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<b>let </b><font color="Maroon" title="c1">A</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">J</font> be    <a href="valuat_1.html#M1" title="VALUAT_1:mode.1">interpretation</a> of <font color="Maroon" title="c1">A</font>;<br/>



<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K7" title="CQC_LANG:func.7">CQC-WFF</a> ] <b>means </b>for <font color="Olive" title="b1">v</font> being   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="valuat_1.html#K2" title="VALUAT_1:func.2">Valuations_in</a> <font color="Maroon" title="c1">A</font><br/> for <font color="Olive" title="b2">vS</font>, <font color="Olive" title="b3">vS1</font>, <font color="Olive" title="b4">vS2</font> being  <a href="sublemma.html#NM1" title="SUBLEMMA:NM.1">Val_Sub</a> of <font color="Maroon" title="c1">A</font>  st ( for <font color="Olive" title="b5">y</font> being  <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a>  st <font color="Olive" title="b5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Olive" title="b3">vS1</font> holds <br/> not <font color="Olive" title="b5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> $1 ) &amp; ( for <font color="Olive" title="b5">y</font> being  <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a>  st <font color="Olive" title="b5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Olive" title="b4">vS2</font> holds <br/><font color="Olive" title="b4">vS2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b5">y</font> ) &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Olive" title="b2">vS</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Olive" title="b4">vS2</font> holds <br/>( <font color="Maroon" title="c2">J</font>,<font color="Olive" title="b1">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <font color="Olive" title="b2">vS</font> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> $1 iff <font color="Maroon" title="c2">J</font>,<font color="Olive" title="b1">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b2">vS</font> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Olive" title="b3">vS1</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Olive" title="b4">vS2</font></span>)</span> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> $1 );<br/>
<a NAME="E1:98"/><i><font color="Green" title="E81">A1</font></i>: 
for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K7" title="CQC_LANG:func.7">CQC-WFF</a> <br/> for <font color="Olive" title="b3">x</font> being  <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a><br/> for <font color="Olive" title="b4">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/> for <font color="Olive" title="b5">l</font> being  <a href="cqc_lang.html#NM2" title="CQC_LANG:NM.2">CQC-variable_list</a> of <font color="Olive" title="b4">k</font><br/> for <font color="Olive" title="b6">P</font> being  <a href="qc_lang1.html#NM6" title="QC_LANG1:NM.6">QC-pred_symbol</a> of <font color="Olive" title="b4">k</font> holds <br/> ( <font color="Maroon">S<sub>1</sub></font>[ <a href="cqc_lang.html#K9" title="CQC_LANG:func.9">VERUM</a> ] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b6">P</font> <a href="cqc_lang.html#K8" title="CQC_LANG:func.8">!</a> <font color="Olive" title="b5">l</font>] &amp; ( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">p</font>] implies <font color="Maroon">S<sub>1</sub></font>[ <a href="cqc_lang.html#K10" title="CQC_LANG:func.10">'not'</a> <font color="Olive" title="b1">p</font>] ) &amp; ( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">p</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">q</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">p</font> <a href="cqc_lang.html#K11" title="CQC_LANG:func.11">'&amp;'</a> <font color="Olive" title="b2">q</font>] ) &amp; ( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">p</font>] implies <font color="Maroon">S<sub>1</sub></font>[ <a href="cqc_lang.html#K15" title="CQC_LANG:func.15">All</a> <font color="Olive" title="b3">x</font>,<font color="Olive" title="b1">p</font>] ) )
 
<b>by </b><i><a class="ref" href="sublemma.html#T79" target="_self" title="SUBLEMMA:th.79">Th79</a>, <a class="ref" href="sublemma.html#T80" target="_self" title="SUBLEMMA:th.80">Th80</a>, <a class="ref" href="sublemma.html#T81" target="_self" title="SUBLEMMA:th.81">Th81</a>, <a class="ref" href="sublemma.html#T84" target="_self" title="SUBLEMMA:th.84">Th84</a>, <a class="ref" href="valuat_1.html#T44" title="VALUAT_1:th.44">VALUAT_1:44</a></i>;<br/>
<b>thus </b><a NAME="E2:98"/>
for <font color="Olive" title="b1">p</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K7" title="CQC_LANG:func.7">CQC-WFF</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">p</font>]
 <b>from </b><i><a class="ref" href="cqc_lang.html#S1" title="CQC_LANG:sch.1">CQC_LANG:sch 1</a>(<a class="txt" href="sublemma.html#E1:98"><i><font color="Green" title="E81">A1</font></i></a>);</i><br/>


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