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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>, <font color="Maroon">c<sub>4</sub></font> be    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">c<sub>1</sub></font>;<br/>



<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:48_1_1"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font> or <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_1"><b>case </b></a><a NAME="E1:48_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><div class="add"><a NAME="E2:48_1_1_1"/><b>then </b>
<span class="p1"><a href="euclid_4.html#K4">|.</a><span class="default"><span class="p2">(<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">.|</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="txt" href="euclidlp.html#E15"><i><font color="Green">Lemma15</font></i></a></i>;<br/><a NAME="E3:48_1_1_1"/><b>then </b><i><font color="Green">E48</font></i>: 
<span class="p1"><a href="euclid_4.html#K4">|.</a><span class="default"><span class="p2">(<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">.|</a></span> <a href="square_1.html#K5">^2</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="ref" href="square_1.html#T74">SQUARE_1:74</a></i>;<br/><b>set </b><font color="Maroon">c<sub>5</sub></font> =  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><span class="p2"><a href="euclid_4.html#K5">|(</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 class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="euclid_4.html#K5">)|</a></span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><span class="p3"><a href="euclid_4.html#K4">|.</a><span class="default"><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><a href="euclid_4.html#K4">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span>;<br/><b>set </b><font color="Maroon">c<sub>6</sub></font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p4">(<span class="default"><span class="p5"><a href="euclid_4.html#K5">|(</a><span class="default"><span class="p0">(<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 class="p0">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="euclid_4.html#K5">)|</a></span> <a href="real_1.html#K6">/</a> <span class="p5">(<span class="default"><span class="p0"><a href="euclid_4.html#K4">|.</a><span class="default"><span class="p1">(<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">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><span class="p4"><a href="euclid_4.html#K5">|(</a><span class="default"><span class="p5">(<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 class="p5">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="euclid_4.html#K5">)|</a></span> <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><span class="p5"><a href="euclid_4.html#K4">|.</a><span class="default"><span class="p0">(<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">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>;<br/><span class="p1"><a href="euclid_4.html#K5">|(</a><span class="default"><span class="p2">(<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 class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K8">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p0">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><span class="p2"><a href="euclid_4.html#K5">|(</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 class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="euclid_4.html#K5">)|</a></span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><span class="p3"><a href="euclid_4.html#K4">|.</a><span class="default"><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><a href="euclid_4.html#K4">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="euclid.html#K7">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p0">(<span class="default"><span class="p1"><a href="euclid_4.html#K5">|(</a><span class="default"><span class="p2">(<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 class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="euclid_4.html#K5">)|</a></span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2"><a href="euclid_4.html#K4">|.</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">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span></span><a href="euclid_4.html#K5">)|</a></span> = 
<span class="p1"><a href="euclid_4.html#K5">|(</a><span class="default"><span class="p2">(<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 class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K8">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="real_1.html#K3">+</a> <span class="p0">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><span class="p3"><a href="euclid_4.html#K5">|(</a><span class="default"><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 class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="euclid_4.html#K5">)|</a></span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><span class="p4"><a href="euclid_4.html#K4">|.</a><span class="default"><span class="p5">(<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">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="euclid.html#K8">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p5">(<span class="default"><span class="p0"><a href="euclid_4.html#K5">|(</a><span class="default"><span class="p1">(<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 class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="euclid_4.html#K5">)|</a></span> <a href="real_1.html#K6">/</a> <span class="p0">(<span class="default"><span class="p1"><a href="euclid_4.html#K4">|.</a><span class="default"><span class="p2">(<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">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span><a href="euclid_4.html#K5">)|</a></span>
<b>by </b><i><a class="ref" href="euclidlp.html#T5" target="_self">Th5</a></i>
<br/>.= 
<span class="p1"><a href="euclid_4.html#K5">|(</a><span class="default"><span class="p2">(<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 class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K8">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="euclid.html#K7">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><span class="p3"><a href="euclid_4.html#K5">|(</a><span class="default"><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 class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="euclid_4.html#K5">)|</a></span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><span class="p4"><a href="euclid_4.html#K4">|.</a><span class="default"><span class="p5">(<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">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K8">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p5">(<span class="default"><span class="p0"><a href="euclid_4.html#K5">|(</a><span class="default"><span class="p1">(<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 class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="euclid_4.html#K5">)|</a></span> <a href="real_1.html#K6">/</a> <span class="p0">(<span class="default"><span class="p1"><a href="euclid_4.html#K4">|.</a><span class="default"><span class="p2">(<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">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span><a href="euclid_4.html#K5">)|</a></span>
<b>by </b><i><a class="ref" href="euclid_4.html#T7">EUCLID_4:7</a></i>
<br/>.= 
<span class="p1"><a href="euclid_4.html#K5">|(</a><span class="default"><span class="p2">(<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 class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K8">-</a> <span class="p5">(<span class="default">1 <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="euclid.html#K8">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p0">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><span class="p2"><a href="euclid_4.html#K5">|(</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 class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="euclid_4.html#K5">)|</a></span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><span class="p3"><a href="euclid_4.html#K4">|.</a><span class="default"><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><a href="euclid_4.html#K4">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="euclid.html#K8">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p5">(<span class="default"><span class="p0"><a href="euclid_4.html#K5">|(</a><span class="default"><span class="p1">(<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 class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="euclid_4.html#K5">)|</a></span> <a href="real_1.html#K6">/</a> <span class="p0">(<span class="default"><span class="p1"><a href="euclid_4.html#K4">|.</a><span class="default"><span class="p2">(<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">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span><a href="euclid_4.html#K5">)|</a></span>
<b>by </b><i><a class="ref" href="euclidlp.html#T5" target="_self">Th5</a></i>
<br/>.= 
<span class="p1"><a href="euclid_4.html#K5">|(</a><span class="default"><span class="p2">(<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 class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="euclid.html#K8">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p0">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><span class="p2"><a href="euclid_4.html#K5">|(</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 class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="euclid_4.html#K5">)|</a></span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><span class="p3"><a href="euclid_4.html#K4">|.</a><span class="default"><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><a href="euclid_4.html#K4">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="euclid.html#K8">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p5">(<span class="default"><span class="p0"><a href="euclid_4.html#K5">|(</a><span class="default"><span class="p1">(<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 class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="euclid_4.html#K5">)|</a></span> <a href="real_1.html#K6">/</a> <span class="p0">(<span class="default"><span class="p1"><a href="euclid_4.html#K4">|.</a><span class="default"><span class="p2">(<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">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span><a href="euclid_4.html#K5">)|</a></span>
<b>by </b><i><a class="ref" href="euclid_4.html#T3">EUCLID_4:3</a></i>
<br/>.= 
<span class="p1"><a href="euclid_4.html#K5">|(</a><span class="default"><span class="p2">(<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 class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="euclid.html#K8">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p0">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><span class="p2"><a href="euclid_4.html#K5">|(</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 class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="euclid_4.html#K5">)|</a></span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><span class="p3"><a href="euclid_4.html#K4">|.</a><span class="default"><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><a href="euclid_4.html#K4">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="euclid.html#K7">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p0">(<span class="default"><span class="p1"><a href="euclid_4.html#K5">|(</a><span class="default"><span class="p2">(<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 class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="euclid_4.html#K5">)|</a></span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2"><a href="euclid_4.html#K4">|.</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">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span></span><a href="euclid_4.html#K5">)|</a></span>
<b>by </b><i><a class="ref" href="euclidlp.html#T5" target="_self">Th5</a></i>
<br/>.= 
<span class="p1"><a href="euclid_4.html#K5">|(</a><span class="default"><span class="p2">(<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 class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="euclid.html#K8">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p0">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><span class="p2"><a href="euclid_4.html#K5">|(</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 class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="euclid_4.html#K5">)|</a></span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><span class="p3"><a href="euclid_4.html#K4">|.</a><span class="default"><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><a href="euclid_4.html#K4">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="euclid.html#K7">+</a> <span class="p4">(<span class="default"><a href="euclid.html#K6">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><span class="p3"><a href="euclid_4.html#K5">|(</a><span class="default"><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 class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="euclid_4.html#K5">)|</a></span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><span class="p4"><a href="euclid_4.html#K4">|.</a><span class="default"><span class="p5">(<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">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid_4.html#K5">)|</a></span>
<b>by </b><i><a class="ref" href="euclidlp.html#T8" target="_self">Th8</a></i>
<br/>.= 
<span class="p1"><a href="euclid_4.html#K5">|(</a><span class="default"><span class="p2">(<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 class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="euclid.html#K8">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p0">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><span class="p2"><a href="euclid_4.html#K5">|(</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 class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="euclid_4.html#K5">)|</a></span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><span class="p3"><a href="euclid_4.html#K4">|.</a><span class="default"><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><a href="euclid_4.html#K4">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="euclid.html#K7">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p0">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><span class="p2"><a href="euclid_4.html#K5">|(</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 class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="euclid_4.html#K5">)|</a></span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><span class="p3"><a href="euclid_4.html#K4">|.</a><span class="default"><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><a href="euclid_4.html#K4">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K9">*</a> <span class="p5">(<span class="default"><a href="euclid.html#K6">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid_4.html#K5">)|</a></span>
<b>by </b><i><a class="ref" href="euclidlp.html#T8" target="_self">Th8</a></i>
<br/>.= 
<span class="p1"><a href="euclid_4.html#K5">|(</a><span class="default"><span class="p2">(<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 class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="euclid.html#K8">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p5">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p0">(<span class="default"><span class="p1"><a href="euclid_4.html#K5">|(</a><span class="default"><span class="p2">(<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 class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="euclid_4.html#K5">)|</a></span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2"><a href="euclid_4.html#K4">|.</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">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K9">*</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K7">+</a> <span class="p5">(<span class="default"><a href="euclid.html#K6">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="euclid_4.html#K5">)|</a></span>
<b>by </b><i><a class="ref" href="euclid_4.html#T6">EUCLID_4:6</a></i>
<br/>.= 
<span class="p1"><a href="euclid_4.html#K5">|(</a><span class="default"><span class="p2">(<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 class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="euclid.html#K8">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p5">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p0">(<span class="default"><span class="p1"><a href="euclid_4.html#K5">|(</a><span class="default"><span class="p2">(<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 class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="euclid_4.html#K5">)|</a></span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2"><a href="euclid_4.html#K4">|.</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">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K9">*</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>)</span></span><a href="euclid_4.html#K5">)|</a></span>
<b>by </b><i><a class="ref" href="euclidlp.html#T6" target="_self">Th6</a></i>
<br/>.= 
<span class="p1"><a href="euclid_4.html#K5">|(</a><span class="default"><span class="p2">(<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 class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="euclid_4.html#K5">)|</a></span> <a href="real_1.html#K5">-</a> <span class="p1"><a href="euclid_4.html#K5">|(</a><span class="default"><span class="p2">(<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 class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p5">(<span class="default"><span class="p0"><a href="euclid_4.html#K5">|(</a><span class="default"><span class="p1">(<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 class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="euclid_4.html#K5">)|</a></span> <a href="real_1.html#K6">/</a> <span class="p0">(<span class="default"><span class="p1"><a href="euclid_4.html#K4">|.</a><span class="default"><span class="p2">(<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">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K9">*</a> <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>)</span></span><a href="euclid_4.html#K5">)|</a></span>
<b>by </b><i><a class="ref" href="euclid_4.html#T31">EUCLID_4:31</a></i>
<br/>.= 
<span class="p1"><a href="euclid_4.html#K5">|(</a><span class="default"><span class="p2">(<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 class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="euclid_4.html#K5">)|</a></span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p4">(<span class="default"><span class="p5"><a href="euclid_4.html#K5">|(</a><span class="default"><span class="p0">(<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 class="p0">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="euclid_4.html#K5">)|</a></span> <a href="real_1.html#K6">/</a> <span class="p5">(<span class="default"><span class="p0"><a href="euclid_4.html#K4">|.</a><span class="default"><span class="p1">(<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">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2"><a href="euclid_4.html#K5">|(</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 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#K5">)|</a></span></span>)</span>
<b>by </b><i><a class="ref" href="euclid_4.html#T26">EUCLID_4:26</a></i>
<br/>.= 
<span class="p1"><a href="euclid_4.html#K5">|(</a><span class="default"><span class="p2">(<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 class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="euclid_4.html#K5">)|</a></span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><span class="p4"><a href="euclid_4.html#K5">|(</a><span class="default"><span class="p5">(<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 class="p5">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="euclid_4.html#K5">)|</a></span> <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><span class="p5"><a href="euclid_4.html#K4">|.</a><span class="default"><span class="p0">(<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">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2"><a href="euclid_4.html#K5">|(</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 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#K5">)|</a></span></span>)</span>

<br/>.= 
<span class="p1"><a href="euclid_4.html#K5">|(</a><span class="default"><span class="p2">(<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 class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="euclid_4.html#K5">)|</a></span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><span class="p4"><a href="euclid_4.html#K5">|(</a><span class="default"><span class="p5">(<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 class="p5">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="euclid_4.html#K5">)|</a></span> <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><span class="p5"><a href="euclid_4.html#K4">|.</a><span class="default"><span class="p0">(<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">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3"><a href="euclid_4.html#K4">|.</a><span class="default"><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><a href="euclid_4.html#K4">.|</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#T12">EUCLID_2:12</a></i>
<br/>.= 
<span class="p1"><a href="euclid_4.html#K5">|(</a><span class="default"><span class="p2">(<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 class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="euclid_4.html#K5">)|</a></span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p4"><a href="euclid_4.html#K5">|(</a><span class="default"><span class="p5">(<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 class="p5">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="euclid_4.html#K5">)|</a></span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><span class="p4"><a href="euclid_4.html#K4">|.</a><span class="default"><span class="p5">(<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">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3"><a href="euclid_4.html#K4">|.</a><span class="default"><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><a href="euclid_4.html#K4">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T188">XCMPLX_1:188</a></i>
<br/>.= 
<span class="p1"><a href="euclid_4.html#K5">|(</a><span class="default"><span class="p2">(<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 class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="euclid_4.html#K5">)|</a></span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2"><a href="euclid_4.html#K5">|(</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 class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="euclid_4.html#K5">)|</a></span></span>)</span>
<b>by </b><i><a class="txt" href="euclidlp.html#E3:48_1_1_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a></i>
<br/>.= 
0

;<br/><b>hence </b><a NAME="E5:48_1_1_1"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p4">(<span class="default"><span class="p5"><a href="euclid_4.html#K5">|(</a><span class="default"><span class="p0">(<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 class="p0">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="euclid_4.html#K5">)|</a></span> <a href="real_1.html#K6">/</a> <span class="p5">(<span class="default"><span class="p0"><a href="euclid_4.html#K4">|.</a><span class="default"><span class="p1">(<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">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><span class="p4"><a href="euclid_4.html#K5">|(</a><span class="default"><span class="p5">(<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 class="p5">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="euclid_4.html#K5">)|</a></span> <a href="real_1.html#K6">/</a> <span class="p4">(<span class="default"><span class="p5"><a href="euclid_4.html#K4">|.</a><span class="default"><span class="p0">(<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">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="euclid_4.html#K2">Line</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K8">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p5">(<span class="default"><span class="p0"><a href="euclid_4.html#K5">|(</a><span class="default"><span class="p1">(<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 class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="euclid_4.html#K5">)|</a></span> <a href="real_1.html#K6">/</a> <span class="p0">(<span class="default"><span class="p1"><a href="euclid_4.html#K4">|.</a><span class="default"><span class="p2">(<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">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="euclid.html#K7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p4">(<span class="default"><span class="p5"><a href="euclid_4.html#K5">|(</a><span class="default"><span class="p0">(<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 class="p0">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="euclid_4.html#K5">)|</a></span> <a href="real_1.html#K6">/</a> <span class="p5">(<span class="default"><span class="p0"><a href="euclid_4.html#K4">|.</a><span class="default"><span class="p1">(<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">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="euclid_4.html#R1">are_orthogonal</a>  )
 <b>by </b><i><a class="ref" href="euclid_4.html#D6">EUCLID_4:def 6</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_2"><b>case </b></a><a NAME="E1:48_1_1_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><div class="add"><a NAME="E2:48_1_1_2"/><b>then </b><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K5">0*</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="euclidlp.html#T7" target="_self">Th7</a></i>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>set </b><font color="Maroon">c<sub>6</sub></font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>;<br/><a NAME="E3:48_1_1_2"/><i><font color="Green">E49</font></i>: 
<span class="p1"><a href="euclid_4.html#K5">|(</a><span class="default"><span class="p2">(<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 class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K8">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="euclid.html#K7">+</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span></span><a href="euclid_4.html#K5">)|</a></span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="euclidlp.html#E2:48_1_1_2"><i><font color="Green">E48</font></i></a>, <a class="ref" href="euclid_4.html#T23">EUCLID_4:23</a></i>;<br/><b>take </b><font color="Maroon">c<sub>7</sub></font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>;<br/><b>thus </b><a NAME="E4:48_1_1_2"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="euclid_4.html#K2">Line</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>7</sub></font> <a href="euclid_4.html#R1">are_orthogonal</a>  )
 <b>by </b><i><a class="txt" href="euclidlp.html#E3:48_1_1_2"><i><font color="Green">E49</font></i></a>, <a class="ref" href="euclid_4.html#D6">EUCLID_4:def 6</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:48"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="euclid_4.html#K2">Line</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K8">-</a> <font color="Olive">b<sub>1</sub></font> <a href="euclid_4.html#R1">are_orthogonal</a>  )
 ;<br/>


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