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<b>let </b><font color="Maroon">f</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> ;<br/><b>let </b><font color="Maroon">a</font> be   <a href="real_1.html#NM1">Real</a>;<br/>



<b>assume </b><a NAME="E1:12"/><i><font color="Green">E27</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">a</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 ;<br/>

<a NAME="E2:12"/><i><font color="Green">E28</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="euclid_2.html#T8">EUCLID_2:8</a></i>;<br/>
<a NAME="E3:12"/><i><font color="Green">E30</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="rvsum_1.html#T65">RVSUM_1:65</a></i>;<br/>
<a NAME="E4:12"/><i><font color="Green">E31</font></i>: 
 <a href="rfinseq2.html#K1" target="_self">max_p</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E5:12"/><b>then </b><i><font color="Green">E32</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="rfinseq2.html#K1" target="_self">max_p</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">f</font></span>)</span> &amp;  <a href="rfinseq2.html#K1" target="_self">max_p</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">f</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">f</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E6:12"/><i><font color="Green">E35</font></i>: 
 <a href="rfinseq2.html#K3" target="_self">max</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="rfinseq2.html#K1" target="_self">max_p</a> <span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="rvsum_1.html#T66">RVSUM_1:66</a></i>;<br/>
<a NAME="E7:12"/>
 <a href="rfinseq2.html#K1" target="_self">max_p</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E8:12"/><b>then </b><i><font color="Green">E36</font></i>: 
 <a href="rfinseq2.html#K1" target="_self">max_p</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E9:12"/><b>then </b><i><font color="Green">E37</font></i>: 
<font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K1" target="_self">max_p</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K1" target="_self">max_p</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E10:12"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="rfinseq2.html#K1" target="_self">max_p</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">f</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K1" target="_self">max_p</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i><a class="ref" href="rvsum_1.html#T66">RVSUM_1:66</a></i>;<br/>
<a NAME="E11:12"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="rfinseq2.html#K1" target="_self">max_p</a> <span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">f</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K1" target="_self">max_p</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="rvsum_1.html#T66">RVSUM_1:66</a></i>;<br/>
<a NAME="E12:12"/><i><font color="Green">E40</font></i>: 
 <a href="rfinseq2.html#K1" target="_self">max_p</a> <font color="Maroon">f</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E13:12"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">f</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K1" target="_self">max_p</a> <font color="Maroon">f</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">f</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K1" target="_self">max_p</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E14:12"/><b>then </b>
<font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K1" target="_self">max_p</a> <font color="Maroon">f</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K1" target="_self">max_p</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 
<b>by </b><i>, , , <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/>
<a NAME="E15:12"/><b>then </b><i><font color="Green">E41</font></i>: 
<font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K1" target="_self">max_p</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K1" target="_self">max_p</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/>
<a NAME="E16:12"/>
 <a href="rfinseq2.html#K1" target="_self">max_p</a> <font color="Maroon">f</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E17:12"/><b>then </b><i><font color="Green">E42</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">f</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K1" target="_self">max_p</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">f</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K1" target="_self">max_p</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E18:12"/>
<font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K1" target="_self">max_p</a> <font color="Maroon">f</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K1" target="_self">max_p</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E19:12"/><b>then </b>
<font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="rfinseq2.html#K1" target="_self">max_p</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="rfinseq2.html#K1" target="_self">max_p</a> <span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/>
<a NAME="E20:12"/><b>then </b><i><font color="Green">E43</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">f</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K1" target="_self">max_p</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">f</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K1" target="_self">max_p</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i>, , , <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/>
<a NAME="E21:12"/><i><font color="Green">E44</font></i>: 
 <a href="rfinseq2.html#K1" target="_self">max_p</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">f</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="rfinseq2.html#K1" target="_self">max_p</a> <font color="Maroon">f</font>
 
<b>by </b><i>, , , , </i>;<br/>
<a NAME="E22:12"/>
 <a href="rfinseq2.html#K1" target="_self">max_p</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">f</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="rfinseq2.html#K1" target="_self">max_p</a> <font color="Maroon">f</font>
 
<b>by </b><i>, , , , </i>;<br/>
<b>hence </b><a NAME="E23:12"/>
(  <a href="rfinseq2.html#K3" target="_self">max</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K3" target="_self">max</a> <font color="Maroon">f</font></span>)</span> &amp;  <a href="rfinseq2.html#K1" target="_self">max_p</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rvsum_1.html#K9">*</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rfinseq2.html#K1" target="_self">max_p</a> <font color="Maroon">f</font> )
 <b>by </b><i>, , <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/>


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