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<span class="kw">let </span><font color="Maroon" title="c1">P</font>, <font color="Maroon" title="c2">R</font>, <font color="Maroon" title="c3">Q</font> be   <a href="relat_1.html#NM1" title="RELAT_1:NM.1">Relation</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c1">P</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">*</a> <font color="Maroon" title="c2">R</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">P</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">*</a> <font color="Maroon" title="c3">Q</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">P</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">R</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c3">Q</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c4">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="relat_1.html#D3" target="_self" title="RELAT_1:def.3">RELAT_1:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">a</font>,<font color="Olive" title="b1">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">P</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">*</a> <font color="Maroon" title="c2">R</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">P</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">*</a> <font color="Maroon" title="c3">Q</font></span>)</span> holds <br/><span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">a</font>,<font color="Olive" title="b1">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">R</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c3">Q</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c5">b</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">P</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">*</a> <font color="Maroon" title="c2">R</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">P</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">*</a> <font color="Maroon" title="c3">Q</font></span>)</span> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">R</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c3">Q</font></span>)</span> )</span><br/>









<span class="kw">assume </span><a NAME="E1:51"/><span class="lab"><font color="Green" title="E19">A1</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">P</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">*</a> <font color="Maroon" title="c2">R</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">P</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">*</a> <font color="Maroon" title="c3">Q</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">R</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c3">Q</font></span>)</span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:51"/><span class="lab"><font color="Green" title="E20">A2</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c6">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P</font>
 <span class="kw">and </span><br/><a NAME="E4:51"/><span class="lab"><font color="Green" title="E21">A3</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">y</font>,<font color="Maroon" title="c5">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D8" target="_self" title="RELAT_1:def.8">Def8</a></span>;<br/>
<a NAME="E5:51"/>
 not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">*</a> <font color="Maroon" title="c3">Q</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E6:51"/><span class="kw">then </span>
(  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c6">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P</font> or  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">y</font>,<font color="Maroon" title="c5">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Q</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D8" target="_self" title="RELAT_1:def.8">Def8</a></span>;<br/>
<a NAME="E7:51"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">y</font>,<font color="Maroon" title="c5">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c3">Q</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:51"><span class="lab"><font color="Green" title="E20">A2</font></span></a>, <a class="txt" href="#E4:51"><span class="lab"><font color="Green" title="E21">A3</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:51"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">R</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c3">Q</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:51"><span class="lab"><font color="Green" title="E20">A2</font></span></a>, <a class="ref" href="relat_1.html#D8" target="_self" title="RELAT_1:def.8">Def8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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