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<b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="quatern2.html#K3" title="QUATERN2:func.3">|.</a><span class="default"><font color="Maroon" title="c1">z1</font></span><a href="quatern2.html#K3" title="QUATERN2:func.3">.|</a></span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1"><a href="quatern2.html#K3" title="QUATERN2:func.3">|.</a><span class="default"><font color="Maroon" title="c2">z2</font></span><a href="quatern2.html#K3" title="QUATERN2:func.3">.|</a></span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="quatern2.html#K3" title="QUATERN2:func.3">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">z1</font> <a href="quaterni.html#K26" title="QUATERNI:func.26">+</a> <font color="Maroon" title="c2">z2</font></span>)</span></span><a href="quatern2.html#K3" title="QUATERN2:func.3">.|</a></span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2"><a href="quatern2.html#K3" title="QUATERN2:func.3">|.</a><span class="default"><span class="p3">(<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="quatern2.html#K3" title="QUATERN2:func.3">.|</a></span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span><br/>



<a NAME="E1:54"/><i><font color="Green" title="E14">A1</font></i>: 
<span class="p1"><a href="quatern2.html#K3" title="QUATERN2:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">z1</font> <a href="quaterni.html#K26" title="QUATERNI:func.26">+</a> <font color="Maroon" title="c2">z2</font></span>)</span></span><a href="quatern2.html#K3" title="QUATERN2:func.3">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="quatern2.html#K3" title="QUATERN2:func.3">|.</a><span class="default"><font color="Maroon" title="c1">z1</font></span><a href="quatern2.html#K3" title="QUATERN2:func.3">.|</a></span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1"><a href="quatern2.html#K3" title="QUATERN2:func.3">|.</a><span class="default"><font color="Maroon" title="c2">z2</font></span><a href="quatern2.html#K3" title="QUATERN2:func.3">.|</a></span>
 
<b>by </b><i><a class="ref" href="quaterni.html#T79" title="QUATERNI:th.79">QUATERNI:79</a></i>;<br/>
<a NAME="E2:54"/>
<span class="p1"><a href="quatern2.html#K3" title="QUATERN2:func.3">|.</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="quatern2.html#K3" title="QUATERN2:func.3">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="quatern2.html#K3" title="QUATERN2:func.3">|.</a><span class="default"><font color="Maroon" title="c1">z1</font></span><a href="quatern2.html#K3" title="QUATERN2:func.3">.|</a></span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1"><a href="quatern2.html#K3" title="QUATERN2:func.3">|.</a><span class="default"><font color="Maroon" title="c2">z2</font></span><a href="quatern2.html#K3" title="QUATERN2:func.3">.|</a></span>
 
<b>by </b><i><a class="ref" href="quaterni.html#T80" title="QUATERNI:th.80">QUATERNI:80</a></i>;<br/>
<a NAME="E3:54"/><b>then </b>
<span class="p1"><a href="quatern2.html#K3" title="QUATERN2:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">z1</font> <a href="quaterni.html#K26" title="QUATERNI:func.26">+</a> <font color="Maroon" title="c2">z2</font></span>)</span></span><a href="quatern2.html#K3" title="QUATERN2:func.3">.|</a></span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1"><a href="quatern2.html#K3" title="QUATERN2:func.3">|.</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="quatern2.html#K3" title="QUATERN2:func.3">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="quatern2.html#K3" title="QUATERN2:func.3">|.</a><span class="default"><font color="Maroon" title="c1">z1</font></span><a href="quatern2.html#K3" title="QUATERN2:func.3">.|</a></span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2"><a href="quatern2.html#K3" title="QUATERN2:func.3">|.</a><span class="default"><font color="Maroon" title="c2">z2</font></span><a href="quatern2.html#K3" title="QUATERN2:func.3">.|</a></span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2"><a href="quatern2.html#K3" title="QUATERN2:func.3">|.</a><span class="default"><font color="Maroon" title="c1">z1</font></span><a href="quatern2.html#K3" title="QUATERN2:func.3">.|</a></span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2"><a href="quatern2.html#K3" title="QUATERN2:func.3">|.</a><span class="default"><font color="Maroon" title="c2">z2</font></span><a href="quatern2.html#K3" title="QUATERN2:func.3">.|</a></span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E1:54"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/>
<a NAME="E4:54"/><b>then </b>
<span class="p1"><a href="quatern2.html#K3" title="QUATERN2:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">z1</font> <a href="quaterni.html#K26" title="QUATERNI:func.26">+</a> <font color="Maroon" title="c2">z2</font></span>)</span></span><a href="quatern2.html#K3" title="QUATERN2:func.3">.|</a></span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1"><a href="quatern2.html#K3" title="QUATERN2:func.3">|.</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="quatern2.html#K3" title="QUATERN2:func.3">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="quatern2.html#K3" title="QUATERN2:func.3">|.</a><span class="default"><font color="Maroon" title="c1">z1</font></span><a href="quatern2.html#K3" title="QUATERN2:func.3">.|</a></span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2"><a href="quatern2.html#K3" title="QUATERN2:func.3">|.</a><span class="default"><font color="Maroon" title="c2">z2</font></span><a href="quatern2.html#K3" title="QUATERN2:func.3">.|</a></span></span>)</span>
 
;<br/>
<b>hence </b><a NAME="E5:54"/>
<span class="p1"><a href="quatern2.html#K3" title="QUATERN2:func.3">|.</a><span class="default"><font color="Maroon" title="c1">z1</font></span><a href="quatern2.html#K3" title="QUATERN2:func.3">.|</a></span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1"><a href="quatern2.html#K3" title="QUATERN2:func.3">|.</a><span class="default"><font color="Maroon" title="c2">z2</font></span><a href="quatern2.html#K3" title="QUATERN2:func.3">.|</a></span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="quatern2.html#K3" title="QUATERN2:func.3">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">z1</font> <a href="quaterni.html#K26" title="QUATERNI:func.26">+</a> <font color="Maroon" title="c2">z2</font></span>)</span></span><a href="quatern2.html#K3" title="QUATERN2:func.3">.|</a></span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2"><a href="quatern2.html#K3" title="QUATERN2:func.3">|.</a><span class="default"><span class="p3">(<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="quatern2.html#K3" title="QUATERN2:func.3">.|</a></span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T81" title="XREAL_1:th.81">XREAL_1:81</a></i>; <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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