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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="henmodel.html#V1">Consistent</a> <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">c<sub>2</sub></font> be    <a href="henmodel.html#M1">Henkin_interpretation</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a> ] means ( <font color="Maroon">c<sub>1</sub></font> is <a href="goedelcp.html#V1" target="_self">negation_faithful</a> &amp; <font color="Maroon">c<sub>1</sub></font> is <a href="goedelcp.html#V2" target="_self">with_examples</a> implies ( <font color="Maroon">c<sub>2</sub></font>, <a href="henmodel.html#K4">valH</a>  <a href="valuat_1.html#R1">|=</a> <font color="Maroon">a<sub>1</sub></font> iff <font color="Maroon">c<sub>1</sub></font> <a href="henmodel.html#R1">|-</a> <font color="Maroon">a<sub>1</sub></font> ) );<br/>
<a NAME="E1:7"/><i><font color="Green">E11</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a> <br/>for <font color="Olive">b<sub>3</sub></font> being  <a href="qc_lang1.html#NM3">bound_QC-variable</a><br/>for <font color="Olive">b<sub>4</sub></font> being  <a href="nat_1.html#NM1">Nat</a><br/>for <font color="Olive">b<sub>5</sub></font> being  <a href="cqc_lang.html#NM2">CQC-variable_list</a> of <font color="Olive">b<sub>4</sub></font><br/>for <font color="Olive">b<sub>6</sub></font> being  <a href="qc_lang1.html#NM6">QC-pred_symbol</a> of <font color="Olive">b<sub>4</sub></font> holds <br/>( <font color="Maroon">S<sub>1</sub></font>[ <a href="cqc_lang.html#K9">VERUM</a> ] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>6</sub></font> <a href="cqc_lang.html#K8">!</a> <font color="Olive">b<sub>5</sub></font>] &amp; ( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] implies <font color="Maroon">S<sub>1</sub></font>[ <a href="cqc_lang.html#K10">'not'</a> <font color="Olive">b<sub>1</sub></font>] ) &amp; ( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>2</sub></font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="cqc_lang.html#K11">'&amp;'</a> <font color="Olive">b<sub>2</sub></font>] ) )
 
<b>by </b><i><a class="ref" href="goedelcp.html#D1" target="_self">Def1</a>, <a class="ref" href="goedelcp.html#T6" target="_self">Th6</a>, <a class="ref" href="henmodel.html#T16">HENMODEL:16</a>, <a class="ref" href="henmodel.html#T17">HENMODEL:17</a>, <a class="ref" href="henmodel.html#D3">HENMODEL:def 3</a>, <a class="ref" href="valuat_1.html#T28">VALUAT_1:28</a>, <a class="ref" href="valuat_1.html#T29">VALUAT_1:29</a></i>;<br/>
<a NAME="E2:7"/><i><font color="Green">E12</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a>  st  <a href="cqc_sim1.html#K6">QuantNbr</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0 holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>from </b><i><a class="ref" href="substut2.html#S3">SUBSTUT2:sch 3</a>(<a class="txt" href="goedelcp.html#E1:7"><i><font color="Green">E11</font></i></a>);<br/></i>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a> ;<br/><b>assume </b><a NAME="E1:7_1"/>
 <a href="cqc_sim1.html#K6">QuantNbr</a> <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 0
 ;<br/><a NAME="E2:7_1"/><b>then </b>
 <a href="cqc_sim1.html#K6">QuantNbr</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="nat_1.html#T18">NAT_1:18</a></i>;<br/><b>hence </b><a NAME="E3:7_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>3</sub></font>]
 <b>by </b><i><a class="txt" href="goedelcp.html#E2:7"><i><font color="Green">E12</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:7"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a>  st  <a href="cqc_sim1.html#K6">QuantNbr</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 0 &amp; <font color="Maroon">c<sub>1</sub></font> is <a href="goedelcp.html#V1" target="_self">negation_faithful</a> &amp; <font color="Maroon">c<sub>1</sub></font> is <a href="goedelcp.html#V2" target="_self">with_examples</a> holds <br/>( <font color="Maroon">c<sub>2</sub></font>, <a href="henmodel.html#K4">valH</a>  <a href="valuat_1.html#R1">|=</a> <font color="Olive">b<sub>1</sub></font> iff <font color="Maroon">c<sub>1</sub></font> <a href="henmodel.html#R1">|-</a> <font color="Olive">b<sub>1</sub></font> )
 ;<br/>


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