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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="b3">R</font> <a href="relset_2.html#K5" title="RELSET_2:func.5">.:^</a> <font color="Olive" title="b1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">R</font> <a href="relset_2.html#K5" title="RELSET_2:func.5">.:^</a> <span class="p1">(<span class="default"><span class="p2">(<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#K5" title="RELSET_2:func.5">.:^</a> <span class="p2">(<span class="default"><font color="Olive" title="b3">R</font> <a href="relset_2.html#K5" title="RELSET_2:func.5">.:^</a> <font color="Olive" title="b1">X</font></span>)</span></span>)</span> &amp; <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#K5" title="RELSET_2:func.5">.:^</a> <font color="Olive" title="b2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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#K5" title="RELSET_2:func.5">.:^</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">R</font> <a href="relset_2.html#K5" title="RELSET_2:func.5">.:^</a> <span class="p2">(<span class="default"><span class="p3">(<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#K5" title="RELSET_2:func.5">.:^</a> <font color="Olive" title="b2">Y</font></span>)</span></span>)</span> )</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="Olive" title="b2">R</font> <a href="relset_2.html#K5" title="RELSET_2:func.5">.:^</a> <font color="Maroon" title="c3">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">R</font> <a href="relset_2.html#K5" title="RELSET_2:func.5">.:^</a> <span class="p1">(<span class="default"><span class="p2">(<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#K5" title="RELSET_2:func.5">.:^</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">R</font> <a href="relset_2.html#K5" title="RELSET_2:func.5">.:^</a> <font color="Maroon" title="c3">X</font></span>)</span></span>)</span> &amp; <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#K5" title="RELSET_2:func.5">.:^</a> <font color="Olive" title="b1">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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#K5" title="RELSET_2:func.5">.:^</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">R</font> <a href="relset_2.html#K5" title="RELSET_2:func.5">.:^</a> <span class="p2">(<span class="default"><span class="p3">(<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#K5" title="RELSET_2:func.5">.:^</a> <font color="Olive" title="b1">Y</font></span>)</span></span>)</span> )</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="Olive" title="b1">R</font> <a href="relset_2.html#K5" title="RELSET_2:func.5">.:^</a> <font color="Maroon" title="c3">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">R</font> <a href="relset_2.html#K5" title="RELSET_2:func.5">.:^</a> <span class="p1">(<span class="default"><span class="p2">(<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#K5" title="RELSET_2:func.5">.:^</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">R</font> <a href="relset_2.html#K5" title="RELSET_2:func.5">.:^</a> <font color="Maroon" title="c3">X</font></span>)</span></span>)</span> &amp; <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#K5" title="RELSET_2:func.5">.:^</a> <font color="Maroon" title="c4">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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#K5" title="RELSET_2:func.5">.:^</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">R</font> <a href="relset_2.html#K5" title="RELSET_2:func.5">.:^</a> <span class="p2">(<span class="default"><span class="p3">(<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#K5" title="RELSET_2:func.5">.:^</a> <font color="Maroon" title="c4">Y</font></span>)</span></span>)</span> )</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="c5">R</font> <a href="relset_2.html#K5" title="RELSET_2:func.5">.:^</a> <font color="Maroon" title="c3">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">R</font> <a href="relset_2.html#K5" title="RELSET_2:func.5">.:^</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#K5" title="RELSET_2:func.5">.:^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">R</font> <a href="relset_2.html#K5" title="RELSET_2:func.5">.:^</a> <font color="Maroon" title="c3">X</font></span>)</span></span>)</span> &amp; <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#K5" title="RELSET_2:func.5">.:^</a> <font color="Maroon" title="c4">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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#K5" title="RELSET_2:func.5">.:^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">R</font> <a href="relset_2.html#K5" title="RELSET_2:func.5">.:^</a> <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="relset_2.html#K5" title="RELSET_2:func.5">.:^</a> <font color="Maroon" title="c4">Y</font></span>)</span></span>)</span> )</span><br/>









<b>thus </b><a NAME="E1:71"/>
<font color="Maroon" title="c5">R</font> <a href="relset_2.html#K5" title="RELSET_2:func.5">.:^</a> <font color="Maroon" title="c3">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">R</font> <a href="relset_2.html#K5" title="RELSET_2:func.5">.:^</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#K5" title="RELSET_2:func.5">.:^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">R</font> <a href="relset_2.html#K5" title="RELSET_2:func.5">.:^</a> <font color="Maroon" title="c3">X</font></span>)</span></span>)</span>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <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#K5" title="RELSET_2:func.5">.:^</a> <font color="Maroon" title="c4">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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#K5" title="RELSET_2:func.5">.:^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">R</font> <a href="relset_2.html#K5" title="RELSET_2:func.5">.:^</a> <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="relset_2.html#K5" title="RELSET_2:func.5">.:^</a> <font color="Maroon" title="c4">Y</font></span>)</span></span>)</span></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:71_1"/>
<font color="Maroon" title="c5">R</font> <a href="relset_2.html#K5" title="RELSET_2:func.5">.:^</a> <font color="Maroon" title="c3">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">R</font> <a href="relset_2.html#K5" title="RELSET_2:func.5">.:^</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#K5" title="RELSET_2:func.5">.:^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">R</font> <a href="relset_2.html#K5" title="RELSET_2:func.5">.:^</a> <font color="Maroon" title="c3">X</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="relset_2.html#T61" target="_self" title="RELSET_2:th.61">Th61</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">R</font> <a href="relset_2.html#K5" title="RELSET_2:func.5">.:^</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#K5" title="RELSET_2:func.5">.:^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">R</font> <a href="relset_2.html#K5" title="RELSET_2:func.5">.:^</a> <font color="Maroon" title="c3">X</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">R</font> <a href="relset_2.html#K5" title="RELSET_2:func.5">.:^</a> <font color="Maroon" title="c3">X</font></span><br/>

<a NAME="E2:71_1"/>
<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#K5" title="RELSET_2:func.5">.:^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">R</font> <a href="relset_2.html#K5" title="RELSET_2:func.5">.:^</a> <font color="Maroon" title="c3">X</font></span>)</span>
 
<b>by </b><i><a class="ref" href="relset_2.html#T61" target="_self" title="RELSET_2:th.61">Th61</a></i>;<br/>
<b>hence </b><a NAME="E3:71_1"/>
<font color="Maroon" title="c5">R</font> <a href="relset_2.html#K5" title="RELSET_2:func.5">.:^</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#K5" title="RELSET_2:func.5">.:^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">R</font> <a href="relset_2.html#K5" title="RELSET_2:func.5">.:^</a> <font color="Maroon" title="c3">X</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">R</font> <a href="relset_2.html#K5" title="RELSET_2:func.5">.:^</a> <font color="Maroon" title="c3">X</font>
 <b>by </b><i><a class="ref" href="relset_2.html#T32" target="_self" title="RELSET_2:th.32">Th32</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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

<b>thus </b><a NAME="E2:71"/>
<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#K5" title="RELSET_2:func.5">.:^</a> <font color="Maroon" title="c4">Y</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#K5" title="RELSET_2:func.5">.:^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">R</font> <a href="relset_2.html#K5" title="RELSET_2:func.5">.:^</a> <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="relset_2.html#K5" title="RELSET_2:func.5">.:^</a> <font color="Maroon" title="c4">Y</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="relset_2.html#T61" target="_self" title="RELSET_2:th.61">Th61</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <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#K5" title="RELSET_2:func.5">.:^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">R</font> <a href="relset_2.html#K5" title="RELSET_2:func.5">.:^</a> <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="relset_2.html#K5" title="RELSET_2:func.5">.:^</a> <font color="Maroon" title="c4">Y</font></span>)</span></span>)</span> <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#K5" title="RELSET_2:func.5">.:^</a> <font color="Maroon" title="c4">Y</font></span><br/>

<a NAME="E3:71"/>
<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#K5" title="RELSET_2:func.5">.:^</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#K5" title="RELSET_2:func.5">.:^</a> <font color="Maroon" title="c4">Y</font></span>)</span>
 
<b>by </b><i><a class="ref" href="relset_2.html#T61" target="_self" title="RELSET_2:th.61">Th61</a></i>;<br/>
<b>hence </b><a NAME="E4:71"/>
<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#K5" title="RELSET_2:func.5">.:^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">R</font> <a href="relset_2.html#K5" title="RELSET_2:func.5">.:^</a> <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="relset_2.html#K5" title="RELSET_2:func.5">.:^</a> <font color="Maroon" title="c4">Y</font></span>)</span></span>)</span> <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#K5" title="RELSET_2:func.5">.:^</a> <font color="Maroon" title="c4">Y</font>
 <b>by </b><i><a class="ref" href="relset_2.html#T32" target="_self" title="RELSET_2:th.32">Th32</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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