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<span class="kw">let </span><font color="Maroon" title="c1">x</font>, <font color="Maroon" title="c2">y</font>, <font color="Maroon" title="c3">w</font>, <font color="Maroon" title="c4">z</font>, <font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c6">b</font>, <font color="Maroon" title="c7">c</font>, <font color="Maroon" title="c8">d</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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">w</font>,<font color="Maroon" title="c4">z</font> <a href="zfmisc_1.html#R2" title="ZFMISC_1:pred.2">are_mutually_different</a>  implies ( <span class="p1">(<span class="default">(<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">w</font>,<font color="Maroon" title="c4">z</font>) <a href="quaterni.html#K2" title="QUATERNI:func.2">--&gt;</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c8">d</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> &amp; <span class="p1">(<span class="default">(<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">w</font>,<font color="Maroon" title="c4">z</font>) <a href="quaterni.html#K2" title="QUATERNI:func.2">--&gt;</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c8">d</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font> &amp; <span class="p1">(<span class="default">(<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">w</font>,<font color="Maroon" title="c4">z</font>) <a href="quaterni.html#K2" title="QUATERNI:func.2">--&gt;</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c8">d</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">c</font> &amp; <span class="p1">(<span class="default">(<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">w</font>,<font color="Maroon" title="c4">z</font>) <a href="quaterni.html#K2" title="QUATERNI:func.2">--&gt;</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c8">d</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">d</font> ) )</span><br/>















<span class="kw">assume </span><a NAME="E1:8"/><span class="lab"><font color="Green" title="E6">A1</font></span>: 
<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">w</font>,<font color="Maroon" title="c4">z</font> <a href="zfmisc_1.html#R2" title="ZFMISC_1:pred.2">are_mutually_different</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default">(<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">w</font>,<font color="Maroon" title="c4">z</font>) <a href="quaterni.html#K2" title="QUATERNI:func.2">--&gt;</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c8">d</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> &amp; <span class="p1">(<span class="default">(<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">w</font>,<font color="Maroon" title="c4">z</font>) <a href="quaterni.html#K2" title="QUATERNI:func.2">--&gt;</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c8">d</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font> &amp; <span class="p1">(<span class="default">(<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">w</font>,<font color="Maroon" title="c4">z</font>) <a href="quaterni.html#K2" title="QUATERNI:func.2">--&gt;</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c8">d</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">c</font> &amp; <span class="p1">(<span class="default">(<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">w</font>,<font color="Maroon" title="c4">z</font>) <a href="quaterni.html#K2" title="QUATERNI:func.2">--&gt;</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c8">d</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">d</font> )</span><br/>

<a NAME="E2:8"/><span class="kw">then </span><span class="lab"><font color="Green" title="E7">A2</font></span>: 
<font color="Maroon" title="c1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">w</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#D6" title="ZFMISC_1:def.6">ZFMISC_1:def 6</a></span>;<br/>
<a NAME="E3:8"/><span class="lab"><font color="Green" title="E8">A3</font></span>: 
<font color="Maroon" title="c1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">z</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:8"><span class="lab"><font color="Green" title="E6">A1</font></span></a>, <a class="ref" href="zfmisc_1.html#D6" title="ZFMISC_1:def.6">ZFMISC_1:def 6</a></span>;<br/>
<a NAME="E4:8"/><span class="lab"><font color="Green" title="E9">A4</font></span>: 
<font color="Maroon" title="c2">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">w</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:8"><span class="lab"><font color="Green" title="E6">A1</font></span></a>, <a class="ref" href="zfmisc_1.html#D6" title="ZFMISC_1:def.6">ZFMISC_1:def 6</a></span>;<br/>
<a NAME="E5:8"/><span class="lab"><font color="Green" title="E10">A5</font></span>: 
<font color="Maroon" title="c2">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">z</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:8"><span class="lab"><font color="Green" title="E6">A1</font></span></a>, <a class="ref" href="zfmisc_1.html#D6" title="ZFMISC_1:def.6">ZFMISC_1:def 6</a></span>;<br/>
<a NAME="E6:8"/><span class="lab"><font color="Green" title="E11">A6</font></span>: 
<font color="Maroon" title="c1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:8"><span class="lab"><font color="Green" title="E6">A1</font></span></a>, <a class="ref" href="zfmisc_1.html#D6" title="ZFMISC_1:def.6">ZFMISC_1:def 6</a></span>;<br/>
<a NAME="E7:8"/><span class="lab"><font color="Green" title="E12">A7</font></span>: 
<font color="Maroon" title="c3">w</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">z</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:8"><span class="lab"><font color="Green" title="E6">A1</font></span></a>, <a class="ref" href="zfmisc_1.html#D6" title="ZFMISC_1:def.6">ZFMISC_1:def 6</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c9">f</font> = (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>) <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>);<br/>
<span class="kw">set </span><font color="Maroon" title="c10">g</font> = (<font color="Maroon" title="c3">w</font>,<font color="Maroon" title="c4">z</font>) <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> (<font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c8">d</font>);<br/>
<a NAME="E8:8"/><span class="lab"><font color="Green" title="E13">A8</font></span>: 
<span class="p1">(<span class="default">(<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>) <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:8"><span class="lab"><font color="Green" title="E11">A6</font></span></a>, <a class="ref" href="funct_4.html#T63" title="FUNCT_4:th.63">FUNCT_4:63</a></span>;<br/>
<a NAME="E9:8"/><span class="lab"><font color="Green" title="E14">A9</font></span>: 
<span class="p1">(<span class="default">(<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>) <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#T63" title="FUNCT_4:th.63">FUNCT_4:63</a></span>;<br/>
<a NAME="E10:8"/><span class="lab"><font color="Green" title="E15">A10</font></span>: 
<span class="p1">(<span class="default">(<font color="Maroon" title="c3">w</font>,<font color="Maroon" title="c4">z</font>) <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> (<font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c8">d</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:8"><span class="lab"><font color="Green" title="E12">A7</font></span></a>, <a class="ref" href="funct_4.html#T63" title="FUNCT_4:th.63">FUNCT_4:63</a></span>;<br/>
<a NAME="E11:8"/><span class="lab"><font color="Green" title="E16">A11</font></span>: 
<span class="p1">(<span class="default">(<font color="Maroon" title="c3">w</font>,<font color="Maroon" title="c4">z</font>) <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> (<font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c8">d</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">d</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#T63" title="FUNCT_4:th.63">FUNCT_4:63</a></span>;<br/>
<a NAME="E12:8"/><span class="lab"><font color="Green" title="E17">A12</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c3">w</font>,<font color="Maroon" title="c4">z</font>) <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> (<font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c8">d</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c3">w</font>,<font color="Maroon" title="c4">z</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#T62" title="FUNCT_4:th.62">FUNCT_4:62</a></span>;<br/>
<a NAME="E13:8"/><span class="kw">then </span><span class="lab"><font color="Green" title="E18">A13</font></span>: 
 not <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c3">w</font>,<font color="Maroon" title="c4">z</font>) <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> (<font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c8">d</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:8"><span class="lab"><font color="Green" title="E7">A2</font></span></a>, <a class="txt" href="#E3:8"><span class="lab"><font color="Green" title="E8">A3</font></span></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/>
<a NAME="E14:8"/><span class="lab"><font color="Green" title="E19">A14</font></span>: 
 not <font color="Maroon" title="c2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c3">w</font>,<font color="Maroon" title="c4">z</font>) <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> (<font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c8">d</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:8"><span class="lab"><font color="Green" title="E9">A4</font></span></a>, <a class="txt" href="#E5:8"><span class="lab"><font color="Green" title="E10">A5</font></span></a>, <a class="txt" href="#E12:8"><span class="lab"><font color="Green" title="E17">A12</font></span></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/>
<a NAME="E15:8"/><span class="lab"><font color="Green" title="E20">A15</font></span>: 
<font color="Maroon" title="c3">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c3">w</font>,<font color="Maroon" title="c4">z</font>) <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> (<font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c8">d</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:8"><span class="lab"><font color="Green" title="E17">A12</font></span></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/>
<a NAME="E16:8"/>
<font color="Maroon" title="c4">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default">(<font color="Maroon" title="c3">w</font>,<font color="Maroon" title="c4">z</font>) <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> (<font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c8">d</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:8"><span class="lab"><font color="Green" title="E17">A12</font></span></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E17:8"/>
( <span class="p1">(<span class="default">(<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">w</font>,<font color="Maroon" title="c4">z</font>) <a href="quaterni.html#K2" title="QUATERNI:func.2">--&gt;</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c8">d</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> &amp; <span class="p1">(<span class="default">(<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">w</font>,<font color="Maroon" title="c4">z</font>) <a href="quaterni.html#K2" title="QUATERNI:func.2">--&gt;</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c8">d</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font> &amp; <span class="p1">(<span class="default">(<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">w</font>,<font color="Maroon" title="c4">z</font>) <a href="quaterni.html#K2" title="QUATERNI:func.2">--&gt;</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c8">d</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">c</font> &amp; <span class="p1">(<span class="default">(<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">w</font>,<font color="Maroon" title="c4">z</font>) <a href="quaterni.html#K2" title="QUATERNI:func.2">--&gt;</a> (<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c8">d</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">d</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:8"><span class="lab"><font color="Green" title="E13">A8</font></span></a>, <a class="txt" href="#E9:8"><span class="lab"><font color="Green" title="E14">A9</font></span></a>, <a class="txt" href="#E10:8"><span class="lab"><font color="Green" title="E15">A10</font></span></a>, <a class="txt" href="#E11:8"><span class="lab"><font color="Green" title="E16">A11</font></span></a>, <a class="txt" href="#E13:8"><span class="lab"><font color="Green" title="E18">A13</font></span></a>, <a class="txt" href="#E14:8"><span class="lab"><font color="Green" title="E19">A14</font></span></a>, <a class="txt" href="#E15:8"><span class="lab"><font color="Green" title="E20">A15</font></span></a>, <a class="ref" href="funct_4.html#T11" title="FUNCT_4:th.11">FUNCT_4:11</a>, <a class="ref" href="funct_4.html#T13" title="FUNCT_4:th.13">FUNCT_4:13</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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