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<span class="kw">let </span><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">x9</font>, <font color="Maroon" title="c10">y9</font>, <font color="Maroon" title="c11">z9</font>, <font color="Maroon" title="c12">w9</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</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">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">x9</font>,<font color="Maroon" title="c10">y9</font>,<font color="Maroon" title="c11">z9</font>,<font color="Maroon" title="c12">w9</font> implies ( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x9</font> &amp; <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">y9</font> &amp; <font color="Maroon" title="c7">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">z9</font> &amp; <font color="Maroon" title="c8">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">w9</font> ) )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:26"/><span class="lab"><font color="Green" title="E18">A1</font></span>: 
<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> 
 <span class="kw">and </span><br/><a NAME="E2:26"/><span class="lab"><font color="Green" title="E19">A2</font></span>: 
<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">x9</font>,<font color="Maroon" title="c10">y9</font>,<font color="Maroon" title="c11">z9</font>,<font color="Maroon" title="c12">w9</font>
 ; <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="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x9</font> &amp; <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">y9</font> &amp; <font color="Maroon" title="c7">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">z9</font> &amp; <font color="Maroon" title="c8">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">w9</font> )</span><br/>

<span class="lab"><font color="Green" title="E20">A3</font></span>: <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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26"><span class="lab"><font color="Green" title="E18">A1</font></span></a>, <a class="ref" href="quaterni.html#T3" target="_self" title="QUATERNI:th.3">Th3</a></span>
<br/>.= 
<font color="Maroon" title="c9">x9</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26"><span class="lab"><font color="Green" title="E18">A1</font></span></a>, <a class="txt" href="#E2:26"><span class="lab"><font color="Green" title="E19">A2</font></span></a>, <a class="ref" href="quaterni.html#T3" target="_self" title="QUATERNI:th.3">Th3</a></span>
;<br/>
<span class="lab"><font color="Green" title="E21">A4</font></span>: <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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26"><span class="lab"><font color="Green" title="E18">A1</font></span></a>, <a class="ref" href="quaterni.html#T3" target="_self" title="QUATERNI:th.3">Th3</a></span>
<br/>.= 
<font color="Maroon" title="c10">y9</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26"><span class="lab"><font color="Green" title="E18">A1</font></span></a>, <a class="txt" href="#E2:26"><span class="lab"><font color="Green" title="E19">A2</font></span></a>, <a class="ref" href="quaterni.html#T3" target="_self" title="QUATERNI:th.3">Th3</a></span>
;<br/>
<span class="lab"><font color="Green" title="E22">A5</font></span>: <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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26"><span class="lab"><font color="Green" title="E18">A1</font></span></a>, <a class="ref" href="quaterni.html#T3" target="_self" title="QUATERNI:th.3">Th3</a></span>
<br/>.= 
<font color="Maroon" title="c11">z9</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26"><span class="lab"><font color="Green" title="E18">A1</font></span></a>, <a class="txt" href="#E2:26"><span class="lab"><font color="Green" title="E19">A2</font></span></a>, <a class="ref" href="quaterni.html#T3" target="_self" title="QUATERNI:th.3">Th3</a></span>
;<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26"><span class="lab"><font color="Green" title="E18">A1</font></span></a>, <a class="ref" href="quaterni.html#T3" target="_self" title="QUATERNI:th.3">Th3</a></span>
<br/>.= 
<font color="Maroon" title="c12">w9</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26"><span class="lab"><font color="Green" title="E18">A1</font></span></a>, <a class="txt" href="#E2:26"><span class="lab"><font color="Green" title="E19">A2</font></span></a>, <a class="ref" href="quaterni.html#T3" target="_self" title="QUATERNI:th.3">Th3</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E7:26"/>
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x9</font> &amp; <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">y9</font> &amp; <font color="Maroon" title="c7">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">z9</font> &amp; <font color="Maroon" title="c8">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">w9</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:26"><span class="lab"><font color="Green" title="E20">A3</font></span></a>, <a class="txt" href="#E4:26"><span class="lab"><font color="Green" title="E21">A4</font></span></a>, <a class="txt" href="#E5:26"><span class="lab"><font color="Green" title="E22">A5</font></span></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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