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<div class="add">

<b>let </b><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> ;<br/>



<b>thus </b><a NAME="E1:141"/>
( <span class="p1"><a href="quaterni.html#K32" title="QUATERNI:func.32">|.</a><span class="default"><span class="p2">(<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></span><a href="quaterni.html#K32" title="QUATERNI:func.32">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 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> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="141_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:141_1"/>
<span class="p1"><a href="quaterni.html#K32" title="QUATERNI:func.32">|.</a><span class="default"><span class="p2">(<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></span><a href="quaterni.html#K32" title="QUATERNI:func.32">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/>

<a NAME="E2:141_1"/><b>then </b>
<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> 0
 
<b>by </b><i><a class="ref" href="quaterni.html#T66" target="_self" title="QUATERNI:th.66">Th66</a></i>;<br/>
<b>hence </b><a NAME="E3:141_1"/>
<font color="Maroon" title="c1">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">z2</font>
 <b>by </b><i><a class="txt" href="quaterni.html#E143"><i><font color="Green" title="E77">Lm36</font></i></a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:141"/>
<font color="Maroon" title="c1">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">z2</font>
 ;<br/>

<b>hence </b><a NAME="E3:141"/>
<span class="p1"><a href="quaterni.html#K32" title="QUATERNI:func.32">|.</a><span class="default"><span class="p2">(<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></span><a href="quaterni.html#K32" title="QUATERNI:func.32">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="txt" href="quaterni.html#E24"><i><font color="Green" title="E18">Lm5</font></i></a>, <a class="txt" href="quaterni.html#E140"><i><font color="Green" title="E74">Lm33</font></i></a>, <a class="ref" href="quaterni.html#T65" target="_self" title="QUATERNI:th.65">Th65</a></i>;<br/>


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