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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>2</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>3</sub></font> be    <a href="fraenkel.html#M2">Element</a> of  <a href="valuat_1.html#K2" target="_self">Valuations_in</a> <font color="Maroon">c<sub>2</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="cqc_lang.html#NM2">CQC-variable_list</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="valuat_1.html#M1" target="_self">interpretation</a> of <font color="Maroon">c<sub>2</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="qc_lang1.html#NM6">QC-pred_symbol</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>











<div><i><font color="Green">E21</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:31_1"/>
<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="valuat_1.html#R1" target="_self">|=</a> <font color="Maroon">c<sub>6</sub></font> <a href="cqc_lang.html#K8">!</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><a NAME="E2:31_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="cqc_lang.html#K8">!</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 <b>by </b><i><a class="ref" href="valuat_1.html#D12" target="_self">Def12</a></i>;<br/><b>hence </b><a NAME="E3:31_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="valuat_1.html#K9" target="_self">'in'</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="valuat_1.html#K11" target="_self">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 <b>by </b><i><a class="txt" href="valuat_1.html#E20"><i><font color="Green">Lemma13</font></i></a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:31_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="valuat_1.html#K9" target="_self">'in'</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="valuat_1.html#K11" target="_self">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 ;<br/><a NAME="E2:31_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="cqc_lang.html#K8">!</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 <b>by </b><i><a class="txt" href="valuat_1.html#E20"><i><font color="Green">Lemma13</font></i></a></i>;<br/><b>hence </b><a NAME="E3:31_2"/>
<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="valuat_1.html#R1" target="_self">|=</a> <font color="Maroon">c<sub>6</sub></font> <a href="cqc_lang.html#K8">!</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="valuat_1.html#D12" target="_self">Def12</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:31"/>
( <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="valuat_1.html#R1" target="_self">|=</a> <font color="Maroon">c<sub>6</sub></font> <a href="cqc_lang.html#K8">!</a> <font color="Maroon">c<sub>4</sub></font> iff <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="valuat_1.html#K9" target="_self">'in'</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="valuat_1.html#K11" target="_self">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  )
 <b>by </b><i><a class="txt" href="valuat_1.html#E1:31"><i><font color="Green">E21</font></i></a></i>;<br/>


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