<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">z1</font> be  <a href="quaterni.html#V1" target="_self">quaternion</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">z2</font> be  <a href="quaterni.html#V1" target="_self">quaternion</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>



<b>assume </b><a NAME="E1:81"/><i><font color="Green">E37</font></i>: 
(  <a href="quaterni.html#K17" target="_self">Rea</a> <font color="Maroon">z1</font> <a href="hidden.html#R1">=</a> 0 &amp;  <a href="quaterni.html#K17" target="_self">Rea</a> <font color="Maroon">z2</font> <a href="hidden.html#R1">=</a> 0 )
 ;<br/>

<b>thus </b> <a href="quaterni.html#K17" target="_self">Rea</a> <span class="p1">(<span class="default"><font color="Maroon">z1</font> <a href="quaterni.html#K10" target="_self">*</a> <font color="Maroon">z2</font></span>)</span> = 
<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">z1</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><a href="quaterni.html#K17" target="_self">Rea</a> <font color="Maroon">z2</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</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">z1</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><a href="quaterni.html#K18" target="_self">Im1</a> <font color="Maroon">z2</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</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">z1</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="quaterni.html#K19" target="_self">Im2</a> <font color="Maroon">z2</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</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">z1</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="quaterni.html#K20" target="_self">Im3</a> <font color="Maroon">z2</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<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"><span class="p4">(<span class="default"><a href="quaterni.html#K18" target="_self">Im1</a> <font color="Maroon">z1</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><a href="quaterni.html#K18" target="_self">Im1</a> <font color="Maroon">z2</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</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">z1</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="quaterni.html#K19" target="_self">Im2</a> <font color="Maroon">z2</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</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">z1</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="quaterni.html#K20" target="_self">Im3</a> <font color="Maroon">z2</font></span>)</span></span>)</span>
<b>by </b><i/>
;<br/>

<b>thus </b> <a href="quaterni.html#K18" target="_self">Im1</a> <span class="p1">(<span class="default"><font color="Maroon">z1</font> <a href="quaterni.html#K10" target="_self">*</a> <font color="Maroon">z2</font></span>)</span> = 
<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">z1</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><a href="quaterni.html#K18" target="_self">Im1</a> <font color="Maroon">z2</font></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#K18" target="_self">Im1</a> <font color="Maroon">z1</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><a href="quaterni.html#K17" target="_self">Rea</a> <font color="Maroon">z2</font></span>)</span></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">z1</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="quaterni.html#K20" target="_self">Im3</a> <font color="Maroon">z2</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</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">z1</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="quaterni.html#K19" target="_self">Im2</a> <font color="Maroon">z2</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="quaterni.html#K19" target="_self">Im2</a> <font color="Maroon">z1</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="quaterni.html#K20" target="_self">Im3</a> <font color="Maroon">z2</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</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">z1</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="quaterni.html#K19" target="_self">Im2</a> <font color="Maroon">z2</font></span>)</span></span>)</span>
<b>by </b><i/>
;<br/>

<b>thus </b> <a href="quaterni.html#K19" target="_self">Im2</a> <span class="p1">(<span class="default"><font color="Maroon">z1</font> <a href="quaterni.html#K10" target="_self">*</a> <font color="Maroon">z2</font></span>)</span> = 
<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">z1</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><a href="quaterni.html#K19" target="_self">Im2</a> <font color="Maroon">z2</font></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">z1</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><a href="quaterni.html#K17" target="_self">Rea</a> <font color="Maroon">z2</font></span>)</span></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">z1</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="quaterni.html#K18" target="_self">Im1</a> <font color="Maroon">z2</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="quaterni.html#K18" target="_self">Im1</a> <font color="Maroon">z1</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="quaterni.html#K20" target="_self">Im3</a> <font color="Maroon">z2</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="quaterni.html#K20" target="_self">Im3</a> <font color="Maroon">z1</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="quaterni.html#K18" target="_self">Im1</a> <font color="Maroon">z2</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="quaterni.html#K18" target="_self">Im1</a> <font color="Maroon">z1</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="quaterni.html#K20" target="_self">Im3</a> <font color="Maroon">z2</font></span>)</span></span>)</span>
<b>by </b><i/>
;<br/>

<b>thus </b> <a href="quaterni.html#K20" target="_self">Im3</a> <span class="p1">(<span class="default"><font color="Maroon">z1</font> <a href="quaterni.html#K10" target="_self">*</a> <font color="Maroon">z2</font></span>)</span> = 
<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">z1</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><a href="quaterni.html#K20" target="_self">Im3</a> <font color="Maroon">z2</font></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#K20" target="_self">Im3</a> <font color="Maroon">z1</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><a href="quaterni.html#K17" target="_self">Rea</a> <font color="Maroon">z2</font></span>)</span></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#K18" target="_self">Im1</a> <font color="Maroon">z1</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="quaterni.html#K19" target="_self">Im2</a> <font color="Maroon">z2</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</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">z1</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="quaterni.html#K18" target="_self">Im1</a> <font color="Maroon">z2</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="quaterni.html#K18" target="_self">Im1</a> <font color="Maroon">z1</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="quaterni.html#K19" target="_self">Im2</a> <font color="Maroon">z2</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</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">z1</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="quaterni.html#K18" target="_self">Im1</a> <font color="Maroon">z2</font></span>)</span></span>)</span>
<b>by </b><i/>
;<br/>


</div>
