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<span class="kw">let </span><font color="Maroon" title="c1">z</font>, <font color="Maroon" title="c2">z1</font>, <font color="Maroon" title="c3">z2</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="c2">z1</font> <a href="quaterni.html#K29" title="QUATERNI:func.29">-</a> <font color="Maroon" title="c3">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">z1</font> <a href="quaterni.html#K29" title="QUATERNI:func.29">-</a> <font color="Maroon" title="c1">z</font></span>)</span> <a href="quaterni.html#K26" title="QUATERNI:func.26">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="quaterni.html#K29" title="QUATERNI:func.29">-</a> <font color="Maroon" title="c3">z2</font></span>)</span></span><br/>

<span class="lab"><font color="Green" title="E83">A1</font></span>:  <a href="quaterni.html#K17" title="QUATERNI:func.17">Rea</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">z1</font> <a href="quaterni.html#K29" title="QUATERNI:func.29">-</a> <font color="Maroon" title="c1">z</font></span>)</span> <a href="quaterni.html#K26" title="QUATERNI:func.26">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="quaterni.html#K29" title="QUATERNI:func.29">-</a> <font color="Maroon" title="c3">z2</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="quaterni.html#K17" title="QUATERNI:func.17">Rea</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">z1</font> <a href="quaterni.html#K29" title="QUATERNI:func.29">-</a> <font color="Maroon" title="c1">z</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="quaterni.html#K17" title="QUATERNI:func.17">Rea</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="quaterni.html#K29" title="QUATERNI:func.29">-</a> <font color="Maroon" title="c3">z2</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="quaterni.html#E62"><span class="lab"><font color="Green" title="E39">Lm13</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="quaterni.html#K17" title="QUATERNI:func.17">Rea</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">z1</font> <a href="quaterni.html#K29" title="QUATERNI:func.29">-</a> <font color="Maroon" title="c1">z</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="quaterni.html#K17" title="QUATERNI:func.17">Rea</a> <font color="Maroon" title="c1">z</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="quaterni.html#K17" title="QUATERNI:func.17">Rea</a> <font color="Maroon" title="c3">z2</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="quaterni.html#T42" target="_self" title="QUATERNI:th.42">Th42</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="quaterni.html#K17" title="QUATERNI:func.17">Rea</a> <font color="Maroon" title="c2">z1</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="quaterni.html#K17" title="QUATERNI:func.17">Rea</a> <font color="Maroon" title="c1">z</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="quaterni.html#K17" title="QUATERNI:func.17">Rea</a> <font color="Maroon" title="c1">z</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="quaterni.html#K17" title="QUATERNI:func.17">Rea</a> <font color="Maroon" title="c3">z2</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="quaterni.html#T42" target="_self" title="QUATERNI:th.42">Th42</a></span>
<br/>.= 
 <a href="quaterni.html#K17" title="QUATERNI:func.17">Rea</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">z1</font> <a href="quaterni.html#K29" title="QUATERNI:func.29">-</a> <font color="Maroon" title="c3">z2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="quaterni.html#E143"><span class="lab"><font color="Green" title="E77">Lm36</font></span></a></span>
;<br/>
<span class="lab"><font color="Green" title="E84">A2</font></span>:  <a href="quaterni.html#K18" title="QUATERNI:func.18">Im1</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">z1</font> <a href="quaterni.html#K29" title="QUATERNI:func.29">-</a> <font color="Maroon" title="c1">z</font></span>)</span> <a href="quaterni.html#K26" title="QUATERNI:func.26">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="quaterni.html#K29" title="QUATERNI:func.29">-</a> <font color="Maroon" title="c3">z2</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="quaterni.html#K18" title="QUATERNI:func.18">Im1</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">z1</font> <a href="quaterni.html#K29" title="QUATERNI:func.29">-</a> <font color="Maroon" title="c1">z</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="quaterni.html#K18" title="QUATERNI:func.18">Im1</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="quaterni.html#K29" title="QUATERNI:func.29">-</a> <font color="Maroon" title="c3">z2</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="quaterni.html#E62"><span class="lab"><font color="Green" title="E39">Lm13</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="quaterni.html#K18" title="QUATERNI:func.18">Im1</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">z1</font> <a href="quaterni.html#K29" title="QUATERNI:func.29">-</a> <font color="Maroon" title="c1">z</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="quaterni.html#K18" title="QUATERNI:func.18">Im1</a> <font color="Maroon" title="c1">z</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="quaterni.html#K18" title="QUATERNI:func.18">Im1</a> <font color="Maroon" title="c3">z2</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="quaterni.html#T42" target="_self" title="QUATERNI:th.42">Th42</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="quaterni.html#K18" title="QUATERNI:func.18">Im1</a> <font color="Maroon" title="c2">z1</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="quaterni.html#K18" title="QUATERNI:func.18">Im1</a> <font color="Maroon" title="c1">z</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="quaterni.html#K18" title="QUATERNI:func.18">Im1</a> <font color="Maroon" title="c1">z</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="quaterni.html#K18" title="QUATERNI:func.18">Im1</a> <font color="Maroon" title="c3">z2</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="quaterni.html#T42" target="_self" title="QUATERNI:th.42">Th42</a></span>
<br/>.= 
 <a href="quaterni.html#K18" title="QUATERNI:func.18">Im1</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">z1</font> <a href="quaterni.html#K29" title="QUATERNI:func.29">-</a> <font color="Maroon" title="c3">z2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="quaterni.html#E143"><span class="lab"><font color="Green" title="E77">Lm36</font></span></a></span>
;<br/>
<span class="lab"><font color="Green" title="E85">A3</font></span>:  <a href="quaterni.html#K19" title="QUATERNI:func.19">Im2</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">z1</font> <a href="quaterni.html#K29" title="QUATERNI:func.29">-</a> <font color="Maroon" title="c1">z</font></span>)</span> <a href="quaterni.html#K26" title="QUATERNI:func.26">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="quaterni.html#K29" title="QUATERNI:func.29">-</a> <font color="Maroon" title="c3">z2</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="quaterni.html#K19" title="QUATERNI:func.19">Im2</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">z1</font> <a href="quaterni.html#K29" title="QUATERNI:func.29">-</a> <font color="Maroon" title="c1">z</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="quaterni.html#K19" title="QUATERNI:func.19">Im2</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="quaterni.html#K29" title="QUATERNI:func.29">-</a> <font color="Maroon" title="c3">z2</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="quaterni.html#E62"><span class="lab"><font color="Green" title="E39">Lm13</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="quaterni.html#K19" title="QUATERNI:func.19">Im2</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">z1</font> <a href="quaterni.html#K29" title="QUATERNI:func.29">-</a> <font color="Maroon" title="c1">z</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="quaterni.html#K19" title="QUATERNI:func.19">Im2</a> <font color="Maroon" title="c1">z</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="quaterni.html#K19" title="QUATERNI:func.19">Im2</a> <font color="Maroon" title="c3">z2</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="quaterni.html#T42" target="_self" title="QUATERNI:th.42">Th42</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="quaterni.html#K19" title="QUATERNI:func.19">Im2</a> <font color="Maroon" title="c2">z1</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="quaterni.html#K19" title="QUATERNI:func.19">Im2</a> <font color="Maroon" title="c1">z</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="quaterni.html#K19" title="QUATERNI:func.19">Im2</a> <font color="Maroon" title="c1">z</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="quaterni.html#K19" title="QUATERNI:func.19">Im2</a> <font color="Maroon" title="c3">z2</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="quaterni.html#T42" target="_self" title="QUATERNI:th.42">Th42</a></span>
<br/>.= 
 <a href="quaterni.html#K19" title="QUATERNI:func.19">Im2</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">z1</font> <a href="quaterni.html#K29" title="QUATERNI:func.29">-</a> <font color="Maroon" title="c3">z2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="quaterni.html#E143"><span class="lab"><font color="Green" title="E77">Lm36</font></span></a></span>
;<br/>
 <a href="quaterni.html#K20" title="QUATERNI:func.20">Im3</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">z1</font> <a href="quaterni.html#K29" title="QUATERNI:func.29">-</a> <font color="Maroon" title="c1">z</font></span>)</span> <a href="quaterni.html#K26" title="QUATERNI:func.26">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="quaterni.html#K29" title="QUATERNI:func.29">-</a> <font color="Maroon" title="c3">z2</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="quaterni.html#K20" title="QUATERNI:func.20">Im3</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">z1</font> <a href="quaterni.html#K29" title="QUATERNI:func.29">-</a> <font color="Maroon" title="c1">z</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="quaterni.html#K20" title="QUATERNI:func.20">Im3</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="quaterni.html#K29" title="QUATERNI:func.29">-</a> <font color="Maroon" title="c3">z2</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="quaterni.html#E62"><span class="lab"><font color="Green" title="E39">Lm13</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="quaterni.html#K20" title="QUATERNI:func.20">Im3</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">z1</font> <a href="quaterni.html#K29" title="QUATERNI:func.29">-</a> <font color="Maroon" title="c1">z</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="quaterni.html#K20" title="QUATERNI:func.20">Im3</a> <font color="Maroon" title="c1">z</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="quaterni.html#K20" title="QUATERNI:func.20">Im3</a> <font color="Maroon" title="c3">z2</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="quaterni.html#T42" target="_self" title="QUATERNI:th.42">Th42</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="quaterni.html#K20" title="QUATERNI:func.20">Im3</a> <font color="Maroon" title="c2">z1</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="quaterni.html#K20" title="QUATERNI:func.20">Im3</a> <font color="Maroon" title="c1">z</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="quaterni.html#K20" title="QUATERNI:func.20">Im3</a> <font color="Maroon" title="c1">z</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><a href="quaterni.html#K20" title="QUATERNI:func.20">Im3</a> <font color="Maroon" title="c3">z2</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="quaterni.html#T42" target="_self" title="QUATERNI:th.42">Th42</a></span>
<br/>.= 
 <a href="quaterni.html#K20" title="QUATERNI:func.20">Im3</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">z1</font> <a href="quaterni.html#K29" title="QUATERNI:func.29">-</a> <font color="Maroon" title="c3">z2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="quaterni.html#E143"><span class="lab"><font color="Green" title="E77">Lm36</font></span></a></span>
;<br/>
<span class="kw">hence </span><a NAME="E5:140"/>
<font color="Maroon" title="c2">z1</font> <a href="quaterni.html#K29" title="QUATERNI:func.29">-</a> <font color="Maroon" title="c3">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">z1</font> <a href="quaterni.html#K29" title="QUATERNI:func.29">-</a> <font color="Maroon" title="c1">z</font></span>)</span> <a href="quaterni.html#K26" title="QUATERNI:func.26">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="quaterni.html#K29" title="QUATERNI:func.29">-</a> <font color="Maroon" title="c3">z2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:140"><span class="lab"><font color="Green" title="E83">A1</font></span></a>, <a class="txt" href="#E2:140"><span class="lab"><font color="Green" title="E84">A2</font></span></a>, <a class="txt" href="#E3:140"><span class="lab"><font color="Green" title="E85">A3</font></span></a>, <a class="ref" href="quaterni.html#T25" target="_self" title="QUATERNI:th.25">Th25</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/>


</div>
