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<span class="kw">let </span><font color="Maroon" title="c1">z1</font>, <font color="Maroon" title="c2">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="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">z1</font> <a href="quaterni.html#K29" title="QUATERNI:func.29">-</a> <font color="Maroon" title="c2">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="quaterni.html#K28" title="QUATERNI:func.28">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">z2</font> <a href="quaterni.html#K29" title="QUATERNI:func.29">-</a> <font color="Maroon" title="c1">z1</font></span>)</span></span><br/>



<a NAME="E1:140"/><span class="lab"><font color="Green" title="E81">A1</font></span>: 
 <a href="quaterni.html#K17" title="QUATERNI:func.17">Rea</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">z2</font> <a href="quaterni.html#K29" title="QUATERNI:func.29">-</a> <font color="Maroon" title="c1">z1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="quaterni.html#K17" title="QUATERNI:func.17">Rea</a> <font color="Maroon" title="c2">z2</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="quaterni.html#K17" title="QUATERNI:func.17">Rea</a> <font color="Maroon" title="c1">z1</font></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 NAME="E2:140"/><span class="lab"><font color="Green" title="E82">A2</font></span>: 
 <a href="quaterni.html#K18" title="QUATERNI:func.18">Im1</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">z2</font> <a href="quaterni.html#K29" title="QUATERNI:func.29">-</a> <font color="Maroon" title="c1">z1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="quaterni.html#K18" title="QUATERNI:func.18">Im1</a> <font color="Maroon" title="c2">z2</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="quaterni.html#K18" title="QUATERNI:func.18">Im1</a> <font color="Maroon" title="c1">z1</font></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 NAME="E3:140"/><span class="lab"><font color="Green" title="E83">A3</font></span>: 
 <a href="quaterni.html#K19" title="QUATERNI:func.19">Im2</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">z2</font> <a href="quaterni.html#K29" title="QUATERNI:func.29">-</a> <font color="Maroon" title="c1">z1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="quaterni.html#K19" title="QUATERNI:func.19">Im2</a> <font color="Maroon" title="c2">z2</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="quaterni.html#K19" title="QUATERNI:func.19">Im2</a> <font color="Maroon" title="c1">z1</font></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 NAME="E4:140"/><span class="lab"><font color="Green" title="E84">A4</font></span>: 
 <a href="quaterni.html#K20" title="QUATERNI:func.20">Im3</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">z2</font> <a href="quaterni.html#K29" title="QUATERNI:func.29">-</a> <font color="Maroon" title="c1">z1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="quaterni.html#K20" title="QUATERNI:func.20">Im3</a> <font color="Maroon" title="c2">z2</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="quaterni.html#K20" title="QUATERNI:func.20">Im3</a> <font color="Maroon" title="c1">z1</font></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 NAME="E5:140"/><span class="lab"><font color="Green" title="E85">A5</font></span>: 
 <a href="quaterni.html#K17" title="QUATERNI:func.17">Rea</a> <span class="p1">(<span class="default"><a href="quaterni.html#K28" title="QUATERNI:func.28">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">z2</font> <a href="quaterni.html#K29" title="QUATERNI:func.29">-</a> <font color="Maroon" title="c1">z1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</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="c2">z2</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">z1</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:140"><span class="lab"><font color="Green" title="E81">A1</font></span></a>, <a class="ref" href="quaterni.html#T41" target="_self" title="QUATERNI:th.41">Th41</a></span>;<br/>
<a NAME="E6:140"/><span class="lab"><font color="Green" title="E86">A6</font></span>: 
 <a href="quaterni.html#K18" title="QUATERNI:func.18">Im1</a> <span class="p1">(<span class="default"><a href="quaterni.html#K28" title="QUATERNI:func.28">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">z2</font> <a href="quaterni.html#K29" title="QUATERNI:func.29">-</a> <font color="Maroon" title="c1">z1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</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="c2">z2</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">z1</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:140"><span class="lab"><font color="Green" title="E82">A2</font></span></a>, <a class="ref" href="quaterni.html#T41" target="_self" title="QUATERNI:th.41">Th41</a></span>;<br/>
<a NAME="E7:140"/><span class="lab"><font color="Green" title="E87">A7</font></span>: 
 <a href="quaterni.html#K19" title="QUATERNI:func.19">Im2</a> <span class="p1">(<span class="default"><a href="quaterni.html#K28" title="QUATERNI:func.28">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">z2</font> <a href="quaterni.html#K29" title="QUATERNI:func.29">-</a> <font color="Maroon" title="c1">z1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</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="c2">z2</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">z1</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:140"><span class="lab"><font color="Green" title="E83">A3</font></span></a>, <a class="ref" href="quaterni.html#T41" target="_self" title="QUATERNI:th.41">Th41</a></span>;<br/>
<a NAME="E8:140"/><span class="lab"><font color="Green" title="E88">A8</font></span>: 
 <a href="quaterni.html#K20" title="QUATERNI:func.20">Im3</a> <span class="p1">(<span class="default"><a href="quaterni.html#K28" title="QUATERNI:func.28">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">z2</font> <a href="quaterni.html#K29" title="QUATERNI:func.29">-</a> <font color="Maroon" title="c1">z1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</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="c2">z2</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">z1</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:140"><span class="lab"><font color="Green" title="E84">A4</font></span></a>, <a class="ref" href="quaterni.html#T41" target="_self" title="QUATERNI:th.41">Th41</a></span>;<br/>
<a NAME="E9:140"/><span class="lab"><font color="Green" title="E89">A9</font></span>: 
 <a href="quaterni.html#K17" title="QUATERNI:func.17">Rea</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">z1</font> <a href="quaterni.html#K29" title="QUATERNI:func.29">-</a> <font color="Maroon" title="c2">z2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="quaterni.html#K17" title="QUATERNI:func.17">Rea</a> <font color="Maroon" title="c1">z1</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="quaterni.html#K17" title="QUATERNI:func.17">Rea</a> <font color="Maroon" title="c2">z2</font></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 NAME="E10:140"/><span class="lab"><font color="Green" title="E90">A10</font></span>: 
 <a href="quaterni.html#K18" title="QUATERNI:func.18">Im1</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">z1</font> <a href="quaterni.html#K29" title="QUATERNI:func.29">-</a> <font color="Maroon" title="c2">z2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="quaterni.html#K18" title="QUATERNI:func.18">Im1</a> <font color="Maroon" title="c1">z1</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="quaterni.html#K18" title="QUATERNI:func.18">Im1</a> <font color="Maroon" title="c2">z2</font></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 NAME="E11:140"/><span class="lab"><font color="Green" title="E91">A11</font></span>: 
 <a href="quaterni.html#K19" title="QUATERNI:func.19">Im2</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">z1</font> <a href="quaterni.html#K29" title="QUATERNI:func.29">-</a> <font color="Maroon" title="c2">z2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="quaterni.html#K19" title="QUATERNI:func.19">Im2</a> <font color="Maroon" title="c1">z1</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="quaterni.html#K19" title="QUATERNI:func.19">Im2</a> <font color="Maroon" title="c2">z2</font></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 NAME="E12:140"/>
 <a href="quaterni.html#K20" title="QUATERNI:func.20">Im3</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">z1</font> <a href="quaterni.html#K29" title="QUATERNI:func.29">-</a> <font color="Maroon" title="c2">z2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="quaterni.html#K20" title="QUATERNI:func.20">Im3</a> <font color="Maroon" title="c1">z1</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="quaterni.html#K20" title="QUATERNI:func.20">Im3</a> <font color="Maroon" title="c2">z2</font></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="kw">hence </span><a NAME="E13:140"/>
<font color="Maroon" title="c1">z1</font> <a href="quaterni.html#K29" title="QUATERNI:func.29">-</a> <font color="Maroon" title="c2">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="quaterni.html#K28" title="QUATERNI:func.28">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">z2</font> <a href="quaterni.html#K29" title="QUATERNI:func.29">-</a> <font color="Maroon" title="c1">z1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="quaterni.html#T25" target="_self" title="QUATERNI:th.25">Th25</a>, <a class="txt" href="#E9:140"><span class="lab"><font color="Green" title="E89">A9</font></span></a>, <a class="txt" href="#E10:140"><span class="lab"><font color="Green" title="E90">A10</font></span></a>, <a class="txt" href="#E11:140"><span class="lab"><font color="Green" title="E91">A11</font></span></a>, <a class="txt" href="#E5:140"><span class="lab"><font color="Green" title="E85">A5</font></span></a>, <a class="txt" href="#E6:140"><span class="lab"><font color="Green" title="E86">A6</font></span></a>, <a class="txt" href="#E7:140"><span class="lab"><font color="Green" title="E87">A7</font></span></a>, <a class="txt" href="#E8:140"><span class="lab"><font color="Green" title="E88">A8</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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