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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> ;<br/>

<b>assume </b><a NAME="E1:10"/><i><font color="Green">E5</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 ;<br/>

<a NAME="E2:10"/><b>then </b><i><font color="Green">E6</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="rvsum_1.html#K3">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="euclid_2.html#T6">EUCLID_2:6</a></i>;<br/>
<a NAME="E3:10"/><b>then </b><i><font color="Green">E7</font></i>: 
 <a href="rfinseq2.html#K1" target="_self">max_p</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="rvsum_1.html#K3">+</a> <font color="Maroon">c<sub>2</sub></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">c<sub>1</sub></font> <a href="rvsum_1.html#K3">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="rfinseq2.html#E1:10"><i><font color="Green">E5</font></i></a>, <a class="ref" href="rfinseq2.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E4:10"/><b>then </b><i><font color="Green">E8</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">c<sub>1</sub></font> <a href="rvsum_1.html#K3">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp;  <a href="rfinseq2.html#K1" target="_self">max_p</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="rvsum_1.html#K3">+</a> <font color="Maroon">c<sub>2</sub></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">c<sub>1</sub></font> <a href="rvsum_1.html#K3">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> )
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<i><font color="Green">E9</font></i>:  <a href="rfinseq2.html#K3" target="_self">max</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="rvsum_1.html#K3">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="rvsum_1.html#K3">+</a> <font color="Maroon">c<sub>2</sub></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">c<sub>1</sub></font> <a href="rvsum_1.html#K3">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></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">c<sub>1</sub></font> <a href="rvsum_1.html#K3">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></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">c<sub>1</sub></font> <a href="rvsum_1.html#K3">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="rfinseq2.html#E3:10"><i><font color="Green">E7</font></i></a>, <a class="ref" href="rvsum_1.html#T26">RVSUM_1:26</a></i>
;<br/>
<a NAME="E6:10"/><i><font color="Green">E10</font></i>: 
 <a href="rfinseq2.html#K1" target="_self">max_p</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="rvsum_1.html#K3">+</a> <font color="Maroon">c<sub>2</sub></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">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="rfinseq2.html#E2:10"><i><font color="Green">E6</font></i></a>, <a class="txt" href="rfinseq2.html#E4:10"><i><font color="Green">E8</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E7:10"/><b>then </b>
 <a href="rfinseq2.html#K1" target="_self">max_p</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="rvsum_1.html#K3">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E8:10"/><b>then </b><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>1</sub></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">c<sub>1</sub></font> <a href="rvsum_1.html#K3">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></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">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="rfinseq2.html#E1:10"><i><font color="Green">E5</font></i></a>, <a class="ref" href="rfinseq2.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E9:10"/>
 <a href="rfinseq2.html#K1" target="_self">max_p</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="rvsum_1.html#K3">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="rfinseq2.html#E1:10"><i><font color="Green">E5</font></i></a>, <a class="txt" href="rfinseq2.html#E6:10"><i><font color="Green">E10</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E10:10"/><b>then </b><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>2</sub></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">c<sub>1</sub></font> <a href="rvsum_1.html#K3">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></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">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="rfinseq2.html#E1:10"><i><font color="Green">E5</font></i></a>, <a class="ref" href="rfinseq2.html#D1" target="_self">Def1</a></i>;<br/>
<b>thus </b><a NAME="E11:10"/>
 <a href="rfinseq2.html#K3" target="_self">max</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="rvsum_1.html#K3">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K3" target="_self">max</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K3" target="_self">max</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="rfinseq2.html#E5:10"><i><font color="Green">E9</font></i></a>, <a class="txt" href="rfinseq2.html#E8:10"><i><font color="Green">E11</font></i></a>, <a class="txt" href="rfinseq2.html#E10:10"><i><font color="Green">E12</font></i></a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>


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