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

<b>let </b><font color="Maroon" title="c1">H</font> be   <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</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">H</font> is <a href="zf_lang.html#V2" title="ZF_LANG:attr.2">being_equality</a> or <font color="Maroon" title="c1">H</font> is <a href="zf_lang.html#V3" title="ZF_LANG:attr.3">being_membership</a> or <font color="Maroon" title="c1">H</font> is <a href="zf_lang.html#V4" title="ZF_LANG:attr.4">negative</a> or <font color="Maroon" title="c1">H</font> is <a href="zf_lang.html#V5" title="ZF_LANG:attr.5">conjunctive</a> or <font color="Maroon" title="c1">H</font> is <a href="zf_lang.html#V6" title="ZF_LANG:attr.6">universal</a> )</span><br/>

<a NAME="E1:28"/><i><font color="Green" title="E15">A1</font></i>: 
<font color="Maroon" title="c1">H</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zf_lang.html#K9" title="ZF_LANG:func.9">WFF</a> 
 
<b>by </b><i><a class="ref" href="zf_lang.html#D9" target="_self" title="ZF_LANG:def.9">Def9</a></i>;<br/>
<b>assume </b><a NAME="E2:28"/><i><font color="Green" title="E16">A2</font></i>: 
( not <font color="Maroon" title="c1">H</font> is <a href="zf_lang.html#V2" title="ZF_LANG:attr.2">being_equality</a> &amp; not <font color="Maroon" title="c1">H</font> is <a href="zf_lang.html#V3" title="ZF_LANG:attr.3">being_membership</a> &amp; not <font color="Maroon" title="c1">H</font> is <a href="zf_lang.html#V4" title="ZF_LANG:attr.4">negative</a> &amp; not <font color="Maroon" title="c1">H</font> is <a href="zf_lang.html#V5" title="ZF_LANG:attr.5">conjunctive</a> &amp; not <font color="Maroon" title="c1">H</font> is <a href="zf_lang.html#V6" title="ZF_LANG:attr.6">universal</a> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E3:28"/><i><font color="Green" title="E17">A3</font></i>: 
<a href="zf_lang.html#K9" title="ZF_LANG:func.9">WFF</a>  <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">H</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> is  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> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1"><b>proof </b></a><div class="add">

<a NAME="E1:28_1"/><i><font color="Green" title="E18">A4</font></i>: 
<span class="p1">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_lang.html#K9" title="ZF_LANG:func.9">WFF</a> 
 
<b>by </b><i><a class="ref" href="zf_lang.html#D8" target="_self" title="ZF_LANG:def.8">Def8</a></i>;<br/>
<a NAME="E2:28_1"/>
<span class="p1">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 1</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">H</font>
 
<b>by </b><i><a class="txt" href="#E2:28"><i><font color="Green" title="E16">A2</font></i></a>, <a class="ref" href="zf_lang.html#D10" target="_self" title="ZF_LANG:def.10">Def10</a></i>;<br/>
<a NAME="E3:28_1"/><b>then </b>
not <span class="p1">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">H</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E4:28_1"/>
<a href="zf_lang.html#K9" title="ZF_LANG:func.9">WFF</a>  <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">H</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> is  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> 
 <b>by </b><i><a class="txt" href="#E1:28_1"><i><font color="Green" title="E18">A4</font></i></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0: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/>


</div><b>end;</b></div>
<a NAME="E4:28"/><i><font color="Green" title="E18">A5</font></i>: 
for <font color="Olive" title="b1">a</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="zf_lang.html#K9" title="ZF_LANG:func.9">WFF</a>  <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">H</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> holds <br/><font color="Olive" title="b1">a</font> is    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="zf_lang.html#D8" target="_self" title="ZF_LANG:def.8">Def8</a></i>;<br/>
<div><i><font color="Green" title="E19">A6</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="28_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c3">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"> ( <font color="Maroon" title="c2">x</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="zf_lang.html#K9" title="ZF_LANG:func.9">WFF</a>  <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">H</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c2">x</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="zf_lang.html#K9" title="ZF_LANG:func.9">WFF</a>  <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">H</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )</span><br/><a NAME="E1:28_2"/><i><font color="Green" title="E20">A7</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_lang.html#K9" title="ZF_LANG:func.9">WFF</a> 
 <b>by </b><i><a class="ref" href="zf_lang.html#D8" target="_self" title="ZF_LANG:def.8">Def8</a></i>;<br/><a NAME="E2:28_2"/>
<font color="Maroon" title="c2">x</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <font color="Maroon" title="c3">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">H</font>
 <b>by </b><i><a class="txt" href="#E2:28"><i><font color="Green" title="E16">A2</font></i></a>, <a class="ref" href="zf_lang.html#D10" target="_self" title="ZF_LANG:def.10">Def10</a></i>;<br/><a NAME="E3:28_2"/><b>then </b>
not <font color="Maroon" title="c2">x</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">H</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E4:28_2"/>
<font color="Maroon" title="c2">x</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="zf_lang.html#K9" title="ZF_LANG:func.9">WFF</a>  <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">H</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="#E1:28_2"><i><font color="Green" title="E20">A7</font></i></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">x</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="zf_lang.html#K9" title="ZF_LANG:func.9">WFF</a>  <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">H</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><br/><a NAME="E5:28_2"/><i><font color="Green" title="E21">A8</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_lang.html#K9" title="ZF_LANG:func.9">WFF</a> 
 <b>by </b><i><a class="ref" href="zf_lang.html#D8" target="_self" title="ZF_LANG:def.8">Def8</a></i>;<br/><a NAME="E6:28_2"/>
<font color="Maroon" title="c2">x</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c3">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">H</font>
 <b>by </b><i><a class="txt" href="#E2:28"><i><font color="Green" title="E16">A2</font></i></a>, <a class="ref" href="zf_lang.html#D11" target="_self" title="ZF_LANG:def.11">Def11</a></i>;<br/><a NAME="E7:28_2"/><b>then </b>
not <font color="Maroon" title="c2">x</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">H</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E8:28_2"/>
<font color="Maroon" title="c2">x</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="zf_lang.html#K9" title="ZF_LANG:func.9">WFF</a>  <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">H</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="#E5:28_2"><i><font color="Green" title="E21">A8</font></i></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0: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/></div><b>end;</b></div>
<div><i><font color="Green" title="E20">A9</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="28_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c2">p</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="zf_lang.html#K9" title="ZF_LANG:func.9">WFF</a>  <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">H</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> implies  <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="zf_lang.html#K9" title="ZF_LANG:func.9">WFF</a>  <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">H</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )</span><br/><b>assume </b><a NAME="E1:28_3"/><i><font color="Green" title="E21">A10</font></i>: 
<font color="Maroon" title="c2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="zf_lang.html#K9" title="ZF_LANG:func.9">WFF</a>  <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">H</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="zf_lang.html#K9" title="ZF_LANG:func.9">WFF</a>  <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">H</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><br/><a NAME="E2:28_3"/><b>then </b><i><font color="Green" title="E22">A11</font></i>: 
 <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_lang.html#K9" title="ZF_LANG:func.9">WFF</a> 
 <b>by </b><i><a class="ref" href="zf_lang.html#D8" target="_self" title="ZF_LANG:def.8">Def8</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c3">H1</font> = <font color="Maroon" title="c2">p</font> as   <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a> <b>by </b><i><a class="txt" href="#E1:28_3"><i><font color="Green" title="E21">A10</font></i></a>, <a class="ref" href="zf_lang.html#D9" target="_self" title="ZF_LANG:def.9">Def9</a></i>;<br/><a NAME="E4:28_3"/>
 <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <font color="Maroon" title="c3">H1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">H</font>
 <b>by </b><i><a class="txt" href="#E2:28"><i><font color="Green" title="E16">A2</font></i></a>, <a class="ref" href="zf_lang.html#D12" target="_self" title="ZF_LANG:def.12">Def12</a></i>;<br/><a NAME="E5:28_3"/><b>then </b>
not  <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">H</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E6:28_3"/>
 <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="zf_lang.html#K9" title="ZF_LANG:func.9">WFF</a>  <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">H</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="#E2:28_3"><i><font color="Green" title="E22">A11</font></i></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0: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/></div><b>end;</b></div>
<div><i><font color="Green" title="E21">A12</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="28_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c2">p</font>, <font color="Maroon" title="c3">q</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="zf_lang.html#K9" title="ZF_LANG:func.9">WFF</a>  <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">H</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c3">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="zf_lang.html#K9" title="ZF_LANG:func.9">WFF</a>  <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">H</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> implies <font color="Maroon" title="c2">p</font> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c3">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="zf_lang.html#K9" title="ZF_LANG:func.9">WFF</a>  <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">H</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )</span><br/><b>assume </b><a NAME="E1:28_4"/><i><font color="Green" title="E22">A13</font></i>: 
( <font color="Maroon" title="c2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="zf_lang.html#K9" title="ZF_LANG:func.9">WFF</a>  <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">H</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c3">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="zf_lang.html#K9" title="ZF_LANG:func.9">WFF</a>  <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">H</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">p</font> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c3">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="zf_lang.html#K9" title="ZF_LANG:func.9">WFF</a>  <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">H</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><br/><a NAME="E2:28_4"/><b>then </b><i><font color="Green" title="E23">A14</font></i>: 
<font color="Maroon" title="c2">p</font> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c3">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_lang.html#K9" title="ZF_LANG:func.9">WFF</a> 
 <b>by </b><i><a class="ref" href="zf_lang.html#D8" target="_self" title="ZF_LANG:def.8">Def8</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c4">F</font> = <font color="Maroon" title="c2">p</font>, <font color="Maroon" title="c5">G</font> = <font color="Maroon" title="c3">q</font> as   <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a> <b>by </b><i><a class="txt" href="#E1:28_4"><i><font color="Green" title="E22">A13</font></i></a>, <a class="ref" href="zf_lang.html#D9" target="_self" title="ZF_LANG:def.9">Def9</a></i>;<br/><a NAME="E4:28_4"/>
<font color="Maroon" title="c4">F</font> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c5">G</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">H</font>
 <b>by </b><i><a class="txt" href="#E2:28"><i><font color="Green" title="E16">A2</font></i></a>, <a class="ref" href="zf_lang.html#D13" target="_self" title="ZF_LANG:def.13">Def13</a></i>;<br/><a NAME="E5:28_4"/><b>then </b>
not <font color="Maroon" title="c2">p</font> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c3">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">H</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E6:28_4"/>
<font color="Maroon" title="c2">p</font> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c3">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="zf_lang.html#K9" title="ZF_LANG:func.9">WFF</a>  <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">H</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="#E2:28_4"><i><font color="Green" title="E23">A14</font></i></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0: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/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c2">x</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">p</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="zf_lang.html#K9" title="ZF_LANG:func.9">WFF</a>  <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">H</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> holds <br/> <a href="zf_lang.html#K8" title="ZF_LANG:func.8">All</a> <font color="Maroon" title="c2">x</font>,<font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="zf_lang.html#K9" title="ZF_LANG:func.9">WFF</a>  <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">H</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><br/><b>let </b><font color="Maroon" title="c3">p</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</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">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="zf_lang.html#K9" title="ZF_LANG:func.9">WFF</a>  <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">H</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> implies  <a href="zf_lang.html#K8" title="ZF_LANG:func.8">All</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="zf_lang.html#K9" title="ZF_LANG:func.9">WFF</a>  <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">H</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )</span><br/><b>assume </b><a NAME="E1:28_5"/><i><font color="Green" title="E22">A15</font></i>: 
<font color="Maroon" title="c3">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="zf_lang.html#K9" title="ZF_LANG:func.9">WFF</a>  <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">H</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="zf_lang.html#K8" title="ZF_LANG:func.8">All</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="zf_lang.html#K9" title="ZF_LANG:func.9">WFF</a>  <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">H</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><br/><a NAME="E2:28_5"/><b>then </b><i><font color="Green" title="E23">A16</font></i>: 
 <a href="zf_lang.html#K8" title="ZF_LANG:func.8">All</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_lang.html#K9" title="ZF_LANG:func.9">WFF</a> 
 <b>by </b><i><a class="ref" href="zf_lang.html#D8" target="_self" title="ZF_LANG:def.8">Def8</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c4">H1</font> = <font color="Maroon" title="c3">p</font> as   <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a> <b>by </b><i><a class="txt" href="#E1:28_5"><i><font color="Green" title="E22">A15</font></i></a>, <a class="ref" href="zf_lang.html#D9" target="_self" title="ZF_LANG:def.9">Def9</a></i>;<br/><a NAME="E4:28_5"/>
 <a href="zf_lang.html#K8" title="ZF_LANG:func.8">All</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c4">H1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">H</font>
 <b>by </b><i><a class="txt" href="#E2:28"><i><font color="Green" title="E16">A2</font></i></a>, <a class="ref" href="zf_lang.html#D14" target="_self" title="ZF_LANG:def.14">Def14</a></i>;<br/><a NAME="E5:28_5"/><b>then </b>
not  <a href="zf_lang.html#K8" title="ZF_LANG:func.8">All</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">H</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E6:28_5"/>
 <a href="zf_lang.html#K8" title="ZF_LANG:func.8">All</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="zf_lang.html#K9" title="ZF_LANG:func.9">WFF</a>  <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">H</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="#E2:28_5"><i><font color="Green" title="E23">A16</font></i></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0: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/></div><b>end;</b></div>
<a NAME="E9:28"/><b>then </b>
 <a href="zf_lang.html#K9" title="ZF_LANG:func.9">WFF</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <a href="zf_lang.html#K9" title="ZF_LANG:func.9">WFF</a>  <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">H</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="txt" href="#E3:28"><i><font color="Green" title="E17">A3</font></i></a>, <a class="txt" href="#E4:28"><i><font color="Green" title="E18">A5</font></i></a>, <a class="txt" href="#E5:28"><i><font color="Green" title="E19">A6</font></i></a>, <a class="txt" href="#E6:28"><i><font color="Green" title="E20">A9</font></i></a>, <a class="txt" href="#E7:28"><i><font color="Green" title="E21">A12</font></i></a>, <a class="ref" href="zf_lang.html#D8" target="_self" title="ZF_LANG:def.8">Def8</a></i>;<br/>
<a NAME="E10:28"/><b>then </b>
<font color="Maroon" title="c1">H</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="zf_lang.html#K9" title="ZF_LANG:func.9">WFF</a>  <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">H</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="txt" href="#E1:28"><i><font color="Green" title="E15">A1</font></i></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<a NAME="E11:28"/><b>then </b>
( not <font color="Maroon" title="c1">H</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">H</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c1">H</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">H</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></i>;<br/>
<b>hence </b><a NAME="E12:28"/>
contradiction
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
