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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">e</font>, <font color="Maroon" title="c6">i</font>, <font color="Maroon" title="c7">j</font>, <font color="Maroon" title="c8">k</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"> for <font color="Olive" title="b1">g</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st <font color="Maroon" title="c1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">b</font> &amp; <font color="Maroon" title="c3">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">d</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><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></span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span> &amp; <font color="Olive" title="b1">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">e</font> &amp; <font color="Olive" title="b1">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">i</font> &amp; <font color="Olive" title="b1">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">j</font> &amp; <font color="Olive" title="b1">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">k</font> holds <br/><font color="Olive" title="b1">g</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="c5">e</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font>,<font color="Maroon" title="c8">k</font></span><br/><b>let </b><font color="Maroon" title="c9">h</font> be   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</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> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">b</font> &amp; <font color="Maroon" title="c3">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">d</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c9">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><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></span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span> &amp; <font color="Maroon" title="c9">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">e</font> &amp; <font color="Maroon" title="c9">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">i</font> &amp; <font color="Maroon" title="c9">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">j</font> &amp; <font color="Maroon" title="c9">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">k</font> implies <font color="Maroon" title="c9">h</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="c5">e</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font>,<font color="Maroon" title="c8">k</font> )</span><br/>



<b>assume </b><a NAME="E1:18"/><i><font color="Green" title="E9">A1</font></i>: 
( <font color="Maroon" title="c1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">b</font> &amp; <font color="Maroon" title="c3">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">d</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c9">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><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></span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span> &amp; <font color="Maroon" title="c9">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">e</font> &amp; <font color="Maroon" title="c9">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">i</font> &amp; <font color="Maroon" title="c9">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">j</font> &amp; <font color="Maroon" title="c9">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">k</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c9">h</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="c5">e</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font>,<font color="Maroon" title="c8">k</font></span><br/>

<b>set </b><font color="Maroon" title="c10">f</font> = <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font> <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> <font color="Maroon" title="c5">e</font>,<font color="Maroon" title="c6">i</font>;<br/>
<b>set </b><font color="Maroon" title="c11">g</font> = <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font> <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> <font color="Maroon" title="c7">j</font>,<font color="Maroon" title="c8">k</font>;<br/>
<a NAME="E2:18"/><i><font color="Green" title="E10">A2</font></i>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font> <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> <font color="Maroon" title="c5">e</font>,<font color="Maroon" title="c6">i</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="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font> <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> <font color="Maroon" title="c7">j</font>,<font color="Maroon" title="c8">k</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">c</font>,<font color="Maroon" title="c4">d</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> )
 
<b>by </b><i><a class="ref" href="funct_4.html#T65" title="FUNCT_4:th.65">FUNCT_4:65</a></i>;<br/>
<a NAME="E3:18"/><b>then </b><i><font color="Green" title="E11">A3</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c9">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font> <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> <font color="Maroon" title="c5">e</font>,<font color="Maroon" title="c6">i</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font> <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> <font color="Maroon" title="c7">j</font>,<font color="Maroon" title="c8">k</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="quaterni.html#E1:18"><i><font color="Green" title="E9">A1</font></i></a>, <a class="ref" href="enumset1.html#T45" title="ENUMSET1:th.45">ENUMSET1:45</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c12">x</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="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font> <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> <font color="Maroon" title="c5">e</font>,<font color="Maroon" title="c6">i</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font> <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> <font color="Maroon" title="c7">j</font>,<font color="Maroon" title="c8">k</font></span>)</span></span>)</span> implies ( ( <font color="Maroon" title="c12">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">c</font>,<font color="Maroon" title="c4">d</font> <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> <font color="Maroon" title="c7">j</font>,<font color="Maroon" title="c8">k</font></span>)</span> implies <font color="Maroon" title="c9">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font> <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> <font color="Maroon" title="c7">j</font>,<font color="Maroon" title="c8">k</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font> ) &amp; ( not <font color="Maroon" title="c12">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">c</font>,<font color="Maroon" title="c4">d</font> <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> <font color="Maroon" title="c7">j</font>,<font color="Maroon" title="c8">k</font></span>)</span> implies <font color="Maroon" title="c9">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font> <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> <font color="Maroon" title="c5">e</font>,<font color="Maroon" title="c6">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font> ) ) )</span><br/><b>assume </b><a NAME="E1:18_1"/><i><font color="Green" title="E12">A4</font></i>: 
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font> <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> <font color="Maroon" title="c5">e</font>,<font color="Maroon" title="c6">i</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font> <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> <font color="Maroon" title="c7">j</font>,<font color="Maroon" title="c8">k</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( <font color="Maroon" title="c12">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">c</font>,<font color="Maroon" title="c4">d</font> <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> <font color="Maroon" title="c7">j</font>,<font color="Maroon" title="c8">k</font></span>)</span> implies <font color="Maroon" title="c9">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font> <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> <font color="Maroon" title="c7">j</font>,<font color="Maroon" title="c8">k</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font> ) &amp; ( not <font color="Maroon" title="c12">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">c</font>,<font color="Maroon" title="c4">d</font> <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> <font color="Maroon" title="c7">j</font>,<font color="Maroon" title="c8">k</font></span>)</span> implies <font color="Maroon" title="c9">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font> <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> <font color="Maroon" title="c5">e</font>,<font color="Maroon" title="c6">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font> ) )</span><br/><b>thus </b><a NAME="E2:18_1"/>
( <font color="Maroon" title="c12">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">c</font>,<font color="Maroon" title="c4">d</font> <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> <font color="Maroon" title="c7">j</font>,<font color="Maroon" title="c8">k</font></span>)</span> implies <font color="Maroon" title="c9">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font> <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> <font color="Maroon" title="c7">j</font>,<font color="Maroon" title="c8">k</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( not <font color="Maroon" title="c12">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">c</font>,<font color="Maroon" title="c4">d</font> <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> <font color="Maroon" title="c7">j</font>,<font color="Maroon" title="c8">k</font></span>)</span> implies <font color="Maroon" title="c9">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font> <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> <font color="Maroon" title="c5">e</font>,<font color="Maroon" title="c6">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:18_1_1"/><i><font color="Green" title="E13">A5</font></i>: 
<font color="Maroon" title="c12">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">c</font>,<font color="Maroon" title="c4">d</font> <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> <font color="Maroon" title="c7">j</font>,<font color="Maroon" title="c8">k</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c9">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font> <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> <font color="Maroon" title="c7">j</font>,<font color="Maroon" title="c8">k</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:18_1_1_1"/>
( <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c</font> or <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">d</font> )
 </span><b>by </b><i><a class="txt" href="quaterni.html#E2:18"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="quaterni.html#E1:18_1_1"><i><font color="Green" title="E13">A5</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_1_1_1"><b>suppose </b></a><a NAME="E1:18_1_1_1_1"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c9">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font> <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> <font color="Maroon" title="c7">j</font>,<font color="Maroon" title="c8">k</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font></span><br/><div class="add"><b>hence </b><a NAME="E2:18_1_1_1_1"/>
<font color="Maroon" title="c9">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font> <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> <font color="Maroon" title="c7">j</font>,<font color="Maroon" title="c8">k</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font>
 <b>by </b><i><a class="txt" href="quaterni.html#E1:18"><i><font color="Green" title="E9">A1</font></i></a>, <a class="ref" href="funct_4.html#T66" title="FUNCT_4:th.66">FUNCT_4:66</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_1_1_2"><b>suppose </b></a><a NAME="E1:18_1_1_1_2"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">d</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c9">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font> <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> <font color="Maroon" title="c7">j</font>,<font color="Maroon" title="c8">k</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font></span><br/><div class="add"><b>hence </b><a NAME="E2:18_1_1_1_2"/>
<font color="Maroon" title="c9">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font> <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> <font color="Maroon" title="c7">j</font>,<font color="Maroon" title="c8">k</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font>
 <b>by </b><i><a class="txt" href="quaterni.html#E1:18"><i><font color="Green" title="E9">A1</font></i></a>, <a class="ref" href="funct_4.html#T66" title="FUNCT_4:th.66">FUNCT_4:66</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>thus </b><a NAME="E3:18_1"/>
( not <font color="Maroon" title="c12">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">c</font>,<font color="Maroon" title="c4">d</font> <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> <font color="Maroon" title="c7">j</font>,<font color="Maroon" title="c8">k</font></span>)</span> implies <font color="Maroon" title="c9">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font> <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> <font color="Maroon" title="c5">e</font>,<font color="Maroon" title="c6">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:18_1_2"/>
not <font color="Maroon" title="c12">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">c</font>,<font color="Maroon" title="c4">d</font> <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> <font color="Maroon" title="c7">j</font>,<font color="Maroon" title="c8">k</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c9">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font> <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> <font color="Maroon" title="c5">e</font>,<font color="Maroon" title="c6">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font></span><br/>

<a NAME="E2:18_1_2"/><b>then </b><i><font color="Green" title="E13">A6</font></i>: 
<font color="Maroon" title="c12">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="c1">a</font>,<font color="Maroon" title="c2">b</font> <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> <font color="Maroon" title="c5">e</font>,<font color="Maroon" title="c6">i</font></span>)</span>
 
<b>by </b><i><a class="txt" href="quaterni.html#E1:18_1"><i><font color="Green" title="E12">A4</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:18_1_2_1"/>
( <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">a</font> or <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">b</font> )
 </span><b>by </b><i><a class="txt" href="quaterni.html#E2:18"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="quaterni.html#E2:18_1_2"><i><font color="Green" title="E13">A6</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_2_1_1"><b>suppose </b></a><a NAME="E1:18_1_2_1_1"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">a</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c9">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font> <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> <font color="Maroon" title="c5">e</font>,<font color="Maroon" title="c6">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font></span><br/><div class="add"><b>hence </b><a NAME="E2:18_1_2_1_1"/>
<font color="Maroon" title="c9">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font> <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> <font color="Maroon" title="c5">e</font>,<font color="Maroon" title="c6">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font>
 <b>by </b><i><a class="txt" href="quaterni.html#E1:18"><i><font color="Green" title="E9">A1</font></i></a>, <a class="ref" href="funct_4.html#T66" title="FUNCT_4:th.66">FUNCT_4:66</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_2_1_2"><b>suppose </b></a><a NAME="E1:18_1_2_1_2"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c9">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font> <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> <font color="Maroon" title="c5">e</font>,<font color="Maroon" title="c6">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font></span><br/><div class="add"><b>hence </b><a NAME="E2:18_1_2_1_2"/>
<font color="Maroon" title="c9">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font> <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> <font color="Maroon" title="c5">e</font>,<font color="Maroon" title="c6">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font>
 <b>by </b><i><a class="txt" href="quaterni.html#E1:18"><i><font color="Green" title="E9">A1</font></i></a>, <a class="ref" href="funct_4.html#T66" title="FUNCT_4:th.66">FUNCT_4:66</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E5:18"/>
<font color="Maroon" title="c9">h</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="c5">e</font>,<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font>,<font color="Maroon" title="c8">k</font>
 <b>by </b><i><a class="txt" href="quaterni.html#E3:18"><i><font color="Green" title="E11">A3</font></i></a>, <a class="ref" href="funct_4.html#D1" title="FUNCT_4:def.1">FUNCT_4:def 1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
