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<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>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">S1</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="substut1.html#K38">CQC-Sub-WFF</a> ;<br/><b>let </b><font color="Maroon">S2</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="E1:32"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">S1</font> <a href="substut1.html#K19">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">S2</font> <a href="substut1.html#K19">`2</a> 
 ;<br/>

<a NAME="E2:32"/><i><font color="Green">E56</font></i>: 
(  <a href="sublemma.html#K3" target="_self">Val_S</a> <font color="Maroon">v</font>,<font color="Maroon">S1</font> <a href="hidden.html#R1">=</a>  <a href="sublemma.html#K3" target="_self">Val_S</a> <font color="Maroon">v</font>,<span class="p1">(<span class="default"><a href="sublemma.html#K7" target="_self">CQCSub_&amp;</a> <font color="Maroon">S1</font>,<font color="Maroon">S2</font></span>)</span> &amp;  <a href="sublemma.html#K3" target="_self">Val_S</a> <font color="Maroon">v</font>,<font color="Maroon">S2</font> <a href="hidden.html#R1">=</a>  <a href="sublemma.html#K3" target="_self">Val_S</a> <font color="Maroon">v</font>,<span class="p1">(<span class="default"><a href="sublemma.html#K7" target="_self">CQCSub_&amp;</a> <font color="Maroon">S1</font>,<font color="Maroon">S2</font></span>)</span> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E3:32"/><b>then </b><i><font color="Green">E57</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"><a href="sublemma.html#K3" target="_self">Val_S</a> <font color="Maroon">v</font>,<font color="Maroon">S1</font></span>)</span> <a href="valuat_1.html#R1">|=</a> <font color="Maroon">S1</font> <a href="sublemma.html#K2" target="_self">`1</a>  &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"><a href="sublemma.html#K3" target="_self">Val_S</a> <font color="Maroon">v</font>,<font color="Maroon">S1</font></span>)</span> <a href="valuat_1.html#R1">|=</a> <font color="Maroon">S2</font> <a href="sublemma.html#K2" target="_self">`1</a>  ) 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"><a href="sublemma.html#K3" target="_self">Val_S</a> <font color="Maroon">v</font>,<span class="p2">(<span class="default"><a href="sublemma.html#K7" target="_self">CQCSub_&amp;</a> <font color="Maroon">S1</font>,<font color="Maroon">S2</font></span>)</span></span>)</span> <a href="valuat_1.html#R1">|=</a> <span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span> <a href="cqc_lang.html#K11">'&amp;'</a> <span class="p1">(<span class="default"><font color="Maroon">S2</font> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span> )
 
<b>by </b><i><a class="ref" href="valuat_1.html#T29">VALUAT_1:29</a></i>;<br/>
<a NAME="E4:32"/>
( <font color="Maroon">J</font>,<font color="Maroon">v</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">v</font>,<span class="p2">(<span class="default"><a href="sublemma.html#K7" target="_self">CQCSub_&amp;</a> <font color="Maroon">S1</font>,<font color="Maroon">S2</font></span>)</span></span>)</span> <a href="valuat_1.html#R1">|=</a> <span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span> <a href="cqc_lang.html#K11">'&amp;'</a> <span class="p1">(<span class="default"><font color="Maroon">S2</font> <a href="sublemma.html#K2" target="_self">`1</a> </span>)</span> 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"><a href="sublemma.html#K3" target="_self">Val_S</a> <font color="Maroon">v</font>,<span class="p2">(<span class="default"><a href="sublemma.html#K7" target="_self">CQCSub_&amp;</a> <font color="Maroon">S1</font>,<font color="Maroon">S2</font></span>)</span></span>)</span> <a href="valuat_1.html#R1">|=</a> <span class="p1">(<span class="default"><a href="sublemma.html#K7" target="_self">CQCSub_&amp;</a> <font color="Maroon">S1</font>,<font color="Maroon">S2</font></span>)</span> <a href="sublemma.html#K2" target="_self">`1</a>  )
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E5:32"/>
( ( <font color="Maroon">J</font>,<font color="Maroon">v</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">v</font>,<font color="Maroon">S1</font></span>)</span> <a href="sublemma.html#R1" target="_self">|=</a> <font color="Maroon">S1</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"><a href="sublemma.html#K3" target="_self">Val_S</a> <font color="Maroon">v</font>,<font color="Maroon">S2</font></span>)</span> <a href="sublemma.html#R1" target="_self">|=</a> <font color="Maroon">S2</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"><a href="sublemma.html#K3" target="_self">Val_S</a> <font color="Maroon">v</font>,<span class="p2">(<span class="default"><a href="sublemma.html#K7" target="_self">CQCSub_&amp;</a> <font color="Maroon">S1</font>,<font color="Maroon">S2</font></span>)</span></span>)</span> <a href="sublemma.html#R1" target="_self">|=</a>  <a href="sublemma.html#K7" target="_self">CQCSub_&amp;</a> <font color="Maroon">S1</font>,<font color="Maroon">S2</font> )
 <b>by </b><i>, , </i>;<br/>


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