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<b>let </b><font color="Maroon">A</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">B</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">X</font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">A</font>;<br/><b>let </b><font color="Maroon">Y</font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">B</font>;<br/><b>let </b><font color="Maroon">R</font> be    <a href="relset_1.html#M2">Relation</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font>;<br/>









<a NAME="E1:69"/>
( <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="relset_1.html#K6">~</a> </span>)</span> <a href="relset_2.html#K5" target="_self">.:^</a> <font color="Maroon">Y</font> iff <font color="Maroon">X</font> <a href="xboole_0.html#R1">misses</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="relset_1.html#K6">~</a> </span>)</span> <a href="relset_2.html#K5" target="_self">.:^</a> <font color="Maroon">Y</font></span>)</span> <a href="subset_1.html#K3">`</a>  )
 
<b>by </b><i><a class="ref" href="subset_1.html#T44">SUBSET_1:44</a></i>;<br/>
<a NAME="E2:69"/><b>then </b>
( <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="relset_1.html#K6">~</a> </span>)</span> <a href="relset_2.html#K5" target="_self">.:^</a> <font color="Maroon">Y</font> iff <font color="Maroon">X</font> <a href="xboole_0.html#R1">misses</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="relset_1.html#K6">~</a> </span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="relat_1.html#K9">.:</a> <font color="Maroon">Y</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E3:69"/><b>then </b><i><font color="Green">E21</font></i>: 
( <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="relset_1.html#K6">~</a> </span>)</span> <a href="relset_2.html#K5" target="_self">.:^</a> <font color="Maroon">Y</font> iff <font color="Maroon">X</font> <a href="subset_1.html#K8">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">R</font> <a href="relset_1.html#K6">~</a> </span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="relat_1.html#K9">.:</a> <font color="Maroon">Y</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<b>reconsider </b><font color="Maroon">S</font> = <font color="Maroon">R</font> <a href="subset_1.html#K3">`</a>  as    <a href="relset_1.html#M2">Relation</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font> <b>by </b><i><a class="ref" href="relset_1.html#D1">RELSET_1:def 1</a></i>;<br/>
<a NAME="E5:69"/>
( <font color="Maroon">X</font> <a href="xboole_0.html#R1">misses</a> <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="relset_1.html#K6">~</a> </span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">Y</font> iff <font color="Maroon">Y</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">S</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E6:69"/><b>then </b>
( <font color="Maroon">X</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="relset_1.html#K6">~</a> </span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">Y</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  iff <font color="Maroon">Y</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E7:69"/><b>then </b>
( <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="relset_1.html#K6">~</a> </span>)</span> <a href="relset_2.html#K5" target="_self">.:^</a> <font color="Maroon">Y</font> iff <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="relset_2.html#K5" target="_self">.:^</a> <font color="Maroon">X</font></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i>, <a class="ref" href="relset_2.html#T4" target="_self">Th4</a>, </i>;<br/>
<a NAME="E8:69"/><b>then </b>
( <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="relset_1.html#K6">~</a> </span>)</span> <a href="relset_2.html#K5" target="_self">.:^</a> <font color="Maroon">Y</font> iff <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="relset_2.html#K5" target="_self">.:^</a> <font color="Maroon">X</font></span>)</span> <a href="subset_1.html#K3">`</a>  <a href="xboole_0.html#R1">misses</a> <font color="Maroon">Y</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<b>hence </b><a NAME="E9:69"/>
( <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="relset_1.html#K6">~</a> </span>)</span> <a href="relset_2.html#K5" target="_self">.:^</a> <font color="Maroon">Y</font> iff <font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">R</font> <a href="relset_2.html#K5" target="_self">.:^</a> <font color="Maroon">X</font> )
 <b>by </b><i><a class="ref" href="subset_1.html#T44">SUBSET_1:44</a></i>;<br/>


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