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<span class="kw">let </span><font color="Maroon" title="c1">r</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"> <a href="quatern2.html#K1" title="QUATERN2:func.1">0q</a>  <a href="quaterni.html#K27" title="QUATERNI:func.27">*</a> <font color="Maroon" title="c1">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</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>, <font color="Maroon" title="c6">y1</font>, <font color="Maroon" title="c7">y2</font>, <font color="Maroon" title="c8">y3</font>, <font color="Maroon" title="c9">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="E2:20"/><span class="lab"><font color="Green" title="E14">A1</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</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="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">and </span><br/><a NAME="E3:20"/>
<font color="Maroon" title="c1">r</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="c6">y1</font>,<font color="Maroon" title="c7">y2</font>,<font color="Maroon" title="c8">y3</font>,<font color="Maroon" title="c9">y4</font></span><a href="quaterni.html#K6" title="QUATERNI:func.6">*]</a></span>
 <span class="kw">and </span><br/><a NAME="E4:20"/><span class="lab"><font color="Green" title="E15">A2</font></span>: 
<a href="quatern2.html#K1" title="QUATERN2:func.1">0q</a>  <a href="quaterni.html#K27" title="QUATERNI:func.27">*</a> <font color="Maroon" title="c1">r</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"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<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="c6">y1</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p5">(<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="c7">y2</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="c4">x3</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c8">y3</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<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="c9">y4</font></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"><font color="Maroon" title="c2">x1</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c7">y2</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p5">(<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="c6">y1</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="c4">x3</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c9">y4</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<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="c8">y3</font></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"><font color="Maroon" title="c2">x1</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c8">y3</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p5">(<span class="default"><font color="Maroon" title="c6">y1</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="c7">y2</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="p3">(<span class="default"><font color="Maroon" title="c9">y4</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c3">x2</font></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"><font color="Maroon" title="c2">x1</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c9">y4</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p5">(<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="c6">y1</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="c3">x2</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c8">y3</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<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="c7">y2</font></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="ref" href="quaterni.html#D10" title="QUATERNI:def.10">QUATERNI:def 10</a></span>;<br/>
<a NAME="E5:20"/><span class="lab"><font color="Green" title="E16">A3</font></span>: 
<font color="Maroon" title="c2">x1</font> <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="#E2:20"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="ref" href="quatern2.html#T11" target="_self" title="QUATERN2:th.11">Th11</a>, <a class="ref" href="quaterni.html#T12" title="QUATERNI:th.12">QUATERNI:12</a></span>;<br/>
<a NAME="E6:20"/><span class="lab"><font color="Green" title="E17">A4</font></span>: 
<font color="Maroon" title="c3">x2</font> <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="#E2:20"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="ref" href="quatern2.html#T11" target="_self" title="QUATERN2:th.11">Th11</a>, <a class="ref" href="quaterni.html#T12" title="QUATERNI:th.12">QUATERNI:12</a></span>;<br/>
<a NAME="E7:20"/><span class="lab"><font color="Green" title="E18">A5</font></span>: 
<font color="Maroon" title="c4">x3</font> <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="#E2:20"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="ref" href="quatern2.html#T11" target="_self" title="QUATERN2:th.11">Th11</a>, <a class="ref" href="quaterni.html#T12" title="QUATERNI:th.12">QUATERNI:12</a></span>;<br/>
<a NAME="E8:20"/>
<font color="Maroon" title="c5">x4</font> <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="#E2:20"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="ref" href="quatern2.html#T11" target="_self" title="QUATERN2:th.11">Th11</a>, <a class="ref" href="quaterni.html#T12" title="QUATERNI:th.12">QUATERNI:12</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:20"/>
<a href="quatern2.html#K1" title="QUATERN2:func.1">0q</a>  <a href="quaterni.html#K27" title="QUATERNI:func.27">*</a> <font color="Maroon" title="c1">r</font> <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="#E4:20"><span class="lab"><font color="Green" title="E15">A2</font></span></a>, <a class="txt" href="#E5:20"><span class="lab"><font color="Green" title="E16">A3</font></span></a>, <a class="txt" href="#E6:20"><span class="lab"><font color="Green" title="E17">A4</font></span></a>, <a class="txt" href="#E7:20"><span class="lab"><font color="Green" title="E18">A5</font></span></a>, <a class="ref" href="quatern2.html#T11" target="_self" title="QUATERN2:th.11">Th11</a></span>; <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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