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<b>let </b><font color="Maroon">c<sub>1</sub></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">c<sub>2</sub></font> be    <a href="valuat_1.html#M1">interpretation</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="subset_1.html#M2">Element</a> of  <a href="substut1.html#K38">CQC-Sub-WFF</a> ;<br/>





<b>assume </b><a NAME="E1:27"/><i><font color="Green">E19</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="fraenkel.html#M2">Element</a> of  <a href="valuat_1.html#K2">Valuations_in</a> <font color="Maroon">c<sub>1</sub></font> holds <br/>( <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="valuat_1.html#R1">|=</a>  <a href="substut1.html#K39">CQC_Sub</a> <font color="Maroon">c<sub>3</sub></font> iff <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p1">(<span class="default"><a href="sublemma.html#K3" target="_self">Val_S</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="sublemma.html#R1" target="_self">|=</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/>

<b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="fraenkel.html#M2">Element</a> of  <a href="valuat_1.html#K2">Valuations_in</a> <font color="Maroon">c<sub>1</sub></font>;<br/>

<a NAME="E2:27"/><i><font color="Green">E20</font></i>: 
( <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="valuat_1.html#R1">|=</a>  <a href="cqc_lang.html#K10">'not'</a> <span class="p1">(<span class="default"><a href="substut1.html#K39">CQC_Sub</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> iff not <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="valuat_1.html#R1">|=</a>  <a href="substut1.html#K39">CQC_Sub</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="ref" href="valuat_1.html#T28">VALUAT_1:28</a></i>;<br/>
<a NAME="E3:27"/>
( not <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p1">(<span class="default"><a href="sublemma.html#K3" target="_self">Val_S</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="sublemma.html#R1" target="_self">|=</a> <font color="Maroon">c<sub>3</sub></font> iff <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p1">(<span class="default"><a href="sublemma.html#K3" target="_self">Val_S</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="sublemma.html#R1" target="_self">|=</a>  <a href="sublemma.html#K6" target="_self">Sub_not</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="ref" href="sublemma.html#T18" target="_self">Th18</a></i>;<br/>
<b>hence </b><a NAME="E4:27"/>
( <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="valuat_1.html#R1">|=</a>  <a href="substut1.html#K39">CQC_Sub</a> <span class="p1">(<span class="default"><a href="sublemma.html#K6" target="_self">Sub_not</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> iff <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p1">(<span class="default"><a href="sublemma.html#K3" target="_self">Val_S</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p2">(<span class="default"><a href="sublemma.html#K6" target="_self">Sub_not</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="sublemma.html#R1" target="_self">|=</a>  <a href="sublemma.html#K6" target="_self">Sub_not</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="sublemma.html#E1:27"><i><font color="Green">E19</font></i></a>, <a class="txt" href="sublemma.html#E2:27"><i><font color="Green">E20</font></i></a>, <a class="ref" href="sublemma.html#T19" target="_self">Th19</a>, <a class="ref" href="substut1.html#T29">SUBSTUT1:29</a></i>;<br/>


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