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<b>let </b><font color="Maroon">x</font> be   <a href="qc_lang1.html#NM3">bound_QC-variable</a>;<br/><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">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">S</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="E1:18"/>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="substut1.html#K19">`2</a> </span>)</span>
 ;<br/>

<a NAME="E2:18"/><b>then </b><i><font color="Green">E55</font></i>: 
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="substut1.html#K2">@</a> <span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="substut1.html#K19">`2</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="substut1.html#D2">SUBSTUT1:def 2</a></i>;<br/>
<a NAME="E3:18"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="substut1.html#K2">@</a> <span class="p3">(<span class="default"><font color="Maroon">S</font> <a href="substut1.html#K19">`2</a> </span>)</span></span>)</span> <a href="relset_1.html#K7">*</a> <font color="Maroon">v</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="substut1.html#K2">@</a> <span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="substut1.html#K19">`2</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T44">RELAT_1:44</a></i>;<br/>
<a NAME="E4:18"/><b>then </b>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="substut1.html#K2">@</a> <span class="p3">(<span class="default"><font color="Maroon">S</font> <a href="substut1.html#K19">`2</a> </span>)</span></span>)</span> <a href="relset_1.html#K7">*</a> <font color="Maroon">v</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E5:18"/><b>then </b>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</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">S</font></span>)</span>
 
;<br/>
<a NAME="E6:18"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><a href="sublemma.html#K3" target="_self">Val_S</a> <font color="Maroon">v</font>,<font color="Maroon">S</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>;<br/>
<b>hence </b><a NAME="E7:18"/>
<span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p2">(<span class="default"><a href="sublemma.html#K3" target="_self">Val_S</a> <font color="Maroon">v</font>,<font color="Maroon">S</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
 ;<br/>


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