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<b>let </b><font color="Maroon">c1</font> be  <a href="quaterni.html#V1" target="_self">quaternion</a>  <a href="ordinal1.html#NM1" target="_self">number</a> , <font color="Maroon">c2</font> be  <a href="quaterni.html#V1" target="_self">quaternion</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:29_1_2"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">z</font> <a href="quaterni.html#K7" target="_self">+</a> <font color="Maroon">c1</font> <a href="hidden.html#R1">=</a> 0
 <b>and </b><br/><a NAME="E2:29_1_2"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">z</font> <a href="quaterni.html#K7" target="_self">+</a> <font color="Maroon">c2</font> <a href="hidden.html#R1">=</a> 0
 ;<br/>

<b>consider </b><font color="Maroon">x1</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Maroon">x2</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Maroon">x3</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Maroon">x4</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Maroon">y1</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Maroon">y2</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Maroon">y3</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Maroon">y4</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E4:29_1_2"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="quaterni.html#K6" target="_self">[*</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font>,<font color="Maroon">x3</font>,<font color="Maroon">x4</font></span><a href="quaterni.html#K6" target="_self">*]</a></span>
 <b>and </b><br/><a NAME="E5:29_1_2"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">c1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="quaterni.html#K6" target="_self">[*</a><span class="default"><font color="Maroon">y1</font>,<font color="Maroon">y2</font>,<font color="Maroon">y3</font>,<font color="Maroon">y4</font></span><a href="quaterni.html#K6" target="_self">*]</a></span>
 <b>and </b><br/><a NAME="E6:29_1_2"/><i><font color="Green">E44</font></i>: 
0 <a href="hidden.html#R1">=</a> <span class="p1"><a href="quaterni.html#K6" target="_self">[*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x1</font> <a href="real_1.html#K3">+</a> <font color="Maroon">y1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">x2</font> <a href="real_1.html#K3">+</a> <font color="Maroon">y2</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">x3</font> <a href="real_1.html#K3">+</a> <font color="Maroon">y3</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">x4</font> <a href="real_1.html#K3">+</a> <font color="Maroon">y4</font></span>)</span></span><a href="quaterni.html#K6" target="_self">*]</a></span>
 <b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">x1'</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Maroon">x2'</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Maroon">x3'</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Maroon">x4'</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Maroon">y1'</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Maroon">y2'</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Maroon">y3'</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Maroon">y4'</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E8:29_1_2"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="quaterni.html#K6" target="_self">[*</a><span class="default"><font color="Maroon">x1'</font>,<font color="Maroon">x2'</font>,<font color="Maroon">x3'</font>,<font color="Maroon">x4'</font></span><a href="quaterni.html#K6" target="_self">*]</a></span>
 <b>and </b><br/><a NAME="E9:29_1_2"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">c2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="quaterni.html#K6" target="_self">[*</a><span class="default"><font color="Maroon">y1'</font>,<font color="Maroon">y2'</font>,<font color="Maroon">y3'</font>,<font color="Maroon">y4'</font></span><a href="quaterni.html#K6" target="_self">*]</a></span>
 <b>and </b><br/><a NAME="E10:29_1_2"/><i><font color="Green">E70</font></i>: 
0 <a href="hidden.html#R1">=</a> <span class="p1"><a href="quaterni.html#K6" target="_self">[*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x1'</font> <a href="real_1.html#K3">+</a> <font color="Maroon">y1'</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">x2'</font> <a href="real_1.html#K3">+</a> <font color="Maroon">y2'</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">x3'</font> <a href="real_1.html#K3">+</a> <font color="Maroon">y3'</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">x4'</font> <a href="real_1.html#K3">+</a> <font color="Maroon">y4'</font></span>)</span></span><a href="quaterni.html#K6" target="_self">*]</a></span>
 <b>by </b><i>, </i>;<br/>
<a NAME="E11:29_1_2"/><i><font color="Green">E71</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x1'</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2'</font> &amp; <font color="Maroon">x3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x3'</font> &amp; <font color="Maroon">x4</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x4'</font> )
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E12:29_1_2"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">x1</font> <a href="real_1.html#K3">+</a> <font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E13:29_1_2"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">x1</font> <a href="real_1.html#K3">+</a> <font color="Maroon">y1'</font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i>, , , </i>;<br/>
<a NAME="E14:29_1_2"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">x2</font> <a href="real_1.html#K3">+</a> <font color="Maroon">y2</font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E15:29_1_2"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">x2</font> <a href="real_1.html#K3">+</a> <font color="Maroon">y2'</font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i>, , , </i>;<br/>
<a NAME="E16:29_1_2"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">x3</font> <a href="real_1.html#K3">+</a> <font color="Maroon">y3</font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E17:29_1_2"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">x3</font> <a href="real_1.html#K3">+</a> <font color="Maroon">y3'</font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i>, , , </i>;<br/>
<a NAME="E18:29_1_2"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">x4</font> <a href="real_1.html#K3">+</a> <font color="Maroon">y4</font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E19:29_1_2"/>
<font color="Maroon">x4</font> <a href="real_1.html#K3">+</a> <font color="Maroon">y4'</font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i>, , , </i>;<br/>
<b>hence </b><a NAME="E20:29_1_2"/>
<font color="Maroon">c1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c2</font>
 <b>by </b><i>, , , , <a class="ref" href="quaterni.html#D2" target="_self">Def2</a>, <a class="ref" href="quaterni.html#T1" target="_self">Th1</a>, , , <a class="ref" href="quaterni.html#T2" target="_self">Th2</a></i>;<br/>


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