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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="E1:89"/><i><font color="Green">E37</font></i>: 
 <a href="xcmplx_0.html#K1">&lt;i&gt;</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="quaterni.html#K6" target="_self">[*</a><span class="default">0,1,0,0</span><a href="quaterni.html#K6" target="_self">*]</a></span>
 
<b>by </b><i/>;<br/>
<font color="Maroon">z</font> <a href="quaterni.html#K31" target="_self">*'</a>  = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="quaterni.html#K17" target="_self">Rea</a> <font color="Maroon">z</font></span>)</span> <a href="quaterni.html#K23" target="_self">+</a> <span class="p3"><a href="quaterni.html#K6" target="_self">[*</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p0">(<span class="default"><a href="quaterni.html#K18" target="_self">Im1</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> 0</span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p0">(<span class="default"><a href="quaterni.html#K18" target="_self">Im1</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> 1</span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p0">(<span class="default"><a href="quaterni.html#K18" target="_self">Im1</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> 0</span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p0">(<span class="default"><a href="quaterni.html#K18" target="_self">Im1</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> 0</span>)</span></span><a href="quaterni.html#K6" target="_self">*]</a></span></span>)</span> <a href="quaterni.html#K26" target="_self">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p4">(<span class="default"><a href="quaterni.html#K19" target="_self">Im2</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="quaterni.html#K25" target="_self">*</a> <a href="quaterni.html#K11" target="_self">&lt;j&gt;</a> </span>)</span></span>)</span> <a href="quaterni.html#K26" target="_self">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><a href="quaterni.html#K20" target="_self">Im3</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="quaterni.html#K25" target="_self">*</a> <a href="quaterni.html#K12" target="_self">&lt;k&gt;</a> </span>)</span>
<b>by </b><i>, <a class="txt" href="quaterni.html#E8"><i><font color="Green">Lemma45</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="quaterni.html#K17" target="_self">Rea</a> <font color="Maroon">z</font></span>)</span> <a href="quaterni.html#K23" target="_self">+</a> <span class="p3"><a href="quaterni.html#K6" target="_self">[*</a><span class="default">0,<span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p5">(<span class="default"><a href="quaterni.html#K18" target="_self">Im1</a> <font color="Maroon">z</font></span>)</span></span>)</span>,0,0</span><a href="quaterni.html#K6" target="_self">*]</a></span></span>)</span> <a href="quaterni.html#K26" target="_self">+</a> <span class="p2"><a href="quaterni.html#K6" target="_self">[*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p5">(<span class="default"><a href="quaterni.html#K19" target="_self">Im2</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> 0</span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p5">(<span class="default"><a href="quaterni.html#K19" target="_self">Im2</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> 0</span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p5">(<span class="default"><a href="quaterni.html#K19" target="_self">Im2</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> 1</span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p5">(<span class="default"><a href="quaterni.html#K19" target="_self">Im2</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> 0</span>)</span></span><a href="quaterni.html#K6" target="_self">*]</a></span></span>)</span> <a href="quaterni.html#K26" target="_self">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><a href="quaterni.html#K20" target="_self">Im3</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="quaterni.html#K25" target="_self">*</a> <a href="quaterni.html#K12" target="_self">&lt;k&gt;</a> </span>)</span>
<b>by </b><i><a class="txt" href="quaterni.html#E8"><i><font color="Green">Lemma45</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="quaterni.html#K17" target="_self">Rea</a> <font color="Maroon">z</font></span>)</span> <a href="quaterni.html#K23" target="_self">+</a> <span class="p3"><a href="quaterni.html#K6" target="_self">[*</a><span class="default">0,<span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p5">(<span class="default"><a href="quaterni.html#K18" target="_self">Im1</a> <font color="Maroon">z</font></span>)</span></span>)</span>,0,0</span><a href="quaterni.html#K6" target="_self">*]</a></span></span>)</span> <a href="quaterni.html#K26" target="_self">+</a> <span class="p2"><a href="quaterni.html#K6" target="_self">[*</a><span class="default">0,0,<span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p4">(<span class="default"><a href="quaterni.html#K19" target="_self">Im2</a> <font color="Maroon">z</font></span>)</span></span>)</span>,0</span><a href="quaterni.html#K6" target="_self">*]</a></span></span>)</span> <a href="quaterni.html#K26" target="_self">+</a> <span class="p1"><a href="quaterni.html#K6" target="_self">[*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p4">(<span class="default"><a href="quaterni.html#K20" target="_self">Im3</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> 0</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p4">(<span class="default"><a href="quaterni.html#K20" target="_self">Im3</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> 0</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p4">(<span class="default"><a href="quaterni.html#K20" target="_self">Im3</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> 0</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p4">(<span class="default"><a href="quaterni.html#K20" target="_self">Im3</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> 1</span>)</span></span><a href="quaterni.html#K6" target="_self">*]</a></span>
<b>by </b><i><a class="txt" href="quaterni.html#E8"><i><font color="Green">Lemma45</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="quaterni.html#K6" target="_self">[*</a><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="real_1.html#K3">+</a> 0</span>)</span>,<span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p4">(<span class="default"><a href="quaterni.html#K18" target="_self">Im1</a> <font color="Maroon">z</font></span>)</span></span>)</span>,0,0</span><a href="quaterni.html#K6" target="_self">*]</a></span> <a href="quaterni.html#K26" target="_self">+</a> <span class="p2"><a href="quaterni.html#K6" target="_self">[*</a><span class="default">0,0,<span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p4">(<span class="default"><a href="quaterni.html#K19" target="_self">Im2</a> <font color="Maroon">z</font></span>)</span></span>)</span>,0</span><a href="quaterni.html#K6" target="_self">*]</a></span></span>)</span> <a href="quaterni.html#K26" target="_self">+</a> <span class="p1"><a href="quaterni.html#K6" target="_self">[*</a><span class="default">0,0,0,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><a href="quaterni.html#K20" target="_self">Im3</a> <font color="Maroon">z</font></span>)</span></span>)</span></span><a href="quaterni.html#K6" target="_self">*]</a></span>
<b>by </b><i/>
<br/>.= 
<span class="p1"><a href="quaterni.html#K6" target="_self">[*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="quaterni.html#K17" target="_self">Rea</a> <font color="Maroon">z</font></span>)</span> <a href="real_1.html#K3">+</a> 0</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p4">(<span class="default"><a href="quaterni.html#K18" target="_self">Im1</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 0</span>)</span>,<span class="p2">(<span class="default">0 <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p4">(<span class="default"><a href="quaterni.html#K19" target="_self">Im2</a> <font color="Maroon">z</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default">0 <a href="real_1.html#K3">+</a> 0</span>)</span></span><a href="quaterni.html#K6" target="_self">*]</a></span> <a href="quaterni.html#K26" target="_self">+</a> <span class="p1"><a href="quaterni.html#K6" target="_self">[*</a><span class="default">0,0,0,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><a href="quaterni.html#K20" target="_self">Im3</a> <font color="Maroon">z</font></span>)</span></span>)</span></span><a href="quaterni.html#K6" target="_self">*]</a></span>
<b>by </b><i/>
<br/>.= 
<span class="p1"><a href="quaterni.html#K6" target="_self">[*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="quaterni.html#K17" target="_self">Rea</a> <font color="Maroon">z</font></span>)</span> <a href="real_1.html#K3">+</a> 0</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p4">(<span class="default"><a href="quaterni.html#K18" target="_self">Im1</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 0</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p4">(<span class="default"><a href="quaterni.html#K19" target="_self">Im2</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> 0</span>)</span>,<span class="p2">(<span class="default">0 <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p4">(<span class="default"><a href="quaterni.html#K20" target="_self">Im3</a> <font color="Maroon">z</font></span>)</span></span>)</span></span>)</span></span><a href="quaterni.html#K6" target="_self">*]</a></span>
<b>by </b><i/>
;<br/>
<b>hence </b><a NAME="E3:89"/>
<font color="Maroon">z</font> <a href="quaterni.html#K31" target="_self">*'</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="quaterni.html#K6" target="_self">[*</a><span class="default"><span class="p2">(<span class="default"><a href="quaterni.html#K17" target="_self">Rea</a> <font color="Maroon">z</font></span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><a href="quaterni.html#K18" target="_self">Im1</a> <font color="Maroon">z</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><a href="quaterni.html#K19" target="_self">Im2</a> <font color="Maroon">z</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><a href="quaterni.html#K20" target="_self">Im3</a> <font color="Maroon">z</font></span>)</span></span>)</span></span><a href="quaterni.html#K6" target="_self">*]</a></span>
 ;<br/>


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