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<b>let </b><font color="Maroon">X</font> be   <a href="subset_1.html#NM2">Subset</a> of <a href="cqc_lang.html#K7">CQC-WFF</a> ;<br/><b>let </b><font color="Maroon">p</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a> ;<br/><b>let </b><font color="Maroon">x</font> be   <a href="qc_lang1.html#NM3">bound_QC-variable</a>;<br/>





<b>assume </b><a NAME="E1:4"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">X</font> is <a href="goedelcp.html#V2" target="_self">with_examples</a>
 ;<br/>

<b>thus </b><a NAME="E2:4"/>
( <font color="Maroon">X</font> <a href="henmodel.html#R1">|-</a>  <a href="cqc_lang.html#K16">Ex</a> <font color="Maroon">x</font>,<font color="Maroon">p</font> implies  ex <font color="Olive">y</font> being  <a href="qc_lang1.html#NM3">bound_QC-variable</a> st <font color="Maroon">X</font> <a href="henmodel.html#R1">|-</a> <font color="Maroon">p</font> <a href="substut2.html#K4">.</a> <font color="Maroon">x</font>,<font color="Olive">y</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:4_1"/>
<font color="Maroon">X</font> <a href="henmodel.html#R1">|-</a>  <a href="cqc_lang.html#K16">Ex</a> <font color="Maroon">x</font>,<font color="Maroon">p</font>
 ;<br/>

<b>then consider </b><font color="Maroon">f1</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a> <b> such that </b><br/><a NAME="E3:4_1"/><i><font color="Green">E55</font></i>: 
(  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> &amp;  <a href="calcul_1.html#R4">|-</a> <font color="Maroon">f1</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="cqc_lang.html#K16">Ex</a> <font color="Maroon">x</font>,<font color="Maroon">p</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> )
 <b>by </b><i><a class="ref" href="henmodel.html#D2">HENMODEL:def 2</a></i>;<br/>
<b>consider </b><font color="Maroon">y</font> being   <a href="qc_lang1.html#NM3">bound_QC-variable</a><b> such that </b><br/><a NAME="E5:4_1"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">X</font> <a href="henmodel.html#R1">|-</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <span class="p2">(<span class="default"><a href="cqc_lang.html#K16">Ex</a> <font color="Maroon">x</font>,<font color="Maroon">p</font></span>)</span></span>)</span> <a href="cqc_lang.html#K13">'or'</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="substut2.html#K4">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span>
 <b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">f2</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a> <b> such that </b><br/><a NAME="E7:4_1"/><i><font color="Green">E57</font></i>: 
(  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> &amp;  <a href="calcul_1.html#R4">|-</a> <font color="Maroon">f2</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <span class="p4">(<span class="default"><a href="cqc_lang.html#K16">Ex</a> <font color="Maroon">x</font>,<font color="Maroon">p</font></span>)</span></span>)</span> <a href="cqc_lang.html#K13">'or'</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="substut2.html#K4">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> )
 <b>by </b><i>, <a class="ref" href="henmodel.html#D2">HENMODEL:def 2</a></i>;<br/>
<b>take </b>
<font color="Maroon">y</font>
;<br/>

<a NAME="E8:4_1"/>
(  <a href="calcul_1.html#R4">|-</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">f2</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="cqc_lang.html#K16">Ex</a> <font color="Maroon">x</font>,<font color="Maroon">p</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> &amp;  <a href="calcul_1.html#R4">|-</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">f2</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <span class="p4">(<span class="default"><a href="cqc_lang.html#K16">Ex</a> <font color="Maroon">x</font>,<font color="Maroon">p</font></span>)</span></span>)</span> <a href="cqc_lang.html#K13">'or'</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="substut2.html#K4">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> )
 
<b>by </b><i>, , <a class="ref" href="calcul_2.html#T20">CALCUL_2:20</a>, <a class="ref" href="henmodel.html#T5">HENMODEL:5</a></i>;<br/>
<a NAME="E9:4_1"/><b>then </b><i><font color="Green">E58</font></i>: 
 <a href="calcul_1.html#R4">|-</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">f2</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="substut2.html#K4">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
 
<b>by </b><i/>;<br/>
<a NAME="E10:4_1"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">f2</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f1</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T44">FINSEQ_1:44</a></i>;<br/>
<a NAME="E11:4_1"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">f2</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font>
 
<b>by </b><i>, , <a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/>
<b>hence </b><a NAME="E12:4_1"/>
<font color="Maroon">X</font> <a href="henmodel.html#R1">|-</a> <font color="Maroon">p</font> <a href="substut2.html#K4">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>
 <b>by </b><i>, <a class="ref" href="henmodel.html#D2">HENMODEL:def 2</a></i>;<br/>


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

<b>thus </b><a NAME="E3:4"/>
(  ex <font color="Olive">y</font> being  <a href="qc_lang1.html#NM3">bound_QC-variable</a> st <font color="Maroon">X</font> <a href="henmodel.html#R1">|-</a> <font color="Maroon">p</font> <a href="substut2.html#K4">.</a> <font color="Maroon">x</font>,<font color="Olive">y</font> implies <font color="Maroon">X</font> <a href="henmodel.html#R1">|-</a>  <a href="cqc_lang.html#K16">Ex</a> <font color="Maroon">x</font>,<font color="Maroon">p</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_2"><b>proof </b></a><div class="add">

<b>given </b><font color="Maroon">y</font> being   <a href="qc_lang1.html#NM3">bound_QC-variable</a><b> such that </b><a NAME="E1:4_2"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">X</font> <a href="henmodel.html#R1">|-</a> <font color="Maroon">p</font> <a href="substut2.html#K4">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>
 ;<br/>

<b>consider </b><font color="Maroon">f1</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a> <b> such that </b><br/><a NAME="E3:4_2"/><i><font color="Green">E60</font></i>: 
(  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> &amp;  <a href="calcul_1.html#R4">|-</a> <font color="Maroon">f1</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="substut2.html#K4">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> )
 <b>by </b><i>, <a class="ref" href="henmodel.html#D2">HENMODEL:def 2</a></i>;<br/>
<a NAME="E4:4_2"/>
 <a href="calcul_1.html#R4">|-</a> <font color="Maroon">f1</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="cqc_lang.html#K16">Ex</a> <font color="Maroon">x</font>,<font color="Maroon">p</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
 
<b>by </b><i>, <a class="ref" href="calcul_1.html#T58">CALCUL_1:58</a></i>;<br/>
<b>hence </b><a NAME="E5:4_2"/>
<font color="Maroon">X</font> <a href="henmodel.html#R1">|-</a>  <a href="cqc_lang.html#K16">Ex</a> <font color="Maroon">x</font>,<font color="Maroon">p</font>
 <b>by </b><i>, <a class="ref" href="henmodel.html#D2">HENMODEL:def 2</a></i>;<br/>


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


</div>
