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<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">F</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8">QC-WFF</a> ;<br/><b>let </b><font color="Maroon">G</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8">QC-WFF</a> ;<br/>





<b>assume </b><a NAME="E1:30"/><i><font color="Green">E22</font></i>: 
( <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K1" target="_self">list_of_immediate_constituents</a> <font color="Maroon">F</font></span>)</span> &amp; <font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K1" target="_self">list_of_immediate_constituents</a> <font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> )
 ;<br/>

<a NAME="E2:30"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">F</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="qc_lang1.html#K11">VERUM</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:30_1"/>
<font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="qc_lang1.html#K11">VERUM</a> 
 ;<br/>

<a NAME="E2:30_1"/><b>then </b>
 <a href="qc_lang4.html#K1" target="_self">list_of_immediate_constituents</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K6">&lt;*&gt;</a> <a href="qc_lang1.html#K8">QC-WFF</a> 
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:30_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T26">FINSEQ_1:26</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:30"/><i><font color="Green">E30</font></i>: 
not <font color="Maroon">F</font> is <a href="qc_lang1.html#V2">atomic</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:30_2"/>
<font color="Maroon">F</font> is <a href="qc_lang1.html#V2">atomic</a>
 ;<br/>

<a NAME="E2:30_2"/><b>then </b>
 <a href="qc_lang4.html#K1" target="_self">list_of_immediate_constituents</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K6">&lt;*&gt;</a> <a href="qc_lang1.html#K8">QC-WFF</a> 
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:30_2"/>
contradiction
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T26">FINSEQ_1:26</a></i>;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_3"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:30_3"/>
( <font color="Maroon">F</font> is <a href="qc_lang1.html#V3">negative</a> or <font color="Maroon">F</font> is <a href="qc_lang1.html#V4">conjunctive</a> or <font color="Maroon">F</font> is <a href="qc_lang1.html#V5">universal</a> )
 </span><b>by </b><i>, <a class="txt" href="qc_lang4.html#E2:30"><i><font color="Green">E29</font></i></a>, <a class="ref" href="qc_lang1.html#T33">QC_LANG1:33</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_3_1"><b>suppose </b></a><a NAME="E1:30_3_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">F</font> is <a href="qc_lang1.html#V3">negative</a>
 ;<br/><div class="add"><a NAME="E2:30_3_1"/><b>then </b><i><font color="Green">E35</font></i>: 
 <a href="qc_lang4.html#K1" target="_self">list_of_immediate_constituents</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="qc_lang1.html#K17">the_argument_of</a> <font color="Maroon">F</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i/>;<br/><a NAME="E3:30_3_1"/><b>then </b>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 1
 <b>by </b><i>, <a class="ref" href="finseq_1.html#D8">FINSEQ_1:def 8</a></i>;<br/><a NAME="E4:30_3_1"/><b>then </b>
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E5:30_3_1"/><b>then </b>
<font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="qc_lang1.html#K17">the_argument_of</a> <font color="Maroon">F</font>
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#D8">FINSEQ_1:def 8</a></i>;<br/><b>hence </b><a NAME="E6:30_3_1"/>
<font color="Maroon">G</font> <a href="qc_lang2.html#R1">is_immediate_constituent_of</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="txt" href="qc_lang4.html#E3:30"><i><font color="Green">E30</font></i></a>, <a class="ref" href="qc_lang2.html#T65">QC_LANG2:65</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_3_2"><b>suppose </b></a><a NAME="E1:30_3_2"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">F</font> is <a href="qc_lang1.html#V4">conjunctive</a>
 ;<br/><div class="add"><a NAME="E2:30_3_2"/><b>then </b><i><font color="Green">E39</font></i>: 
 <a href="qc_lang4.html#K1" target="_self">list_of_immediate_constituents</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="qc_lang1.html#K18">the_left_argument_of</a> <font color="Maroon">F</font></span>)</span>,<span class="p2">(<span class="default"><a href="qc_lang1.html#K19">the_right_argument_of</a> <font color="Maroon">F</font></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span>
 <b>by </b><i/>;<br/><a NAME="E3:30_3_2"/><i><font color="Green">E40</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="qc_lang1.html#K18">the_left_argument_of</a> <font color="Maroon">F</font></span>)</span>,<span class="p2">(<span class="default"><a href="qc_lang1.html#K19">the_right_argument_of</a> <font color="Maroon">F</font></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="hidden.html#R1">=</a> 2 &amp; <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="qc_lang1.html#K18">the_left_argument_of</a> <font color="Maroon">F</font></span>)</span>,<span class="p2">(<span class="default"><a href="qc_lang1.html#K19">the_right_argument_of</a> <font color="Maroon">F</font></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a>  <a href="qc_lang1.html#K18">the_left_argument_of</a> <font color="Maroon">F</font> &amp; <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="qc_lang1.html#K18">the_left_argument_of</a> <font color="Maroon">F</font></span>)</span>,<span class="p2">(<span class="default"><a href="qc_lang1.html#K19">the_right_argument_of</a> <font color="Maroon">F</font></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R1">=</a>  <a href="qc_lang1.html#K19">the_right_argument_of</a> <font color="Maroon">F</font> )
 <b>by </b><i><a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>;<br/><a NAME="E4:30_3_2"/><b>then </b><i><font color="Green">E42</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">1,2</span><a href="tarski.html#K2">}</a></span>
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_3_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_3_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:30_3_2_1_1"/>
( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 2 )
 </span><b>by </b><i>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_3_2_1_1_1"><b>suppose </b></a><a NAME="E1:30_3_2_1_1_1"/>
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:30_3_2_1_1_1"/>
<font color="Maroon">G</font> <a href="qc_lang2.html#R1">is_immediate_constituent_of</a> <font color="Maroon">F</font>
 <b>by </b><i>, <a class="txt" href="qc_lang4.html#E3:30"><i><font color="Green">E30</font></i></a>, , , <a class="ref" href="qc_lang2.html#T66">QC_LANG2:66</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_3_2_1_1_2"><b>suppose </b></a><a NAME="E1:30_3_2_1_1_2"/>
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 2
 ;<br/><div class="add"><b>hence </b><a NAME="E2:30_3_2_1_1_2"/>
<font color="Maroon">G</font> <a href="qc_lang2.html#R1">is_immediate_constituent_of</a> <font color="Maroon">F</font>
 <b>by </b><i>, <a class="txt" href="qc_lang4.html#E3:30"><i><font color="Green">E30</font></i></a>, , , <a class="ref" href="qc_lang2.html#T66">QC_LANG2:66</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E6:30_3_2"/>
<font color="Maroon">G</font> <a href="qc_lang2.html#R1">is_immediate_constituent_of</a> <font color="Maroon">F</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_3_3"><b>suppose </b></a><a NAME="E1:30_3_3"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">F</font> is <a href="qc_lang1.html#V5">universal</a>
 ;<br/><div class="add"><a NAME="E2:30_3_3"/><b>then </b>
( not <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="qc_lang1.html#K11">VERUM</a>  &amp; not <font color="Maroon">F</font> is <a href="qc_lang1.html#V2">atomic</a> &amp; not <font color="Maroon">F</font> is <a href="qc_lang1.html#V3">negative</a> &amp; not <font color="Maroon">F</font> is <a href="qc_lang1.html#V4">conjunctive</a> )
 <b>by </b><i><a class="ref" href="qc_lang1.html#T51">QC_LANG1:51</a></i>;<br/><a NAME="E3:30_3_3"/><b>then </b><i><font color="Green">E44</font></i>: 
 <a href="qc_lang4.html#K1" target="_self">list_of_immediate_constituents</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="qc_lang1.html#K21">the_scope_of</a> <font color="Maroon">F</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i/>;<br/><a NAME="E4:30_3_3"/><b>then </b>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 1
 <b>by </b><i>, <a class="ref" href="finseq_1.html#D8">FINSEQ_1:def 8</a></i>;<br/><a NAME="E5:30_3_3"/><b>then </b>
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E6:30_3_3"/><b>then </b>
<font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="qc_lang1.html#K21">the_scope_of</a> <font color="Maroon">F</font>
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#D8">FINSEQ_1:def 8</a></i>;<br/><b>hence </b><a NAME="E7:30_3_3"/>
<font color="Maroon">G</font> <a href="qc_lang2.html#R1">is_immediate_constituent_of</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="txt" href="qc_lang4.html#E3:30"><i><font color="Green">E30</font></i></a>, <a class="ref" href="qc_lang2.html#T67">QC_LANG2:67</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
