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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/>



<b>assume </b><a NAME="E1:37"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> is <a href="series_1.html#V1">summable</a> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">c<sub>2</sub></font> <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> 0 ) )
 ;<br/>

<a NAME="E2:37"/><b>then </b><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="seqm_3.html#K1">^\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> is <a href="series_1.html#V1">summable</a>
 
<b>by </b><i><a class="ref" href="series_1.html#T15">SERIES_1:15</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><a NAME="E1:37_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seqm_3.html#K1">^\</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="seqm_3.html#D9">SEQM_3:def 9</a></i>;<br/><b>hence </b><a NAME="E2:37_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seqm_3.html#K1">^\</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="txt" href="irrat_1.html#E1:37"><i><font color="Green">E35</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E4:37"/><b>then </b>
 <a href="series_1.html#K2">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seqm_3.html#K1">^\</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="txt" href="irrat_1.html#E2:37"><i><font color="Green">E36</font></i></a>, <a class="ref" href="series_1.html#T21">SERIES_1:21</a></i>;<br/>
<a NAME="E5:37"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="series_1.html#K2">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="seqm_3.html#K1">^\</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 0
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<b>hence </b><a NAME="E6:37"/>
 <a href="series_1.html#K2">Sum</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="series_1.html#K1">Partial_Sums</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="irrat_1.html#E1:37"><i><font color="Green">E35</font></i></a>, <a class="ref" href="series_1.html#T18">SERIES_1:18</a></i>;<br/>


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