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<b>let </b><font color="Maroon" title="c1">A</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">J</font> be    <a href="valuat_1.html#M1" title="VALUAT_1:mode.1">interpretation</a> of <font color="Maroon" title="c1">A</font>;<br/><b>let </b><font color="Maroon" title="c3">S</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="substut1.html#K38" title="SUBSTUT1:func.38">CQC-Sub-WFF</a> ;<br/>





<b>assume </b><a NAME="E1:9"/><i><font color="Green" title="E6">A1</font></i>: 
<font color="Maroon" title="c3">S</font> is <a href="substut1.html#V2" title="SUBSTUT1:attr.2">Sub_VERUM</a>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c4">Sub</font> being   <a href="substut1.html#NM1" title="SUBSTUT1:NM.1">CQC_Substitution</a><b> such that </b><br/><a NAME="E3:9"/><i><font color="Green" title="E7">A2</font></i>: 
<font color="Maroon" title="c3">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><a href="cqc_lang.html#K6" title="CQC_LANG:func.6">VERUM</a> ,<font color="Maroon" title="c4">Sub</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 <b>by </b><i><a class="ref" href="substut1.html#D19" title="SUBSTUT1:def.19">SUBSTUT1:def 19</a></i>;<br/>
<b>let </b><font color="Maroon" title="c5">v</font> be    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="valuat_1.html#K2" title="VALUAT_1:func.2">Valuations_in</a> <font color="Maroon" title="c1">A</font>;<br/>

<a NAME="E4:9"/>
<font color="Maroon" title="c3">S</font> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cqc_lang.html#K6" title="CQC_LANG:func.6">VERUM</a> 
 
<b>by </b><i><a class="txt" href="sublemma.html#E3:9"><i><font color="Green" title="E7">A2</font></i></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>;<br/>
<a NAME="E5:9"/><b>then </b>
( <font color="Maroon" title="c2">J</font>,<font color="Maroon" title="c5">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><a href="sublemma.html#K3" title="SUBLEMMA:func.3">Val_S</a> <font color="Maroon" title="c5">v</font>,<font color="Maroon" title="c3">S</font></span>)</span> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a>  <a href="cqc_lang.html#K6" title="CQC_LANG:func.6">VERUM</a>  iff <font color="Maroon" title="c2">J</font>,<font color="Maroon" title="c5">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><a href="sublemma.html#K3" title="SUBLEMMA:func.3">Val_S</a> <font color="Maroon" title="c5">v</font>,<font color="Maroon" title="c3">S</font></span>)</span> <a href="sublemma.html#R1" title="SUBLEMMA:pred.1">|=</a> <font color="Maroon" title="c3">S</font> )
 
<b>by </b><i><a class="ref" href="sublemma.html#D3" target="_self" title="SUBLEMMA:def.3">Def3</a></i>;<br/>
<b>hence </b><a NAME="E6:9"/>
( <font color="Maroon" title="c2">J</font>,<font color="Maroon" title="c5">v</font> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a>  <a href="substut1.html#K39" title="SUBSTUT1:func.39">CQC_Sub</a> <font color="Maroon" title="c3">S</font> implies <font color="Maroon" title="c2">J</font>,<font color="Maroon" title="c5">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><a href="sublemma.html#K3" title="SUBLEMMA:func.3">Val_S</a> <font color="Maroon" title="c5">v</font>,<font color="Maroon" title="c3">S</font></span>)</span> <a href="sublemma.html#R1" title="SUBLEMMA:pred.1">|=</a> <font color="Maroon" title="c3">S</font> )
 <b>by </b><i><a class="ref" href="valuat_1.html#T44" title="VALUAT_1:th.44">VALUAT_1:44</a></i>;<br/>

<a NAME="E7:9"/>
( <font color="Maroon" title="c2">J</font>,<font color="Maroon" title="c5">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><a href="sublemma.html#K3" title="SUBLEMMA:func.3">Val_S</a> <font color="Maroon" title="c5">v</font>,<font color="Maroon" title="c3">S</font></span>)</span> <a href="sublemma.html#R1" title="SUBLEMMA:pred.1">|=</a> <font color="Maroon" title="c3">S</font> implies <font color="Maroon" title="c2">J</font>,<font color="Maroon" title="c5">v</font> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a>  <a href="cqc_lang.html#K6" title="CQC_LANG:func.6">VERUM</a>  )
 
<b>by </b><i><a class="ref" href="valuat_1.html#T44" title="VALUAT_1:th.44">VALUAT_1:44</a></i>;<br/>
<b>hence </b><a NAME="E8:9"/>
( <font color="Maroon" title="c2">J</font>,<font color="Maroon" title="c5">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><a href="sublemma.html#K3" title="SUBLEMMA:func.3">Val_S</a> <font color="Maroon" title="c5">v</font>,<font color="Maroon" title="c3">S</font></span>)</span> <a href="sublemma.html#R1" title="SUBLEMMA:pred.1">|=</a> <font color="Maroon" title="c3">S</font> implies <font color="Maroon" title="c2">J</font>,<font color="Maroon" title="c5">v</font> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a>  <a href="substut1.html#K39" title="SUBSTUT1:func.39">CQC_Sub</a> <font color="Maroon" title="c3">S</font> )
 <b>by </b><i><a class="txt" href="sublemma.html#E1:9"><i><font color="Green" title="E6">A1</font></i></a>, <a class="ref" href="sublemma.html#T3" target="_self" title="SUBLEMMA:th.3">Th3</a></i>;<br/>


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