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

<a NAME="E1:50"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="quaterni.html#K1" target="_self">QUATERNION</a> 
 
<b>by </b><i/>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:50_1"/>
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K2">COMPLEX</a>  or  not <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K2">COMPLEX</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1_1"><b>suppose </b></a><a NAME="E1:50_1_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K2">COMPLEX</a> 
 ;<br/><div class="add"><a NAME="E2:50_1_1"/><b>then </b><i><font color="Green">E41</font></i>: 
(  <a href="quaterni.html#K19" target="_self">Im2</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> 0 &amp;  <a href="quaterni.html#K20" target="_self">Im3</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i>, </i>;<br/><b>consider </b><font color="Maroon">z'</font> being  <a href="xcmplx_0.html#V1">complex</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E4:50_1_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z'</font>
 <b>and </b><br/><a NAME="E5:50_1_1"/><i><font color="Green">E43</font></i>: 
 <a href="quaterni.html#K17" target="_self">Rea</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K3">Re</a> <font color="Maroon">z'</font>
 <b>by </b><i><a class="ref" href="quaterni.html#D2" target="_self">Def2</a>, </i>;<br/><b>consider </b><font color="Maroon">z'</font> being  <a href="xcmplx_0.html#V1">complex</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E7:50_1_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z'</font>
 <b>and </b><br/><a NAME="E8:50_1_1"/><i><font color="Green">E68</font></i>: 
 <a href="quaterni.html#K18" target="_self">Im1</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K4">Im</a> <font color="Maroon">z'</font>
 <b>by </b><i><a class="ref" href="quaterni.html#D2" target="_self">Def2</a>, </i>;<br/><a NAME="E9:50_1_1"/>
<span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p2">(<span class="default"><a href="quaterni.html#K17" target="_self">Rea</a> <font color="Maroon">z</font></span>)</span>,<span class="p2">(<span class="default"><a href="quaterni.html#K18" target="_self">Im1</a> <font color="Maroon">z</font></span>)</span></span><a href="arytm_0.html#K5">*]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">z'</font>
 <b>by </b><i>, , , <a class="ref" href="quaterni.html#T2" target="_self">Th2</a>, </i>;<br/><b>hence </b><a NAME="E10:50_1_1"/>
<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"><span class="p2">(<span class="default"><a href="quaterni.html#K17" target="_self">Rea</a> <font color="Maroon">z</font></span>)</span>,<span class="p2">(<span class="default"><a href="quaterni.html#K18" target="_self">Im1</a> <font color="Maroon">z</font></span>)</span>,<span class="p2">(<span class="default"><a href="quaterni.html#K19" target="_self">Im2</a> <font color="Maroon">z</font></span>)</span>,<span class="p2">(<span class="default"><a href="quaterni.html#K20" target="_self">Im3</a> <font color="Maroon">z</font></span>)</span></span><a href="quaterni.html#K6" target="_self">*]</a></span>
 <b>by </b><i><a class="ref" href="quaterni.html#T2" target="_self">Th2</a>, <a class="ref" href="quaterni.html#T1" target="_self">Th1</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1_2"><b>suppose </b></a><a NAME="E1:50_1_2"/><i><font color="Green">E69</font></i>: 
not <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K2">COMPLEX</a> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">f</font> being   <a href="funct_2.html#NM1">Function</a> of 4, <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E3:50_1_2"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E4:50_1_2"/><i><font color="Green">E71</font></i>: 
 <a href="quaterni.html#K18" target="_self">Im1</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> 1
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">f</font> being   <a href="funct_2.html#NM1">Function</a> of 4, <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E6:50_1_2"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E7:50_1_2"/><i><font color="Green">E73</font></i>: 
 <a href="quaterni.html#K17" target="_self">Rea</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> 0
 <b>by </b><i><a class="ref" href="quaterni.html#D2" target="_self">Def2</a>, </i>;<br/><b>consider </b><font color="Maroon">f</font> being   <a href="funct_2.html#NM1">Function</a> of 4, <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E9:50_1_2"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E10:50_1_2"/><i><font color="Green">E76</font></i>: 
 <a href="quaterni.html#K19" target="_self">Im2</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> 2
 <b>by </b><i><a class="ref" href="quaterni.html#D2" target="_self">Def2</a>, </i>;<br/><b>consider </b><font color="Maroon">f</font> being   <a href="funct_2.html#NM1">Function</a> of 4, <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E12:50_1_2"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E13:50_1_2"/><i><font color="Green">E78</font></i>: 
 <a href="quaterni.html#K20" target="_self">Im3</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> 3
 <b>by </b><i><a class="ref" href="quaterni.html#D2" target="_self">Def2</a>, </i>;<br/><b>consider </b><font color="Maroon">a</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Maroon">b</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Maroon">c</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Maroon">d</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="E15:50_1_2"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> 0,1,2,3 <a href="quaterni.html#K3" target="_self">--&gt;</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">d</font>
 <b>by </b><i><a class="ref" href="quaterni.html#T1" target="_self">Th1</a>, </i>;<br/><a NAME="E16:50_1_2"/><i><font color="Green">E80</font></i>: 
 <a href="quaterni.html#K17" target="_self">Rea</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 <b>by </b><i>, <a class="ref" href="quaterni.html#T2" target="_self">Th2</a>, , , </i>;<br/><a NAME="E17:50_1_2"/><i><font color="Green">E81</font></i>: 
 <a href="quaterni.html#K18" target="_self">Im1</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="quaterni.html#T1" target="_self">Th1</a>, , , , </i>;<br/><a NAME="E18:50_1_2"/><i><font color="Green">E82</font></i>: 
 <a href="quaterni.html#K19" target="_self">Im2</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font>
 <b>by </b><i>, <a class="txt" href="quaterni.html#E1:50"><i><font color="Green">E37</font></i></a>, , , </i>;<br/><a NAME="E19:50_1_2"/><i><font color="Green">E83</font></i>: 
 <a href="quaterni.html#K20" target="_self">Im3</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font>
 <b>by </b><i><a class="txt" href="quaterni.html#E6"><i><font color="Green">Lemma38</font></i></a>, <a class="ref" href="quaterni.html#T3" target="_self">Th3</a>, , , </i>;<br/><a NAME="E20:50_1_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="funct_2.html#K1">Funcs</a> 4,<a href="numbers.html#K1">REAL</a> </span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="funct_2.html#K1">Funcs</a> 4,<a href="numbers.html#K1">REAL</a>  : ( <font color="Olive">x</font> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R1">=</a> 0 &amp; <font color="Olive">x</font> <a href="funct_1.html#K1">.</a> 3 <a href="hidden.html#R1">=</a> 0 ) </span>}</span> 
 <b>by </b><i>, <a class="ref" href="quaterni.html#D2" target="_self">Def2</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E21:50_1_2"/><b>then </b><i><font color="Green">E84</font></i>: 
not <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="funct_2.html#K1">Funcs</a> 4,<a href="numbers.html#K1">REAL</a>  : ( <font color="Olive">x</font> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R1">=</a> 0 &amp; <font color="Olive">x</font> <a href="funct_1.html#K1">.</a> 3 <a href="hidden.html#R1">=</a> 0 ) </span>}</span> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><b>reconsider </b><font color="Maroon">f</font> = <font color="Maroon">z</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="funct_2.html#K1">Funcs</a> 4,<a href="numbers.html#K1">REAL</a>  <b>by </b><i>, <a class="ref" href="card_1.html#T90">CARD_1:90</a>, <a class="ref" href="funct_2.html#T11">FUNCT_2:11</a></i>;<br/><a NAME="E23:50_1_2"/>
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 or <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 3 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 <b>by </b><i><a class="txt" href="quaterni.html#E8"><i><font color="Green">Lemma45</font></i></a></i>;<br/><a NAME="E24:50_1_2"/><b>then </b>
( <font color="Maroon">c</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 or <font color="Maroon">d</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E25:50_1_2"/>
<span class="p1"><a href="quaterni.html#K6" target="_self">[*</a><span class="default"><span class="p2">(<span class="default"><a href="quaterni.html#K17" target="_self">Rea</a> <font color="Maroon">z</font></span>)</span>,<span class="p2">(<span class="default"><a href="quaterni.html#K18" target="_self">Im1</a> <font color="Maroon">z</font></span>)</span>,<span class="p2">(<span class="default"><a href="quaterni.html#K19" target="_self">Im2</a> <font color="Maroon">z</font></span>)</span>,<span class="p2">(<span class="default"><a href="quaterni.html#K20" target="_self">Im3</a> <font color="Maroon">z</font></span>)</span></span><a href="quaterni.html#K6" target="_self">*]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font>
 <b>by </b><i>, , , , , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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