<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">x</font>, <font color="Maroon" title="c2">y</font> be   <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</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">M</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">M</font> <a href="zf_model.html#R2" title="ZF_MODEL:pred.2">|=</a>  <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c2">y</font></span>)</span> iff ( <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">y</font> or for <font color="Olive" title="b2">X</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">M</font> holds <br/><font color="Olive" title="b2">X</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b1">M</font> ) )</span><br/><b>let </b><font color="Maroon" title="c3">M</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <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="c3">M</font> <a href="zf_model.html#R2" title="ZF_MODEL:pred.2">|=</a>  <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c2">y</font></span>)</span> iff ( <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">y</font> or for <font color="Olive" title="b1">X</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">M</font> holds <br/><font color="Olive" title="b1">X</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">M</font> ) )</span><br/>





<b>thus </b><a NAME="E1:76"/>
(  not <font color="Maroon" title="c3">M</font> <a href="zf_model.html#R2" title="ZF_MODEL:pred.2">|=</a>  <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c2">y</font></span>)</span> or <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">y</font> or for <font color="Olive" title="b1">X</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">M</font> holds <br/><font color="Olive" title="b1">X</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">M</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">y</font> or for <font color="Olive" title="b1">X</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">M</font> holds <br/><font color="Olive" title="b1">X</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">M</font> ) implies <font color="Maroon" title="c3">M</font> <a href="zf_model.html#R2" title="ZF_MODEL:pred.2">|=</a>  <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c2">y</font></span>)</span> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:76_1"/><i><font color="Green" title="E30">A1</font></i>: 
( ( for <font color="Olive" title="b1">v</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ,<font color="Maroon" title="c3">M</font> holds  <font color="Maroon" title="c3">M</font>,<font color="Olive" title="b1">v</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c2">y</font></span>)</span> ) &amp; <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> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="zf_model.html#D5" title="ZF_MODEL:def.5">ZF_MODEL:def 5</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">X</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">M</font> holds <br/><font color="Olive" title="b1">X</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">M</font></span><br/>

<b>let </b><font color="Maroon" title="c4">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="c4">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">M</font> implies <font color="Maroon" title="c4">X</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">M</font> )</span><br/>

<b>assume </b><a NAME="E2:76_1"/>
<font color="Maroon" title="c4">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">M</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">X</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">M</font></span><br/>

<b>then reconsider </b><font color="Maroon" title="c5">m</font> = <font color="Maroon" title="c4">X</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">M</font> ;<br/>
<b>assume </b><a NAME="E4:76_1"/><i><font color="Green" title="E31">A2</font></i>: 
<font color="Maroon" title="c4">X</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c3">M</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<b>consider </b><font color="Maroon" title="c6">a</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">X</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c3">M</font>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">a</font> = <font color="Maroon" title="c6">a</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">M</font> <b>by </b><i><a class="txt" href="#E4:76_1"><i><font color="Green" title="E31">A2</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">v</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ,<font color="Maroon" title="c3">M</font>;<br/>
<a NAME="E8:76_1"/>
<font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">v</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c7">a</font></span>)</span> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> <font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c5">m</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c2">y</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E1:76_1"><i><font color="Green" title="E30">A1</font></i></a></i>;<br/>
<a NAME="E9:76_1"/><b>then </b>
not <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">v</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c7">a</font></span>)</span> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> <font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c5">m</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c1">x</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c2">y</font>
 
<b>by </b><i><a class="ref" href="zf_model.html#T14" title="ZF_MODEL:th.14">ZF_MODEL:14</a></i>;<br/>
<a NAME="E10:76_1"/><b>then </b>
( not <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">v</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c7">a</font></span>)</span> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> <font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c5">m</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">v</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c7">a</font></span>)</span> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> <font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c5">m</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">y</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c8">v</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c7">a</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">a</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">v</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c7">a</font></span>)</span> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> <font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c5">m</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">m</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">v</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c7">a</font></span>)</span> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> <font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c5">m</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">v</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c7">a</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">x</font> )
 
<b>by </b><i><a class="txt" href="#E1:76_1"><i><font color="Green" title="E30">A1</font></i></a>, <a class="ref" href="funct_7.html#T34" title="FUNCT_7:th.34">FUNCT_7:34</a>, <a class="ref" href="funct_7.html#T130" title="FUNCT_7:th.130">FUNCT_7:130</a>, <a class="ref" href="zf_model.html#T13" title="ZF_MODEL:th.13">ZF_MODEL:13</a></i>;<br/>
<b>hence </b><a NAME="E11:76_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E4:76_1"><i><font color="Green" title="E31">A2</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</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="hs2(this)" href="javascript:()" title="76_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:76_2"/><i><font color="Green" title="E30">A3</font></i>: 
for <font color="Olive" title="b1">X</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">M</font> holds <br/><font color="Olive" title="b1">X</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">M</font>
 ; <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">v</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ,<font color="Maroon" title="c3">M</font> holds  <font color="Maroon" title="c3">M</font>,<font color="Olive" title="b1">v</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c2">y</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c4">v</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a> ,<font color="Maroon" title="c3">M</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">v</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c2">y</font></span>)</span></span><br/><a NAME="E2:76_2"/>
<font color="Maroon" title="c4">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">y</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">M</font>
 <b>by </b><i><a class="txt" href="#E1:76_2"><i><font color="Green" title="E30">A3</font></i></a></i>;<br/><a NAME="E3:76_2"/><b>then </b>
not <font color="Maroon" title="c4">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">y</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/><a NAME="E4:76_2"/><b>then </b>
not <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">v</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c1">x</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c2">y</font>
 <b>by </b><i><a class="ref" href="zf_model.html#T13" title="ZF_MODEL:th.13">ZF_MODEL:13</a></i>;<br/><b>hence </b><a NAME="E5:76_2"/>
<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">v</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c2">y</font></span>)</span>
 <b>by </b><i><a class="ref" href="zf_model.html#T14" title="ZF_MODEL:th.14">ZF_MODEL:14</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>
<b>hence </b><a NAME="E3:76"/>
( ( <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">y</font> or for <font color="Olive" title="b1">X</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">M</font> holds <br/><font color="Olive" title="b1">X</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">M</font> ) implies <font color="Maroon" title="c3">M</font> <a href="zf_model.html#R2" title="ZF_MODEL:pred.2">|=</a>  <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c2">y</font></span>)</span> )
 <b>by </b><i><a class="ref" href="zf_lang1.html#T92" target="_self" title="ZF_LANG1:th.92">Th92</a>, <a class="ref" href="zf_model.html#D5" title="ZF_MODEL:def.5">ZF_MODEL:def 5</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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