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

<span class="kw">set </span><font color="Maroon" title="c1">E</font> =  <a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>;<br/>
<span class="kw">set </span><font color="Maroon" title="c2">R</font> = <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>;<br/>
<a NAME="E1:74_2_1"/><span class="lab"><font color="Green" title="E28">A1</font></span>: 
(  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">rng</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T160" onmouseover="rs('relat_1/T160')" onmouseout="rh()">RELAT_1:160</a></span>;<br/>
<a NAME="E2:74_2_1"/><span class="kw">then </span>
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> is  <a href="proofs_1.html#V1" title="PROOFS_1:attr.1">finitary</a> 
 
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c3">R</font> = <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> as   <a href="proofs_1.html#NM2" title="PROOFS_1:NM.2">Rule</a> ;<br/>
<a NAME="E4:74_2_1"/>
 <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" onmouseover="rs('ordinal1/D12')" onmouseout="rh()">ORDINAL1:def 12</a></span>;<br/>
<a NAME="E5:74_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A2</font></span>: 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/>
<a NAME="E6:74_2_1"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finsub_1.html#D5" onmouseover="rs('finsub_1/D5')" onmouseout="rh()">FINSUB_1:def 5</a></span>;<br/>
<a NAME="E7:74_2_1"/><span class="kw">then </span>
 <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c3">R</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:74_2_1"><span class="lab"><font color="Green" title="E28">A1</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c4">R</font> = <font color="Maroon" title="c3">R</font> as    <a href="proofs_1.html#M1" title="PROOFS_1:mode.1">Rule</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:74_2_1"><span class="lab"><font color="Green" title="E28">A1</font></span></a>, <a class="ref" href="proofs_1.html#D2" target="_self" onmouseover="rs('proofs_1/D2')" onmouseout="rh()">Def2</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c5">P</font> =  <a href="proofs_1.html#G1" title="PROOFS_1:aggr.1">ProofSystem</a>(# <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>,<span class="p1">(<span class="default"><a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span>,<font color="Maroon" title="c4">R</font> #) as   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="proofs_1.html#L1" title="PROOFS_1:struct.1">ProofSystem</a>  ;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c5">P</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">P</font> is  <a href="proofs_1.html#V10" title="PROOFS_1:attr.10">consistent</a>  &amp;  not <font color="Maroon" title="c5">P</font> is  <a href="proofs_1.html#V12" title="PROOFS_1:attr.12">paraconsistent</a>  )</span><br/>

<span class="kw">thus </span><a NAME="E10:74_2_1"/>
<font color="Maroon" title="c5">P</font> is  <a href="proofs_1.html#V10" title="PROOFS_1:attr.10">consistent</a> 
  <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <font color="Maroon" title="c5">P</font> is  <a href="proofs_1.html#V12" title="PROOFS_1:attr.12">paraconsistent</a> </span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="74_2_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">take </span><font color="Maroon" title="c6">a</font> =  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="proofs_1.html#D19" target="_self" onmouseover="rs('proofs_1/D19')" onmouseout="rh()">PROOFS_1:def 19</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">a</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c5">P</font> &amp;  not <font color="Maroon" title="c5">P</font> <a href="proofs_1.html#R3" title="PROOFS_1:pred.3">|-</a> <font color="Maroon" title="c6">a</font> )</span><br/>

<span class="kw">thus </span><a NAME="E1:74_2_1_1"/>
<font color="Maroon" title="c6">a</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c5">P</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <font color="Maroon" title="c5">P</font> <a href="proofs_1.html#R3" title="PROOFS_1:pred.3">|-</a> <font color="Maroon" title="c6">a</font></span><br/>

<span class="kw">assume </span><a NAME="E2:74_2_1_1"/>
<font color="Maroon" title="c5">P</font> <a href="proofs_1.html#R3" title="PROOFS_1:pred.3">|-</a> <font color="Maroon" title="c6">a</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E3:74_2_1_1"/><span class="kw">then </span>
 <a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>,<font color="Maroon" title="c4">R</font> <a href="proofs_1.html#R2" title="PROOFS_1:pred.2">|-</a> <font color="Maroon" title="c6">a</font>
 
;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">p</font> being   <a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>,<font color="Maroon" title="c4">R</font> <a href="proofs_1.html#V2" title="PROOFS_1:attr.2">-correct</a>  <a href="proofs_1.html#NM4" title="PROOFS_1:NM.4">Formula-sequence</a><span class="kw"> such that </span><br/><a NAME="E5:74_2_1_1"/><span class="lab"><font color="Green" title="E30">A10</font></span>: 
<font color="Maroon" title="c6">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="proofs_1.html#K1" title="PROOFS_1:func.1">rng</a> <font color="Maroon" title="c7">p</font>
 ;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">k</font> =  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">p</font>;<br/>
<a NAME="E6:74_2_1_1"/>
 not <font color="Maroon" title="c7">p</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:74_2_1_1"><span class="lab"><font color="Green" title="E30">A10</font></span></a></span>;<br/>
<a NAME="E7:74_2_1_1"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">p</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T20" onmouseover="rs('finseq_1/T20')" onmouseout="rh()">FINSEQ_1:20</a></span>;<br/>
<a NAME="E8:74_2_1_1"/><span class="kw">then </span>
1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">p</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T1" onmouseover="rs('finseq_1/T1')" onmouseout="rh()">FINSEQ_1:1</a></span>;<br/>
<a NAME="E9:74_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A11</font></span>: 
1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c7">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E10:74_2_1_1"/>
<font color="Maroon" title="c7">p</font> is  <a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>,<font color="Maroon" title="c4">R</font> <a href="proofs_1.html#V2" title="PROOFS_1:attr.2">-correct</a> 
 
;<br/>
<a NAME="E11:74_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">p</font>,1 <a href="proofs_1.html#R1" title="PROOFS_1:pred.1">is_a_correct_step_wrt</a>  <a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>,<font color="Maroon" title="c4">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:74_2_1_1"><span class="lab"><font color="Green" title="E31">A11</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">Q</font> being   <a href="proofs_1.html#NM3" title="PROOFS_1:NM.3">Formula-finset</a><span class="kw"> such that </span><br/><a NAME="E13:74_2_1_1"/><span class="lab"><font color="Green" title="E32">A12</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">Q</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">R</font>
 <span class="kw">and </span><br/><a NAME="E14:74_2_1_1"/><span class="lab"><font color="Green" title="E33">A13</font></span>: 
 for <font color="Olive" title="b1">t</font> being    <a href="proofs_1.html#NM1" title="PROOFS_1:NM.1">Formula</a>   st <font color="Olive" title="b1">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Q</font> holds <br/> ex <font color="Olive" title="b2">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/>( <font color="Olive" title="b2">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c7">p</font> &amp; <font color="Olive" title="b2">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 &amp; <font color="Maroon" title="c7">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">t</font> )
 ;<br/>
<a NAME="E15:74_2_1_1"/>
<font color="Maroon" title="c9">Q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:74_2_1_1"><span class="lab"><font color="Green" title="E32">A12</font></span></a>, <a class="ref" href="zfmisc_1.html#T87" onmouseover="rs('zfmisc_1/T87')" onmouseout="rh()">ZFMISC_1:87</a></span>;<br/>
<a NAME="E16:74_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/>
<a NAME="E17:74_2_1_1"/><span class="kw">then </span>
 <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c9">Q</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E19:74_2_1_1"/><span class="lab"><font color="Green" title="E34">A15</font></span>: 
<font color="Maroon" title="c10">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c7">p</font>
 <span class="kw">and </span><br/><a NAME="E20:74_2_1_1"/><span class="lab"><font color="Green" title="E35">A16</font></span>: 
<font color="Maroon" title="c10">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1
 <span class="kw">and </span><br/><a NAME="E21:74_2_1_1"/>
<font color="Maroon" title="c7">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:74_2_1_1"><span class="lab"><font color="Green" title="E33">A13</font></span></a></span>;<br/>
<a NAME="E22:74_2_1_1"/>
<font color="Maroon" title="c10">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">p</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:74_2_1_1"><span class="lab"><font color="Green" title="E34">A15</font></span></a>, <a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E23:74_2_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:74_2_1_1"><span class="lab"><font color="Green" title="E35">A16</font></span></a>, <a class="ref" href="finseq_1.html#T1" onmouseover="rs('finseq_1/T1')" onmouseout="rh()">FINSEQ_1:1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">reconsider </span><font color="Maroon" title="c6">S</font> = <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> as   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c5">P</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:74_2_1"><span class="lab"><font color="Green" title="E29">A2</font></span></a></span>;<br/>
<a NAME="E12:74_2_1"/>
 for <font color="Olive" title="b1">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">a</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c5">P</font> <a href="proofs_1.html#K9" title="PROOFS_1:func.9">\/</a> <font color="Maroon" title="c6">S</font> holds <br/><font color="Maroon" title="c5">P</font> <a href="proofs_1.html#K9" title="PROOFS_1:func.9">\/</a> <font color="Maroon" title="c6">S</font> <a href="proofs_1.html#R3" title="PROOFS_1:pred.3">|-</a> <font color="Olive" title="b1">a</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="74_2_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">a</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c5">P</font> <a href="proofs_1.html#K9" title="PROOFS_1:func.9">\/</a> <font color="Maroon" title="c6">S</font> implies <font color="Maroon" title="c5">P</font> <a href="proofs_1.html#K9" title="PROOFS_1:func.9">\/</a> <font color="Maroon" title="c6">S</font> <a href="proofs_1.html#R3" title="PROOFS_1:pred.3">|-</a> <font color="Maroon" title="c7">a</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:74_2_1_2"/>
<font color="Maroon" title="c7">a</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c5">P</font> <a href="proofs_1.html#K9" title="PROOFS_1:func.9">\/</a> <font color="Maroon" title="c6">S</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">P</font> <a href="proofs_1.html#K9" title="PROOFS_1:func.9">\/</a> <font color="Maroon" title="c6">S</font> <a href="proofs_1.html#R3" title="PROOFS_1:pred.3">|-</a> <font color="Maroon" title="c7">a</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">k</font> = <font color="Maroon" title="c7">a</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  ;<br/>
<span class="kw">set </span><font color="Maroon" title="c9">p</font> = <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a>,<font color="Maroon" title="c8">k</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>;<br/>
<a NAME="E3:74_2_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a>,<font color="Maroon" title="c8">k</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T44" onmouseover="rs('finseq_1/T44')" onmouseout="rh()">FINSEQ_1:44</a></span>;<br/>
<a NAME="E4:74_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A20</font></span>: 
 <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a>,<font color="Maroon" title="c8">k</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></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">1,2</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T2" onmouseover="rs('finseq_1/T2')" onmouseout="rh()">FINSEQ_1:2</a>, <a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E5:74_2_1_2"/><span class="kw">then </span>
( 2 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a>,<font color="Maroon" title="c8">k</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> &amp; <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a>,<font color="Maroon" title="c8">k</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">k</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D2" onmouseover="rs('tarski/D2')" onmouseout="rh()">TARSKI:def 2</a></span>;<br/>
<a NAME="E6:74_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A22</font></span>: 
<font color="Maroon" title="c8">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="proofs_1.html#K1" title="PROOFS_1:func.1">rng</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a>,<font color="Maroon" title="c8">k</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T3" onmouseover="rs('funct_1/T3')" onmouseout="rh()">FUNCT_1:3</a></span>;<br/>
<a NAME="E7:74_2_1_2"/>
<span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a>,<font color="Maroon" title="c8">k</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> is <font color="Maroon" title="c6">S</font>,<font color="Maroon" title="c4">R</font> <a href="proofs_1.html#V2" title="PROOFS_1:attr.2">-correct</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="74_2_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">m</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="proofs_1.html#D7" target="_self" onmouseover="rs('proofs_1/D7')" onmouseout="rh()">PROOFS_1:def 7</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c10">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a>,<font color="Maroon" title="c8">k</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> implies <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a>,<font color="Maroon" title="c8">k</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>,<font color="Maroon" title="c10">m</font> <a href="proofs_1.html#R1" title="PROOFS_1:pred.1">is_a_correct_step_wrt</a> <font color="Maroon" title="c6">S</font>,<font color="Maroon" title="c4">R</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:74_2_1_2_1"/>
<font color="Maroon" title="c10">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a>,<font color="Maroon" title="c8">k</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a>,<font color="Maroon" title="c8">k</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>,<font color="Maroon" title="c10">m</font> <a href="proofs_1.html#R1" title="PROOFS_1:pred.1">is_a_correct_step_wrt</a> <font color="Maroon" title="c6">S</font>,<font color="Maroon" title="c4">R</font></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="74_2_1_2_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:74_2_1_2_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c10">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c10">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:74_2_1_2"><span class="lab"><font color="Green" title="E30">A20</font></span></a>, <a class="ref" href="tarski.html#D2" onmouseover="rs('tarski/D2')" onmouseout="rh()">TARSKI:def 2</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="74_2_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:74_2_1_2_1_1_1"/>
<font color="Maroon" title="c10">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a>,<font color="Maroon" title="c8">k</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>,<font color="Maroon" title="c10">m</font> <a href="proofs_1.html#R1" title="PROOFS_1:pred.1">is_a_correct_step_wrt</a> <font color="Maroon" title="c6">S</font>,<font color="Maroon" title="c4">R</font></span><br/><div class="add"><a NAME="E2:74_2_1_2_1_1_1"/><span class="kw">then </span>
<span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a>,<font color="Maroon" title="c8">k</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a> 
 ;<br/><span class="kw">hence </span><a NAME="E3:74_2_1_2_1_1_1"/>
<span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a>,<font color="Maroon" title="c8">k</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>,<font color="Maroon" title="c10">m</font> <a href="proofs_1.html#R1" title="PROOFS_1:pred.1">is_a_correct_step_wrt</a> <font color="Maroon" title="c6">S</font>,<font color="Maroon" title="c4">R</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="74_2_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:74_2_1_2_1_1_2"/><span class="lab"><font color="Green" title="E32">A25</font></span>: 
<font color="Maroon" title="c10">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a>,<font color="Maroon" title="c8">k</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>,<font color="Maroon" title="c10">m</font> <a href="proofs_1.html#R1" title="PROOFS_1:pred.1">is_a_correct_step_wrt</a> <font color="Maroon" title="c6">S</font>,<font color="Maroon" title="c4">R</font></span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c11">Q</font> = <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> as   <a href="proofs_1.html#NM3" title="PROOFS_1:NM.3">Formula-finset</a> ;<br/><a NAME="E3:74_2_1_2_1_1_2"/>
<font color="Maroon" title="c11">Q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/><a NAME="E4:74_2_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A26</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">Q</font>,<font color="Maroon" title="c8">k</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">R</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T87" onmouseover="rs('zfmisc_1/T87')" onmouseout="rh()">ZFMISC_1:87</a></span>;<br/><a NAME="E5:74_2_1_2_1_1_2"/>
 for <font color="Olive" title="b1">t</font> being    <a href="proofs_1.html#NM1" title="PROOFS_1:NM.1">Formula</a>   st <font color="Olive" title="b1">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">Q</font> holds <br/> ex <font color="Olive" title="b2">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/>( <font color="Olive" title="b2">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a>,<font color="Maroon" title="c8">k</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> &amp; <font color="Olive" title="b2">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">m</font> &amp; <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a>,<font color="Maroon" title="c8">k</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">t</font> )
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="74_2_1_2_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c12">t</font> be    <a href="proofs_1.html#NM1" title="PROOFS_1:NM.1">Formula</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c12">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">Q</font> implies  ex <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/>( <font color="Olive" title="b1">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a>,<font color="Maroon" title="c8">k</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> &amp; <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">m</font> &amp; <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a>,<font color="Maroon" title="c8">k</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">t</font> ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:74_2_1_2_1_1_2_1"/>
<font color="Maroon" title="c12">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">Q</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/>( <font color="Olive" title="b1">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a>,<font color="Maroon" title="c8">k</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> &amp; <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">m</font> &amp; <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a>,<font color="Maroon" title="c8">k</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">t</font> )</span><br/>

<a NAME="E2:74_2_1_2_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A28</font></span>: 
<font color="Maroon" title="c12">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/>
<span class="kw">take </span><font color="Maroon" title="c13">n</font> = 1; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c13">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a>,<font color="Maroon" title="c8">k</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> &amp; <font color="Maroon" title="c13">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">m</font> &amp; <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a>,<font color="Maroon" title="c8">k</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">t</font> )</span><br/>

<span class="kw">thus </span><a NAME="E3:74_2_1_2_1_1_2_1"/>
<font color="Maroon" title="c13">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a>,<font color="Maroon" title="c8">k</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:74_2_1_2"><span class="lab"><font color="Green" title="E30">A20</font></span></a>, <a class="ref" href="tarski.html#D2" onmouseover="rs('tarski/D2')" onmouseout="rh()">TARSKI:def 2</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c13">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">m</font> &amp; <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a>,<font color="Maroon" title="c8">k</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">t</font> )</span><br/>

<span class="kw">thus </span><a NAME="E4:74_2_1_2_1_1_2_1"/>
<font color="Maroon" title="c13">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:74_2_1_2_1_1_2"><span class="lab"><font color="Green" title="E32">A25</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a>,<font color="Maroon" title="c8">k</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">t</font></span><br/>

<span class="kw">thus </span><a NAME="E5:74_2_1_2_1_1_2_1"/>
<span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a>,<font color="Maroon" title="c8">k</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">t</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:74_2_1_2_1_1_2_1"><span class="lab"><font color="Green" title="E34">A28</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E6:74_2_1_2_1_1_2"/>
<span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a>,<font color="Maroon" title="c8">k</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>,<font color="Maroon" title="c10">m</font> <a href="proofs_1.html#R1" title="PROOFS_1:pred.1">is_a_correct_step_wrt</a> <font color="Maroon" title="c6">S</font>,<font color="Maroon" title="c4">R</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:74_2_1_2_1_1_2"><span class="lab"><font color="Green" title="E32">A25</font></span></a>, <a class="txt" href="#E4:74_2_1_2_1_1_2"><span class="lab"><font color="Green" title="E33">A26</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E8:74_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c6">S</font>,<font color="Maroon" title="c4">R</font> <a href="proofs_1.html#R2" title="PROOFS_1:pred.2">|-</a> <font color="Maroon" title="c8">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:74_2_1_2"><span class="lab"><font color="Green" title="E31">A22</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:74_2_1_2"/>
<font color="Maroon" title="c5">P</font> <a href="proofs_1.html#K9" title="PROOFS_1:func.9">\/</a> <font color="Maroon" title="c6">S</font> <a href="proofs_1.html#R3" title="PROOFS_1:pred.3">|-</a> <font color="Maroon" title="c7">a</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E13:74_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">P</font> <a href="proofs_1.html#K9" title="PROOFS_1:func.9">\/</a> <font color="Maroon" title="c6">S</font> is  <a href="proofs_1.html#NV12" title="PROOFS_1:NV.12">inconsistent</a> 
 
;<br/>
<a NAME="E14:74_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">S</font> is  <a href="proofs_1.html#NV14" title="PROOFS_1:NV.14">inconsistent</a> 
 
;<br/>
<span class="kw">hence </span><a NAME="E15:74_2_1"/>
 not <font color="Maroon" title="c5">P</font> is  <a href="proofs_1.html#V12" title="PROOFS_1:attr.12">paraconsistent</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
