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

<b>let </b><font color="Maroon">X</font> be   <a href="bhsp_1.html#NM1">RealUnitarySpace</a>;<br/><b>let </b><font color="Maroon">x</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">y</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>;<br/>





<a NAME="E1:25"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">X</font>) implies  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="bhsp_1.html#K2" target="_self">.|.</a> <font color="Maroon">y</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="bhsp_1.html#K2" target="_self">.|.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="bhsp_1.html#K2" target="_self">.|.</a> <font color="Maroon">y</font></span>)</span></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:25_1"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">X</font>)
 ;<br/>

<a NAME="E2:25_1"/><b>then </b><i><font color="Green">E67</font></i>: 
<font color="Maroon">x</font> <a href="bhsp_1.html#K2" target="_self">.|.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i/>;<br/>
<a NAME="E3:25_1"/>
 <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="bhsp_1.html#K2" target="_self">.|.</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i>, , <a class="ref" href="square_1.html#T82">SQUARE_1:82</a></i>;<br/>
<b>hence </b><a NAME="E4:25_1"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="bhsp_1.html#K2" target="_self">.|.</a> <font color="Maroon">y</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="bhsp_1.html#K2" target="_self">.|.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="bhsp_1.html#K2" target="_self">.|.</a> <font color="Maroon">y</font></span>)</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="E2:25"/>
( <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">X</font>) implies  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="bhsp_1.html#K2" target="_self">.|.</a> <font color="Maroon">y</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="bhsp_1.html#K2" target="_self">.|.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="bhsp_1.html#K2" target="_self">.|.</a> <font color="Maroon">y</font></span>)</span></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:25_2"/>
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">X</font>)
 ;<br/>

<a NAME="E2:25_2"/><b>then </b><i><font color="Green">E68</font></i>: 
<font color="Maroon">x</font> <a href="bhsp_1.html#K2" target="_self">.|.</a> <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i/>;<br/>
<a NAME="E3:25_2"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">x</font> <a href="bhsp_1.html#K2" target="_self">.|.</a> <font color="Maroon">x</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i/>;<br/>
<a NAME="E4:25_2"/><i><font color="Green">E70</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">(<span class="default"><font color="Maroon">x</font> <a href="bhsp_1.html#K2" target="_self">.|.</a> <font color="Maroon">x</font></span>)</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="real_1.html#K4">*</a> <span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="bhsp_1.html#K2" target="_self">.|.</a> <font color="Maroon">y</font></span>)</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">(<span class="default"><font color="Maroon">y</font> <a href="bhsp_1.html#K2" target="_self">.|.</a> <font color="Maroon">y</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_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>reconsider </b><font color="Maroon">t</font> = <font color="Maroon">t</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/>
<a NAME="E2:25_2_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">y</font></span>)</span> <a href="bhsp_1.html#K2" target="_self">.|.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">y</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i/>;<br/>
<a NAME="E3:25_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="rlvect_1.html#K3">*</a> <font color="Maroon">x</font></span>)</span> <a href="bhsp_1.html#K2" target="_self">.|.</a> <span class="p3">(<span class="default"><font color="Maroon">t</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">t</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x</font></span>)</span> <a href="bhsp_1.html#K2" target="_self">.|.</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="bhsp_1.html#K2" target="_self">.|.</a> <font color="Maroon">y</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i/>;<br/>
<a NAME="E4:25_2_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="bhsp_1.html#K2" target="_self">.|.</a> <span class="p4">(<span class="default"><font color="Maroon">t</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">t</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x</font></span>)</span> <a href="bhsp_1.html#K2" target="_self">.|.</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="bhsp_1.html#K2" target="_self">.|.</a> <font color="Maroon">y</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i/>;<br/>
<a NAME="E5:25_2_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">t</font> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><font color="Maroon">t</font> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="bhsp_1.html#K2" target="_self">.|.</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">t</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x</font></span>)</span> <a href="bhsp_1.html#K2" target="_self">.|.</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="bhsp_1.html#K2" target="_self">.|.</a> <font color="Maroon">y</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i/>;<br/>
<a NAME="E6:25_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="real_1.html#K4">*</a> <font color="Maroon">t</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="bhsp_1.html#K2" target="_self">.|.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">t</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x</font></span>)</span> <a href="bhsp_1.html#K2" target="_self">.|.</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="bhsp_1.html#K2" target="_self">.|.</a> <font color="Maroon">y</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
;<br/>
<a NAME="E7:25_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">x</font> <a href="bhsp_1.html#K2" target="_self">.|.</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><font color="Maroon">t</font> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">t</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x</font></span>)</span> <a href="bhsp_1.html#K2" target="_self">.|.</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="bhsp_1.html#K2" target="_self">.|.</a> <font color="Maroon">y</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
;<br/>
<a NAME="E8:25_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">x</font> <a href="bhsp_1.html#K2" target="_self">.|.</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><font color="Maroon">t</font> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="bhsp_1.html#K2" target="_self">.|.</a> <font color="Maroon">y</font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">t</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="bhsp_1.html#K2" target="_self">.|.</a> <font color="Maroon">y</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E9:25_2_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="bhsp_1.html#K2" target="_self">.|.</a> <font color="Maroon">x</font></span>)</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="real_1.html#K4">*</a> <span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="bhsp_1.html#K2" target="_self">.|.</a> <font color="Maroon">y</font></span>)</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">(<span class="default"><font color="Maroon">y</font> <a href="bhsp_1.html#K2" target="_self">.|.</a> <font color="Maroon">y</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 ;<br/>


</div><b>end;</b></div>
<a NAME="E5:25_2"/><i><font color="Green">E72</font></i>: 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="bhsp_1.html#K2" target="_self">.|.</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="xreal_1.html#T65">XREAL_1:65</a></i>;<br/>
<a NAME="E6:25_2"/><i><font color="Green">E73</font></i>: 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="bhsp_1.html#K2" target="_self">.|.</a> <font color="Maroon">y</font></span>)</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="E7:25_2"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">y</font> <a href="bhsp_1.html#K2" target="_self">.|.</a> <font color="Maroon">y</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i/>;<br/>
<a NAME="E8:25_2"/>
 <a href="quin_1.html#K2">delta</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="bhsp_1.html#K2" target="_self">.|.</a> <font color="Maroon">x</font></span>)</span>,<span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="bhsp_1.html#K2" target="_self">.|.</a> <font color="Maroon">y</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="bhsp_1.html#K2" target="_self">.|.</a> <font color="Maroon">y</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 0
 
<b>by </b><i>, , , <a class="ref" href="quin_1.html#T10">QUIN_1:10</a></i>;<br/>
<a NAME="E9:25_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="bhsp_1.html#K2" target="_self">.|.</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">4 <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="bhsp_1.html#K2" target="_self">.|.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="bhsp_1.html#K2" target="_self">.|.</a> <font color="Maroon">y</font></span>)</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="E10:25_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="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="bhsp_1.html#K2" target="_self">.|.</a> <font color="Maroon">y</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">4 <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="bhsp_1.html#K2" target="_self">.|.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="bhsp_1.html#K2" target="_self">.|.</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 0
 
;<br/>
<a NAME="E11:25_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> 2</span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="bhsp_1.html#K2" target="_self">.|.</a> <font color="Maroon">y</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">4 <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="bhsp_1.html#K2" target="_self">.|.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="bhsp_1.html#K2" target="_self">.|.</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 0
 
;<br/>
<a NAME="E12:25_2"/><b>then </b>
4 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="bhsp_1.html#K2" target="_self">.|.</a> <font color="Maroon">y</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="bhsp_1.html#K2" target="_self">.|.</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><font color="Maroon">y</font> <a href="bhsp_1.html#K2" target="_self">.|.</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 0
 
;<br/>
<a NAME="E13:25_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="bhsp_1.html#K2" target="_self">.|.</a> <font color="Maroon">y</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="bhsp_1.html#K2" target="_self">.|.</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="bhsp_1.html#K2" target="_self">.|.</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 0 <a href="real_1.html#K6">/</a> 4
 
<b>by </b><i><a class="ref" href="real_2.html#T177">REAL_2:177</a></i>;<br/>
<a NAME="E14:25_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="bhsp_1.html#K2" target="_self">.|.</a> <font color="Maroon">y</font></span>)</span> <a href="square_1.html#K5">^2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="bhsp_1.html#K2" target="_self">.|.</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="bhsp_1.html#K2" target="_self">.|.</a> <font color="Maroon">y</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T52">XREAL_1:52</a></i>;<br/>
<a NAME="E15:25_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="bhsp_1.html#K2" target="_self">.|.</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="bhsp_1.html#K2" target="_self">.|.</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="bhsp_1.html#K2" target="_self">.|.</a> <font color="Maroon">y</font></span>)</span>
 
<b>by </b><i><a class="ref" href="complex1.html#T161">COMPLEX1:161</a></i>;<br/>
<a NAME="E16:25_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">(<span class="default"><font color="Maroon">x</font> <a href="bhsp_1.html#K2" target="_self">.|.</a> <font color="Maroon">y</font></span>)</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#K7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="bhsp_1.html#K2" target="_self">.|.</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="bhsp_1.html#K2" target="_self">.|.</a> <font color="Maroon">y</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="square_1.html#T94">SQUARE_1:94</a></i>;<br/>
<a NAME="E17:25_2"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="bhsp_1.html#K2" target="_self">.|.</a> <font color="Maroon">y</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="bhsp_1.html#K2" target="_self">.|.</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="bhsp_1.html#K2" target="_self">.|.</a> <font color="Maroon">y</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="square_1.html#T89">SQUARE_1:89</a></i>;<br/>
<b>hence </b><a NAME="E18:25_2"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="bhsp_1.html#K2" target="_self">.|.</a> <font color="Maroon">y</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="bhsp_1.html#K2" target="_self">.|.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="bhsp_1.html#K2" target="_self">.|.</a> <font color="Maroon">y</font></span>)</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>
<b>hence </b><a NAME="E3:25"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="bhsp_1.html#K2" target="_self">.|.</a> <font color="Maroon">y</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="bhsp_1.html#K2" target="_self">.|.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="bhsp_1.html#K2" target="_self">.|.</a> <font color="Maroon">y</font></span>)</span></span>)</span>
 <b>by </b><i/>;<br/>


</div>
