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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="E1:35"/><i><font color="Green">E29</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>
 ;<br/>

<a NAME="E2:35"/><i><font color="Green">E30</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><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><a href="euclid.html#K12">.|</a></span>
 
<b>by </b><i><a class="ref" href="euclid_2.html#T14" target="_self">Th14</a></i>;<br/>
<a NAME="E3:35"/><i><font color="Green">E31</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <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>
 
<b>by </b><i><a class="ref" href="euclid_2.html#T14" target="_self">Th14</a></i>;<br/>
<a NAME="E4:35"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="euclid.html#K12">.|</a></span>
 
<b>by </b><i><a class="ref" href="euclid_2.html#T14" target="_self">Th14</a></i>;<br/>
<a NAME="E5:35"/><b>then </b><i><font color="Green">E32</font></i>: 
0 <a href="xcmplx_0.html#K2">+</a> 0 <a href="xxreal_0.html#R1">&lt;=</a> <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="xcmplx_0.html#K2">+</a> <span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="euclid.html#K12">.|</a></span>
 
<b>by </b><i><a class="txt" href="euclid_2.html#E3:35"><i><font color="Green">E31</font></i></a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<a NAME="E6:35"/><i><font color="Green">E33</font></i>: 
<span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><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><a href="euclid.html#K12">.|</a></span> <a href="square_1.html#K5">^2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><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="square_1.html#K5">^2</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p3"><a href="euclid_2.html#K1" target="_self">|(</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="euclid_2.html#K1" target="_self">)|</a></span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="euclid.html#K12">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="euclid_2.html#E1:35"><i><font color="Green">E29</font></i></a>, <a class="ref" href="euclid_2.html#T33" target="_self">Th33</a></i>;<br/>
<a NAME="E7:35"/><i><font color="Green">E34</font></i>: 
<span class="p1"><a href="euclid_2.html#K1" target="_self">|(</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="euclid_2.html#K1" target="_self">)|</a></span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="complex1.html#K18">abs</a> <span class="p1"><a href="euclid_2.html#K1" target="_self">|(</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="euclid_2.html#K1" target="_self">)|</a></span>
 
<b>by </b><i><a class="ref" href="absvalue.html#T11">ABSVALUE:11</a></i>;<br/>
<a NAME="E8:35"/>
 <a href="complex1.html#K18">abs</a> <span class="p1"><a href="euclid_2.html#K1" target="_self">|(</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="euclid_2.html#K1" target="_self">)|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <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="xcmplx_0.html#K3">*</a> <span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="euclid.html#K12">.|</a></span>
 
<b>by </b><i><a class="txt" href="euclid_2.html#E1:35"><i><font color="Green">E29</font></i></a>, <a class="ref" href="euclid_2.html#T37" target="_self">Th37</a></i>;<br/>
<a NAME="E9:35"/><b>then </b>
<span class="p1"><a href="euclid_2.html#K1" target="_self">|(</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="euclid_2.html#K1" target="_self">)|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <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="xcmplx_0.html#K3">*</a> <span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="euclid.html#K12">.|</a></span>
 
<b>by </b><i><a class="txt" href="euclid_2.html#E7:35"><i><font color="Green">E34</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E10:35"/><b>then </b>
2 <a href="xcmplx_0.html#K3">*</a> <span class="p1"><a href="euclid_2.html#K1" target="_self">|(</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="euclid_2.html#K1" target="_self">)|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2"><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="xcmplx_0.html#K3">*</a> <span class="p2"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="euclid.html#K12">.|</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/>
<a NAME="E11:35"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2"><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="square_1.html#K5">^2</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p2"><a href="euclid_2.html#K1" target="_self">|(</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="euclid_2.html#K1" target="_self">)|</a></span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2"><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="square_1.html#K5">^2</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3"><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="xcmplx_0.html#K3">*</a> <span class="p3"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="euclid.html#K12">.|</a></span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<a NAME="E12:35"/><b>then </b>
<span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><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><a href="euclid.html#K12">.|</a></span> <a href="square_1.html#K5">^2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><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="square_1.html#K5">^2</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p4"><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></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p3"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="euclid.html#K12">.|</a></span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="euclid.html#K12">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="euclid_2.html#E6:35"><i><font color="Green">E33</font></i></a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<a NAME="E13:35"/><b>then </b><i><font color="Green">E35</font></i>: 
<span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><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><a href="euclid.html#K12">.|</a></span> <a href="square_1.html#K5">^2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2"><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="xcmplx_0.html#K2">+</a> <span class="p2"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="euclid.html#K12">.|</a></span></span>)</span> <a href="square_1.html#K3">^2</a> 
 
;<br/>
<a NAME="E14:35"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><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><a href="euclid.html#K12">.|</a></span> <a href="square_1.html#K5">^2</a> 
 
<b>by </b><i><a class="ref" href="xreal_1.html#T65">XREAL_1:65</a></i>;<br/>
<a NAME="E15:35"/><b>then </b>
 <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><span class="p2"><a href="euclid.html#K12">|.</a><span class="default"><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><a href="euclid.html#K12">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="square_1.html#K6">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><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="xcmplx_0.html#K2">+</a> <span class="p3"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="euclid.html#K12">.|</a></span></span>)</span> <a href="square_1.html#K3">^2</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="euclid_2.html#E13:35"><i><font color="Green">E35</font></i></a>, <a class="ref" href="square_1.html#T94">SQUARE_1:94</a></i>;<br/>
<a NAME="E16:35"/><b>then </b>
<span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><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><a href="euclid.html#K12">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="square_1.html#K6">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><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="xcmplx_0.html#K2">+</a> <span class="p3"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="euclid.html#K12">.|</a></span></span>)</span> <a href="square_1.html#K3">^2</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="euclid_2.html#E2:35"><i><font color="Green">E30</font></i></a>, <a class="ref" href="square_1.html#T89">SQUARE_1:89</a></i>;<br/>
<b>hence </b><a NAME="E17:35"/>
<span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><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><a href="euclid.html#K12">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <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="xcmplx_0.html#K2">+</a> <span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="euclid.html#K12">.|</a></span>
 <b>by </b><i><a class="txt" href="euclid_2.html#E5:35"><i><font color="Green">E32</font></i></a>, <a class="ref" href="square_1.html#T89">SQUARE_1:89</a></i>;<br/>


</div>
