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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">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="E1:9"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">S</font> is <a href="substut1.html#V2">Sub_VERUM</a>
 ;<br/>

<b>then consider </b><font color="Maroon">Sub</font> being   <a href="substut1.html#NM1">CQC_Substitution</a><b> such that </b><br/><a NAME="E3:9"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><a href="cqc_lang.html#K9">VERUM</a> ,<font color="Maroon">Sub</font></span><a href="domain_1.html#K1">]</a></span>
 <b>by </b><i><a class="ref" href="substut1.html#D19">SUBSTUT1:def 19</a></i>;<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/>

<a NAME="E4:9"/>
<font color="Maroon">S</font> <a href="sublemma.html#K2" target="_self">`1</a>  <a href="hidden.html#R1">=</a>  <a href="cqc_lang.html#K9">VERUM</a> 
 
<b>by </b><i>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/>
<a NAME="E5:9"/><b>then </b>
( <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">S</font></span>)</span> <a href="valuat_1.html#R1">|=</a>  <a href="cqc_lang.html#K9">VERUM</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>,<font color="Maroon">S</font></span>)</span> <a href="sublemma.html#R1" target="_self">|=</a> <font color="Maroon">S</font> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E6:9"/>
( <font color="Maroon">J</font>,<font color="Maroon">v</font> <a href="valuat_1.html#R1">|=</a>  <a href="substut1.html#K39">CQC_Sub</a> <font color="Maroon">S</font> implies <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">S</font></span>)</span> <a href="sublemma.html#R1" target="_self">|=</a> <font color="Maroon">S</font> )
 <b>by </b><i><a class="ref" href="valuat_1.html#T44">VALUAT_1:44</a></i>;<br/>

<a NAME="E7:9"/>
( <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">S</font></span>)</span> <a href="sublemma.html#R1" target="_self">|=</a> <font color="Maroon">S</font> implies <font color="Maroon">J</font>,<font color="Maroon">v</font> <a href="valuat_1.html#R1">|=</a>  <a href="cqc_lang.html#K9">VERUM</a>  )
 
<b>by </b><i><a class="ref" href="valuat_1.html#T44">VALUAT_1:44</a></i>;<br/>
<b>hence </b><a NAME="E8:9"/>
( <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">S</font></span>)</span> <a href="sublemma.html#R1" target="_self">|=</a> <font color="Maroon">S</font> implies <font color="Maroon">J</font>,<font color="Maroon">v</font> <a href="valuat_1.html#R1">|=</a>  <a href="substut1.html#K39">CQC_Sub</a> <font color="Maroon">S</font> )
 <b>by </b><i>, </i>;<br/>


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