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<b>let </b><font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c2">b</font>, <font color="Maroon" title="c3">c</font>, <font color="Maroon" title="c4">d</font>, <font color="Maroon" title="c5">x</font>, <font color="Maroon" title="c6">y</font>, <font color="Maroon" title="c7">z</font>, <font color="Maroon" title="c8">w</font>, <font color="Maroon" title="c9">x'</font>, <font color="Maroon" title="c10">y'</font>, <font color="Maroon" title="c11">z'</font>, <font color="Maroon" title="c12">w'</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font> <a href="zfmisc_1.html#R2" title="ZFMISC_1:pred.2">are_mutually_different</a>  &amp; <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font> <a href="quaterni.html#K2" title="QUATERNI:func.2">--&gt;</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font>,<font color="Maroon" title="c7">z</font>,<font color="Maroon" title="c8">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font> <a href="quaterni.html#K2" title="QUATERNI:func.2">--&gt;</a> <font color="Maroon" title="c9">x'</font>,<font color="Maroon" title="c10">y'</font>,<font color="Maroon" title="c11">z'</font>,<font color="Maroon" title="c12">w'</font> implies ( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x'</font> &amp; <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">y'</font> &amp; <font color="Maroon" title="c7">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">z'</font> &amp; <font color="Maroon" title="c8">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">w'</font> ) )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:25"/><i><font color="Green" title="E17">A1</font></i>: 
<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font> <a href="zfmisc_1.html#R2" title="ZFMISC_1:pred.2">are_mutually_different</a> 
 <b>and </b><br/><a NAME="E2:25"/><i><font color="Green" title="E18">A2</font></i>: 
<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font> <a href="quaterni.html#K2" title="QUATERNI:func.2">--&gt;</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font>,<font color="Maroon" title="c7">z</font>,<font color="Maroon" title="c8">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font> <a href="quaterni.html#K2" title="QUATERNI:func.2">--&gt;</a> <font color="Maroon" title="c9">x'</font>,<font color="Maroon" title="c10">y'</font>,<font color="Maroon" title="c11">z'</font>,<font color="Maroon" title="c12">w'</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x'</font> &amp; <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">y'</font> &amp; <font color="Maroon" title="c7">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">z'</font> &amp; <font color="Maroon" title="c8">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">w'</font> )</span><br/>

<i><font color="Green" title="E19">A3</font></i>: <font color="Maroon" title="c5">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font> <a href="quaterni.html#K2" title="QUATERNI:func.2">--&gt;</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font>,<font color="Maroon" title="c7">z</font>,<font color="Maroon" title="c8">w</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">a</font>
<b>by </b><i><a class="txt" href="#E1:25"><i><font color="Green" title="E17">A1</font></i></a>, <a class="ref" href="quaterni.html#T3" target="_self" title="QUATERNI:th.3">Th3</a></i>
<br/>.= 
<font color="Maroon" title="c9">x'</font>
<b>by </b><i><a class="txt" href="#E1:25"><i><font color="Green" title="E17">A1</font></i></a>, <a class="txt" href="#E2:25"><i><font color="Green" title="E18">A2</font></i></a>, <a class="ref" href="quaterni.html#T3" target="_self" title="QUATERNI:th.3">Th3</a></i>
;<br/>
<i><font color="Green" title="E20">A4</font></i>: <font color="Maroon" title="c6">y</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font> <a href="quaterni.html#K2" title="QUATERNI:func.2">--&gt;</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font>,<font color="Maroon" title="c7">z</font>,<font color="Maroon" title="c8">w</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">b</font>
<b>by </b><i><a class="txt" href="#E1:25"><i><font color="Green" title="E17">A1</font></i></a>, <a class="ref" href="quaterni.html#T3" target="_self" title="QUATERNI:th.3">Th3</a></i>
<br/>.= 
<font color="Maroon" title="c10">y'</font>
<b>by </b><i><a class="txt" href="#E1:25"><i><font color="Green" title="E17">A1</font></i></a>, <a class="txt" href="#E2:25"><i><font color="Green" title="E18">A2</font></i></a>, <a class="ref" href="quaterni.html#T3" target="_self" title="QUATERNI:th.3">Th3</a></i>
;<br/>
<i><font color="Green" title="E21">A5</font></i>: <font color="Maroon" title="c7">z</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font> <a href="quaterni.html#K2" title="QUATERNI:func.2">--&gt;</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font>,<font color="Maroon" title="c7">z</font>,<font color="Maroon" title="c8">w</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">c</font>
<b>by </b><i><a class="txt" href="#E1:25"><i><font color="Green" title="E17">A1</font></i></a>, <a class="ref" href="quaterni.html#T3" target="_self" title="QUATERNI:th.3">Th3</a></i>
<br/>.= 
<font color="Maroon" title="c11">z'</font>
<b>by </b><i><a class="txt" href="#E1:25"><i><font color="Green" title="E17">A1</font></i></a>, <a class="txt" href="#E2:25"><i><font color="Green" title="E18">A2</font></i></a>, <a class="ref" href="quaterni.html#T3" target="_self" title="QUATERNI:th.3">Th3</a></i>
;<br/>
<font color="Maroon" title="c8">w</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font> <a href="quaterni.html#K2" title="QUATERNI:func.2">--&gt;</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font>,<font color="Maroon" title="c7">z</font>,<font color="Maroon" title="c8">w</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">d</font>
<b>by </b><i><a class="txt" href="#E1:25"><i><font color="Green" title="E17">A1</font></i></a>, <a class="ref" href="quaterni.html#T3" target="_self" title="QUATERNI:th.3">Th3</a></i>
<br/>.= 
<font color="Maroon" title="c12">w'</font>
<b>by </b><i><a class="txt" href="#E1:25"><i><font color="Green" title="E17">A1</font></i></a>, <a class="txt" href="#E2:25"><i><font color="Green" title="E18">A2</font></i></a>, <a class="ref" href="quaterni.html#T3" target="_self" title="QUATERNI:th.3">Th3</a></i>
;<br/>
<b>hence </b><a NAME="E7:25"/>
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x'</font> &amp; <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">y'</font> &amp; <font color="Maroon" title="c7">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">z'</font> &amp; <font color="Maroon" title="c8">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">w'</font> )
 <b>by </b><i><a class="txt" href="#E3:25"><i><font color="Green" title="E19">A3</font></i></a>, <a class="txt" href="#E4:25"><i><font color="Green" title="E20">A4</font></i></a>, <a class="txt" href="#E5:25"><i><font color="Green" title="E21">A5</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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