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<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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">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>   st ( for <font color="Olive" title="b3">v</font> being   <a href="funct_2.html#M2" title="FUNCT_2: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> holds <br/> ( <font color="Olive" title="b1">J</font>,<font color="Olive" title="b3">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="Olive" title="b2">S</font> iff <font color="Olive" title="b1">J</font>,<font color="Olive" title="b3">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="b3">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> ) ) holds <br/>for <font color="Olive" title="b3">v</font> being   <a href="funct_2.html#M2" title="FUNCT_2: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> holds <br/> ( <font color="Olive" title="b1">J</font>,<font color="Olive" title="b3">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> <span class="p1">(<span class="default"><a href="sublemma.html#K6" title="SUBLEMMA:func.6">Sub_not</a> <font color="Olive" title="b2">S</font></span>)</span> iff <font color="Olive" title="b1">J</font>,<font color="Olive" title="b3">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="b3">v</font>,<span class="p2">(<span class="default"><a href="sublemma.html#K6" title="SUBLEMMA:func.6">Sub_not</a> <font color="Olive" title="b2">S</font></span>)</span></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/><span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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>   st ( for <font color="Olive" title="b2">v</font> being   <a href="funct_2.html#M2" title="FUNCT_2: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> holds <br/> ( <font color="Maroon" title="c2">J</font>,<font color="Olive" title="b2">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="Olive" title="b1">S</font> iff <font color="Maroon" title="c2">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="b1">S</font></span>)</span> <a href="sublemma.html#R1" title="SUBLEMMA:pred.1">|=</a> <font color="Olive" title="b1">S</font> ) ) holds <br/>for <font color="Olive" title="b2">v</font> being   <a href="funct_2.html#M2" title="FUNCT_2: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> holds <br/> ( <font color="Maroon" title="c2">J</font>,<font color="Olive" title="b2">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> <span class="p1">(<span class="default"><a href="sublemma.html#K6" title="SUBLEMMA:func.6">Sub_not</a> <font color="Olive" title="b1">S</font></span>)</span> iff <font color="Maroon" title="c2">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>,<span class="p2">(<span class="default"><a href="sublemma.html#K6" title="SUBLEMMA:func.6">Sub_not</a> <font color="Olive" title="b1">S</font></span>)</span></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/><span class="kw">let </span><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( for <font color="Olive" title="b1">v</font> being   <a href="funct_2.html#M2" title="FUNCT_2: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> holds <br/> ( <font color="Maroon" title="c2">J</font>,<font color="Olive" title="b1">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> 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="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 for <font color="Olive" title="b1">v</font> being   <a href="funct_2.html#M2" title="FUNCT_2: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> holds <br/> ( <font color="Maroon" title="c2">J</font>,<font color="Olive" title="b1">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> <span class="p1">(<span class="default"><a href="sublemma.html#K6" title="SUBLEMMA:func.6">Sub_not</a> <font color="Maroon" title="c3">S</font></span>)</span> 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>,<span class="p2">(<span class="default"><a href="sublemma.html#K6" title="SUBLEMMA:func.6">Sub_not</a> <font color="Maroon" title="c3">S</font></span>)</span></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="c3">S</font> ) )</span><br/>





<span class="kw">assume </span><a NAME="E1:29"/><span class="lab"><font color="Green" title="E19">A1</font></span>: 
for <font color="Olive" title="b1">v</font> being   <a href="funct_2.html#M2" title="FUNCT_2: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> holds <br/> ( <font color="Maroon" title="c2">J</font>,<font color="Olive" title="b1">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> 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="Maroon" title="c3">S</font></span>)</span> <a href="sublemma.html#R1" title="SUBLEMMA:pred.1">|=</a> <font color="Maroon" title="c3">S</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">v</font> being   <a href="funct_2.html#M2" title="FUNCT_2: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> holds <br/> ( <font color="Maroon" title="c2">J</font>,<font color="Olive" title="b1">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> <span class="p1">(<span class="default"><a href="sublemma.html#K6" title="SUBLEMMA:func.6">Sub_not</a> <font color="Maroon" title="c3">S</font></span>)</span> 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>,<span class="p2">(<span class="default"><a href="sublemma.html#K6" title="SUBLEMMA:func.6">Sub_not</a> <font color="Maroon" title="c3">S</font></span>)</span></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="c3">S</font> )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c4">v</font> be    <a href="funct_2.html#M2" title="FUNCT_2: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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">J</font>,<font color="Maroon" title="c4">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> <span class="p1">(<span class="default"><a href="sublemma.html#K6" title="SUBLEMMA:func.6">Sub_not</a> <font color="Maroon" title="c3">S</font></span>)</span> iff <font color="Maroon" title="c2">J</font>,<font color="Maroon" title="c4">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="c4">v</font>,<span class="p2">(<span class="default"><a href="sublemma.html#K6" title="SUBLEMMA:func.6">Sub_not</a> <font color="Maroon" title="c3">S</font></span>)</span></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="c3">S</font> )</span><br/>

<a NAME="E2:29"/><span class="lab"><font color="Green" title="E20">A2</font></span>: 
( <font color="Maroon" title="c2">J</font>,<font color="Maroon" title="c4">v</font> <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"><a href="substut1.html#K39" title="SUBSTUT1:func.39">CQC_Sub</a> <font color="Maroon" title="c3">S</font></span>)</span> iff not <font color="Maroon" title="c2">J</font>,<font color="Maroon" title="c4">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> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="valuat_1.html#T28" title="VALUAT_1:th.28">VALUAT_1:28</a></span>;<br/>
<a NAME="E3:29"/>
( not <font color="Maroon" title="c2">J</font>,<font color="Maroon" title="c4">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="c4">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> iff <font color="Maroon" title="c2">J</font>,<font color="Maroon" title="c4">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="c4">v</font>,<font color="Maroon" title="c3">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="c3">S</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="sublemma.html#T18" target="_self" title="SUBLEMMA:th.18">Th18</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:29"/>
( <font color="Maroon" title="c2">J</font>,<font color="Maroon" title="c4">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> <span class="p1">(<span class="default"><a href="sublemma.html#K6" title="SUBLEMMA:func.6">Sub_not</a> <font color="Maroon" title="c3">S</font></span>)</span> iff <font color="Maroon" title="c2">J</font>,<font color="Maroon" title="c4">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="c4">v</font>,<span class="p2">(<span class="default"><a href="sublemma.html#K6" title="SUBLEMMA:func.6">Sub_not</a> <font color="Maroon" title="c3">S</font></span>)</span></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="c3">S</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:29"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="txt" href="#E2:29"><span class="lab"><font color="Green" title="E20">A2</font></span></a>, <a class="ref" href="sublemma.html#T19" target="_self" title="SUBLEMMA:th.19">Th19</a>, <a class="ref" href="substut1.html#T29" title="SUBSTUT1:th.29">SUBSTUT1:29</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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