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<div class="add">

<b>let </b><font color="Maroon">c<sub>1</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="E1:30"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="integra2.html#V1">non-decreasing</a>
 ;<br/>

<a NAME="E2:30"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K5">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp; <font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K5">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="rvsum_1.html#K5">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K5">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:30_1"/><i><font color="Green">E20</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K5">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K5">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> )
 ;<br/>

<a NAME="E2:30_1"/>
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K5">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</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="rvsum_1.html#T34">RVSUM_1:34</a></i>;<br/>
<a NAME="E3:30_1"/><b>then </b><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></font> <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"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="rfinseq2.html#E1:30"><i><font color="Green">E19</font></i></a>, <a class="txt" href="rfinseq2.html#E1:30_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="integra2.html#D1">INTEGRA2:def 1</a></i>;<br/>
<a NAME="E4:30_1"/><i><font color="Green">E22</font></i>: 
<span class="p1">(<span class="default"><a href="rvsum_1.html#K5">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="rvsum_1.html#T35">RVSUM_1:35</a></i>;<br/>
<a NAME="E5:30_1"/>
<span class="p1">(<span class="default"><a href="rvsum_1.html#K5">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> <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"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="rvsum_1.html#T35">RVSUM_1:35</a></i>;<br/>
<b>hence </b><a NAME="E6:30_1"/>
<span class="p1">(<span class="default"><a href="rvsum_1.html#K5">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K5">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="rfinseq2.html#E3:30_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="rfinseq2.html#E4:30_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:30"/>
 <a href="rvsum_1.html#K5">-</a> <font color="Maroon">c<sub>1</sub></font> is <a href="rfinseq.html#V1">non-increasing</a>
 <b>by </b><i><a class="ref" href="rfinseq.html#D4">RFINSEQ:def 4</a></i>;<br/>


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