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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">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><br/> for <font color="Olive" title="b3">S1</font>, <font color="Olive" title="b4">S2</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 <font color="Olive" title="b3">S1</font> <a href="substut1.html#K19" title="SUBSTUT1:func.19">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">S2</font> <a href="substut1.html#K19" title="SUBSTUT1:func.19">`2</a>  holds <br/>( ( <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">S1</font></span>)</span> <a href="sublemma.html#R1" title="SUBLEMMA:pred.1">|=</a> <font color="Olive" title="b3">S1</font> &amp; <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="b4">S2</font></span>)</span> <a href="sublemma.html#R1" title="SUBLEMMA:pred.1">|=</a> <font color="Olive" title="b4">S2</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>,<span class="p2">(<span class="default"><a href="sublemma.html#K7" title="SUBLEMMA:func.7">CQCSub_&amp;</a> <font color="Olive" title="b3">S1</font>,<font color="Olive" title="b4">S2</font></span>)</span></span>)</span> <a href="sublemma.html#R1" title="SUBLEMMA:pred.1">|=</a>  <a href="sublemma.html#K7" title="SUBLEMMA:func.7">CQCSub_&amp;</a> <font color="Olive" title="b3">S1</font>,<font color="Olive" title="b4">S2</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">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><br/> for <font color="Olive" title="b2">S1</font>, <font color="Olive" title="b3">S2</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 <font color="Olive" title="b2">S1</font> <a href="substut1.html#K19" title="SUBSTUT1:func.19">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">S2</font> <a href="substut1.html#K19" title="SUBSTUT1:func.19">`2</a>  holds <br/>( ( <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">S1</font></span>)</span> <a href="sublemma.html#R1" title="SUBLEMMA:pred.1">|=</a> <font color="Olive" title="b2">S1</font> &amp; <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="b3">S2</font></span>)</span> <a href="sublemma.html#R1" title="SUBLEMMA:pred.1">|=</a> <font color="Olive" title="b3">S2</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>,<span class="p2">(<span class="default"><a href="sublemma.html#K7" title="SUBLEMMA:func.7">CQCSub_&amp;</a> <font color="Olive" title="b2">S1</font>,<font color="Olive" title="b3">S2</font></span>)</span></span>)</span> <a href="sublemma.html#R1" title="SUBLEMMA:pred.1">|=</a>  <a href="sublemma.html#K7" title="SUBLEMMA:func.7">CQCSub_&amp;</a> <font color="Olive" title="b2">S1</font>,<font color="Olive" title="b3">S2</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">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"> for <font color="Olive" title="b1">S1</font>, <font color="Olive" title="b2">S2</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 <font color="Olive" title="b1">S1</font> <a href="substut1.html#K19" title="SUBSTUT1:func.19">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">S2</font> <a href="substut1.html#K19" title="SUBSTUT1:func.19">`2</a>  holds <br/>( ( <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">S1</font></span>)</span> <a href="sublemma.html#R1" title="SUBLEMMA:pred.1">|=</a> <font color="Olive" title="b1">S1</font> &amp; <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="b2">S2</font></span>)</span> <a href="sublemma.html#R1" title="SUBLEMMA:pred.1">|=</a> <font color="Olive" title="b2">S2</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>,<span class="p2">(<span class="default"><a href="sublemma.html#K7" title="SUBLEMMA:func.7">CQCSub_&amp;</a> <font color="Olive" title="b1">S1</font>,<font color="Olive" title="b2">S2</font></span>)</span></span>)</span> <a href="sublemma.html#R1" title="SUBLEMMA:pred.1">|=</a>  <a href="sublemma.html#K7" title="SUBLEMMA:func.7">CQCSub_&amp;</a> <font color="Olive" title="b1">S1</font>,<font color="Olive" title="b2">S2</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">S1</font>, <font color="Maroon" title="c5">S2</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"> ( <font color="Maroon" title="c4">S1</font> <a href="substut1.html#K19" title="SUBSTUT1:func.19">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">S2</font> <a href="substut1.html#K19" title="SUBSTUT1:func.19">`2</a>  implies ( ( <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">S1</font></span>)</span> <a href="sublemma.html#R1" title="SUBLEMMA:pred.1">|=</a> <font color="Maroon" title="c4">S1</font> &amp; <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="c5">S2</font></span>)</span> <a href="sublemma.html#R1" title="SUBLEMMA:pred.1">|=</a> <font color="Maroon" title="c5">S2</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>,<span class="p2">(<span class="default"><a href="sublemma.html#K7" title="SUBLEMMA:func.7">CQCSub_&amp;</a> <font color="Maroon" title="c4">S1</font>,<font color="Maroon" title="c5">S2</font></span>)</span></span>)</span> <a href="sublemma.html#R1" title="SUBLEMMA:pred.1">|=</a>  <a href="sublemma.html#K7" title="SUBLEMMA:func.7">CQCSub_&amp;</a> <font color="Maroon" title="c4">S1</font>,<font color="Maroon" title="c5">S2</font> ) )</span><br/>









<span class="kw">assume </span><a NAME="E1:34"/><span class="lab"><font color="Green" title="E24">A1</font></span>: 
<font color="Maroon" title="c4">S1</font> <a href="substut1.html#K19" title="SUBSTUT1:func.19">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">S2</font> <a href="substut1.html#K19" title="SUBSTUT1:func.19">`2</a> 
 ; <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="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">S1</font></span>)</span> <a href="sublemma.html#R1" title="SUBLEMMA:pred.1">|=</a> <font color="Maroon" title="c4">S1</font> &amp; <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="c5">S2</font></span>)</span> <a href="sublemma.html#R1" title="SUBLEMMA:pred.1">|=</a> <font color="Maroon" title="c5">S2</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>,<span class="p2">(<span class="default"><a href="sublemma.html#K7" title="SUBLEMMA:func.7">CQCSub_&amp;</a> <font color="Maroon" title="c4">S1</font>,<font color="Maroon" title="c5">S2</font></span>)</span></span>)</span> <a href="sublemma.html#R1" title="SUBLEMMA:pred.1">|=</a>  <a href="sublemma.html#K7" title="SUBLEMMA:func.7">CQCSub_&amp;</a> <font color="Maroon" title="c4">S1</font>,<font color="Maroon" title="c5">S2</font> )</span><br/>

<a NAME="E2:34"/><span class="kw">then </span>
 <a href="sublemma.html#K3" title="SUBLEMMA:func.3">Val_S</a> <font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c4">S1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="sublemma.html#K3" title="SUBLEMMA:func.3">Val_S</a> <font color="Maroon" title="c3">v</font>,<span class="p1">(<span class="default"><a href="sublemma.html#K7" title="SUBLEMMA:func.7">CQCSub_&amp;</a> <font color="Maroon" title="c4">S1</font>,<font color="Maroon" title="c5">S2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="sublemma.html#T22" target="_self" title="SUBLEMMA:th.22">Th22</a></span>;<br/>
<a NAME="E3:34"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A2</font></span>: 
( ( <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">S1</font></span>)</span> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Maroon" title="c4">S1</font> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a>  &amp; <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">S1</font></span>)</span> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Maroon" title="c5">S2</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>,<span class="p2">(<span class="default"><a href="sublemma.html#K7" title="SUBLEMMA:func.7">CQCSub_&amp;</a> <font color="Maroon" title="c4">S1</font>,<font color="Maroon" title="c5">S2</font></span>)</span></span>)</span> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">S1</font> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span> <a href="cqc_lang.html#K8" title="CQC_LANG:func.8">'&amp;'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">S2</font> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="valuat_1.html#T29" title="VALUAT_1:th.29">VALUAT_1:29</a></span>;<br/>
<a NAME="E4:34"/>
( <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>,<span class="p2">(<span class="default"><a href="sublemma.html#K7" title="SUBLEMMA:func.7">CQCSub_&amp;</a> <font color="Maroon" title="c4">S1</font>,<font color="Maroon" title="c5">S2</font></span>)</span></span>)</span> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">S1</font> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a> </span>)</span> <a href="cqc_lang.html#K8" title="CQC_LANG:func.8">'&amp;'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">S2</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>,<span class="p2">(<span class="default"><a href="sublemma.html#K7" title="SUBLEMMA:func.7">CQCSub_&amp;</a> <font color="Maroon" title="c4">S1</font>,<font color="Maroon" title="c5">S2</font></span>)</span></span>)</span> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <span class="p1">(<span class="default"><a href="sublemma.html#K7" title="SUBLEMMA:func.7">CQCSub_&amp;</a> <font color="Maroon" title="c4">S1</font>,<font color="Maroon" title="c5">S2</font></span>)</span> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:34"><span class="lab"><font color="Green" title="E24">A1</font></span></a>, <a class="ref" href="sublemma.html#T21" target="_self" title="SUBLEMMA:th.21">Th21</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:34"/>
( ( <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">S1</font></span>)</span> <a href="sublemma.html#R1" title="SUBLEMMA:pred.1">|=</a> <font color="Maroon" title="c4">S1</font> &amp; <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="c5">S2</font></span>)</span> <a href="sublemma.html#R1" title="SUBLEMMA:pred.1">|=</a> <font color="Maroon" title="c5">S2</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>,<span class="p2">(<span class="default"><a href="sublemma.html#K7" title="SUBLEMMA:func.7">CQCSub_&amp;</a> <font color="Maroon" title="c4">S1</font>,<font color="Maroon" title="c5">S2</font></span>)</span></span>)</span> <a href="sublemma.html#R1" title="SUBLEMMA:pred.1">|=</a>  <a href="sublemma.html#K7" title="SUBLEMMA:func.7">CQCSub_&amp;</a> <font color="Maroon" title="c4">S1</font>,<font color="Maroon" title="c5">S2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:34"><span class="lab"><font color="Green" title="E24">A1</font></span></a>, <a class="txt" href="#E3:34"><span class="lab"><font color="Green" title="E25">A2</font></span></a>, <a class="ref" href="sublemma.html#D3" target="_self" title="SUBLEMMA:def.3">Def3</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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