<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">p</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span>, <font color="Maroon">q</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span>;<br/>



<b>set </b><font color="Maroon">x</font> = <font color="Maroon">p</font>;<br/>
<b>set </b><font color="Maroon">y</font> = <font color="Maroon">q</font>;<br/>
<a NAME="E1:64"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K16">0.REAL</a> <font color="Maroon">n</font> implies  <a href="complex1.html#K18">abs</a> <span class="p1"><a href="euclid_2.html#K2" target="_self">|(</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">q</font></span><a href="euclid_2.html#K2" target="_self">)|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p2"><a href="euclid_2.html#K2" target="_self">|(</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">p</font></span><a href="euclid_2.html#K2" target="_self">)|</a></span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p2"><a href="euclid_2.html#K2" target="_self">|(</a><span class="default"><font color="Maroon">q</font>,<font color="Maroon">q</font></span><a href="euclid_2.html#K2" target="_self">)|</a></span></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:64_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K16">0.REAL</a> <font color="Maroon">n</font>
 ;<br/>

<a NAME="E2:64_1"/><b>then </b><i><font color="Green">E37</font></i>: 
<span class="p1"><a href="euclid_2.html#K2" target="_self">|(</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">q</font></span><a href="euclid_2.html#K2" target="_self">)|</a></span> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i/>;<br/>
<a NAME="E3:64_1"/>
 <a href="square_1.html#K6">sqrt</a> <span class="p1"><a href="euclid_2.html#K2" target="_self">|(</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">p</font></span><a href="euclid_2.html#K2" target="_self">)|</a></span> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="euclid_2.html#T6" target="_self">Th6</a>, , <a class="ref" href="square_1.html#T82">SQUARE_1:82</a></i>;<br/>
<b>hence </b><a NAME="E4:64_1"/>
 <a href="complex1.html#K18">abs</a> <span class="p1"><a href="euclid_2.html#K2" target="_self">|(</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">q</font></span><a href="euclid_2.html#K2" target="_self">)|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p2"><a href="euclid_2.html#K2" target="_self">|(</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">p</font></span><a href="euclid_2.html#K2" target="_self">)|</a></span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p2"><a href="euclid_2.html#K2" target="_self">|(</a><span class="default"><font color="Maroon">q</font>,<font color="Maroon">q</font></span><a href="euclid_2.html#K2" target="_self">)|</a></span></span>)</span>
 <b>by </b><i>, <a class="ref" href="absvalue.html#T7">ABSVALUE:7</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:64"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="euclid.html#K16">0.REAL</a> <font color="Maroon">n</font> implies  <a href="complex1.html#K18">abs</a> <span class="p1"><a href="euclid_2.html#K2" target="_self">|(</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">q</font></span><a href="euclid_2.html#K2" target="_self">)|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p2"><a href="euclid_2.html#K2" target="_self">|(</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">p</font></span><a href="euclid_2.html#K2" target="_self">)|</a></span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p2"><a href="euclid_2.html#K2" target="_self">|(</a><span class="default"><font color="Maroon">q</font>,<font color="Maroon">q</font></span><a href="euclid_2.html#K2" target="_self">)|</a></span></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:64_2"/>
<font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="euclid.html#K16">0.REAL</a> <font color="Maroon">n</font>
 ;<br/>

<a NAME="E2:64_2"/><b>then </b><i><font color="Green">E43</font></i>: 
<span class="p1"><a href="euclid_2.html#K2" target="_self">|(</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">p</font></span><a href="euclid_2.html#K2" target="_self">)|</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i/>;<br/>
<a NAME="E3:64_2"/><i><font color="Green">E61</font></i>: 
<span class="p1"><a href="euclid_2.html#K2" target="_self">|(</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">p</font></span><a href="euclid_2.html#K2" target="_self">)|</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i/>;<br/>
<a NAME="E4:64_2"/><b>then </b><i><font color="Green">E87</font></i>: 
<span class="p1"><a href="euclid_2.html#K2" target="_self">|(</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">p</font></span><a href="euclid_2.html#K2" target="_self">)|</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/>
<a NAME="E5:64_2"/><i><font color="Green">E88</font></i>: 
for <font color="Olive">t</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="euclid_2.html#K2" target="_self">|(</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">p</font></span><a href="euclid_2.html#K2" target="_self">)|</a></span> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><font color="Olive">t</font> <a href="square_1.html#K3">^2</a> </span>)</span></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_2.html#K2" target="_self">|(</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">q</font></span><a href="euclid_2.html#K2" target="_self">)|</a></span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Olive">t</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1"><a href="euclid_2.html#K2" target="_self">|(</a><span class="default"><font color="Maroon">q</font>,<font color="Maroon">q</font></span><a href="euclid_2.html#K2" target="_self">)|</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">t</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>set </b><font color="Maroon">s</font> = <font color="Maroon">t</font> <a href="square_1.html#K3">^2</a> ;<br/>
<a NAME="E1:64_2_1"/>
<span class="p1"><a href="euclid_2.html#K2" target="_self">|(</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p</font></span>)</span> <a href="euclid.html#K17">+</a> <font color="Maroon">q</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p</font></span>)</span> <a href="euclid.html#K17">+</a> <font color="Maroon">q</font></span>)</span></span><a href="euclid_2.html#K2" target="_self">)|</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i/>;<br/>
<a NAME="E2:64_2_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2"><a href="euclid_2.html#K2" target="_self">|(</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p</font></span>)</span></span><a href="euclid_2.html#K2" target="_self">)|</a></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#K2" target="_self">|(</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p</font></span>)</span>,<font color="Maroon">q</font></span><a href="euclid_2.html#K2" target="_self">)|</a></span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1"><a href="euclid_2.html#K2" target="_self">|(</a><span class="default"><font color="Maroon">q</font>,<font color="Maroon">q</font></span><a href="euclid_2.html#K2" target="_self">)|</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i/>;<br/>
<a NAME="E3:64_2_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="xcmplx_0.html#K3">*</a> <span class="p3"><a href="euclid_2.html#K2" target="_self">|(</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p</font></span>)</span>,<font color="Maroon">p</font></span><a href="euclid_2.html#K2" target="_self">)|</a></span></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#K2" target="_self">|(</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p</font></span>)</span>,<font color="Maroon">q</font></span><a href="euclid_2.html#K2" target="_self">)|</a></span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1"><a href="euclid_2.html#K2" target="_self">|(</a><span class="default"><font color="Maroon">q</font>,<font color="Maroon">q</font></span><a href="euclid_2.html#K2" target="_self">)|</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i/>;<br/>
<a NAME="E4:64_2_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">t</font> <a href="xcmplx_0.html#K3">*</a> <span class="p4"><a href="euclid_2.html#K2" target="_self">|(</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">p</font></span><a href="euclid_2.html#K2" target="_self">)|</a></span></span>)</span></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#K2" target="_self">|(</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p</font></span>)</span>,<font color="Maroon">q</font></span><a href="euclid_2.html#K2" target="_self">)|</a></span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1"><a href="euclid_2.html#K2" target="_self">|(</a><span class="default"><font color="Maroon">q</font>,<font color="Maroon">q</font></span><a href="euclid_2.html#K2" target="_self">)|</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i/>;<br/>
<a NAME="E5:64_2_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">t</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">t</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p3"><a href="euclid_2.html#K2" target="_self">|(</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">p</font></span><a href="euclid_2.html#K2" target="_self">)|</a></span></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#K2" target="_self">|(</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p</font></span>)</span>,<font color="Maroon">q</font></span><a href="euclid_2.html#K2" target="_self">)|</a></span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1"><a href="euclid_2.html#K2" target="_self">|(</a><span class="default"><font color="Maroon">q</font>,<font color="Maroon">q</font></span><a href="euclid_2.html#K2" target="_self">)|</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
;<br/>
<a NAME="E6:64_2_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="euclid_2.html#K2" target="_self">|(</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">p</font></span><a href="euclid_2.html#K2" target="_self">)|</a></span> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">t</font> <a href="square_1.html#K3">^2</a> </span>)</span></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#K2" target="_self">|(</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">t</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p</font></span>)</span>,<font color="Maroon">q</font></span><a href="euclid_2.html#K2" target="_self">)|</a></span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1"><a href="euclid_2.html#K2" target="_self">|(</a><span class="default"><font color="Maroon">q</font>,<font color="Maroon">q</font></span><a href="euclid_2.html#K2" target="_self">)|</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
;<br/>
<a NAME="E7:64_2_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="euclid_2.html#K2" target="_self">|(</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">p</font></span><a href="euclid_2.html#K2" target="_self">)|</a></span> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">t</font> <a href="square_1.html#K3">^2</a> </span>)</span></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">(<span class="default"><span class="p4"><a href="euclid_2.html#K2" target="_self">|(</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">q</font></span><a href="euclid_2.html#K2" target="_self">)|</a></span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">t</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1"><a href="euclid_2.html#K2" target="_self">|(</a><span class="default"><font color="Maroon">q</font>,<font color="Maroon">q</font></span><a href="euclid_2.html#K2" target="_self">)|</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E8:64_2_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="euclid_2.html#K2" target="_self">|(</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">p</font></span><a href="euclid_2.html#K2" target="_self">)|</a></span> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">t</font> <a href="square_1.html#K3">^2</a> </span>)</span></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_2.html#K2" target="_self">|(</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">q</font></span><a href="euclid_2.html#K2" target="_self">)|</a></span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">t</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1"><a href="euclid_2.html#K2" target="_self">|(</a><span class="default"><font color="Maroon">q</font>,<font color="Maroon">q</font></span><a href="euclid_2.html#K2" target="_self">)|</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 ;<br/>


</div><b>end;</b></div>
<b>set </b><font color="Maroon">w</font> =  <a href="complex1.html#K18">abs</a> <span class="p1"><a href="euclid_2.html#K2" target="_self">|(</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">q</font></span><a href="euclid_2.html#K2" target="_self">)|</a></span>;<br/>
<b>set </b><font color="Maroon">u</font> = <span class="p1"><a href="euclid_2.html#K2" target="_self">|(</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">q</font></span><a href="euclid_2.html#K2" target="_self">)|</a></span>;<br/>
<a NAME="E6:64_2"/><i><font color="Green">E89</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2"><a href="euclid_2.html#K2" target="_self">|(</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">q</font></span><a href="euclid_2.html#K2" target="_self">)|</a></span></span>)</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="E7:64_2"/><i><font color="Green">E92</font></i>: 
 <a href="complex1.html#K18">abs</a> <span class="p1"><a href="euclid_2.html#K2" target="_self">|(</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">q</font></span><a href="euclid_2.html#K2" target="_self">)|</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="complex1.html#T132">COMPLEX1:132</a></i>;<br/>
<a NAME="E8:64_2"/><i><font color="Green">E93</font></i>: 
<span class="p1"><a href="euclid_2.html#K2" target="_self">|(</a><span class="default"><font color="Maroon">q</font>,<font color="Maroon">q</font></span><a href="euclid_2.html#K2" target="_self">)|</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i/>;<br/>
<a NAME="E9:64_2"/>
 <a href="quin_1.html#K1">delta</a> <span class="p1"><a href="euclid_2.html#K2" target="_self">|(</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">p</font></span><a href="euclid_2.html#K2" target="_self">)|</a></span>,<span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p2"><a href="euclid_2.html#K2" target="_self">|(</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">q</font></span><a href="euclid_2.html#K2" target="_self">)|</a></span></span>)</span>,<span class="p1"><a href="euclid_2.html#K2" target="_self">|(</a><span class="default"><font color="Maroon">q</font>,<font color="Maroon">q</font></span><a href="euclid_2.html#K2" target="_self">)|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> 0
 
<b>by </b><i>, <a class="ref" href="euclid_2.html#T7" target="_self">Th7</a>, <a class="ref" href="quin_1.html#T10">QUIN_1:10</a></i>;<br/>
<a NAME="E10:64_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p3"><a href="euclid_2.html#K2" target="_self">|(</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">q</font></span><a href="euclid_2.html#K2" target="_self">)|</a></span></span>)</span> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">4 <a href="xcmplx_0.html#K3">*</a> <span class="p3"><a href="euclid_2.html#K2" target="_self">|(</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">p</font></span><a href="euclid_2.html#K2" target="_self">)|</a></span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2"><a href="euclid_2.html#K2" target="_self">|(</a><span class="default"><font color="Maroon">q</font>,<font color="Maroon">q</font></span><a href="euclid_2.html#K2" target="_self">)|</a></span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 0
 
<b>by </b><i><a class="ref" href="quin_1.html#D1">QUIN_1:def 1</a></i>;<br/>
<a NAME="E11:64_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3"><a href="euclid_2.html#K2" target="_self">|(</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">q</font></span><a href="euclid_2.html#K2" target="_self">)|</a></span> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">4 <a href="xcmplx_0.html#K3">*</a> <span class="p3"><a href="euclid_2.html#K2" target="_self">|(</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">p</font></span><a href="euclid_2.html#K2" target="_self">)|</a></span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2"><a href="euclid_2.html#K2" target="_self">|(</a><span class="default"><font color="Maroon">q</font>,<font color="Maroon">q</font></span><a href="euclid_2.html#K2" target="_self">)|</a></span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 0
 
;<br/>
<a NAME="E12:64_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> 2</span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3"><a href="euclid_2.html#K2" target="_self">|(</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">q</font></span><a href="euclid_2.html#K2" target="_self">)|</a></span> <a href="square_1.html#K3">^2</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">4 <a href="xcmplx_0.html#K3">*</a> <span class="p3"><a href="euclid_2.html#K2" target="_self">|(</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">p</font></span><a href="euclid_2.html#K2" target="_self">)|</a></span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2"><a href="euclid_2.html#K2" target="_self">|(</a><span class="default"><font color="Maroon">q</font>,<font color="Maroon">q</font></span><a href="euclid_2.html#K2" target="_self">)|</a></span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 0
 
;<br/>
<a NAME="E13:64_2"/><b>then </b>
4 <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="euclid_2.html#K2" target="_self">|(</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">q</font></span><a href="euclid_2.html#K2" target="_self">)|</a></span> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><span class="p3"><a href="euclid_2.html#K2" target="_self">|(</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">p</font></span><a href="euclid_2.html#K2" target="_self">)|</a></span> <a href="xcmplx_0.html#K3">*</a> <span class="p3"><a href="euclid_2.html#K2" target="_self">|(</a><span class="default"><font color="Maroon">q</font>,<font color="Maroon">q</font></span><a href="euclid_2.html#K2" target="_self">)|</a></span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 0
 
;<br/>
<a NAME="E14:64_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2"><a href="euclid_2.html#K2" target="_self">|(</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">q</font></span><a href="euclid_2.html#K2" target="_self">)|</a></span> <a href="square_1.html#K3">^2</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2"><a href="euclid_2.html#K2" target="_self">|(</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">p</font></span><a href="euclid_2.html#K2" target="_self">)|</a></span> <a href="xcmplx_0.html#K3">*</a> <span class="p2"><a href="euclid_2.html#K2" target="_self">|(</a><span class="default"><font color="Maroon">q</font>,<font color="Maroon">q</font></span><a href="euclid_2.html#K2" target="_self">)|</a></span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 0 <a href="xcmplx_0.html#K7">/</a> 4
 
<b>by </b><i><a class="ref" href="real_2.html#T177">REAL_2:177</a></i>;<br/>
<a NAME="E15:64_2"/><b>then </b>
<span class="p1"><a href="euclid_2.html#K2" target="_self">|(</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">q</font></span><a href="euclid_2.html#K2" target="_self">)|</a></span> <a href="square_1.html#K3">^2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="euclid_2.html#K2" target="_self">|(</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">p</font></span><a href="euclid_2.html#K2" target="_self">)|</a></span> <a href="xcmplx_0.html#K3">*</a> <span class="p1"><a href="euclid_2.html#K2" target="_self">|(</a><span class="default"><font color="Maroon">q</font>,<font color="Maroon">q</font></span><a href="euclid_2.html#K2" target="_self">)|</a></span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T52">XREAL_1:52</a></i>;<br/>
<a NAME="E16:64_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2"><a href="euclid_2.html#K2" target="_self">|(</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">q</font></span><a href="euclid_2.html#K2" target="_self">)|</a></span></span>)</span> <a href="square_1.html#K5">^2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="euclid_2.html#K2" target="_self">|(</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">p</font></span><a href="euclid_2.html#K2" target="_self">)|</a></span> <a href="xcmplx_0.html#K3">*</a> <span class="p1"><a href="euclid_2.html#K2" target="_self">|(</a><span class="default"><font color="Maroon">q</font>,<font color="Maroon">q</font></span><a href="euclid_2.html#K2" target="_self">)|</a></span>
 
<b>by </b><i><a class="ref" href="complex1.html#T161">COMPLEX1:161</a></i>;<br/>
<a NAME="E17:64_2"/><b>then </b>
 <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p3"><a href="euclid_2.html#K2" target="_self">|(</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">q</font></span><a href="euclid_2.html#K2" target="_self">)|</a></span></span>)</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"><a href="euclid_2.html#K2" target="_self">|(</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">p</font></span><a href="euclid_2.html#K2" target="_self">)|</a></span> <a href="xcmplx_0.html#K3">*</a> <span class="p2"><a href="euclid_2.html#K2" target="_self">|(</a><span class="default"><font color="Maroon">q</font>,<font color="Maroon">q</font></span><a href="euclid_2.html#K2" target="_self">)|</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="euclid_2.html#T8" target="_self">Th8</a>, <a class="ref" href="square_1.html#T94">SQUARE_1:94</a></i>;<br/>
<a NAME="E18:64_2"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <span class="p1"><a href="euclid_2.html#K2" target="_self">|(</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">q</font></span><a href="euclid_2.html#K2" target="_self">)|</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"><a href="euclid_2.html#K2" target="_self">|(</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">p</font></span><a href="euclid_2.html#K2" target="_self">)|</a></span> <a href="xcmplx_0.html#K3">*</a> <span class="p2"><a href="euclid_2.html#K2" target="_self">|(</a><span class="default"><font color="Maroon">q</font>,<font color="Maroon">q</font></span><a href="euclid_2.html#K2" target="_self">)|</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="euclid_2.html#T9" target="_self">Th9</a>, <a class="ref" href="square_1.html#T89">SQUARE_1:89</a></i>;<br/>
<b>hence </b><a NAME="E19:64_2"/>
 <a href="complex1.html#K18">abs</a> <span class="p1"><a href="euclid_2.html#K2" target="_self">|(</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">q</font></span><a href="euclid_2.html#K2" target="_self">)|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p2"><a href="euclid_2.html#K2" target="_self">|(</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">p</font></span><a href="euclid_2.html#K2" target="_self">)|</a></span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="square_1.html#K6">sqrt</a> <span class="p2"><a href="euclid_2.html#K2" target="_self">|(</a><span class="default"><font color="Maroon">q</font>,<font color="Maroon">q</font></span><a href="euclid_2.html#K2" target="_self">)|</a></span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="square_1.html#T97">SQUARE_1:97</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:64"/>
 <a href="square_1.html#K6">sqrt</a> <span class="p1"><a href="euclid_2.html#K2" target="_self">|(</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">p</font></span><a href="euclid_2.html#K2" target="_self">)|</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">p</font></span><a href="toprns_1.html#K5">.|</a></span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E4:64"/>
 <a href="complex1.html#K18">abs</a> <span class="p1"><a href="euclid_2.html#K2" target="_self">|(</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">q</font></span><a href="euclid_2.html#K2" target="_self">)|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">p</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xcmplx_0.html#K3">*</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">q</font></span><a href="toprns_1.html#K5">.|</a></span>
 <b>by </b><i><a class="ref" href="euclid_2.html#T5" target="_self">Th5</a>, <a class="ref" href="euclid_2.html#T6" target="_self">Th6</a>, </i>;<br/>


</div>
