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<b>let </b><font color="Maroon" title="c1">A</font>, <font color="Maroon" title="c2">B</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">X</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">A</font><br/> for <font color="Olive" title="b2">Y</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">B</font><br/> for <font color="Olive" title="b3">R</font> being   <a href="relset_1.html#M2" title="RELSET_1:mode.2">Relation</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font> holds <br/> ( <font color="Olive" title="b1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">R</font> <a href="relset_1.html#K6" title="RELSET_1:func.6">~</a> </span>)</span> <a href="relset_2.html#K6" title="RELSET_2:func.6">.:^</a> <font color="Olive" title="b2">Y</font> iff <font color="Olive" title="b2">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b3">R</font> <a href="relset_2.html#K6" title="RELSET_2:func.6">.:^</a> <font color="Olive" title="b1">X</font> )</span><br/><b>let </b><font color="Maroon" title="c3">X</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">A</font>; <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">Y</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">B</font><br/> for <font color="Olive" title="b2">R</font> being   <a href="relset_1.html#M2" title="RELSET_1:mode.2">Relation</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font> holds <br/> ( <font color="Maroon" title="c3">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">R</font> <a href="relset_1.html#K6" title="RELSET_1:func.6">~</a> </span>)</span> <a href="relset_2.html#K6" title="RELSET_2:func.6">.:^</a> <font color="Olive" title="b1">Y</font> iff <font color="Olive" title="b1">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">R</font> <a href="relset_2.html#K6" title="RELSET_2:func.6">.:^</a> <font color="Maroon" title="c3">X</font> )</span><br/><b>let </b><font color="Maroon" title="c4">Y</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">B</font>; <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">R</font> being   <a href="relset_1.html#M2" title="RELSET_1:mode.2">Relation</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font> holds <br/> ( <font color="Maroon" title="c3">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">R</font> <a href="relset_1.html#K6" title="RELSET_1:func.6">~</a> </span>)</span> <a href="relset_2.html#K6" title="RELSET_2:func.6">.:^</a> <font color="Maroon" title="c4">Y</font> iff <font color="Maroon" title="c4">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">R</font> <a href="relset_2.html#K6" title="RELSET_2:func.6">.:^</a> <font color="Maroon" title="c3">X</font> )</span><br/><b>let </b><font color="Maroon" title="c5">R</font> be    <a href="relset_1.html#M2" title="RELSET_1:mode.2">Relation</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">R</font> <a href="relset_1.html#K6" title="RELSET_1:func.6">~</a> </span>)</span> <a href="relset_2.html#K6" title="RELSET_2:func.6">.:^</a> <font color="Maroon" title="c4">Y</font> iff <font color="Maroon" title="c4">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">R</font> <a href="relset_2.html#K6" title="RELSET_2:func.6">.:^</a> <font color="Maroon" title="c3">X</font> )</span><br/>









<a NAME="E1:68"/>
( <font color="Maroon" title="c3">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">R</font> <a href="relset_1.html#K6" title="RELSET_1:func.6">~</a> </span>)</span> <a href="relset_2.html#K6" title="RELSET_2:func.6">.:^</a> <font color="Maroon" title="c4">Y</font> iff <font color="Maroon" title="c3">X</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">R</font> <a href="relset_1.html#K6" title="RELSET_1:func.6">~</a> </span>)</span> <a href="relset_2.html#K6" title="RELSET_2:func.6">.:^</a> <font color="Maroon" title="c4">Y</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  )
 
<b>by </b><i><a class="ref" href="subset_1.html#T44" title="SUBSET_1:th.44">SUBSET_1:44</a></i>;<br/>
<a NAME="E2:68"/><b>then </b>
( <font color="Maroon" title="c3">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">R</font> <a href="relset_1.html#K6" title="RELSET_1:func.6">~</a> </span>)</span> <a href="relset_2.html#K6" title="RELSET_2:func.6">.:^</a> <font color="Maroon" title="c4">Y</font> iff <font color="Maroon" title="c3">X</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">R</font> <a href="relset_1.html#K6" title="RELSET_1:func.6">~</a> </span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c4">Y</font> )
 
<b>by </b><i><a class="ref" href="relset_2.html#T49" target="_self" title="RELSET_2:th.49">Th49</a></i>;<br/>
<a NAME="E3:68"/><b>then </b><i><font color="Green" title="E32">A1</font></i>: 
( <font color="Maroon" title="c3">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">R</font> <a href="relset_1.html#K6" title="RELSET_1:func.6">~</a> </span>)</span> <a href="relset_2.html#K6" title="RELSET_2:func.6">.:^</a> <font color="Maroon" title="c4">Y</font> iff <font color="Maroon" title="c3">X</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">R</font> <a href="relset_1.html#K6" title="RELSET_1:func.6">~</a> </span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c4">Y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">S</font> = <font color="Maroon" title="c5">R</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  as    <a href="relset_1.html#M2" title="RELSET_1:mode.2">Relation</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font> <b>by </b><i><a class="ref" href="relset_1.html#D1" title="RELSET_1:def.1">RELSET_1:def 1</a></i>;<br/>
<a NAME="E5:68"/>
( <font color="Maroon" title="c3">X</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">S</font> <a href="relset_1.html#K6" title="RELSET_1:func.6">~</a> </span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c4">Y</font> iff <font color="Maroon" title="c4">Y</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c6">S</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c3">X</font> )
 
<b>by </b><i><a class="ref" href="relset_2.html#T46" target="_self" title="RELSET_2:th.46">Th46</a></i>;<br/>
<a NAME="E6:68"/><b>then </b>
( <font color="Maroon" title="c3">X</font> <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="c6">S</font> <a href="relset_1.html#K6" title="RELSET_1:func.6">~</a> </span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c4">Y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  iff <font color="Maroon" title="c4">Y</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">S</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E7:68"/><b>then </b>
( <font color="Maroon" title="c3">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">R</font> <a href="relset_1.html#K6" title="RELSET_1:func.6">~</a> </span>)</span> <a href="relset_2.html#K6" title="RELSET_2:func.6">.:^</a> <font color="Maroon" title="c4">Y</font> iff <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">R</font> <a href="relset_2.html#K6" title="RELSET_2:func.6">.:^</a> <font color="Maroon" title="c3">X</font></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> <font color="Maroon" title="c4">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 
<b>by </b><i><a class="txt" href="#E3:68"><i><font color="Green" title="E32">A1</font></i></a>, <a class="ref" href="relset_2.html#T49" target="_self" title="RELSET_2:th.49">Th49</a>, <a class="ref" href="relset_2.html#T60" target="_self" title="RELSET_2:th.60">Th60</a></i>;<br/>
<a NAME="E8:68"/><b>then </b>
( <font color="Maroon" title="c3">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">R</font> <a href="relset_1.html#K6" title="RELSET_1:func.6">~</a> </span>)</span> <a href="relset_2.html#K6" title="RELSET_2:func.6">.:^</a> <font color="Maroon" title="c4">Y</font> iff <span class="p1">(<span class="default"><font color="Maroon" title="c5">R</font> <a href="relset_2.html#K6" title="RELSET_2:func.6">.:^</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c4">Y</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/>
<b>hence </b><a NAME="E9:68"/>
( <font color="Maroon" title="c3">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">R</font> <a href="relset_1.html#K6" title="RELSET_1:func.6">~</a> </span>)</span> <a href="relset_2.html#K6" title="RELSET_2:func.6">.:^</a> <font color="Maroon" title="c4">Y</font> iff <font color="Maroon" title="c4">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">R</font> <a href="relset_2.html#K6" title="RELSET_2:func.6">.:^</a> <font color="Maroon" title="c3">X</font> )
 <b>by </b><i><a class="ref" href="subset_1.html#T44" title="SUBSET_1:th.44">SUBSET_1:44</a></i>; <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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