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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>, <font color="Maroon">c<sub>3</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">c<sub>1</sub></font>;<br/>



<a NAME="E1:45"/>
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="euclid_4.html#K3" target="_self">Rn2Fin</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="euclid_4.html#K3" target="_self">Rn2Fin</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="txt" href="euclid_4.html#E35"><i><font color="Green">Lemma24</font></i></a></i>;<br/>
<a NAME="E2:45"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2"><a href="euclid.html#K12">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="euclid_4.html#K3" target="_self">Rn2Fin</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="rvsum_1.html#K3">+</a> <span class="p4">(<span class="default"><a href="euclid_4.html#K3" target="_self">Rn2Fin</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span><a href="euclid.html#K12">.|</a></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"><a href="euclid.html#K12">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="euclid_4.html#K3" target="_self">Rn2Fin</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="rvsum_1.html#K7">-</a> <span class="p4">(<span class="default"><a href="euclid_4.html#K3" target="_self">Rn2Fin</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span><a href="euclid.html#K12">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="hidden.html#R1">=</a> 2 <a href="real_1.html#K4">*</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"><span class="p4">(<span class="default"><a href="euclid_4.html#K3" target="_self">Rn2Fin</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="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K12">|.</a><span class="default"><span class="p4">(<span class="default"><a href="euclid_4.html#K3" target="_self">Rn2Fin</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="euclid.html#K12">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="euclid_2.html#T35">EUCLID_2:35</a></i>;<br/>
<a NAME="E3:45"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2"><a href="euclid.html#K12">|.</a><span class="default"><span class="p3">(<span class="default"><a href="euclid_4.html#K3" target="_self">Rn2Fin</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K7">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span><a href="euclid.html#K12">.|</a></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"><a href="euclid.html#K12">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="euclid_4.html#K3" target="_self">Rn2Fin</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="rvsum_1.html#K7">-</a> <span class="p4">(<span class="default"><a href="euclid_4.html#K3" target="_self">Rn2Fin</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span><a href="euclid.html#K12">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="hidden.html#R1">=</a> 2 <a href="real_1.html#K4">*</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"><span class="p4">(<span class="default"><a href="euclid_4.html#K3" target="_self">Rn2Fin</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="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K12">|.</a><span class="default"><span class="p4">(<span class="default"><a href="euclid_4.html#K3" target="_self">Rn2Fin</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="euclid.html#K12">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span>
 
;<br/>
<a NAME="E4:45"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2"><a href="euclid.html#K12">|.</a><span class="default"><span class="p3">(<span class="default"><a href="euclid_4.html#K3" target="_self">Rn2Fin</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K7">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span><a href="euclid.html#K12">.|</a></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"><a href="euclid.html#K12">|.</a><span class="default"><span class="p3">(<span class="default"><a href="euclid_4.html#K3" target="_self">Rn2Fin</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span><a href="euclid.html#K12">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="hidden.html#R1">=</a> 2 <a href="real_1.html#K4">*</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"><span class="p4">(<span class="default"><a href="euclid_4.html#K3" target="_self">Rn2Fin</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="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K12">|.</a><span class="default"><span class="p4">(<span class="default"><a href="euclid_4.html#K3" target="_self">Rn2Fin</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="euclid.html#K12">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span>
 
;<br/>
<a NAME="E5:45"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2"><a href="euclid_4.html#K4" target="_self">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K7">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="euclid_4.html#K4" target="_self">.|</a></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"><a href="euclid.html#K12">|.</a><span class="default"><span class="p3">(<span class="default"><a href="euclid_4.html#K3" target="_self">Rn2Fin</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span><a href="euclid.html#K12">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="hidden.html#R1">=</a> 2 <a href="real_1.html#K4">*</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"><span class="p4">(<span class="default"><a href="euclid_4.html#K3" target="_self">Rn2Fin</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="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K12">|.</a><span class="default"><span class="p4">(<span class="default"><a href="euclid_4.html#K3" target="_self">Rn2Fin</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="euclid.html#K12">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span>
 
;<br/>
<a NAME="E6:45"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2"><a href="euclid_4.html#K4" target="_self">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K7">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="euclid_4.html#K4" target="_self">.|</a></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"><a href="euclid_4.html#K4" target="_self">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="euclid_4.html#K4" target="_self">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="hidden.html#R1">=</a> 2 <a href="real_1.html#K4">*</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"><span class="p4">(<span class="default"><a href="euclid_4.html#K3" target="_self">Rn2Fin</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="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K12">|.</a><span class="default"><span class="p4">(<span class="default"><a href="euclid_4.html#K3" target="_self">Rn2Fin</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="euclid.html#K12">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span>
 
;<br/>
<a NAME="E7:45"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2"><a href="euclid_4.html#K4" target="_self">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K7">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="euclid_4.html#K4" target="_self">.|</a></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"><a href="euclid_4.html#K4" target="_self">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="euclid_4.html#K4" target="_self">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="hidden.html#R1">=</a> 2 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="euclid_4.html#K4" target="_self">|.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="euclid_4.html#K4" target="_self">.|</a></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"><a href="euclid.html#K12">|.</a><span class="default"><span class="p4">(<span class="default"><a href="euclid_4.html#K3" target="_self">Rn2Fin</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="euclid.html#K12">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span>
 
;<br/>
<b>hence </b><a NAME="E8:45"/>
<span class="p1">(<span class="default"><span class="p2"><a href="euclid_4.html#K4" target="_self">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K7">+</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="euclid_4.html#K4" target="_self">.|</a></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"><a href="euclid_4.html#K4" target="_self">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="euclid_4.html#K4" target="_self">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="hidden.html#R1">=</a> 2 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="euclid_4.html#K4" target="_self">|.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="euclid_4.html#K4" target="_self">.|</a></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"><a href="euclid_4.html#K4" target="_self">|.</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="euclid_4.html#K4" target="_self">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span>
 ;<br/>


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