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<div class="add">

<b>thus </b><a NAME="E1:26"/>
 <a href="scmfsa_1.html#K5" title="SCMFSA_1:func.5">NAT</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> 
 <b>by </b><i><a class="ref" href="numbers.html#T27" title="NUMBERS:th.27">NUMBERS:27</a></i>;<br/>

<a NAME="E2:26"/><i><font color="Green" title="E15">A1</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds not 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
;<br/>
<a NAME="E3:26"/>
2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="scmfsa_1.html#K5" title="SCMFSA_1:func.5">NAT</a> 
 
;<br/>
<b>hence </b><a NAME="E4:26"/>
 <a href="scmfsa_1.html#K6" title="SCMFSA_1:func.6">SCM+FSA-Instr</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> 
 <b>by </b><i><a class="txt" href="scmfsa_1.html#E2:26"><i><font color="Green" title="E15">A1</font></i></a>, <a class="ref" href="numbers.html#T17" title="NUMBERS:th.17">NUMBERS:17</a>, <a class="ref" href="relat_1.html#D1" title="RELAT_1:def.1">RELAT_1:def 1</a></i>;<br/>

<b>thus </b><a NAME="E5:26"/>
 <a href="scmfsa_1.html#K5" title="SCMFSA_1:func.5">NAT</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="scmfsa_1.html#K6" title="SCMFSA_1:func.6">SCM+FSA-Instr</a> 
 <b>by </b><i><a class="txt" href="scmfsa_1.html#E2:26"><i><font color="Green" title="E15">A1</font></i></a>, <a class="ref" href="relat_1.html#D1" title="RELAT_1:def.1">RELAT_1:def 1</a></i>;<br/>

<a NAME="E6:26"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> 
 
<b>by </b><i><a class="ref" href="int_1.html#D1" title="INT_1:def.1">INT_1:def 1</a></i>;<br/>
<a NAME="E7:26"/><b>then </b>
<span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> is    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#T95" title="FINSEQ_1:th.95">FINSEQ_1:95</a></i>;<br/>
<a NAME="E8:26"/><b>then </b><i><font color="Green" title="E16">A2</font></i>: 
<span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>  <a href="finseq_1.html#K13" title="FINSEQ_1:func.13">*</a> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#D11" title="FINSEQ_1:def.11">FINSEQ_1:def 11</a></i>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:26_1"/>
 <a href="scmfsa_1.html#K5" title="SCMFSA_1:func.5">NAT</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>  <a href="finseq_1.html#K13" title="FINSEQ_1:func.13">*</a> 
 ;<br/><a NAME="E2:26_1"/><b>then </b><i><font color="Green" title="E17">A3</font></i>: 
 ex <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>  st <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">k</font>
 <b>by </b><i><a class="txt" href="scmfsa_1.html#E8:26"><i><font color="Green" title="E16">A2</font></i></a></i>;<br/><a NAME="E3:26_1"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="ref" href="card_1.html#T79" title="CARD_1:th.79">CARD_1:79</a></i>;<br/><a NAME="E4:26_1"/><b>then </b>
<span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</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"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="scmfsa_1.html#E2:26_1"><i><font color="Green" title="E17">A3</font></i></a>, <a class="ref" href="card_1.html#T66" title="CARD_1:th.66">CARD_1:66</a>, <a class="ref" href="card_1.html#T87" title="CARD_1:th.87">CARD_1:87</a></i>;<br/><b>hence </b><a NAME="E5:26_1"/>
contradiction
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T6" title="ZFMISC_1:th.6">ZFMISC_1:6</a></i>;<br/></div>
<b>end;</b></div>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:26_2"/>
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="scmfsa_1.html#K6" title="SCMFSA_1:func.6">SCM+FSA-Instr</a> 
 ;<br/><b>then consider </b><font color="Maroon" title="c1">x</font>, <font color="Maroon" title="c2">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:26_2"/><i><font color="Green" title="E17">A4</font></i>: 
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="ref" href="relat_1.html#D1" title="RELAT_1:def.1">RELAT_1:def 1</a></i>;<br/><b>thus </b><a NAME="E4:26_2"/>
contradiction
 <b>by </b><i><a class="txt" href="scmfsa_1.html#E3:26_2"><i><font color="Green" title="E17">A4</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E11:26"/>
 <a href="scmfsa_1.html#K6" title="SCMFSA_1:func.6">SCM+FSA-Instr</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>  <a href="finseq_1.html#K13" title="FINSEQ_1:func.13">*</a> 
 <b>by </b><i><a class="ref" href="finseq_1.html#T66" title="FINSEQ_1:th.66">FINSEQ_1:66</a></i>;<br/>


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