfunct_3.html#K19" target="_self">max+</a> <span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">y</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="rfunct_3.html#T9" target="_self">Th9</a>, <a class="ref" href="rfunct_3.html#T10" target="_self">Th10</a>, <a class="ref" href="rfunct_3.html#T11" target="_self">Th11</a>, <a class="ref" href="rfunct_3.html#T13" target="_self">Th13</a>, <a class="txt" href="rfunct_3.html#E11:82"><i><font color="Green">E35</font></i></a></i>;<br/>


</div>
