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<b>let </b><font color="Maroon">a</font> be   <a href="real_1.html#NM1">Real</a>;<br/><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/>





<a NAME="E1:35"/><i><font color="Green">E51</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font> <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="E2:35"/><i><font color="Green">E57</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> <a href="bhsp_1.html#K2" target="_self">.|.</a> <font color="Maroon">x</font>
 
<b>by </b><i/>;<br/>
<span class="p1"><a href="bhsp_1.html#K3" target="_self">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x</font></span>)</span></span><a href="bhsp_1.html#K3" target="_self">.||</a></span> = 
 <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <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> <span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">a</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> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="square_1.html#K5">^2</a> </span>)</span> <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">x</font></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <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">(<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>
<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="complex1.html#K18">abs</a> <font color="Maroon">a</font></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">x</font> <a href="bhsp_1.html#K2" target="_self">.|.</a> <font color="Maroon">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T158">COMPLEX1:158</a></i>
;<br/>
<b>hence </b><a NAME="E4:35"/>
<span class="p1"><a href="bhsp_1.html#K3" target="_self">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x</font></span>)</span></span><a href="bhsp_1.html#K3" target="_self">.||</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">a</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="bhsp_1.html#K3" target="_self">||.</a><span class="default"><font color="Maroon">x</font></span><a href="bhsp_1.html#K3" target="_self">.||</a></span>
 ;<br/>


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