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

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="csspace.html#NM1">ComplexUnitarySpace</a>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>4</sub></font>;<br/>



<b>thus </b><a NAME="E1:67"/>
( <span class="p1"><a href="csspace.html#K13" target="_self">||.</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="csspace.html#K13" target="_self">.||</a></span> <a href="hidden.html#R1">=</a> 0 implies <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>4</sub></font>) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:67_1"/>
<span class="p1"><a href="csspace.html#K13" target="_self">||.</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="csspace.html#K13" target="_self">.||</a></span> <a href="hidden.html#R1">=</a> 0
 ;<br/>

<a NAME="E2:67_1"/><b>then </b><i><font color="Green">E56</font></i>: 
 <a href="square_1.html#K7">sqrt</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="csspace.html#K12" target="_self">.|.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="hidden.html#R1">=</a> 0
 
;<br/>
<a NAME="E3:67_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="complex1.html#K3">Re</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="csspace.html#K12" target="_self">.|.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="csspace.html#D13" target="_self">Def13</a></i>;<br/>
<a NAME="E4:67_1"/><b>then </b>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="csspace.html#K12" target="_self">.|.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="complex1.html#K17">.|</a></span>
 
<b>by </b><i><a class="ref" href="csspace.html#T36" target="_self">Th36</a></i>;<br/>
<a NAME="E5:67_1"/><b>then </b>
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="csspace.html#K12" target="_self">.|.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="txt" href="csspace.html#E2:67_1"><i><font color="Green">E56</font></i></a>, <a class="ref" href="square_1.html#T92">SQUARE_1:92</a></i>;<br/>
<a NAME="E6:67_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="csspace.html#K12" target="_self">.|.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K5">0c</a> 
 
<b>by </b><i><a class="ref" href="complex1.html#T131">COMPLEX1:131</a></i>;<br/>
<b>hence </b><a NAME="E7:67_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>4</sub></font>)
 <b>by </b><i><a class="ref" href="csspace.html#D13" target="_self">Def13</a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:67"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>4</sub></font>)
 ;<br/>

<a NAME="E3:67"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="csspace.html#K12" target="_self">.|.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K5">0c</a> 
 
<b>by </b><i><a class="ref" href="csspace.html#D13" target="_self">Def13</a></i>;<br/>
<b>hence </b><a NAME="E4:67"/>
<span class="p1"><a href="csspace.html#K13" target="_self">||.</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="csspace.html#K13" target="_self">.||</a></span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="complex1.html#T130">COMPLEX1:130</a>, <a class="ref" href="square_1.html#T82">SQUARE_1:82</a></i>;<br/>


</div>
