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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="quaterni.html#K29" title="QUATERNI:func.29">-</a> <a href="quatern2.html#K1" title="QUATERN2:func.1">0q</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">z</font></span><br/>

 <a href="quatern2.html#K1" title="QUATERN2:func.1">0q</a>  = 
<span class="p1"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span>
<b>by </b><i><a class="ref" href="arytm_0.html#D7" title="ARYTM_0:def.7">ARYTM_0:def 7</a></i>
<br/>.= 
<span class="p1"><a href="quaterni.html#K6" title="QUATERNI:func.6">[*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="quaterni.html#K6" title="QUATERNI:func.6">*]</a></span>
<b>by </b><i><a class="ref" href="quaterni.html#D6" title="QUATERNI:def.6">QUATERNI:def 6</a></i>
;<br/>
<a NAME="E2:9"/><b>then </b><i><font color="Green" title="E3">A1</font></i>: 
 <a href="quaterni.html#K28" title="QUATERNI:func.28">-</a> <a href="quatern2.html#K1" title="QUATERN2:func.1">0q</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="quaterni.html#K6" title="QUATERNI:func.6">[*</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span><a href="quaterni.html#K6" title="QUATERNI:func.6">*]</a></span>
 
<b>by </b><i><a class="ref" href="quatern2.html#T4" title="QUATERN2:th.4">QUATERN2:4</a></i>;<br/>
<a NAME="E3:9"/>
<font color="Maroon" title="c1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="quaterni.html#K1" title="QUATERNI:func.1">QUATERNION</a> 
 
<b>by </b><i><a class="ref" href="quaterni.html#D2" title="QUATERNI:def.2">QUATERNI:def 2</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c3">y</font>, <font color="Maroon" title="c4">w</font>, <font color="Maroon" title="c5">v</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <b> such that </b><br/><a NAME="E5:9"/><i><font color="Green" title="E4">A2</font></i>: 
<font color="Maroon" title="c1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="quaterni.html#K6" title="QUATERNI:func.6">[*</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c4">w</font>,<font color="Maroon" title="c5">v</font></span><a href="quaterni.html#K6" title="QUATERNI:func.6">*]</a></span>
 <b>by </b><i><a class="ref" href="quaterni.html#T7" title="QUATERNI:th.7">QUATERNI:7</a></i>;<br/>
<font color="Maroon" title="c1">z</font> <a href="quaterni.html#K29" title="QUATERNI:func.29">-</a> <a href="quatern2.html#K1" title="QUATERN2:func.1">0q</a>  = 
<span class="p1"><a href="quaterni.html#K6" title="QUATERNI:func.6">[*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">y</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">w</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span><a href="quaterni.html#K6" title="QUATERNI:func.6">*]</a></span>
<b>by </b><i><a class="txt" href="#E2:9"><i><font color="Green" title="E3">A1</font></i></a>, <a class="txt" href="#E5:9"><i><font color="Green" title="E4">A2</font></i></a>, <a class="ref" href="quaterni.html#D7" title="QUATERNI:def.7">QUATERNI:def 7</a></i>
<br/>.= 
<span class="p1"><a href="quaterni.html#K6" title="QUATERNI:func.6">[*</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c4">w</font>,<font color="Maroon" title="c5">v</font></span><a href="quaterni.html#K6" title="QUATERNI:func.6">*]</a></span>

;<br/>
<b>hence </b><a NAME="E7:9"/>
<font color="Maroon" title="c1">z</font> <a href="quaterni.html#K29" title="QUATERNI:func.29">-</a> <a href="quatern2.html#K1" title="QUATERN2:func.1">0q</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">z</font>
 <b>by </b><i><a class="txt" href="#E5:9"><i><font color="Green" title="E4">A2</font></i></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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