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<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">x</font> be   <a href="real_1.html#NM1">Real</a>;<br/>



<b>assume </b><a NAME="E1:69"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">z1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 ;<br/>

<a NAME="E2:69"/>
(  <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> <a href="quaterni.html#K11" target="_self">&lt;j&gt;</a> </span>)</span> <a href="hidden.html#R1">=</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">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> <a href="quaterni.html#K11" target="_self">&lt;j&gt;</a> </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> <a href="quaterni.html#K11" target="_self">&lt;j&gt;</a> </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> <a href="quaterni.html#K11" target="_self">&lt;j&gt;</a> </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> <a href="quaterni.html#K11" target="_self">&lt;j&gt;</a> </span>)</span></span>)</span> &amp;  <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> <a href="quaterni.html#K11" target="_self">&lt;j&gt;</a> </span>)</span> <a href="hidden.html#R1">=</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">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> <a href="quaterni.html#K11" target="_self">&lt;j&gt;</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#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> <a href="quaterni.html#K11" target="_self">&lt;j&gt;</a> </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> <a href="quaterni.html#K11" target="_self">&lt;j&gt;</a> </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> <a href="quaterni.html#K11" target="_self">&lt;j&gt;</a> </span>)</span></span>)</span> &amp;  <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> <a href="quaterni.html#K11" target="_self">&lt;j&gt;</a> </span>)</span> <a href="hidden.html#R1">=</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">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> <a href="quaterni.html#K11" target="_self">&lt;j&gt;</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">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> <a href="quaterni.html#K11" target="_self">&lt;j&gt;</a> </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> <a href="quaterni.html#K11" target="_self">&lt;j&gt;</a> </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> <a href="quaterni.html#K11" target="_self">&lt;j&gt;</a> </span>)</span></span>)</span> &amp;  <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> <a href="quaterni.html#K11" target="_self">&lt;j&gt;</a> </span>)</span> <a href="hidden.html#R1">=</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">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> <a href="quaterni.html#K11" target="_self">&lt;j&gt;</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#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> <a href="quaterni.html#K11" target="_self">&lt;j&gt;</a> </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> <a href="quaterni.html#K11" target="_self">&lt;j&gt;</a> </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> <a href="quaterni.html#K11" target="_self">&lt;j&gt;</a> </span>)</span></span>)</span> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:69"/>
(  <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> <a href="quaterni.html#K11" target="_self">&lt;j&gt;</a> </span>)</span> <a href="hidden.html#R1">=</a> 0 &amp;  <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> <a href="quaterni.html#K11" target="_self">&lt;j&gt;</a> </span>)</span> <a href="hidden.html#R1">=</a> 0 &amp;  <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> <a href="quaterni.html#K11" target="_self">&lt;j&gt;</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp;  <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> <a href="quaterni.html#K11" target="_self">&lt;j&gt;</a> </span>)</span> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i>, , <a class="ref" href="quaterni.html#T2" target="_self">Th2</a></i>;<br/>


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