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<b>let </b><font color="Maroon">z</font> be  <a href="xcmplx_0.html#V1">complex</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>assume </b><a NAME="E1:73"/>
<span class="p1"><a href="complex1.html#K17" target="_self">|.</a><span class="default"><font color="Maroon">z</font></span><a href="complex1.html#K17" target="_self">.|</a></span> <a href="hidden.html#R1">=</a> 0
 ;<br/>

<a NAME="E2:73"/><b>then </b>
(  <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K3" target="_self">Re</a> <font color="Maroon">z</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K4" target="_self">Im</a> <font color="Maroon">z</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0 &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K3" target="_self">Re</a> <font color="Maroon">z</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K4" target="_self">Im</a> <font color="Maroon">z</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> )
 
<b>by </b><i/>;<br/>
<a NAME="E3:73"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K3" target="_self">Re</a> <font color="Maroon">z</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K4" target="_self">Im</a> <font color="Maroon">z</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="square_1.html#T92">SQUARE_1:92</a></i>;<br/>
<b>hence </b><a NAME="E4:73"/>
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i/>;<br/>


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