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<b>let </b><font color="Maroon" title="c1">z</font> be  <a href="quaterni.html#V1" title="QUATERNI:attr.1">quaternion</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">z</font> <a href="quatern3.html#K7" title="QUATERN3:func.7">^3</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="quaterni.html#K28" title="QUATERNI:func.28">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="quaterni.html#K28" title="QUATERNI:func.28">-</a> <font color="Maroon" title="c1">z</font></span>)</span> <a href="quatern3.html#K8" title="QUATERN3:func.8">^3</a> </span>)</span></span><br/>

<a NAME="E1:106"/><i><font color="Green" title="E23">A1</font></i>: 
<font color="Maroon" title="c1">z</font> <a href="quatern3.html#K7" title="QUATERN3:func.7">^3</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">z</font> <a href="quaterni.html#K27" title="QUATERNI:func.27">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="quatern3.html#K5" title="QUATERN3:func.5">^2</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="quatern2.html#T16" title="QUATERN2:th.16">QUATERN2:16</a></i>;<br/>
<span class="p1">(<span class="default"><a href="quaterni.html#K28" title="QUATERNI:func.28">-</a> <font color="Maroon" title="c1">z</font></span>)</span> <a href="quatern3.html#K8" title="QUATERN3:func.8">^3</a>  = 
<span class="p1">(<span class="default"><a href="quaterni.html#K28" title="QUATERNI:func.28">-</a> <font color="Maroon" title="c1">z</font></span>)</span> <a href="quaterni.html#K27" title="QUATERNI:func.27">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="quaterni.html#K28" title="QUATERNI:func.28">-</a> <font color="Maroon" title="c1">z</font></span>)</span> <a href="quatern3.html#K6" title="QUATERN3:func.6">^2</a> </span>)</span>
<b>by </b><i><a class="ref" href="quatern2.html#T16" title="QUATERN2:th.16">QUATERN2:16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="quaterni.html#K28" title="QUATERNI:func.28">-</a> <font color="Maroon" title="c1">z</font></span>)</span> <a href="quaterni.html#K27" title="QUATERNI:func.27">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="quatern3.html#K5" title="QUATERN3:func.5">^2</a> </span>)</span>
<b>by </b><i><a class="ref" href="quatern3.html#T87" target="_self" title="QUATERN3:th.87">th1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="quaterni.html#K28" title="QUATERNI:func.28">-</a> <a href="quatern2.html#K2" title="QUATERN2:func.2">1q</a> </span>)</span> <a href="quaterni.html#K27" title="QUATERNI:func.27">*</a> <font color="Maroon" title="c1">z</font></span>)</span> <a href="quaterni.html#K27" title="QUATERNI:func.27">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="quatern3.html#K5" title="QUATERN3:func.5">^2</a> </span>)</span>
<b>by </b><i><a class="ref" href="quatern2.html#T19" title="QUATERN2:th.19">QUATERN2:19</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="quaterni.html#K28" title="QUATERNI:func.28">-</a> <a href="quatern2.html#K2" title="QUATERN2:func.2">1q</a> </span>)</span> <a href="quaterni.html#K27" title="QUATERNI:func.27">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="quaterni.html#K27" title="QUATERNI:func.27">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="quatern3.html#K5" title="QUATERN3:func.5">^2</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="quatern2.html#T16" title="QUATERN2:th.16">QUATERN2:16</a></i>
<br/>.= 
 <a href="quaterni.html#K28" title="QUATERNI:func.28">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="quatern3.html#K7" title="QUATERN3:func.7">^3</a> </span>)</span>
<b>by </b><i><a class="txt" href="#E1:106"><i><font color="Green" title="E23">A1</font></i></a>, <a class="ref" href="quatern2.html#T19" title="QUATERN2:th.19">QUATERN2:19</a></i>
;<br/>
<b>hence </b><a NAME="E3:106"/>
<font color="Maroon" title="c1">z</font> <a href="quatern3.html#K7" title="QUATERN3:func.7">^3</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="quaterni.html#K28" title="QUATERNI:func.28">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="quaterni.html#K28" title="QUATERNI:func.28">-</a> <font color="Maroon" title="c1">z</font></span>)</span> <a href="quatern3.html#K8" title="QUATERN3:func.8">^3</a> </span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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