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<span class="kw">let </span><font color="Maroon" title="c1">c</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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="quatern2.html#K1" title="QUATERN2:func.1">0q</a>  implies <font color="Maroon" title="c1">c</font> <a href="quatern2.html#K6" title="QUATERN2:func.6">/</a> <font color="Maroon" title="c1">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="quatern2.html#K2" title="QUATERN2:func.2">1q</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:46"/><span class="lab"><font color="Green" title="E29">A1</font></span>: 
<font color="Maroon" title="c1">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="quatern2.html#K1" title="QUATERN2:func.1">0q</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">c</font> <a href="quatern2.html#K6" title="QUATERN2:func.6">/</a> <font color="Maroon" title="c1">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="quatern2.html#K2" title="QUATERN2:func.2">1q</a> </span><br/>

<span class="kw">consider </span><font color="Maroon" title="c2">x1</font>, <font color="Maroon" title="c3">x2</font>, <font color="Maroon" title="c4">x3</font>, <font color="Maroon" title="c5">x4</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="E3:46"/><span class="lab"><font color="Green" title="E30">A2</font></span>: 
<font color="Maroon" title="c1">c</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">x1</font>,<font color="Maroon" title="c3">x2</font>,<font color="Maroon" title="c4">x3</font>,<font color="Maroon" title="c5">x4</font></span><a href="quaterni.html#K6" title="QUATERNI:func.6">*]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="quatern2.html#E1"><span class="lab"><font color="Green" title="E1">Lm1</font></span></a></span>;<br/>
<a NAME="E4:46"/>
<span class="p1"><a href="quatern2.html#K3" title="QUATERN2:func.3">|.</a><span class="default"><font color="Maroon" title="c1">c</font></span><a href="quatern2.html#K3" title="QUATERN2:func.3">.|</a></span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:46"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="ref" href="quatern2.html#T10" target="_self" title="QUATERN2:th.10">Th10</a></span>;<br/>
<a NAME="E5:46"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A3</font></span>: 
<span class="p1"><a href="quatern2.html#K3" title="QUATERN2:func.3">|.</a><span class="default"><font color="Maroon" title="c1">c</font></span><a href="quatern2.html#K3" title="QUATERN2:func.3">.|</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="square_1.html#T74" title="SQUARE_1:th.74">SQUARE_1:74</a></span>;<br/>
<a NAME="E6:46"/><span class="lab"><font color="Green" title="E32">A4</font></span>: 
 <a href="quaterni.html#K17" title="QUATERNI:func.17">Rea</a> <font color="Maroon" title="c1">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:46"><span class="lab"><font color="Green" title="E30">A2</font></span></a>, <a class="ref" href="quaterni.html#T23" title="QUATERNI:th.23">QUATERNI:23</a></span>;<br/>
<a NAME="E7:46"/><span class="lab"><font color="Green" title="E33">A5</font></span>: 
 <a href="quaterni.html#K18" title="QUATERNI:func.18">Im1</a> <font color="Maroon" title="c1">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:46"><span class="lab"><font color="Green" title="E30">A2</font></span></a>, <a class="ref" href="quaterni.html#T23" title="QUATERNI:th.23">QUATERNI:23</a></span>;<br/>
<a NAME="E8:46"/><span class="lab"><font color="Green" title="E34">A6</font></span>: 
 <a href="quaterni.html#K19" title="QUATERNI:func.19">Im2</a> <font color="Maroon" title="c1">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:46"><span class="lab"><font color="Green" title="E30">A2</font></span></a>, <a class="ref" href="quaterni.html#T23" title="QUATERNI:th.23">QUATERNI:23</a></span>;<br/>
<a NAME="E9:46"/><span class="lab"><font color="Green" title="E35">A7</font></span>: 
 <a href="quaterni.html#K20" title="QUATERNI:func.20">Im3</a> <font color="Maroon" title="c1">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x4</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:46"><span class="lab"><font color="Green" title="E30">A2</font></span></a>, <a class="ref" href="quaterni.html#T23" title="QUATERNI:th.23">QUATERNI:23</a></span>;<br/>
<a NAME="E10:46"/><span class="lab"><font color="Green" title="E36">A8</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">x1</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">x2</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">x3</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">x4</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="quatern2.html#E2"><span class="lab"><font color="Green" title="E2">Lm2</font></span></a></span>;<br/>
<font color="Maroon" title="c1">c</font> <a href="quatern2.html#K6" title="QUATERN2:func.6">/</a> <font color="Maroon" title="c1">c</font> = 
<span class="p1"><a href="quaterni.html#K6" title="QUATERNI:func.6">[*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c2">x1</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c2">x1</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p0">(<span class="default"><font color="Maroon" title="c3">x2</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c3">x2</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">x3</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c4">x3</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">x4</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c5">x4</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p3">(<span class="default"><span class="p4"><a href="quatern2.html#K3" title="QUATERN2:func.3">|.</a><span class="default"><font color="Maroon" title="c1">c</font></span><a href="quatern2.html#K3" title="QUATERN2:func.3">.|</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c2">x1</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c3">x2</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c3">x2</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c2">x1</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">x3</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c5">x4</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">x4</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c4">x3</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p3">(<span class="default"><span class="p4"><a href="quatern2.html#K3" title="QUATERN2:func.3">|.</a><span class="default"><font color="Maroon" title="c1">c</font></span><a href="quatern2.html#K3" title="QUATERN2:func.3">.|</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c2">x1</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c4">x3</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p0">(<span class="default"><font color="Maroon" title="c3">x2</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c5">x4</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">x3</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c2">x1</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">x4</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c3">x2</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p3">(<span class="default"><span class="p4"><a href="quatern2.html#K3" title="QUATERN2:func.3">|.</a><span class="default"><font color="Maroon" title="c1">c</font></span><a href="quatern2.html#K3" title="QUATERN2:func.3">.|</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c2">x1</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c5">x4</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p0">(<span class="default"><font color="Maroon" title="c3">x2</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c4">x3</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">x3</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c3">x2</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">x4</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c2">x1</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p3">(<span class="default"><span class="p4"><a href="quatern2.html#K3" title="QUATERN2:func.3">|.</a><span class="default"><font color="Maroon" title="c1">c</font></span><a href="quatern2.html#K3" title="QUATERN2:func.3">.|</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span></span><a href="quaterni.html#K6" title="QUATERNI:func.6">*]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:46"><span class="lab"><font color="Green" title="E30">A2</font></span></a>, <a class="ref" href="quatern2.html#D1" target="_self" title="QUATERN2:def.1">Def1</a></span>
<br/>.= 
<span class="p1"><a href="quaterni.html#K6" title="QUATERNI:func.6">[*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="quatern2.html#K3" title="QUATERN2:func.3">|.</a><span class="default"><font color="Maroon" title="c1">c</font></span><a href="quatern2.html#K3" title="QUATERN2:func.3">.|</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p3">(<span class="default"><span class="p4"><a href="quatern2.html#K3" title="QUATERN2:func.3">|.</a><span class="default"><font color="Maroon" title="c1">c</font></span><a href="quatern2.html#K3" title="QUATERN2:func.3">.|</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span>,<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:46"><span class="lab"><font color="Green" title="E32">A4</font></span></a>, <a class="txt" href="#E7:46"><span class="lab"><font color="Green" title="E33">A5</font></span></a>, <a class="txt" href="#E8:46"><span class="lab"><font color="Green" title="E34">A6</font></span></a>, <a class="txt" href="#E9:46"><span class="lab"><font color="Green" title="E35">A7</font></span></a>, <a class="txt" href="#E10:46"><span class="lab"><font color="Green" title="E36">A8</font></span></a>, <a class="ref" href="square_1.html#D4" title="SQUARE_1:def.4">SQUARE_1:def 4</a></span>
<br/>.= 
<span class="p1"><a href="quaterni.html#K6" title="QUATERNI:func.6">[*</a><span class="default">1,<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:46"><span class="lab"><font color="Green" title="E31">A3</font></span></a>, <a class="ref" href="xcmplx_1.html#T60" title="XCMPLX_1:th.60">XCMPLX_1:60</a></span>
<br/>.= 
<span class="p1"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default">1,<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="ref" href="quaterni.html#T91" title="QUATERNI:th.91">QUATERNI:91</a></span>
<br/>.= 
1
<span class="kw">by</span> <span class="lab"><a class="ref" href="arytm_0.html#D7" title="ARYTM_0:def.7">ARYTM_0:def 7</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E12:46"/>
<font color="Maroon" title="c1">c</font> <a href="quatern2.html#K6" title="QUATERN2:func.6">/</a> <font color="Maroon" title="c1">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="quatern2.html#K2" title="QUATERN2:func.2">1q</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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