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<b>let </b><font color="Maroon">a</font> be   <a href="cfunct_1.html#NM1">Complex</a>;<br/><b>let </b><font color="Maroon">X</font> be   <a href="csspace.html#NM1">ComplexUnitarySpace</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/>





<a NAME="E1:68"/><i><font color="Green">E31</font></i>: 
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">a</font> <a href="complex1.html#K9">*</a> <font color="Maroon">a</font></span>)</span></span><a href="complex1.html#K17">.|</a></span>
 
<b>by </b><i><a class="ref" href="complex1.html#T132">COMPLEX1:132</a></i>;<br/>
<a NAME="E2:68"/>
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">x</font> <a href="csspace.html#K12" target="_self">.|.</a> <font color="Maroon">x</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E3:68"/><b>then </b><i><font color="Green">E33</font></i>: 
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">x</font> <a href="csspace.html#K12" target="_self">.|.</a> <font color="Maroon">x</font></span>)</span></span><a href="complex1.html#K17">.|</a></span>
 
<b>by </b><i><a class="ref" href="csspace.html#D4" target="_self">Def4</a></i>;<br/>
<a NAME="E4:68"/><i><font color="Green">E36</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">a</font></span><a href="complex1.html#K17">.|</a></span>
 
<b>by </b><i><a class="ref" href="complex1.html#T132">COMPLEX1:132</a></i>;<br/>
<span class="p1"><a href="csspace.html#K13" target="_self">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">x</font></span>)</span></span><a href="csspace.html#K13" target="_self">.||</a></span> = 
 <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">a</font> <a href="complex1.html#K9">*</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="csspace.html#K12" target="_self">.|.</a> <span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span>
<b>by </b><i/>
<br/>.= 
 <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">a</font> <a href="complex1.html#K9">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="complex1.html#K15">*'</a> </span>)</span> <a href="complex1.html#K9">*</a> <span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="csspace.html#K12" target="_self">.|.</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span>
<b>by </b><i/>
<br/>.= 
 <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"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="complex1.html#K9">*</a> <span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="complex1.html#K15">*'</a> </span>)</span></span>)</span> <a href="complex1.html#K9">*</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="csspace.html#K12" target="_self">.|.</a> <font color="Maroon">x</font></span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span>

<br/>.= 
 <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="complex1.html#K9">*</a> <span class="p4">(<span class="default"><font color="Maroon">a</font> <a href="complex1.html#K15">*'</a> </span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K4">*</a> <span class="p2"><a href="complex1.html#K17">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="csspace.html#K12" target="_self">.|.</a> <font color="Maroon">x</font></span>)</span></span><a href="complex1.html#K17">.|</a></span></span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T151">COMPLEX1:151</a></i>
<br/>.= 
 <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="complex1.html#K9">*</a> <font color="Maroon">a</font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K4">*</a> <span class="p2"><a href="complex1.html#K17">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="csspace.html#K12" target="_self">.|.</a> <font color="Maroon">x</font></span>)</span></span><a href="complex1.html#K17">.|</a></span></span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T155">COMPLEX1:155</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p2"><a href="complex1.html#K17">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="complex1.html#K9">*</a> <font color="Maroon">a</font></span>)</span></span><a href="complex1.html#K17">.|</a></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"><a href="complex1.html#K17">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="csspace.html#K12" target="_self">.|.</a> <font color="Maroon">x</font></span>)</span></span><a href="complex1.html#K17">.|</a></span></span>)</span>
<b>by </b><i>, , <a class="ref" href="square_1.html#T97">SQUARE_1:97</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p2">(<span class="default"><span class="p3"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">a</font></span><a href="complex1.html#K17">.|</a></span> <a href="square_1.html#K5">^2</a> </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"><a href="complex1.html#K17">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="csspace.html#K12" target="_self">.|.</a> <font color="Maroon">x</font></span>)</span></span><a href="complex1.html#K17">.|</a></span></span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T151">COMPLEX1:151</a></i>
<br/>.= 
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">a</font></span><a href="complex1.html#K17">.|</a></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"><a href="complex1.html#K17">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="csspace.html#K12" target="_self">.|.</a> <font color="Maroon">x</font></span>)</span></span><a href="complex1.html#K17">.|</a></span></span>)</span>
<b>by </b><i><a class="ref" href="csspace.html#D6" target="_self">Def6</a>, <a class="ref" href="square_1.html#T89">SQUARE_1:89</a></i>
;<br/>
<b>hence </b><a NAME="E6:68"/>
<span class="p1"><a href="csspace.html#K13" target="_self">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="clvect_1.html#K1">*</a> <font color="Maroon">x</font></span>)</span></span><a href="csspace.html#K13" target="_self">.||</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">a</font></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="csspace.html#K13" target="_self">||.</a><span class="default"><font color="Maroon">x</font></span><a href="csspace.html#K13" target="_self">.||</a></span>
 ;<br/>


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