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

<a NAME="E1:122"/><i><font color="Green">E37</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="quaterni.html#K17" target="_self">Rea</a> <font color="Maroon">z</font></span>)</span> <a href="square_1.html#K5">^2</a>  &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="quaterni.html#K18" target="_self">Im1</a> <font color="Maroon">z</font></span>)</span> <a href="square_1.html#K5">^2</a>  &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="quaterni.html#K19" target="_self">Im2</a> <font color="Maroon">z</font></span>)</span> <a href="square_1.html#K5">^2</a>  &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="quaterni.html#K20" target="_self">Im3</a> <font color="Maroon">z</font></span>)</span> <a href="square_1.html#K5">^2</a>  )
 
<b>by </b><i><a class="ref" href="square_1.html#T72">SQUARE_1:72</a></i>;<br/>
<a NAME="E2:122"/><i><font color="Green">E40</font></i>: 
<span class="p1">(<span class="default"><a href="quaterni.html#K19" target="_self">Im2</a> <font color="Maroon">z</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"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="quaterni.html#K17" target="_self">Rea</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="p3">(<span class="default"><span class="p4">(<span class="default"><a href="quaterni.html#K18" target="_self">Im1</a> <font color="Maroon">z</font></span>)</span> <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"><span class="p3">(<span class="default"><a href="quaterni.html#K19" target="_self">Im2</a> <font color="Maroon">z</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="quaterni.html#K20" target="_self">Im3</a> <font color="Maroon">z</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="quaterni.html#T9" target="_self">Th9</a></i>;<br/>
<a NAME="E3:122"/><i><font color="Green">E41</font></i>: 
 <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="quaterni.html#K19" target="_self">Im2</a> <font color="Maroon">z</font></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"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="quaterni.html#K17" target="_self">Rea</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="p4">(<span class="default"><span class="p5">(<span class="default"><a href="quaterni.html#K18" target="_self">Im1</a> <font color="Maroon">z</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="quaterni.html#K19" target="_self">Im2</a> <font color="Maroon">z</font></span>)</span> <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"><span class="p3">(<span class="default"><a href="quaterni.html#K20" target="_self">Im3</a> <font color="Maroon">z</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="square_1.html#T94">SQUARE_1:94</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="122_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:122_1"/>
(  <a href="quaterni.html#K19" target="_self">Im2</a> <font color="Maroon">z</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 or  <a href="quaterni.html#K19" target="_self">Im2</a> <font color="Maroon">z</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="122_1_1"><b>suppose </b></a><a NAME="E1:122_1_1"/>
 <a href="quaterni.html#K19" target="_self">Im2</a> <font color="Maroon">z</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:122_1_1"/>
 <a href="quaterni.html#K19" target="_self">Im2</a> <font color="Maroon">z</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="quaterni.html#K32" target="_self">|.</a><span class="default"><font color="Maroon">z</font></span><a href="quaterni.html#K32" target="_self">.|</a></span>
 <b>by </b><i>, <a class="ref" href="square_1.html#T89">SQUARE_1:89</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="122_1_2"><b>suppose </b></a><a NAME="E1:122_1_2"/><i><font color="Green">E42</font></i>: 
 <a href="quaterni.html#K19" target="_self">Im2</a> <font color="Maroon">z</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 ;<br/><div class="add"><a NAME="E2:122_1_2"/><b>then </b><i><font color="Green">E43</font></i>: 
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><a href="quaterni.html#K19" target="_self">Im2</a> <font color="Maroon">z</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="quaterni.html#K32" target="_self">|.</a><span class="default"><font color="Maroon">z</font></span><a href="quaterni.html#K32" target="_self">.|</a></span>
 <b>by </b><i>, <a class="ref" href="square_1.html#T90">SQUARE_1:90</a></i>;<br/><a NAME="E3:122_1_2"/>
0 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="quaterni.html#K19" target="_self">Im2</a> <font color="Maroon">z</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T71">XREAL_1:71</a></i>;<br/><b>hence </b><a NAME="E4:122_1_2"/>
 <a href="quaterni.html#K19" target="_self">Im2</a> <font color="Maroon">z</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="quaterni.html#K32" target="_self">|.</a><span class="default"><font color="Maroon">z</font></span><a href="quaterni.html#K32" target="_self">.|</a></span>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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