<?xml version="1.0"?>
<div class="add">

<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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">J</font> being   <a href="valuat_1.html#M1" title="VALUAT_1:mode.1">interpretation</a> of <font color="Maroon" title="c1">A</font><br/> for <font color="Olive" title="b2">v</font> being   <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/> for <font color="Olive" title="b3">S</font> being   <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>  holds <br/> ( not <font color="Olive" title="b1">J</font>,<font color="Olive" title="b2">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="Olive" title="b2">v</font>,<font color="Olive" title="b3">S</font></span>)</span> <a href="sublemma.html#R1" title="SUBLEMMA:pred.1">|=</a> <font color="Olive" title="b3">S</font> iff <font color="Olive" title="b1">J</font>,<font color="Olive" title="b2">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="Olive" title="b2">v</font>,<font color="Olive" title="b3">S</font></span>)</span> <a href="sublemma.html#R1" title="SUBLEMMA:pred.1">|=</a>  <a href="sublemma.html#K6" title="SUBLEMMA:func.6">Sub_not</a> <font color="Olive" title="b3">S</font> )</span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">v</font> being   <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/> for <font color="Olive" title="b2">S</font> being   <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>  holds <br/> ( not <font color="Maroon" title="c2">J</font>,<font color="Olive" title="b1">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="Olive" title="b1">v</font>,<font color="Olive" title="b2">S</font></span>)</span> <a href="sublemma.html#R1" title="SUBLEMMA:pred.1">|=</a> <font color="Olive" title="b2">S</font> iff <font color="Maroon" title="c2">J</font>,<font color="Olive" title="b1">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="Olive" title="b1">v</font>,<font color="Olive" title="b2">S</font></span>)</span> <a href="sublemma.html#R1" title="SUBLEMMA:pred.1">|=</a>  <a href="sublemma.html#K6" title="SUBLEMMA:func.6">Sub_not</a> <font color="Olive" title="b2">S</font> )</span><br/><b>let </b><font color="Maroon" title="c3">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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">S</font> being   <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>  holds <br/> ( not <font color="Maroon" title="c2">J</font>,<font color="Maroon" title="c3">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="c3">v</font>,<font color="Olive" title="b1">S</font></span>)</span> <a href="sublemma.html#R1" title="SUBLEMMA:pred.1">|=</a> <font color="Olive" title="b1">S</font> iff <font color="Maroon" title="c2">J</font>,<font color="Maroon" title="c3">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="c3">v</font>,<font color="Olive" title="b1">S</font></span>)</span> <a href="sublemma.html#R1" title="SUBLEMMA:pred.1">|=</a>  <a href="sublemma.html#K6" title="SUBLEMMA:func.6">Sub_not</a> <font color="Olive" title="b1">S</font> )</span><br/><b>let </b><font color="Maroon" title="c4">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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( not <font color="Maroon" title="c2">J</font>,<font color="Maroon" title="c3">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="c3">v</font>,<font color="Maroon" title="c4">S</font></span>)</span> <a href="sublemma.html#R1" title="SUBLEMMA:pred.1">|=</a> <font color="Maroon" title="c4">S</font> iff <font color="Maroon" title="c2">J</font>,<font color="Maroon" title="c3">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="c3">v</font>,<font color="Maroon" title="c4">S</font></span>)</span> <a href="sublemma.html#R1" title="SUBLEMMA:pred.1">|=</a>  <a href="sublemma.html#K6" title="SUBLEMMA:func.6">Sub_not</a> <font color="Maroon" title="c4">S</font> )</span><br/>







<a NAME="E1:26"/><i><font color="Green" title="E17">A1</font></i>: 
( not <font color="Maroon" title="c2">J</font>,<font color="Maroon" title="c3">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="c3">v</font>,<font color="Maroon" title="c4">S</font></span>)</span> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Maroon" title="c4">S</font> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a>  iff <font color="Maroon" title="c2">J</font>,<font color="Maroon" title="c3">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="c3">v</font>,<font color="Maroon" title="c4">S</font></span>)</span> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a>  <a href="cqc_lang.html#K7" title="CQC_LANG:func.7">'not'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span> )
 
<b>by </b><i><a class="ref" href="valuat_1.html#T28" title="VALUAT_1:th.28">VALUAT_1:28</a></i>;<br/>
<a NAME="E2:26"/>
( <font color="Maroon" title="c2">J</font>,<font color="Maroon" title="c3">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="c3">v</font>,<font color="Maroon" title="c4">S</font></span>)</span> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a>  <a href="cqc_lang.html#K7" title="CQC_LANG:func.7">'not'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">S</font> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span> iff <font color="Maroon" title="c2">J</font>,<font color="Maroon" title="c3">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="c3">v</font>,<font color="Maroon" title="c4">S</font></span>)</span> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <span class="p1">(<span class="default"><a href="sublemma.html#K6" title="SUBLEMMA:func.6">Sub_not</a> <font color="Maroon" title="c4">S</font></span>)</span> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a>  )
 
<b>by </b><i><a class="ref" href="sublemma.html#T17" target="_self" title="SUBLEMMA:th.17">Th17</a></i>;<br/>
<b>hence </b><a NAME="E3:26"/>
( not <font color="Maroon" title="c2">J</font>,<font color="Maroon" title="c3">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="c3">v</font>,<font color="Maroon" title="c4">S</font></span>)</span> <a href="sublemma.html#R1" title="SUBLEMMA:pred.1">|=</a> <font color="Maroon" title="c4">S</font> iff <font color="Maroon" title="c2">J</font>,<font color="Maroon" title="c3">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="c3">v</font>,<font color="Maroon" title="c4">S</font></span>)</span> <a href="sublemma.html#R1" title="SUBLEMMA:pred.1">|=</a>  <a href="sublemma.html#K6" title="SUBLEMMA:func.6">Sub_not</a> <font color="Maroon" title="c4">S</font> )
 <b>by </b><i><a class="txt" href="#E1:26"><i><font color="Green" title="E17">A1</font></i></a>, <a class="ref" href="sublemma.html#D3" target="_self" title="SUBLEMMA:def.3">Def3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
