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<span class="kw">set </span><font color="Maroon" title="c1">z1</font> =  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c2">z2</font> =  <a href="xcmplx_0.html#K1" title="XCMPLX_0:func.1">&lt;i&gt;</a> ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c3">y1</font>, <font color="Maroon" title="c4">y2</font>, <font color="Maroon" title="c5">y3</font>, <font color="Maroon" title="c6">y4</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> <span class="kw"> such that </span><br/><a NAME="E2:84"/><span class="lab"><font color="Green" title="E54">A1</font></span>: 
(  <a href="xcmplx_0.html#K1" title="XCMPLX_0:func.1">&lt;i&gt;</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"><font color="Maroon" title="c3">y1</font>,<font color="Maroon" title="c4">y2</font>,<font color="Maroon" title="c5">y3</font>,<font color="Maroon" title="c6">y4</font></span><a href="quaterni.html#K6" title="QUATERNI:func.6">*]</a></span> &amp; <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="quaterni.html#K25" title="QUATERNI:func.25">*</a> <a href="xcmplx_0.html#K1" title="XCMPLX_0:func.1">&lt;i&gt;</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="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c3">y1</font></span>)</span>,<span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c4">y2</font></span>)</span>,<span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c5">y3</font></span>)</span>,<span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c6">y4</font></span>)</span></span><a href="quaterni.html#K6" title="QUATERNI:func.6">*]</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="quaterni.html#D21" target="_self" title="QUATERNI:def.21">Def21</a></span>;<br/>
<span class="lab"><font color="Green" title="E55">A2</font></span>: <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="quaterni.html#K25" title="QUATERNI:func.25">*</a> <a href="xcmplx_0.html#K1" title="XCMPLX_0:func.1">&lt;i&gt;</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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:84"><span class="lab"><font color="Green" title="E54">A1</font></span></a>, <a class="txt" href="quaterni.html#E15"><span class="lab"><font color="Green" title="E9">Lm4</font></span></a></span>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_0.html#D5" title="ARYTM_0:def.5">ARYTM_0:def 5</a></span>
;<br/>
<span class="kw">consider </span><font color="Maroon" title="c7">y1</font>, <font color="Maroon" title="c8">y2</font>, <font color="Maroon" title="c9">y3</font>, <font color="Maroon" title="c10">y4</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> <span class="kw"> such that </span><br/><a NAME="E5:84"/><span class="lab"><font color="Green" title="E56">A3</font></span>: 
(  <a href="quaterni.html#K11" title="QUATERNI:func.11">&lt;j&gt;</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"><font color="Maroon" title="c7">y1</font>,<font color="Maroon" title="c8">y2</font>,<font color="Maroon" title="c9">y3</font>,<font color="Maroon" title="c10">y4</font></span><a href="quaterni.html#K6" title="QUATERNI:func.6">*]</a></span> &amp; <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="quaterni.html#K25" title="QUATERNI:func.25">*</a> <a href="quaterni.html#K11" title="QUATERNI:func.11">&lt;j&gt;</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="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c7">y1</font></span>)</span>,<span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c8">y2</font></span>)</span>,<span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c9">y3</font></span>)</span>,<span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c10">y4</font></span>)</span></span><a href="quaterni.html#K6" title="QUATERNI:func.6">*]</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="quaterni.html#D21" target="_self" title="QUATERNI:def.21">Def21</a></span>;<br/>
<span class="lab"><font color="Green" title="E57">A4</font></span>: <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="quaterni.html#K25" title="QUATERNI:func.25">*</a> <a href="quaterni.html#K11" title="QUATERNI:func.11">&lt;j&gt;</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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:84"><span class="lab"><font color="Green" title="E56">A3</font></span></a>, <a class="txt" href="quaterni.html#E15"><span class="lab"><font color="Green" title="E9">Lm4</font></span></a></span>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_0.html#D5" title="ARYTM_0:def.5">ARYTM_0:def 5</a></span>
;<br/>
<span class="kw">consider </span><font color="Maroon" title="c11">y1</font>, <font color="Maroon" title="c12">y2</font>, <font color="Maroon" title="c13">y3</font>, <font color="Maroon" title="c14">y4</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> <span class="kw"> such that </span><br/><a NAME="E8:84"/><span class="lab"><font color="Green" title="E58">A5</font></span>: 
(  <a href="quaterni.html#K12" title="QUATERNI:func.12">&lt;k&gt;</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"><font color="Maroon" title="c11">y1</font>,<font color="Maroon" title="c12">y2</font>,<font color="Maroon" title="c13">y3</font>,<font color="Maroon" title="c14">y4</font></span><a href="quaterni.html#K6" title="QUATERNI:func.6">*]</a></span> &amp; <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="quaterni.html#K25" title="QUATERNI:func.25">*</a> <a href="quaterni.html#K12" title="QUATERNI:func.12">&lt;k&gt;</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="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c11">y1</font></span>)</span>,<span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c12">y2</font></span>)</span>,<span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c13">y3</font></span>)</span>,<span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c14">y4</font></span>)</span></span><a href="quaterni.html#K6" title="QUATERNI:func.6">*]</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="quaterni.html#D21" target="_self" title="QUATERNI:def.21">Def21</a></span>;<br/>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="quaterni.html#K25" title="QUATERNI:func.25">*</a> <a href="quaterni.html#K12" title="QUATERNI:func.12">&lt;k&gt;</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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:84"><span class="lab"><font color="Green" title="E58">A5</font></span></a>, <a class="txt" href="quaterni.html#E15"><span class="lab"><font color="Green" title="E9">Lm4</font></span></a></span>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_0.html#D5" title="ARYTM_0:def.5">ARYTM_0:def 5</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E10:84"/>
( <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="quaterni.html#K25" title="QUATERNI:func.25">*</a> <a href="xcmplx_0.html#K1" title="XCMPLX_0:func.1">&lt;i&gt;</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="quaterni.html#K25" title="QUATERNI:func.25">*</a> <a href="quaterni.html#K11" title="QUATERNI:func.11">&lt;j&gt;</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="quaterni.html#K25" title="QUATERNI:func.25">*</a> <a href="quaterni.html#K12" title="QUATERNI:func.12">&lt;k&gt;</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:84"><span class="lab"><font color="Green" title="E55">A2</font></span></a>, <a class="txt" href="#E6:84"><span class="lab"><font color="Green" title="E57">A4</font></span></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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