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<b>let </b><font color="Maroon" title="c1">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"> for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="orders_1.html#R9" title="ORDERS_1:pred.9">is_inferior_of</a> <font color="Maroon" title="c1">R</font> &amp; <font color="Maroon" title="c1">R</font> is <a href="relat_2.html#V4" title="RELAT_2:attr.4">antisymmetric</a> holds <br/><font color="Olive" title="b1">x</font> <a href="orders_1.html#R7" title="ORDERS_1:pred.7">is_minimal_in</a> <font color="Maroon" title="c1">R</font></span><br/><b>let </b><font color="Maroon" title="c2">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"> ( <font color="Maroon" title="c2">x</font> <a href="orders_1.html#R9" title="ORDERS_1:pred.9">is_inferior_of</a> <font color="Maroon" title="c1">R</font> &amp; <font color="Maroon" title="c1">R</font> is <a href="relat_2.html#V4" title="RELAT_2:attr.4">antisymmetric</a> implies <font color="Maroon" title="c2">x</font> <a href="orders_1.html#R7" title="ORDERS_1:pred.7">is_minimal_in</a> <font color="Maroon" title="c1">R</font> )</span><br/>



<b>assume </b><b>that </b><br/><a NAME="E1:74"/><i><font color="Green" title="E46">A1</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="orders_1.html#R9" title="ORDERS_1:pred.9">is_inferior_of</a> <font color="Maroon" title="c1">R</font>
 <b>and </b><br/><a NAME="E2:74"/><i><font color="Green" title="E47">A2</font></i>: 
<font color="Maroon" title="c1">R</font> is <a href="relat_2.html#V4" title="RELAT_2:attr.4">antisymmetric</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">x</font> <a href="orders_1.html#R7" title="ORDERS_1:pred.7">is_minimal_in</a> <font color="Maroon" title="c1">R</font></span><br/>

<b>thus </b><a NAME="E3:74"/><i><font color="Green" title="E48">A3</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c1">R</font>
 <b>by </b><i><a class="txt" href="orders_1.html#E1:74"><i><font color="Green" title="E46">A1</font></i></a>, <a class="ref" href="orders_1.html#D14" target="_self" title="ORDERS_1:def.14">Def14</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="orders_1.html#D12" target="_self" title="ORDERS_1:def.12">ORDERS_1:def 12</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">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c1">R</font> or  not <font color="Olive" title="b1">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">x</font> or  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">y</font>,<font color="Maroon" title="c2">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="c1">R</font> )</span><br/>

<b>let </b><font color="Maroon" title="c3">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"> (  not <font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c1">R</font> or  not <font color="Maroon" title="c3">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">x</font> or  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c2">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="c1">R</font> )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E4:74"/><i><font color="Green" title="E49">A4</font></i>: 
<font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c1">R</font>
 <b>and </b><br/><a NAME="E5:74"/><i><font color="Green" title="E50">A5</font></i>: 
<font color="Maroon" title="c3">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">x</font>
 <b>and </b><br/><a NAME="E6:74"/><i><font color="Green" title="E51">A6</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c2">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="c1">R</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="E7:74"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">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">R</font> &amp; <font color="Maroon" title="c1">R</font> <a href="relat_2.html#R4" title="RELAT_2:pred.4">is_antisymmetric_in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c1">R</font> )
 
<b>by </b><i><a class="txt" href="orders_1.html#E1:74"><i><font color="Green" title="E46">A1</font></i></a>, <a class="txt" href="orders_1.html#E2:74"><i><font color="Green" title="E47">A2</font></i></a>, <a class="txt" href="orders_1.html#E4:74"><i><font color="Green" title="E49">A4</font></i></a>, <a class="txt" href="orders_1.html#E5:74"><i><font color="Green" title="E50">A5</font></i></a>, <a class="ref" href="orders_1.html#D14" target="_self" title="ORDERS_1:def.14">Def14</a>, <a class="ref" href="relat_2.html#D12" title="RELAT_2:def.12">RELAT_2:def 12</a></i>;<br/>
<b>hence </b><a NAME="E8:74"/>
contradiction
 <b>by </b><i><a class="txt" href="orders_1.html#E3:74"><i><font color="Green" title="E48">A3</font></i></a>, <a class="txt" href="orders_1.html#E4:74"><i><font color="Green" title="E49">A4</font></i></a>, <a class="txt" href="orders_1.html#E5:74"><i><font color="Green" title="E50">A5</font></i></a>, <a class="txt" href="orders_1.html#E6:74"><i><font color="Green" title="E51">A6</font></i></a>, <a class="ref" href="relat_2.html#D4" title="RELAT_2:def.4">RELAT_2:def 4</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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