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<b>let </b><font color="Maroon" title="c1">X</font> be    <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">A</font>, <font color="Olive" title="b2">B</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> holds   <a href="prob_1.html#K4" title="PROB_1:func.4">Intersection</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">A</font> <a href="funct_7.html#K12" title="FUNCT_7:func.12">followed_by</a> <font color="Olive" title="b2">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Olive" title="b2">B</font></span><br/><b>let </b><font color="Maroon" title="c2">A</font>, <font color="Maroon" title="c3">B</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="prob_1.html#K4" title="PROB_1:func.4">Intersection</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="funct_7.html#K12" title="FUNCT_7:func.12">followed_by</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">B</font></span><br/>



<b>set </b><font color="Maroon" title="c4">A1</font> = <font color="Maroon" title="c2">A</font> <a href="funct_7.html#K12" title="FUNCT_7:func.12">followed_by</a> <font color="Maroon" title="c3">B</font>;<br/>
<a NAME="E1:26"/>
 <a href="prob_1.html#K3" title="PROB_1:func.3">Complement</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="funct_7.html#K12" title="FUNCT_7:func.12">followed_by</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="funct_7.html#K12" title="FUNCT_7:func.12">followed_by</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="prob_1.html#E36"><i><font color="Green" title="E15">Lmx1</font></i></a></i>;<br/>
<a NAME="E2:26"/><b>then </b>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="prob_1.html#K3" title="PROB_1:func.3">Complement</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="funct_7.html#K12" title="FUNCT_7:func.12">followed_by</a> <font color="Maroon" title="c3">B</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 
<b>by </b><i><a class="ref" href="funct_7.html#T128" title="FUNCT_7:th.128">FUNCT_7:128</a></i>;<br/>
<a NAME="E3:26"/><b>then </b>
 <a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <span class="p1">(<span class="default"><a href="prob_1.html#K3" title="PROB_1:func.3">Complement</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="funct_7.html#K12" title="FUNCT_7:func.12">followed_by</a> <font color="Maroon" title="c3">B</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T93" title="ZFMISC_1:th.93">ZFMISC_1:93</a></i>;<br/>
<a NAME="E4:26"/><b>then </b>
<span class="p1">(<span class="default"><a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <span class="p2">(<span class="default"><a href="prob_1.html#K3" title="PROB_1:func.3">Complement</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="funct_7.html#K12" title="FUNCT_7:func.12">followed_by</a> <font color="Maroon" title="c3">B</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T53" title="XBOOLE_1:th.53">XBOOLE_1:53</a></i>;<br/>
<b>hence </b><a NAME="E5:26"/>
 <a href="prob_1.html#K4" title="PROB_1:func.4">Intersection</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="funct_7.html#K12" title="FUNCT_7:func.12">followed_by</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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