<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <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> <a href="relat_1.html#K4" title="RELAT_1:func.4">~</a>  iff <font color="Olive" title="b1">x</font> <a href="orders_1.html#R8" title="ORDERS_1:pred.8">is_superior_of</a> <font color="Maroon" title="c1">R</font> )</span><br/><span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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> <a href="relat_1.html#K4" title="RELAT_1:func.4">~</a>  iff <font color="Maroon" title="c2">x</font> <a href="orders_1.html#R8" title="ORDERS_1:pred.8">is_superior_of</a> <font color="Maroon" title="c1">R</font> )</span><br/>



<a NAME="E1:187"/><span class="lab"><font color="Green" title="E44">A1</font></span>: 
 <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c1">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="relat_1.html#K4" title="RELAT_1:func.4">~</a> </span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T38" title="RELAT_1:th.38">RELAT_1:38</a></span>;<br/>
<span class="kw">thus </span><a NAME="E2:187"/>
( <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> <a href="relat_1.html#K4" title="RELAT_1:func.4">~</a>  implies <font color="Maroon" title="c2">x</font> <a href="orders_1.html#R8" title="ORDERS_1:pred.8">is_superior_of</a> <font color="Maroon" title="c1">R</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">x</font> <a href="orders_1.html#R8" title="ORDERS_1:pred.8">is_superior_of</a> <font color="Maroon" title="c1">R</font> implies <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> <a href="relat_1.html#K4" title="RELAT_1:func.4">~</a>  )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="187_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:187_1"/><span class="lab"><font color="Green" title="E45">A2</font></span>: 
<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> <span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="relat_1.html#K4" title="RELAT_1:func.4">~</a> </span>)</span>
 <span class="kw">and </span><br/><a NAME="E2:187_1"/><span class="lab"><font color="Green" title="E46">A3</font></span>: 
for <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <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> <span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="relat_1.html#K4" title="RELAT_1:func.4">~</a> </span>)</span> &amp; <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> holds <br/><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="Olive" title="b1">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> <a href="relat_1.html#K4" title="RELAT_1:func.4">~</a> 
 ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="orders_1.html#D14" target="_self" title="ORDERS_1:def.14">ORDERS_1:def 14</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">x</font> <a href="orders_1.html#R8" title="ORDERS_1:pred.8">is_superior_of</a> <font color="Maroon" title="c1">R</font></span><br/>

<span class="kw">thus </span><a NAME="E3:187_1"/>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:187_1"><span class="lab"><font color="Green" title="E45">A2</font></span></a>, <a class="ref" href="relat_1.html#T38" title="RELAT_1:th.38">RELAT_1:38</a></span>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="orders_1.html#D13" target="_self" title="ORDERS_1:def.13">ORDERS_1:def 13</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <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> &amp; <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> holds <br/><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/>

<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <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> &amp; <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> implies <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/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E4:187_1"/><span class="lab"><font color="Green" title="E47">A4</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E5:187_1"/><span class="lab"><font color="Green" title="E48">A5</font></span>: 
<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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="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/>

<a NAME="E6:187_1"/>
<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> <a href="relat_1.html#K4" title="RELAT_1:func.4">~</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:187"><span class="lab"><font color="Green" title="E44">A1</font></span></a>, <a class="txt" href="#E2:187_1"><span class="lab"><font color="Green" title="E46">A3</font></span></a>, <a class="txt" href="#E4:187_1"><span class="lab"><font color="Green" title="E47">A4</font></span></a>, <a class="txt" href="#E5:187_1"><span class="lab"><font color="Green" title="E48">A5</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:187_1"/>
<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 class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D7" title="RELAT_1:def.7">RELAT_1:def 7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E3:187"/><span class="lab"><font color="Green" title="E45">A6</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E4:187"/><span class="lab"><font color="Green" title="E46">A7</font></span>: 
for <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <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> &amp; <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> holds <br/><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 class="comment"><font color="Red">:: according to </font></span><a class="ref" href="orders_1.html#D13" target="_self" title="ORDERS_1:def.13">ORDERS_1:def 13</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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> <a href="relat_1.html#K4" title="RELAT_1:func.4">~</a> </span><br/>

<span class="kw">thus </span><a NAME="E5:187"/>
<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> <span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="relat_1.html#K4" title="RELAT_1:func.4">~</a> </span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:187"><span class="lab"><font color="Green" title="E45">A6</font></span></a>, <a class="ref" href="relat_1.html#T38" title="RELAT_1:th.38">RELAT_1:38</a></span>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="orders_1.html#D14" target="_self" title="ORDERS_1:def.14">ORDERS_1:def 14</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <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> <span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="relat_1.html#K4" title="RELAT_1:func.4">~</a> </span>)</span> &amp; <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> holds <br/><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="Olive" title="b1">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> <a href="relat_1.html#K4" title="RELAT_1:func.4">~</a> </span><br/>

<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <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> <span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="relat_1.html#K4" title="RELAT_1:func.4">~</a> </span>)</span> &amp; <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> implies <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> <a href="relat_1.html#K4" title="RELAT_1:func.4">~</a>  )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E6:187"/><span class="lab"><font color="Green" title="E47">A8</font></span>: 
<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> <span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="relat_1.html#K4" title="RELAT_1:func.4">~</a> </span>)</span>
 <span class="kw">and </span><br/><a NAME="E7:187"/><span class="lab"><font color="Green" title="E48">A9</font></span>: 
<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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="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> <a href="relat_1.html#K4" title="RELAT_1:func.4">~</a> </span><br/>

<a NAME="E8:187"/>
<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 class="kw">by</span> <span class="lab"><a class="txt" href="#E1:187"><span class="lab"><font color="Green" title="E44">A1</font></span></a>, <a class="txt" href="#E4:187"><span class="lab"><font color="Green" title="E46">A7</font></span></a>, <a class="txt" href="#E6:187"><span class="lab"><font color="Green" title="E47">A8</font></span></a>, <a class="txt" href="#E7:187"><span class="lab"><font color="Green" title="E48">A9</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:187"/>
<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> <a href="relat_1.html#K4" title="RELAT_1:func.4">~</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D7" title="RELAT_1:def.7">RELAT_1:def 7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
