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

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

<b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <a href="qc_lang1.html#K11" target="_self">VERUM</a> </span>)</span> <a href="funct_1.html#K1">.</a> 1</span>)</span> <a href="mcart_1.html#K1">`1</a>  = 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,0</span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K1">`1</a> 
<b>by </b><i><a class="ref" href="finseq_1.html#D8">FINSEQ_1:def 8</a></i>
<br/>.= 
0
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
;<br/>

<b>thus </b><a NAME="E2:43"/>
( <font color="Maroon">F</font> is <a href="qc_lang1.html#V2" target="_self">atomic</a> implies  ex <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> 1 is   <a href="qc_lang1.html#NM6">QC-pred_symbol</a> of <font color="Olive">k</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:43_2"/>
<font color="Maroon">F</font> is <a href="qc_lang1.html#V2" target="_self">atomic</a>
 ;<br/>

<b>then consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">P</font> being   <a href="qc_lang1.html#NM6">QC-pred_symbol</a> of <font color="Maroon">k</font>, <font color="Maroon">ll</font> being    <a href="qc_lang1.html#M1" target="_self">QC-variable_list</a> of <font color="Maroon">k</font><b> such that </b><br/><a NAME="E3:43_2"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="qc_lang1.html#K9" target="_self">!</a> <font color="Maroon">ll</font>
 <b>by </b><i/>;<br/>
<a NAME="E4:43_2"/>
 <a href="qc_lang1.html#K10" target="_self">@</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"><font color="Maroon">P</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">ll</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:43_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">P</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T58">FINSEQ_1:58</a></i>;<br/>
<b>hence </b><a NAME="E6:43_2"/>
 ex <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> 1 is   <a href="qc_lang1.html#NM6">QC-pred_symbol</a> of <font color="Olive">k</font>
 ;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E3:43"/>
( <font color="Maroon">F</font> is <a href="qc_lang1.html#V3" target="_self">negative</a> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> 1</span>)</span> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> 1 )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:43_3"/>
<font color="Maroon">F</font> is <a href="qc_lang1.html#V3" target="_self">negative</a>
 ;<br/>

<b>then consider </b><font color="Maroon">p</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8" target="_self">QC-WFF</a> <b> such that </b><br/><a NAME="E3:43_3"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="qc_lang1.html#K12" target="_self">'not'</a> <font color="Maroon">p</font>
 <b>by </b><i/>;<br/>
<a NAME="E4:43_3"/>
 <a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default">1,0</span><a href="tarski.html#K4">]</a></span></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">p</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E5:43_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,0</span><a href="tarski.html#K4">]</a></span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T58">FINSEQ_1:58</a></i>;<br/>
<b>hence </b><a NAME="E6:43_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> 1</span>)</span> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E4:43"/>
( <font color="Maroon">F</font> is <a href="qc_lang1.html#V4" target="_self">conjunctive</a> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> 1</span>)</span> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> 2 )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_4"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:43_4"/>
<font color="Maroon">F</font> is <a href="qc_lang1.html#V4" target="_self">conjunctive</a>
 ;<br/>

<b>then consider </b><font color="Maroon">p</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8" target="_self">QC-WFF</a> , <font color="Maroon">q</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8" target="_self">QC-WFF</a> <b> such that </b><br/><a NAME="E3:43_4"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="qc_lang1.html#K13" target="_self">'&amp;'</a> <font color="Maroon">q</font>
 <b>by </b><i/>;<br/>
<a NAME="E4:43_4"/>
 <a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default">2,0</span><a href="tarski.html#K4">]</a></span></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">p</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">q</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T45">FINSEQ_1:45</a></i>;<br/>
<a NAME="E5:43_4"/><b>then </b>
<span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">2,0</span><a href="tarski.html#K4">]</a></span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T58">FINSEQ_1:58</a></i>;<br/>
<b>hence </b><a NAME="E6:43_4"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> 1</span>)</span> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> 2
 <b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E5:43"/>
( <font color="Maroon">F</font> is <a href="qc_lang1.html#V5" target="_self">universal</a> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> 1</span>)</span> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> 3 )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_5"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:43_5"/>
<font color="Maroon">F</font> is <a href="qc_lang1.html#V5" target="_self">universal</a>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being   <a href="qc_lang1.html#NM3">bound_QC-variable</a>, <font color="Maroon">p</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8" target="_self">QC-WFF</a> <b> such that </b><br/><a NAME="E3:43_5"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="qc_lang1.html#K14" target="_self">All</a> <font color="Maroon">x</font>,<font color="Maroon">p</font>
 <b>by </b><i/>;<br/>
<a NAME="E4:43_5"/>
 <a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default">3,0</span><a href="tarski.html#K4">]</a></span></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">p</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T45">FINSEQ_1:45</a></i>;<br/>
<a NAME="E5:43_5"/><b>then </b>
<span class="p1">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">3,0</span><a href="tarski.html#K4">]</a></span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T58">FINSEQ_1:58</a></i>;<br/>
<b>hence </b><a NAME="E6:43_5"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="qc_lang1.html#K10" target="_self">@</a> <font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> 1</span>)</span> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> 3
 <b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/>


</div><b>end;</b></div>


</div>
