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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">R</font> being  <a href="relat_1.html#NM1" title="RELAT_1:NM.1">Relation</a> holds <br/> ( <font color="Olive" title="b1">R</font> <a href="relat_2.html#R4" title="RELAT_2:pred.4">is_antisymmetric_in</a> <font color="Maroon" title="c1">X</font> iff <font color="Olive" title="b1">R</font> <a href="xboole_0.html#K4" title="XBOOLE_0:func.4">\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="relat_2.html#R5" title="RELAT_2:pred.5">is_asymmetric_in</a> <font color="Maroon" title="c1">X</font> )</span><br/><b>let </b><font color="Maroon" title="c2">R</font> be   <a href="relat_1.html#NM1" title="RELAT_1:NM.1">Relation</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">R</font> <a href="relat_2.html#R4" title="RELAT_2:pred.4">is_antisymmetric_in</a> <font color="Maroon" title="c1">X</font> iff <font color="Maroon" title="c2">R</font> <a href="xboole_0.html#K4" title="XBOOLE_0:func.4">\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="relat_2.html#R5" title="RELAT_2:pred.5">is_asymmetric_in</a> <font color="Maroon" title="c1">X</font> )</span><br/>



<b>thus </b><a NAME="E1:5"/>
( <font color="Maroon" title="c2">R</font> <a href="relat_2.html#R4" title="RELAT_2:pred.4">is_antisymmetric_in</a> <font color="Maroon" title="c1">X</font> implies <font color="Maroon" title="c2">R</font> <a href="xboole_0.html#K4" title="XBOOLE_0:func.4">\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="relat_2.html#R5" title="RELAT_2:pred.5">is_asymmetric_in</a> <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"> ( <font color="Maroon" title="c2">R</font> <a href="xboole_0.html#K4" title="XBOOLE_0:func.4">\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="relat_2.html#R5" title="RELAT_2:pred.5">is_asymmetric_in</a> <font color="Maroon" title="c1">X</font> implies <font color="Maroon" title="c2">R</font> <a href="relat_2.html#R4" title="RELAT_2:pred.4">is_antisymmetric_in</a> <font color="Maroon" title="c1">X</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:5_1"/><i><font color="Green" title="E17">A1</font></i>: 
<font color="Maroon" title="c2">R</font> <a href="relat_2.html#R4" title="RELAT_2:pred.4">is_antisymmetric_in</a> <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"> <font color="Maroon" title="c2">R</font> <a href="xboole_0.html#K4" title="XBOOLE_0:func.4">\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="relat_2.html#R5" title="RELAT_2:pred.5">is_asymmetric_in</a> <font color="Maroon" title="c1">X</font></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c4">y</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"> ( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">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="c2">R</font> <a href="xboole_0.html#K4" title="XBOOLE_0:func.4">\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c1">X</font></span>)</span> implies  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c3">x</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="xboole_0.html#K4" title="XBOOLE_0:func.4">\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c1">X</font></span>)</span> )</span><br/><b>assume </b><a NAME="E1:5_1_1"/><i><font color="Green" title="E18">A2</font></i>: 
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">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="c2">R</font> <a href="xboole_0.html#K4" title="XBOOLE_0:func.4">\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c1">X</font></span>)</span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c3">x</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="xboole_0.html#K4" title="XBOOLE_0:func.4">\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c1">X</font></span>)</span></span><br/><a NAME="E2:5_1_1"/><b>then </b><i><font color="Green" title="E19">A3</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">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="c2">R</font> &amp; not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">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>  <a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c1">X</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></i>;<br/><a NAME="E3:5_1_1"/><b>then </b>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">y</font>
 <b>by </b><i><a class="txt" href="#E1:5_1_1"><i><font color="Green" title="E18">A2</font></i></a>, <a class="ref" href="relat_1.html#D10" title="RELAT_1:def.10">RELAT_1:def 10</a></i>;<br/><a NAME="E4:5_1_1"/><b>then </b>
not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c3">x</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>
 <b>by </b><i><a class="txt" href="#E1:5_1"><i><font color="Green" title="E17">A1</font></i></a>, <a class="txt" href="#E1:5_1_1"><i><font color="Green" title="E18">A2</font></i></a>, <a class="txt" href="#E2:5_1_1"><i><font color="Green" title="E19">A3</font></i></a>, <a class="ref" href="relat_2.html#D4" target="_self" title="RELAT_2:def.4">Def4</a></i>;<br/><b>hence </b><a NAME="E5:5_1_1"/>
not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c3">x</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="xboole_0.html#K4" title="XBOOLE_0:func.4">\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c1">X</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</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>hence </b><a NAME="E3:5_1"/>
<font color="Maroon" title="c2">R</font> <a href="xboole_0.html#K4" title="XBOOLE_0:func.4">\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="relat_2.html#R5" title="RELAT_2:pred.5">is_asymmetric_in</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="ref" href="relat_2.html#D5" target="_self" title="RELAT_2:def.5">Def5</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="E2:5"/>
( <font color="Maroon" title="c2">R</font> <a href="xboole_0.html#K4" title="XBOOLE_0:func.4">\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="relat_2.html#R5" title="RELAT_2:pred.5">is_asymmetric_in</a> <font color="Maroon" title="c1">X</font> implies <font color="Maroon" title="c2">R</font> <a href="relat_2.html#R4" title="RELAT_2:pred.4">is_antisymmetric_in</a> <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"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:5_2"/><i><font color="Green" title="E17">A4</font></i>: 
<font color="Maroon" title="c2">R</font> <a href="xboole_0.html#K4" title="XBOOLE_0:func.4">\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="relat_2.html#R5" title="RELAT_2:pred.5">is_asymmetric_in</a> <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"> <font color="Maroon" title="c2">R</font> <a href="relat_2.html#R4" title="RELAT_2:pred.4">is_antisymmetric_in</a> <font color="Maroon" title="c1">X</font></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c4">y</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"> ( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">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="c2">R</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c3">x</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> implies <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y</font> )</span><br/><b>assume </b><a NAME="E1:5_2_1"/><i><font color="Green" title="E18">A5</font></i>: 
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">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="c2">R</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c3">x</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 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="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y</font></span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_2_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:5_2_1_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:5_2_1_1"/><b>then </b>
( not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">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>  <a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c1">X</font> &amp; not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c3">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c1">X</font> )
 <b>by </b><i><a class="ref" href="relat_1.html#D10" title="RELAT_1:def.10">RELAT_1:def 10</a></i>;<br/><a NAME="E3:5_2_1_1"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">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="c2">R</font> <a href="xboole_0.html#K4" title="XBOOLE_0:func.4">\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c1">X</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c3">x</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="xboole_0.html#K4" title="XBOOLE_0:func.4">\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c1">X</font></span>)</span> )
 <b>by </b><i><a class="txt" href="#E1:5_2_1"><i><font color="Green" title="E18">A5</font></i></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></i>;<br/><b>hence </b><a NAME="E4:5_2_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:5_2"><i><font color="Green" title="E17">A4</font></i></a>, <a class="txt" href="#E1:5_2_1"><i><font color="Green" title="E18">A5</font></i></a>, <a class="ref" href="relat_2.html#D5" target="_self" title="RELAT_2:def.5">Def5</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>hence </b><a NAME="E3:5_2_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y</font>
 ; <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>hence </b><a NAME="E3:5_2"/>
<font color="Maroon" title="c2">R</font> <a href="relat_2.html#R4" title="RELAT_2:pred.4">is_antisymmetric_in</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="ref" href="relat_2.html#D4" target="_self" title="RELAT_2:def.4">Def4</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>


</div>
