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<span class="kw">let </span><font color="Maroon" title="c1">z1</font>, <font color="Maroon" title="c2">z2</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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="quaterni.html#K32" title="QUATERNI:func.32">|.</a><span class="default"><font color="Maroon" title="c1">z1</font></span><a href="quaterni.html#K32" title="QUATERNI:func.32">.|</a></span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3"><a href="quaterni.html#K32" title="QUATERNI:func.32">|.</a><span class="default"><font color="Maroon" title="c2">z2</font></span><a href="quaterni.html#K32" title="QUATERNI:func.32">.|</a></span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="quaterni.html#K32" title="QUATERNI:func.32">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">z1</font> <a href="quaterni.html#K29" title="QUATERNI:func.29">-</a> <font color="Maroon" title="c2">z2</font></span>)</span></span><a href="quaterni.html#K32" title="QUATERNI:func.32">.|</a></span></span><br/>



<a NAME="E1:148"/>
<span class="p1"><a href="quaterni.html#K32" title="QUATERNI:func.32">|.</a><span class="default"><font color="Maroon" title="c2">z2</font></span><a href="quaterni.html#K32" title="QUATERNI:func.32">.|</a></span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1"><a href="quaterni.html#K32" title="QUATERNI:func.32">|.</a><span class="default"><font color="Maroon" title="c1">z1</font></span><a href="quaterni.html#K32" title="QUATERNI:func.32">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="quaterni.html#K32" title="QUATERNI:func.32">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">z2</font> <a href="quaterni.html#K29" title="QUATERNI:func.29">-</a> <font color="Maroon" title="c1">z1</font></span>)</span></span><a href="quaterni.html#K32" title="QUATERNI:func.32">.|</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="quaterni.html#T82" target="_self" title="QUATERNI:th.82">Th82</a></span>;<br/>
<a NAME="E2:148"/><span class="kw">then </span>
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><span class="p2"><a href="quaterni.html#K32" title="QUATERNI:func.32">|.</a><span class="default"><font color="Maroon" title="c1">z1</font></span><a href="quaterni.html#K32" title="QUATERNI:func.32">.|</a></span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2"><a href="quaterni.html#K32" title="QUATERNI:func.32">|.</a><span class="default"><font color="Maroon" title="c2">z2</font></span><a href="quaterni.html#K32" title="QUATERNI:func.32">.|</a></span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="quaterni.html#K32" title="QUATERNI:func.32">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">z1</font> <a href="quaterni.html#K29" title="QUATERNI:func.29">-</a> <font color="Maroon" title="c2">z2</font></span>)</span></span><a href="quaterni.html#K32" title="QUATERNI:func.32">.|</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="quaterni.html#T83" target="_self" title="QUATERNI:th.83">Th83</a></span>;<br/>
<a NAME="E3:148"/><span class="kw">then </span><span class="lab"><font color="Green" title="E85">A1</font></span>: 
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1"><a href="quaterni.html#K32" title="QUATERNI:func.32">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">z1</font> <a href="quaterni.html#K29" title="QUATERNI:func.29">-</a> <font color="Maroon" title="c2">z2</font></span>)</span></span><a href="quaterni.html#K32" title="QUATERNI:func.32">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p2">(<span class="default"><span class="p3"><a href="quaterni.html#K32" title="QUATERNI:func.32">|.</a><span class="default"><font color="Maroon" title="c1">z1</font></span><a href="quaterni.html#K32" title="QUATERNI:func.32">.|</a></span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3"><a href="quaterni.html#K32" title="QUATERNI:func.32">|.</a><span class="default"><font color="Maroon" title="c2">z2</font></span><a href="quaterni.html#K32" title="QUATERNI:func.32">.|</a></span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T24" title="XREAL_1:th.24">XREAL_1:24</a></span>;<br/>
<a NAME="E4:148"/>
<span class="p1"><a href="quaterni.html#K32" title="QUATERNI:func.32">|.</a><span class="default"><font color="Maroon" title="c1">z1</font></span><a href="quaterni.html#K32" title="QUATERNI:func.32">.|</a></span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1"><a href="quaterni.html#K32" title="QUATERNI:func.32">|.</a><span class="default"><font color="Maroon" title="c2">z2</font></span><a href="quaterni.html#K32" title="QUATERNI:func.32">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="quaterni.html#K32" title="QUATERNI:func.32">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">z1</font> <a href="quaterni.html#K29" title="QUATERNI:func.29">-</a> <font color="Maroon" title="c2">z2</font></span>)</span></span><a href="quaterni.html#K32" title="QUATERNI:func.32">.|</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="quaterni.html#T82" target="_self" title="QUATERNI:th.82">Th82</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:148"/>
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="quaterni.html#K32" title="QUATERNI:func.32">|.</a><span class="default"><font color="Maroon" title="c1">z1</font></span><a href="quaterni.html#K32" title="QUATERNI:func.32">.|</a></span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3"><a href="quaterni.html#K32" title="QUATERNI:func.32">|.</a><span class="default"><font color="Maroon" title="c2">z2</font></span><a href="quaterni.html#K32" title="QUATERNI:func.32">.|</a></span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="quaterni.html#K32" title="QUATERNI:func.32">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">z1</font> <a href="quaterni.html#K29" title="QUATERNI:func.29">-</a> <font color="Maroon" title="c2">z2</font></span>)</span></span><a href="quaterni.html#K32" title="QUATERNI:func.32">.|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:148"><span class="lab"><font color="Green" title="E85">A1</font></span></a>, <a class="ref" href="absvalue.html#T5" title="ABSVALUE:th.5">ABSVALUE:5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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