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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="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="quatern2.html#K3" title="QUATERN2:func.3">|.</a><span class="default"><font color="Maroon" title="c1">z1</font></span><a href="quatern2.html#K3" title="QUATERN2:func.3">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="quatern2.html#K3" title="QUATERN2:func.3">|.</a><span class="default"><font color="Maroon" title="c2">z2</font></span><a href="quatern2.html#K3" title="QUATERN2:func.3">.|</a></span> &amp; <span class="p1"><a href="quatern2.html#K3" title="QUATERN2:func.3">|.</a><span class="default"><font color="Maroon" title="c1">z1</font></span><a href="quatern2.html#K3" title="QUATERN2:func.3">.|</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c1">z1</font> <a href="quatern2.html#K8" title="QUATERN2:func.8">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">z2</font> <a href="quatern2.html#K8" title="QUATERN2:func.8">"</a>  implies <font color="Maroon" title="c1">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">z2</font> )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:51"/><span class="lab"><font color="Green" title="E11">A1</font></span>: 
<span class="p1"><a href="quatern2.html#K3" title="QUATERN2:func.3">|.</a><span class="default"><font color="Maroon" title="c1">z1</font></span><a href="quatern2.html#K3" title="QUATERN2:func.3">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="quatern2.html#K3" title="QUATERN2:func.3">|.</a><span class="default"><font color="Maroon" title="c2">z2</font></span><a href="quatern2.html#K3" title="QUATERN2:func.3">.|</a></span>
 <span class="kw">and </span><br/><a NAME="E2:51"/><span class="lab"><font color="Green" title="E12">A2</font></span>: 
<span class="p1"><a href="quatern2.html#K3" title="QUATERN2:func.3">|.</a><span class="default"><font color="Maroon" title="c1">z1</font></span><a href="quatern2.html#K3" title="QUATERN2:func.3">.|</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">and </span><br/><a NAME="E3:51"/><span class="lab"><font color="Green" title="E13">A3</font></span>: 
<font color="Maroon" title="c1">z1</font> <a href="quatern2.html#K8" title="QUATERN2:func.8">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">z2</font> <a href="quatern2.html#K8" title="QUATERN2:func.8">"</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="c1">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">z2</font></span><br/>

<a NAME="E4:51"/><span class="lab"><font color="Green" title="E14">A4</font></span>: 
<span class="p1"><a href="quatern2.html#K3" title="QUATERN2:func.3">|.</a><span class="default"><font color="Maroon" title="c1">z1</font></span><a href="quatern2.html#K3" title="QUATERN2:func.3">.|</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:51"><span class="lab"><font color="Green" title="E12">A2</font></span></a>, <a class="ref" href="xcmplx_1.html#T6" title="XCMPLX_1:th.6">XCMPLX_1:6</a></span>;<br/>
<a NAME="E5:51"/><span class="lab"><font color="Green" title="E15">A5</font></span>: 
 <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="quatern2.html#K8" title="QUATERN2:func.8">"</a> </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="c1">z1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default"><span class="p3"><a href="quatern2.html#K3" title="QUATERN2:func.3">|.</a><span class="default"><font color="Maroon" title="c1">z1</font></span><a href="quatern2.html#K3" title="QUATERN2:func.3">.|</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="quatern2.html#T29" title="QUATERN2:th.29">QUATERN2:29</a></span>;<br/>
<a NAME="E6:51"/><span class="lab"><font color="Green" title="E16">A6</font></span>: 
 <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="c1">z1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default"><span class="p3"><a href="quatern2.html#K3" title="QUATERN2:func.3">|.</a><span class="default"><font color="Maroon" title="c1">z1</font></span><a href="quatern2.html#K3" title="QUATERN2:func.3">.|</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </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#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default"><span class="p3"><a href="quatern2.html#K3" title="QUATERN2:func.3">|.</a><span class="default"><font color="Maroon" title="c2">z2</font></span><a href="quatern2.html#K3" title="QUATERN2:func.3">.|</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:51"><span class="lab"><font color="Green" title="E13">A3</font></span></a>, <a class="txt" href="#E5:51"><span class="lab"><font color="Green" title="E15">A5</font></span></a>, <a class="ref" href="quatern2.html#T29" title="QUATERN2:th.29">QUATERN2:29</a></span>;<br/>
<a NAME="E7:51"/><span class="lab"><font color="Green" title="E17">A7</font></span>: 
 <a href="quaterni.html#K20" title="QUATERNI:func.20">Im3</a> <font color="Maroon" title="c1">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="quaterni.html#K20" title="QUATERNI:func.20">Im3</a> <font color="Maroon" title="c2">z2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51"><span class="lab"><font color="Green" title="E11">A1</font></span></a>, <a class="txt" href="#E6:51"><span class="lab"><font color="Green" title="E16">A6</font></span></a>, <a class="txt" href="#E4:51"><span class="lab"><font color="Green" title="E14">A4</font></span></a>, <a class="ref" href="xcmplx_1.html#T53" title="XCMPLX_1:th.53">XCMPLX_1:53</a></span>;<br/>
<a NAME="E8:51"/><span class="lab"><font color="Green" title="E18">A8</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="quatern2.html#K8" title="QUATERN2:func.8">"</a> </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="c1">z1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default"><span class="p3"><a href="quatern2.html#K3" title="QUATERN2:func.3">|.</a><span class="default"><font color="Maroon" title="c1">z1</font></span><a href="quatern2.html#K3" title="QUATERN2:func.3">.|</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="quatern2.html#T29" title="QUATERN2:th.29">QUATERN2:29</a></span>;<br/>
<a NAME="E9:51"/><span class="lab"><font color="Green" title="E19">A9</font></span>: 
 <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="c1">z1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default"><span class="p3"><a href="quatern2.html#K3" title="QUATERN2:func.3">|.</a><span class="default"><font color="Maroon" title="c1">z1</font></span><a href="quatern2.html#K3" title="QUATERN2:func.3">.|</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </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#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default"><span class="p3"><a href="quatern2.html#K3" title="QUATERN2:func.3">|.</a><span class="default"><font color="Maroon" title="c2">z2</font></span><a href="quatern2.html#K3" title="QUATERN2:func.3">.|</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:51"><span class="lab"><font color="Green" title="E13">A3</font></span></a>, <a class="txt" href="#E8:51"><span class="lab"><font color="Green" title="E18">A8</font></span></a>, <a class="ref" href="quatern2.html#T29" title="QUATERN2:th.29">QUATERN2:29</a></span>;<br/>
<a NAME="E10:51"/><span class="lab"><font color="Green" title="E20">A10</font></span>: 
 <a href="quaterni.html#K19" title="QUATERNI:func.19">Im2</a> <font color="Maroon" title="c1">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="quaterni.html#K19" title="QUATERNI:func.19">Im2</a> <font color="Maroon" title="c2">z2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51"><span class="lab"><font color="Green" title="E11">A1</font></span></a>, <a class="txt" href="#E9:51"><span class="lab"><font color="Green" title="E19">A9</font></span></a>, <a class="txt" href="#E4:51"><span class="lab"><font color="Green" title="E14">A4</font></span></a>, <a class="ref" href="xcmplx_1.html#T53" title="XCMPLX_1:th.53">XCMPLX_1:53</a></span>;<br/>
<a NAME="E11:51"/><span class="lab"><font color="Green" title="E21">A11</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="quatern2.html#K8" title="QUATERN2:func.8">"</a> </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="c1">z1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default"><span class="p3"><a href="quatern2.html#K3" title="QUATERN2:func.3">|.</a><span class="default"><font color="Maroon" title="c1">z1</font></span><a href="quatern2.html#K3" title="QUATERN2:func.3">.|</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="quatern2.html#T29" title="QUATERN2:th.29">QUATERN2:29</a></span>;<br/>
<a NAME="E12:51"/><span class="lab"><font color="Green" title="E22">A12</font></span>: 
 <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="c1">z1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default"><span class="p3"><a href="quatern2.html#K3" title="QUATERN2:func.3">|.</a><span class="default"><font color="Maroon" title="c1">z1</font></span><a href="quatern2.html#K3" title="QUATERN2:func.3">.|</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </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#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default"><span class="p3"><a href="quatern2.html#K3" title="QUATERN2:func.3">|.</a><span class="default"><font color="Maroon" title="c2">z2</font></span><a href="quatern2.html#K3" title="QUATERN2:func.3">.|</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:51"><span class="lab"><font color="Green" title="E13">A3</font></span></a>, <a class="txt" href="#E11:51"><span class="lab"><font color="Green" title="E21">A11</font></span></a>, <a class="ref" href="quatern2.html#T29" title="QUATERN2:th.29">QUATERN2:29</a></span>;<br/>
<a NAME="E13:51"/><span class="lab"><font color="Green" title="E23">A13</font></span>: 
 <a href="quaterni.html#K18" title="QUATERNI:func.18">Im1</a> <font color="Maroon" title="c1">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="quaterni.html#K18" title="QUATERNI:func.18">Im1</a> <font color="Maroon" title="c2">z2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51"><span class="lab"><font color="Green" title="E11">A1</font></span></a>, <a class="txt" href="#E12:51"><span class="lab"><font color="Green" title="E22">A12</font></span></a>, <a class="txt" href="#E4:51"><span class="lab"><font color="Green" title="E14">A4</font></span></a>, <a class="ref" href="xcmplx_1.html#T53" title="XCMPLX_1:th.53">XCMPLX_1:53</a></span>;<br/>
<a NAME="E14:51"/><span class="lab"><font color="Green" title="E24">A14</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="quatern2.html#K8" title="QUATERN2:func.8">"</a> </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#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><span class="p2"><a href="quatern2.html#K3" title="QUATERN2:func.3">|.</a><span class="default"><font color="Maroon" title="c1">z1</font></span><a href="quatern2.html#K3" title="QUATERN2:func.3">.|</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="quatern2.html#T29" title="QUATERN2:th.29">QUATERN2:29</a></span>;<br/>
<a NAME="E15:51"/><span class="lab"><font color="Green" title="E25">A15</font></span>: 
<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#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><span class="p2"><a href="quatern2.html#K3" title="QUATERN2:func.3">|.</a><span class="default"><font color="Maroon" title="c1">z1</font></span><a href="quatern2.html#K3" title="QUATERN2:func.3">.|</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </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#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><span class="p2"><a href="quatern2.html#K3" title="QUATERN2:func.3">|.</a><span class="default"><font color="Maroon" title="c2">z2</font></span><a href="quatern2.html#K3" title="QUATERN2:func.3">.|</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:51"><span class="lab"><font color="Green" title="E13">A3</font></span></a>, <a class="txt" href="#E14:51"><span class="lab"><font color="Green" title="E24">A14</font></span></a>, <a class="ref" href="quatern2.html#T29" title="QUATERN2:th.29">QUATERN2:29</a></span>;<br/>
<a NAME="E16:51"/><span class="lab"><font color="Green" title="E26">A16</font></span>: 
 <a href="quaterni.html#K17" title="QUATERNI:func.17">Rea</a> <font color="Maroon" title="c1">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="quaterni.html#K17" title="QUATERNI:func.17">Rea</a> <font color="Maroon" title="c2">z2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51"><span class="lab"><font color="Green" title="E11">A1</font></span></a>, <a class="txt" href="#E15:51"><span class="lab"><font color="Green" title="E25">A15</font></span></a>, <a class="txt" href="#E4:51"><span class="lab"><font color="Green" title="E14">A4</font></span></a>, <a class="ref" href="xcmplx_1.html#T53" title="XCMPLX_1:th.53">XCMPLX_1:53</a></span>;<br/>
<span class="kw">thus </span><a NAME="E17:51"/>
<font color="Maroon" title="c1">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">z2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:51"><span class="lab"><font color="Green" title="E26">A16</font></span></a>, <a class="txt" href="#E13:51"><span class="lab"><font color="Green" title="E23">A13</font></span></a>, <a class="txt" href="#E10:51"><span class="lab"><font color="Green" title="E20">A10</font></span></a>, <a class="txt" href="#E7:51"><span class="lab"><font color="Green" title="E17">A7</font></span></a>, <a class="ref" href="quaterni.html#T25" title="QUATERNI:th.25">QUATERNI:25</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/>


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