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<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="E1:24"/><i><font color="Green">E20</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 2
 ;<br/>

<a NAME="E2:24"/><i><font color="Green">E21</font></i>: 
<span class="p1">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> 1</span>)</span> <a href="square_1.html#K5">^2</a> 
 
<b>by </b><i><a class="ref" href="rvsum_1.html#T78">RVSUM_1:78</a></i>;<br/>
<a NAME="E3:24"/><i><font color="Green">E22</font></i>: 
<span class="p1">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> 2 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> 2</span>)</span> <a href="square_1.html#K5">^2</a> 
 
<b>by </b><i><a class="ref" href="rvsum_1.html#T78">RVSUM_1:78</a></i>;<br/>
<a NAME="E4:24"/><i><font color="Green">E23</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K11">sqr</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#T77">RVSUM_1:77</a></i>;<br/>
<a NAME="E5:24"/>
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E6:24"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <font color="Maroon">c<sub>1</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="txt" href="euclid_3.html#E4:24"><i><font color="Green">E23</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E7:24"/><b>then </b>
 <a href="rvsum_1.html#K11">sqr</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> 1</span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> 2</span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span><a href="finseq_1.html#K10">*&gt;</a></span>
 
<b>by </b><i><a class="txt" href="euclid_3.html#E1:24"><i><font color="Green">E20</font></i></a>, <a class="txt" href="euclid_3.html#E2:24"><i><font color="Green">E21</font></i></a>, <a class="txt" href="euclid_3.html#E3:24"><i><font color="Green">E22</font></i></a>, <a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>;<br/>
<b>then </b> <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K11">sqr</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> = 
 <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> 1</span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> 2</span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D9">FINSEQ_1:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> 1</span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> 2</span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T104">RVSUM_1:104</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> 1</span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> 2</span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T103">RVSUM_1:103</a></i>
;<br/>
<b>hence </b><a NAME="E9:24"/>
<span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="euclid.html#K12">.|</a></span> <a href="hidden.html#R1">=</a>  <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> 1</span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> 2</span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span>
 ;<br/>


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