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<b>let </b><font color="Maroon" title="c1">z1</font>, <font color="Maroon" title="c2">z2</font>, <font color="Maroon" title="c3">z3</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"><span class="p2">(<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> <a href="quaterni.html#K29" title="QUATERNI:func.29">-</a> <font color="Maroon" title="c3">z3</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"><a href="quatern2.html#K3" title="QUATERN2:func.3">|.</a><span class="default"><font color="Maroon" title="c3">z3</font></span><a href="quatern2.html#K3" title="QUATERN2:func.3">.|</a></span></span><br/>





<a NAME="E1:46"/><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"><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> <a href="quaterni.html#K29" title="QUATERNI:func.29">-</a> <font color="Maroon" title="c3">z3</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"><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="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="c3">z3</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="E2:46"/>
<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="E3:46"/><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"><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> <a href="quaterni.html#K29" title="QUATERNI:func.29">-</a> <font color="Maroon" title="c3">z3</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"><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> <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="c3">z3</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="#E1:46"><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="E4:46"/><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"><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> <a href="quaterni.html#K29" title="QUATERNI:func.29">-</a> <font color="Maroon" title="c3">z3</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">(<span class="default"><span class="p3"><a href="quatern2.html#K3" title="QUATERN2:func.3">|.</a><span class="default"><span class="p4">(<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="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3"><a href="quatern2.html#K3" title="QUATERN2:func.3">|.</a><span class="default"><font color="Maroon" title="c3">z3</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="p2">(<span class="default"><span class="p3"><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="p3"><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></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</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>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></i>;<br/>
<b>hence </b><a NAME="E5:46"/>
<span class="p1"><a href="quatern2.html#K3" title="QUATERN2:func.3">|.</a><span class="default"><span class="p2">(<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> <a href="quaterni.html#K29" title="QUATERNI:func.29">-</a> <font color="Maroon" title="c3">z3</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"><a href="quatern2.html#K3" title="QUATERN2:func.3">|.</a><span class="default"><font color="Maroon" title="c3">z3</font></span><a href="quatern2.html#K3" title="QUATERN2:func.3">.|</a></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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