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

<b>let </b><font color="Maroon" title="c1">X</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="intpro_1.html#K1" title="INTPRO_1:func.1">MC-wff</a> ;<br/><b>let </b><font color="Maroon" title="c2">p</font>, <font color="Maroon" title="c3">q</font>, <font color="Maroon" title="c4">r</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="intpro_1.html#K1" title="INTPRO_1:func.1">MC-wff</a> ;<br/>







<b>thus </b><a NAME="E1:133"/>
<font color="Maroon" title="c2">p</font> <a href="intpro_1.html#K3" title="INTPRO_1:func.3">=&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">q</font> <a href="intpro_1.html#K3" title="INTPRO_1:func.3">=&gt;</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="intpro_1.html#K13" title="INTPRO_1:func.13">CnS4</a> <font color="Maroon" title="c1">X</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="133_1"><b>proof </b></a><div class="add">

<a NAME="E1:133_1"/><i><font color="Green" title="E97">A1</font></i>: 
<font color="Maroon" title="c2">p</font> <a href="intpro_1.html#K3" title="INTPRO_1:func.3">=&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">q</font> <a href="intpro_1.html#K3" title="INTPRO_1:func.3">=&gt;</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="intpro_1.html#K7" title="INTPRO_1:func.7">CnIPC</a> <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="ref" href="intpro_1.html#T1" target="_self" title="INTPRO_1:th.1">Th1</a></i>;<br/>
<a NAME="E2:133_1"/>
 <a href="intpro_1.html#K7" title="INTPRO_1:func.7">CnIPC</a> <font color="Maroon" title="c1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="intpro_1.html#K13" title="INTPRO_1:func.13">CnS4</a> <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="ref" href="intpro_1.html#T81" target="_self" title="INTPRO_1:th.81">Th81</a></i>;<br/>
<b>hence </b><a NAME="E3:133_1"/>
<font color="Maroon" title="c2">p</font> <a href="intpro_1.html#K3" title="INTPRO_1:func.3">=&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">q</font> <a href="intpro_1.html#K3" title="INTPRO_1:func.3">=&gt;</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="intpro_1.html#K13" title="INTPRO_1:func.13">CnS4</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="intpro_1.html#E1:133_1"><i><font color="Green" title="E97">A1</font></i></a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E2:133"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="intpro_1.html#K3" title="INTPRO_1:func.3">=&gt;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">q</font> <a href="intpro_1.html#K3" title="INTPRO_1:func.3">=&gt;</a> <font color="Maroon" title="c4">r</font></span>)</span></span>)</span> <a href="intpro_1.html#K3" title="INTPRO_1:func.3">=&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="intpro_1.html#K3" title="INTPRO_1:func.3">=&gt;</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="intpro_1.html#K3" title="INTPRO_1:func.3">=&gt;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="intpro_1.html#K3" title="INTPRO_1:func.3">=&gt;</a> <font color="Maroon" title="c4">r</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="intpro_1.html#K13" title="INTPRO_1:func.13">CnS4</a> <font color="Maroon" title="c1">X</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="133_2"><b>proof </b></a><div class="add">

<a NAME="E1:133_2"/><i><font color="Green" title="E97">A2</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="intpro_1.html#K3" title="INTPRO_1:func.3">=&gt;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">q</font> <a href="intpro_1.html#K3" title="INTPRO_1:func.3">=&gt;</a> <font color="Maroon" title="c4">r</font></span>)</span></span>)</span> <a href="intpro_1.html#K3" title="INTPRO_1:func.3">=&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="intpro_1.html#K3" title="INTPRO_1:func.3">=&gt;</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="intpro_1.html#K3" title="INTPRO_1:func.3">=&gt;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="intpro_1.html#K3" title="INTPRO_1:func.3">=&gt;</a> <font color="Maroon" title="c4">r</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="intpro_1.html#K7" title="INTPRO_1:func.7">CnIPC</a> <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="ref" href="intpro_1.html#T2" target="_self" title="INTPRO_1:th.2">Th2</a></i>;<br/>
<a NAME="E2:133_2"/>
 <a href="intpro_1.html#K7" title="INTPRO_1:func.7">CnIPC</a> <font color="Maroon" title="c1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="intpro_1.html#K13" title="INTPRO_1:func.13">CnS4</a> <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="ref" href="intpro_1.html#T81" target="_self" title="INTPRO_1:th.81">Th81</a></i>;<br/>
<b>hence </b><a NAME="E3:133_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="intpro_1.html#K3" title="INTPRO_1:func.3">=&gt;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">q</font> <a href="intpro_1.html#K3" title="INTPRO_1:func.3">=&gt;</a> <font color="Maroon" title="c4">r</font></span>)</span></span>)</span> <a href="intpro_1.html#K3" title="INTPRO_1:func.3">=&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="intpro_1.html#K3" title="INTPRO_1:func.3">=&gt;</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="intpro_1.html#K3" title="INTPRO_1:func.3">=&gt;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="intpro_1.html#K3" title="INTPRO_1:func.3">=&gt;</a> <font color="Maroon" title="c4">r</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="intpro_1.html#K13" title="INTPRO_1:func.13">CnS4</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="intpro_1.html#E1:133_2"><i><font color="Green" title="E97">A2</font></i></a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E3:133"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="intpro_1.html#K4" title="INTPRO_1:func.4">'&amp;'</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="intpro_1.html#K3" title="INTPRO_1:func.3">=&gt;</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="intpro_1.html#K13" title="INTPRO_1:func.13">CnS4</a> <font color="Maroon" title="c1">X</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="133_3"><b>proof </b></a><div class="add">

<a NAME="E1:133_3"/><i><font color="Green" title="E97">A3</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="intpro_1.html#K4" title="INTPRO_1:func.4">'&amp;'</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="intpro_1.html#K3" title="INTPRO_1:func.3">=&gt;</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="intpro_1.html#K7" title="INTPRO_1:func.7">CnIPC</a> <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="ref" href="intpro_1.html#T3" target="_self" title="INTPRO_1:th.3">Th3</a></i>;<br/>
<a NAME="E2:133_3"/>
 <a href="intpro_1.html#K7" title="INTPRO_1:func.7">CnIPC</a> <font color="Maroon" title="c1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="intpro_1.html#K13" title="INTPRO_1:func.13">CnS4</a> <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="ref" href="intpro_1.html#T81" target="_self" title="INTPRO_1:th.81">Th81</a></i>;<br/>
<b>hence </b><a NAME="E3:133_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="intpro_1.html#K4" title="INTPRO_1:func.4">'&amp;'</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="intpro_1.html#K3" title="INTPRO_1:func.3">=&gt;</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="intpro_1.html#K13" title="INTPRO_1:func.13">CnS4</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="intpro_1.html#E1:133_3"><i><font color="Green" title="E97">A3</font></i></a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E4:133"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="intpro_1.html#K4" title="INTPRO_1:func.4">'&amp;'</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="intpro_1.html#K3" title="INTPRO_1:func.3">=&gt;</a> <font color="Maroon" title="c3">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="intpro_1.html#K13" title="INTPRO_1:func.13">CnS4</a> <font color="Maroon" title="c1">X</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="133_4"><b>proof </b></a><div class="add">

<a NAME="E1:133_4"/><i><font color="Green" title="E97">A4</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="intpro_1.html#K4" title="INTPRO_1:func.4">'&amp;'</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="intpro_1.html#K3" title="INTPRO_1:func.3">=&gt;</a> <font color="Maroon" title="c3">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="intpro_1.html#K7" title="INTPRO_1:func.7">CnIPC</a> <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="ref" href="intpro_1.html#T4" target="_self" title="INTPRO_1:th.4">Th4</a></i>;<br/>
<a NAME="E2:133_4"/>
 <a href="intpro_1.html#K7" title="INTPRO_1:func.7">CnIPC</a> <font color="Maroon" title="c1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="intpro_1.html#K13" title="INTPRO_1:func.13">CnS4</a> <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="ref" href="intpro_1.html#T81" target="_self" title="INTPRO_1:th.81">Th81</a></i>;<br/>
<b>hence </b><a NAME="E3:133_4"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="intpro_1.html#K4" title="INTPRO_1:func.4">'&amp;'</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="intpro_1.html#K3" title="INTPRO_1:func.3">=&gt;</a> <font color="Maroon" title="c3">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="intpro_1.html#K13" title="INTPRO_1:func.13">CnS4</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="intpro_1.html#E1:133_4"><i><font color="Green" title="E97">A4</font></i></a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E5:133"/>
<font color="Maroon" title="c2">p</font> <a href="intpro_1.html#K3" title="INTPRO_1:func.3">=&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">q</font> <a href="intpro_1.html#K3" title="INTPRO_1:func.3">=&gt;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="intpro_1.html#K4" title="INTPRO_1:func.4">'&amp;'</a> <font color="Maroon" title="c3">q</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="intpro_1.html#K13" title="INTPRO_1:func.13">CnS4</a> <font color="Maroon" title="c1">X</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="133_5"><b>proof </b></a><div class="add">

<a NAME="E1:133_5"/><i><font color="Green" title="E97">A5</font></i>: 
<font color="Maroon" title="c2">p</font> <a href="intpro_1.html#K3" title="INTPRO_1:func.3">=&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">q</font> <a href="intpro_1.html#K3" title="INTPRO_1:func.3">=&gt;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="intpro_1.html#K4" title="INTPRO_1:func.4">'&amp;'</a> <font color="Maroon" title="c3">q</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="intpro_1.html#K7" title="INTPRO_1:func.7">CnIPC</a> <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="ref" href="intpro_1.html#T5" target="_self" title="INTPRO_1:th.5">Th5</a></i>;<br/>
<a NAME="E2:133_5"/>
 <a href="intpro_1.html#K7" title="INTPRO_1:func.7">CnIPC</a> <font color="Maroon" title="c1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="intpro_1.html#K13" title="INTPRO_1:func.13">CnS4</a> <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="ref" href="intpro_1.html#T81" target="_self" title="INTPRO_1:th.81">Th81</a></i>;<br/>
<b>hence </b><a NAME="E3:133_5"/>
<font color="Maroon" title="c2">p</font> <a href="intpro_1.html#K3" title="INTPRO_1:func.3">=&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">q</font> <a href="intpro_1.html#K3" title="INTPRO_1:func.3">=&gt;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="intpro_1.html#K4" title="INTPRO_1:func.4">'&amp;'</a> <font color="Maroon" title="c3">q</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="intpro_1.html#K13" title="INTPRO_1:func.13">CnS4</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="intpro_1.html#E1:133_5"><i><font color="Green" title="E97">A5</font></i></a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E6:133"/>
<font color="Maroon" title="c2">p</font> <a href="intpro_1.html#K3" title="INTPRO_1:func.3">=&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="intpro_1.html#K5" title="INTPRO_1:func.5">'or'</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="intpro_1.html#K13" title="INTPRO_1:func.13">CnS4</a> <font color="Maroon" title="c1">X</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="133_6"><b>proof </b></a><div class="add">

<a NAME="E1:133_6"/><i><font color="Green" title="E97">A6</font></i>: 
<font color="Maroon" title="c2">p</font> <a href="intpro_1.html#K3" title="INTPRO_1:func.3">=&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="intpro_1.html#K5" title="INTPRO_1:func.5">'or'</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="intpro_1.html#K7" title="INTPRO_1:func.7">CnIPC</a> <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="ref" href="intpro_1.html#T6" target="_self" title="INTPRO_1:th.6">Th6</a></i>;<br/>
<a NAME="E2:133_6"/>
 <a href="intpro_1.html#K7" title="INTPRO_1:func.7">CnIPC</a> <font color="Maroon" title="c1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="intpro_1.html#K13" title="INTPRO_1:func.13">CnS4</a> <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="ref" href="intpro_1.html#T81" target="_self" title="INTPRO_1:th.81">Th81</a></i>;<br/>
<b>hence </b><a NAME="E3:133_6"/>
<font color="Maroon" title="c2">p</font> <a href="intpro_1.html#K3" title="INTPRO_1:func.3">=&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="intpro_1.html#K5" title="INTPRO_1:func.5">'or'</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="intpro_1.html#K13" title="INTPRO_1:func.13">CnS4</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="intpro_1.html#E1:133_6"><i><font color="Green" title="E97">A6</font></i></a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E7:133"/>
<font color="Maroon" title="c3">q</font> <a href="intpro_1.html#K3" title="INTPRO_1:func.3">=&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="intpro_1.html#K5" title="INTPRO_1:func.5">'or'</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="intpro_1.html#K13" title="INTPRO_1:func.13">CnS4</a> <font color="Maroon" title="c1">X</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="133_7"><b>proof </b></a><div class="add">

<a NAME="E1:133_7"/><i><font color="Green" title="E97">A7</font></i>: 
<font color="Maroon" title="c3">q</font> <a href="intpro_1.html#K3" title="INTPRO_1:func.3">=&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="intpro_1.html#K5" title="INTPRO_1:func.5">'or'</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="intpro_1.html#K7" title="INTPRO_1:func.7">CnIPC</a> <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="ref" href="intpro_1.html#T7" target="_self" title="INTPRO_1:th.7">Th7</a></i>;<br/>
<a NAME="E2:133_7"/>
 <a href="intpro_1.html#K7" title="INTPRO_1:func.7">CnIPC</a> <font color="Maroon" title="c1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="intpro_1.html#K13" title="INTPRO_1:func.13">CnS4</a> <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="ref" href="intpro_1.html#T81" target="_self" title="INTPRO_1:th.81">Th81</a></i>;<br/>
<b>hence </b><a NAME="E3:133_7"/>
<font color="Maroon" title="c3">q</font> <a href="intpro_1.html#K3" title="INTPRO_1:func.3">=&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="intpro_1.html#K5" title="INTPRO_1:func.5">'or'</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="intpro_1.html#K13" title="INTPRO_1:func.13">CnS4</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="intpro_1.html#E1:133_7"><i><font color="Green" title="E97">A7</font></i></a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E8:133"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="intpro_1.html#K3" title="INTPRO_1:func.3">=&gt;</a> <font color="Maroon" title="c4">r</font></span>)</span> <a href="intpro_1.html#K3" title="INTPRO_1:func.3">=&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">q</font> <a href="intpro_1.html#K3" title="INTPRO_1:func.3">=&gt;</a> <font color="Maroon" title="c4">r</font></span>)</span> <a href="intpro_1.html#K3" title="INTPRO_1:func.3">=&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="intpro_1.html#K5" title="INTPRO_1:func.5">'or'</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="intpro_1.html#K3" title="INTPRO_1:func.3">=&gt;</a> <font color="Maroon" title="c4">r</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="intpro_1.html#K13" title="INTPRO_1:func.13">CnS4</a> <font color="Maroon" title="c1">X</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="133_8"><b>proof </b></a><div class="add">

<a NAME="E1:133_8"/><i><font color="Green" title="E97">A8</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="intpro_1.html#K3" title="INTPRO_1:func.3">=&gt;</a> <font color="Maroon" title="c4">r</font></span>)</span> <a href="intpro_1.html#K3" title="INTPRO_1:func.3">=&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">q</font> <a href="intpro_1.html#K3" title="INTPRO_1:func.3">=&gt;</a> <font color="Maroon" title="c4">r</font></span>)</span> <a href="intpro_1.html#K3" title="INTPRO_1:func.3">=&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="intpro_1.html#K5" title="INTPRO_1:func.5">'or'</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="intpro_1.html#K3" title="INTPRO_1:func.3">=&gt;</a> <font color="Maroon" title="c4">r</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="intpro_1.html#K7" title="INTPRO_1:func.7">CnIPC</a> <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="ref" href="intpro_1.html#T8" target="_self" title="INTPRO_1:th.8">Th8</a></i>;<br/>
<a NAME="E2:133_8"/>
 <a href="intpro_1.html#K7" title="INTPRO_1:func.7">CnIPC</a> <font color="Maroon" title="c1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="intpro_1.html#K13" title="INTPRO_1:func.13">CnS4</a> <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="ref" href="intpro_1.html#T81" target="_self" title="INTPRO_1:th.81">Th81</a></i>;<br/>
<b>hence </b><a NAME="E3:133_8"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="intpro_1.html#K3" title="INTPRO_1:func.3">=&gt;</a> <font color="Maroon" title="c4">r</font></span>)</span> <a href="intpro_1.html#K3" title="INTPRO_1:func.3">=&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">q</font> <a href="intpro_1.html#K3" title="INTPRO_1:func.3">=&gt;</a> <font color="Maroon" title="c4">r</font></span>)</span> <a href="intpro_1.html#K3" title="INTPRO_1:func.3">=&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="intpro_1.html#K5" title="INTPRO_1:func.5">'or'</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="intpro_1.html#K3" title="INTPRO_1:func.3">=&gt;</a> <font color="Maroon" title="c4">r</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="intpro_1.html#K13" title="INTPRO_1:func.13">CnS4</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="intpro_1.html#E1:133_8"><i><font color="Green" title="E97">A8</font></i></a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E9:133"/>
<a href="intpro_1.html#K2" title="INTPRO_1:func.2">FALSUM</a>  <a href="intpro_1.html#K3" title="INTPRO_1:func.3">=&gt;</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="intpro_1.html#K13" title="INTPRO_1:func.13">CnS4</a> <font color="Maroon" title="c1">X</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="133_9"><b>proof </b></a><div class="add">

<a NAME="E1:133_9"/><i><font color="Green" title="E97">A9</font></i>: 
<a href="intpro_1.html#K2" title="INTPRO_1:func.2">FALSUM</a>  <a href="intpro_1.html#K3" title="INTPRO_1:func.3">=&gt;</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="intpro_1.html#K7" title="INTPRO_1:func.7">CnIPC</a> <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="ref" href="intpro_1.html#T9" target="_self" title="INTPRO_1:th.9">Th9</a></i>;<br/>
<a NAME="E2:133_9"/>
 <a href="intpro_1.html#K7" title="INTPRO_1:func.7">CnIPC</a> <font color="Maroon" title="c1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="intpro_1.html#K13" title="INTPRO_1:func.13">CnS4</a> <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="ref" href="intpro_1.html#T81" target="_self" title="INTPRO_1:th.81">Th81</a></i>;<br/>
<b>hence </b><a NAME="E3:133_9"/>
<a href="intpro_1.html#K2" title="INTPRO_1:func.2">FALSUM</a>  <a href="intpro_1.html#K3" title="INTPRO_1:func.3">=&gt;</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="intpro_1.html#K13" title="INTPRO_1:func.13">CnS4</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="intpro_1.html#E1:133_9"><i><font color="Green" title="E97">A9</font></i></a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E10:133"/>
<font color="Maroon" title="c2">p</font> <a href="intpro_1.html#K5" title="INTPRO_1:func.5">'or'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="intpro_1.html#K3" title="INTPRO_1:func.3">=&gt;</a> <a href="intpro_1.html#K2" title="INTPRO_1:func.2">FALSUM</a> </span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="intpro_1.html#K13" title="INTPRO_1:func.13">CnS4</a> <font color="Maroon" title="c1">X</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="133_10"><b>proof </b></a><div class="add">

<a NAME="E1:133_10"/>
for <font color="Olive" title="b1">T</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="intpro_1.html#K1" title="INTPRO_1:func.1">MC-wff</a>   st <font color="Olive" title="b1">T</font> is <a href="intpro_1.html#V10" title="INTPRO_1:attr.10">S4_theory</a> &amp; <font color="Maroon" title="c1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">T</font> holds <br/><font color="Maroon" title="c2">p</font> <a href="intpro_1.html#K5" title="INTPRO_1:func.5">'or'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="intpro_1.html#K3" title="INTPRO_1:func.3">=&gt;</a> <a href="intpro_1.html#K2" title="INTPRO_1:func.2">FALSUM</a> </span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">T</font>
 
<b>by </b><i><a class="ref" href="intpro_1.html#D22" target="_self" title="INTPRO_1:def.22">Def22</a></i>;<br/>
<b>hence </b><a NAME="E2:133_10"/>
<font color="Maroon" title="c2">p</font> <a href="intpro_1.html#K5" title="INTPRO_1:func.5">'or'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="intpro_1.html#K3" title="INTPRO_1:func.3">=&gt;</a> <a href="intpro_1.html#K2" title="INTPRO_1:func.2">FALSUM</a> </span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="intpro_1.html#K13" title="INTPRO_1:func.13">CnS4</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="ref" href="intpro_1.html#D23" target="_self" title="INTPRO_1:def.23">Def23</a></i>;<br/>


</div><b>end;</b></div>


</div>
