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

<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">q</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">A</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">J</font> be    <a href="valuat_1.html#M1">interpretation</a> of <font color="Maroon">A</font>;<br/>







<b>assume </b><a NAME="E1:94"/><i><font color="Green">E55</font></i>: 
( ( for <font color="Olive">v</font> being   <a href="fraenkel.html#M2">Element</a> of  <a href="valuat_1.html#K2">Valuations_in</a> <font color="Maroon">A</font><br/> for <font color="Olive">vS</font>, <font color="Olive">vS1</font>, <font color="Olive">vS2</font> being  <a href="sublemma.html#NM1">Val_Sub</a> of <font color="Maroon">A</font>  st ( for <font color="Olive">y</font> being  <a href="qc_lang1.html#NM3">bound_QC-variable</a>  st <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Olive">vS1</font> holds <br/> not <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">p</font> ) &amp; ( for <font color="Olive">y</font> being  <a href="qc_lang1.html#NM3">bound_QC-variable</a>  st <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Olive">vS2</font> holds <br/><font color="Olive">vS2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">v</font> <a href="funct_2.html#K3">.</a> <font color="Olive">y</font> ) &amp;  <a href="relset_1.html#K4">dom</a> <font color="Olive">vS</font> <a href="xboole_0.html#R1">misses</a>  <a href="relset_1.html#K4">dom</a> <font color="Olive">vS2</font> holds <br/>( <font color="Maroon">J</font>,<font color="Olive">v</font> <a href="sublemma.html#K1" target="_self">.</a> <font color="Olive">vS</font> <a href="valuat_1.html#R1">|=</a> <font color="Maroon">p</font> iff <font color="Maroon">J</font>,<font color="Olive">v</font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive">vS</font> <a href="sublemma.html#K15" target="_self">+*</a> <font color="Olive">vS1</font></span>)</span> <a href="sublemma.html#K15" target="_self">+*</a> <font color="Olive">vS2</font></span>)</span> <a href="valuat_1.html#R1">|=</a> <font color="Maroon">p</font> ) ) &amp; ( for <font color="Olive">v</font> being   <a href="fraenkel.html#M2">Element</a> of  <a href="valuat_1.html#K2">Valuations_in</a> <font color="Maroon">A</font><br/> for <font color="Olive">vS</font>, <font color="Olive">vS1</font>, <font color="Olive">vS2</font> being  <a href="sublemma.html#NM1">Val_Sub</a> of <font color="Maroon">A</font>  st ( for <font color="Olive">y</font> being  <a href="qc_lang1.html#NM3">bound_QC-variable</a>  st <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Olive">vS1</font> holds <br/> not <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">q</font> ) &amp; ( for <font color="Olive">y</font> being  <a href="qc_lang1.html#NM3">bound_QC-variable</a>  st <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Olive">vS2</font> holds <br/><font color="Olive">vS2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">v</font> <a href="funct_2.html#K3">.</a> <font color="Olive">y</font> ) &amp;  <a href="relset_1.html#K4">dom</a> <font color="Olive">vS</font> <a href="xboole_0.html#R1">misses</a>  <a href="relset_1.html#K4">dom</a> <font color="Olive">vS2</font> holds <br/>( <font color="Maroon">J</font>,<font color="Olive">v</font> <a href="sublemma.html#K1" target="_self">.</a> <font color="Olive">vS</font> <a href="valuat_1.html#R1">|=</a> <font color="Maroon">q</font> iff <font color="Maroon">J</font>,<font color="Olive">v</font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive">vS</font> <a href="sublemma.html#K15" target="_self">+*</a> <font color="Olive">vS1</font></span>)</span> <a href="sublemma.html#K15" target="_self">+*</a> <font color="Olive">vS2</font></span>)</span> <a href="valuat_1.html#R1">|=</a> <font color="Maroon">q</font> ) ) )
 ;<br/>

<b>let </b><font color="Maroon">v</font> be    <a href="fraenkel.html#M2">Element</a> of  <a href="valuat_1.html#K2">Valuations_in</a> <font color="Maroon">A</font>;<br/><b>let </b><font color="Maroon">vS</font> be   <a href="sublemma.html#NM1">Val_Sub</a> of <font color="Maroon">A</font>;<br/><b>let </b><font color="Maroon">vS1</font> be   <a href="sublemma.html#NM1">Val_Sub</a> of <font color="Maroon">A</font>;<br/><b>let </b><font color="Maroon">vS2</font> be   <a href="sublemma.html#NM1">Val_Sub</a> of <font color="Maroon">A</font>;<br/>







<b>assume </b><a NAME="E2:94"/><i><font color="Green">E56</font></i>: 
( ( for <font color="Olive">y</font> being  <a href="qc_lang1.html#NM3">bound_QC-variable</a>  st <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">vS1</font> holds <br/> not <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="cqc_lang.html#K11">'&amp;'</a> <font color="Maroon">q</font></span>)</span> ) &amp; ( for <font color="Olive">y</font> being  <a href="qc_lang1.html#NM3">bound_QC-variable</a>  st <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">vS2</font> holds <br/><font color="Maroon">vS2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> <a href="funct_2.html#K3">.</a> <font color="Olive">y</font> ) &amp;  <a href="relset_1.html#K4">dom</a> <font color="Maroon">vS</font> <a href="xboole_0.html#R1">misses</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">vS2</font> )
 ;<br/>

<a NAME="E3:94"/><i><font color="Green">E57</font></i>: 
for <font color="Olive">y</font> being  <a href="qc_lang1.html#NM3">bound_QC-variable</a>  st <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">vS1</font> holds <br/>( not <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">p</font> &amp; not <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">q</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="94_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</font> be   <a href="qc_lang1.html#NM3">bound_QC-variable</a>;<br/>

<b>assume </b><a NAME="E1:94_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">vS1</font>
 ;<br/>

<a NAME="E2:94_1"/><b>then </b>
not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="cqc_lang.html#K11">'&amp;'</a> <font color="Maroon">q</font></span>)</span>
 
<b>by </b><i><a class="ref" href="sublemma.html#T6" target="_self">Th6</a></i>;<br/>
<a NAME="E3:94_1"/><b>then </b>
not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">p</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">q</font></span>)</span>
 
<b>by </b><i><a class="ref" href="qc_lang3.html#T14">QC_LANG3:14</a></i>;<br/>
<b>hence </b><a NAME="E4:94_1"/>
( not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">p</font> &amp; not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">q</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:94"/>
( <font color="Maroon">J</font>,<font color="Maroon">v</font> <a href="sublemma.html#K1" target="_self">.</a> <font color="Maroon">vS</font> <a href="valuat_1.html#R1">|=</a> <font color="Maroon">p</font> &amp; <font color="Maroon">J</font>,<font color="Maroon">v</font> <a href="sublemma.html#K1" target="_self">.</a> <font color="Maroon">vS</font> <a href="valuat_1.html#R1">|=</a> <font color="Maroon">q</font> iff ( <font color="Maroon">J</font>,<font color="Maroon">v</font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">vS</font> <a href="sublemma.html#K15" target="_self">+*</a> <font color="Maroon">vS1</font></span>)</span> <a href="sublemma.html#K15" target="_self">+*</a> <font color="Maroon">vS2</font></span>)</span> <a href="valuat_1.html#R1">|=</a> <font color="Maroon">p</font> &amp; <font color="Maroon">J</font>,<font color="Maroon">v</font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">vS</font> <a href="sublemma.html#K15" target="_self">+*</a> <font color="Maroon">vS1</font></span>)</span> <a href="sublemma.html#K15" target="_self">+*</a> <font color="Maroon">vS2</font></span>)</span> <a href="valuat_1.html#R1">|=</a> <font color="Maroon">q</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="94_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:94_2"/>
( <font color="Maroon">J</font>,<font color="Maroon">v</font> <a href="sublemma.html#K1" target="_self">.</a> <font color="Maroon">vS</font> <a href="valuat_1.html#R1">|=</a> <font color="Maroon">p</font> &amp; <font color="Maroon">J</font>,<font color="Maroon">v</font> <a href="sublemma.html#K1" target="_self">.</a> <font color="Maroon">vS</font> <a href="valuat_1.html#R1">|=</a> <font color="Maroon">q</font> implies ( <font color="Maroon">J</font>,<font color="Maroon">v</font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">vS</font> <a href="sublemma.html#K15" target="_self">+*</a> <font color="Maroon">vS1</font></span>)</span> <a href="sublemma.html#K15" target="_self">+*</a> <font color="Maroon">vS2</font></span>)</span> <a href="valuat_1.html#R1">|=</a> <font color="Maroon">p</font> &amp; <font color="Maroon">J</font>,<font color="Maroon">v</font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">vS</font> <a href="sublemma.html#K15" target="_self">+*</a> <font color="Maroon">vS1</font></span>)</span> <a href="sublemma.html#K15" target="_self">+*</a> <font color="Maroon">vS2</font></span>)</span> <a href="valuat_1.html#R1">|=</a> <font color="Maroon">q</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="94_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:94_2_1"/><i><font color="Green">E58</font></i>: 
( <font color="Maroon">J</font>,<font color="Maroon">v</font> <a href="sublemma.html#K1" target="_self">.</a> <font color="Maroon">vS</font> <a href="valuat_1.html#R1">|=</a> <font color="Maroon">p</font> &amp; <font color="Maroon">J</font>,<font color="Maroon">v</font> <a href="sublemma.html#K1" target="_self">.</a> <font color="Maroon">vS</font> <a href="valuat_1.html#R1">|=</a> <font color="Maroon">q</font> )
 ;<br/>

<a NAME="E2:94_2_1"/><i><font color="Green">E59</font></i>: 
for <font color="Olive">y</font> being  <a href="qc_lang1.html#NM3">bound_QC-variable</a>  st <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">vS1</font> holds <br/> not <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">p</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:94_2_1"/>
for <font color="Olive">y</font> being  <a href="qc_lang1.html#NM3">bound_QC-variable</a>  st <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">vS1</font> holds <br/> not <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">q</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E4:94_2_1"/>
( <font color="Maroon">J</font>,<font color="Maroon">v</font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">vS</font> <a href="sublemma.html#K15" target="_self">+*</a> <font color="Maroon">vS1</font></span>)</span> <a href="sublemma.html#K15" target="_self">+*</a> <font color="Maroon">vS2</font></span>)</span> <a href="valuat_1.html#R1">|=</a> <font color="Maroon">p</font> &amp; <font color="Maroon">J</font>,<font color="Maroon">v</font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">vS</font> <a href="sublemma.html#K15" target="_self">+*</a> <font color="Maroon">vS1</font></span>)</span> <a href="sublemma.html#K15" target="_self">+*</a> <font color="Maroon">vS2</font></span>)</span> <a href="valuat_1.html#R1">|=</a> <font color="Maroon">q</font> )
 <b>by </b><i><a class="ref" href="sublemma.html#T5" target="_self">Th5</a>, <a class="ref" href="sublemma.html#T6" target="_self">Th6</a>, , </i>;<br/>


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

<b>thus </b><a NAME="E2:94_2"/>
( <font color="Maroon">J</font>,<font color="Maroon">v</font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">vS</font> <a href="sublemma.html#K15" target="_self">+*</a> <font color="Maroon">vS1</font></span>)</span> <a href="sublemma.html#K15" target="_self">+*</a> <font color="Maroon">vS2</font></span>)</span> <a href="valuat_1.html#R1">|=</a> <font color="Maroon">p</font> &amp; <font color="Maroon">J</font>,<font color="Maroon">v</font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">vS</font> <a href="sublemma.html#K15" target="_self">+*</a> <font color="Maroon">vS1</font></span>)</span> <a href="sublemma.html#K15" target="_self">+*</a> <font color="Maroon">vS2</font></span>)</span> <a href="valuat_1.html#R1">|=</a> <font color="Maroon">q</font> implies ( <font color="Maroon">J</font>,<font color="Maroon">v</font> <a href="sublemma.html#K1" target="_self">.</a> <font color="Maroon">vS</font> <a href="valuat_1.html#R1">|=</a> <font color="Maroon">p</font> &amp; <font color="Maroon">J</font>,<font color="Maroon">v</font> <a href="sublemma.html#K1" target="_self">.</a> <font color="Maroon">vS</font> <a href="valuat_1.html#R1">|=</a> <font color="Maroon">q</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="94_2_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:94_2_2"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">J</font>,<font color="Maroon">v</font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">vS</font> <a href="sublemma.html#K15" target="_self">+*</a> <font color="Maroon">vS1</font></span>)</span> <a href="sublemma.html#K15" target="_self">+*</a> <font color="Maroon">vS2</font></span>)</span> <a href="valuat_1.html#R1">|=</a> <font color="Maroon">p</font> &amp; <font color="Maroon">J</font>,<font color="Maroon">v</font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">vS</font> <a href="sublemma.html#K15" target="_self">+*</a> <font color="Maroon">vS1</font></span>)</span> <a href="sublemma.html#K15" target="_self">+*</a> <font color="Maroon">vS2</font></span>)</span> <a href="valuat_1.html#R1">|=</a> <font color="Maroon">q</font> )
 ;<br/>

<a NAME="E2:94_2_2"/><i><font color="Green">E61</font></i>: 
for <font color="Olive">y</font> being  <a href="qc_lang1.html#NM3">bound_QC-variable</a>  st <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">vS1</font> holds <br/> not <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">p</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:94_2_2"/>
for <font color="Olive">y</font> being  <a href="qc_lang1.html#NM3">bound_QC-variable</a>  st <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">vS1</font> holds <br/> not <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">q</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E4:94_2_2"/>
( <font color="Maroon">J</font>,<font color="Maroon">v</font> <a href="sublemma.html#K1" target="_self">.</a> <font color="Maroon">vS</font> <a href="valuat_1.html#R1">|=</a> <font color="Maroon">p</font> &amp; <font color="Maroon">J</font>,<font color="Maroon">v</font> <a href="sublemma.html#K1" target="_self">.</a> <font color="Maroon">vS</font> <a href="valuat_1.html#R1">|=</a> <font color="Maroon">q</font> )
 <b>by </b><i><a class="ref" href="sublemma.html#T5" target="_self">Th5</a>, <a class="ref" href="sublemma.html#T6" target="_self">Th6</a>, , </i>;<br/>


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


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:94"/>
( <font color="Maroon">J</font>,<font color="Maroon">v</font> <a href="sublemma.html#K1" target="_self">.</a> <font color="Maroon">vS</font> <a href="valuat_1.html#R1">|=</a> <font color="Maroon">p</font> <a href="cqc_lang.html#K11">'&amp;'</a> <font color="Maroon">q</font> iff <font color="Maroon">J</font>,<font color="Maroon">v</font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">vS</font> <a href="sublemma.html#K15" target="_self">+*</a> <font color="Maroon">vS1</font></span>)</span> <a href="sublemma.html#K15" target="_self">+*</a> <font color="Maroon">vS2</font></span>)</span> <a href="valuat_1.html#R1">|=</a> <font color="Maroon">p</font> <a href="cqc_lang.html#K11">'&amp;'</a> <font color="Maroon">q</font> )
 <b>by </b><i><a class="ref" href="valuat_1.html#T29">VALUAT_1:29</a></i>;<br/>


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