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<b>let </b><font color="Maroon">m</font> be  <a href="quaterni.html#V1" target="_self">quaternion</a>  <a href="ordinal1.html#NM1" target="_self">number</a> , <font color="Maroon">n</font> be  <a href="quaterni.html#V1" target="_self">quaternion</a>  <a href="ordinal1.html#NM1" target="_self">number</a> , <font color="Maroon">x</font> be  <a href="quaterni.html#V1" target="_self">quaternion</a>  <a href="ordinal1.html#NM1" target="_self">number</a> , <font color="Maroon">y</font> be  <a href="quaterni.html#V1" target="_self">quaternion</a>  <a href="ordinal1.html#NM1" target="_self">number</a> , <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/>

<b>assume </b><a NAME="E1:59"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="quaterni.html#K7" target="_self">+</a> <font color="Maroon">n</font></span>)</span> <a href="quaterni.html#K7" target="_self">+</a> <font color="Maroon">x</font></span>)</span> <a href="quaterni.html#K7" target="_self">+</a> <font color="Maroon">y</font>
 ;<br/>

<b>consider </b><font color="Maroon">m1</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Maroon">m2</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Maroon">m3</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Maroon">m4</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Maroon">n1</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Maroon">n2</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Maroon">n3</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Maroon">n4</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="E3:59"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="quaterni.html#K6" target="_self">[*</a><span class="default"><font color="Maroon">m1</font>,<font color="Maroon">m2</font>,<font color="Maroon">m3</font>,<font color="Maroon">m4</font></span><a href="quaterni.html#K6" target="_self">*]</a></span>
 <b>and </b><br/><a NAME="E4:59"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="quaterni.html#K6" target="_self">[*</a><span class="default"><font color="Maroon">n1</font>,<font color="Maroon">n2</font>,<font color="Maroon">n3</font>,<font color="Maroon">n4</font></span><a href="quaterni.html#K6" target="_self">*]</a></span>
 <b>and </b><br/><a NAME="E5:59"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">m</font> <a href="quaterni.html#K7" target="_self">+</a> <font color="Maroon">n</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"><font color="Maroon">m1</font> <a href="real_1.html#K3">+</a> <font color="Maroon">n1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">m2</font> <a href="real_1.html#K3">+</a> <font color="Maroon">n2</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">m3</font> <a href="real_1.html#K3">+</a> <font color="Maroon">n3</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">m4</font> <a href="real_1.html#K3">+</a> <font color="Maroon">n4</font></span>)</span></span><a href="quaterni.html#K6" target="_self">*]</a></span>
 <b>by </b><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="E7:59"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">x</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="E8:59"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">y</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="E9:59"/>
<font color="Maroon">x</font> <a href="quaterni.html#K7" target="_self">+</a> <font color="Maroon">y</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"><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/>;<br/>
<a NAME="E10:59"/><i><font color="Green">E68</font></i>: 
(  <a href="quaterni.html#K17" target="_self">Rea</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m1</font> &amp;  <a href="quaterni.html#K17" target="_self">Rea</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n1</font> &amp;  <a href="quaterni.html#K17" target="_self">Rea</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> &amp;  <a href="quaterni.html#K17" target="_self">Rea</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y1</font> &amp;  <a href="quaterni.html#K18" target="_self">Im1</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m2</font> &amp;  <a href="quaterni.html#K18" target="_self">Im1</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n2</font> &amp;  <a href="quaterni.html#K18" target="_self">Im1</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font> &amp;  <a href="quaterni.html#K18" target="_self">Im1</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y2</font> &amp;  <a href="quaterni.html#K19" target="_self">Im2</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m3</font> &amp;  <a href="quaterni.html#K19" target="_self">Im2</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n3</font> &amp;  <a href="quaterni.html#K19" target="_self">Im2</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x3</font> &amp;  <a href="quaterni.html#K19" target="_self">Im2</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y3</font> &amp;  <a href="quaterni.html#K20" target="_self">Im3</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m4</font> &amp;  <a href="quaterni.html#K20" target="_self">Im3</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n4</font> &amp;  <a href="quaterni.html#K20" target="_self">Im3</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x4</font> &amp;  <a href="quaterni.html#K20" target="_self">Im3</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y4</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>, <a class="txt" href="quaterni.html#E4:59"><i><font color="Green">E41</font></i></a>, <a class="txt" href="quaterni.html#E5:59"><i><font color="Green">E42</font></i></a>, </i>;<br/>
<a NAME="E11:59"/>
<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="quaterni.html#K7" target="_self">+</a> <font color="Maroon">n</font></span>)</span> <a href="quaterni.html#K7" target="_self">+</a> <font color="Maroon">x</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"><span class="p3">(<span class="default"><font color="Maroon">m1</font> <a href="real_1.html#K3">+</a> <font color="Maroon">n1</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">x1</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">m2</font> <a href="real_1.html#K3">+</a> <font color="Maroon">n2</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">x2</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">m3</font> <a href="real_1.html#K3">+</a> <font color="Maroon">n3</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">x3</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">m4</font> <a href="real_1.html#K3">+</a> <font color="Maroon">n4</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">x4</font></span>)</span></span><a href="quaterni.html#K6" target="_self">*]</a></span>
 
<b>by </b><i><a class="txt" href="quaterni.html#E3:59"><i><font color="Green">E40</font></i></a>, <a class="txt" href="quaterni.html#E4:59"><i><font color="Green">E41</font></i></a>, </i>;<br/>
<a NAME="E12:59"/><b>then </b>
<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"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">m1</font> <a href="real_1.html#K3">+</a> <font color="Maroon">n1</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">x1</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">y1</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">m2</font> <a href="real_1.html#K3">+</a> <font color="Maroon">n2</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">x2</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">y2</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">m3</font> <a href="real_1.html#K3">+</a> <font color="Maroon">n3</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">x3</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">y3</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">m4</font> <a href="real_1.html#K3">+</a> <font color="Maroon">n4</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">x4</font></span>)</span> <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><a class="txt" href="quaterni.html#E1:59"><i><font color="Green">E37</font></i></a>, <a class="txt" href="quaterni.html#E5:59"><i><font color="Green">E42</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E13:59"/>
(  <a href="quaterni.html#K17" target="_self">Rea</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="quaterni.html#K17" target="_self">Rea</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="quaterni.html#K17" target="_self">Rea</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="quaterni.html#K17" target="_self">Rea</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="quaterni.html#K17" target="_self">Rea</a> <font color="Maroon">y</font></span>)</span> &amp;  <a href="quaterni.html#K18" target="_self">Im1</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="quaterni.html#K18" target="_self">Im1</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="quaterni.html#K18" target="_self">Im1</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="quaterni.html#K18" target="_self">Im1</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="quaterni.html#K18" target="_self">Im1</a> <font color="Maroon">y</font></span>)</span> &amp;  <a href="quaterni.html#K19" target="_self">Im2</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="quaterni.html#K19" target="_self">Im2</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="quaterni.html#K19" target="_self">Im2</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="quaterni.html#K19" target="_self">Im2</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="quaterni.html#K19" target="_self">Im2</a> <font color="Maroon">y</font></span>)</span> &amp;  <a href="quaterni.html#K20" target="_self">Im3</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="quaterni.html#K20" target="_self">Im3</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="quaterni.html#K20" target="_self">Im3</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="quaterni.html#K20" target="_self">Im3</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="quaterni.html#K20" target="_self">Im3</a> <font color="Maroon">y</font></span>)</span> )
 <b>by </b><i>, <a class="txt" href="quaterni.html#E7:59"><i><font color="Green">E43</font></i></a></i>;<br/>


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