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<b>let </b><font color="Maroon" title="c1">X</font>, <font color="Maroon" title="c2">Y</font>, <font color="Maroon" title="c3">Z</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="relset_1.html#M2" title="RELSET_1:mode.2">Relation</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font>  st <font color="Olive" title="b1">R</font> is <a href="relat_2.html#V3" title="RELAT_2:attr.3">symmetric</a> holds <br/><font color="Olive" title="b1">R</font> <a href="toler_1.html#K1" title="TOLER_1:func.1">|_2</a> <font color="Maroon" title="c3">Z</font> is <a href="relat_2.html#V3" title="RELAT_2:attr.3">symmetric</a></span><br/><b>let </b><font color="Maroon" title="c4">R</font> be    <a href="relset_1.html#M2" title="RELSET_1:mode.2">Relation</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">R</font> is <a href="relat_2.html#V3" title="RELAT_2:attr.3">symmetric</a> implies <font color="Maroon" title="c4">R</font> <a href="toler_1.html#K1" title="TOLER_1:func.1">|_2</a> <font color="Maroon" title="c3">Z</font> is <a href="relat_2.html#V3" title="RELAT_2:attr.3">symmetric</a> )</span><br/>







<b>assume </b><a NAME="E1:21"/>
<font color="Maroon" title="c4">R</font> is <a href="relat_2.html#V3" title="RELAT_2:attr.3">symmetric</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">R</font> <a href="toler_1.html#K1" title="TOLER_1:func.1">|_2</a> <font color="Maroon" title="c3">Z</font> is <a href="relat_2.html#V3" title="RELAT_2:attr.3">symmetric</a></span><br/>

<a NAME="E2:21"/><b>then </b><i><font color="Green" title="E12">A1</font></i>: 
<font color="Maroon" title="c4">R</font> <a href="relat_2.html#R3" title="RELAT_2:pred.3">is_symmetric_in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c4">R</font>
 
<b>by </b><i><a class="ref" href="relat_2.html#D11" title="RELAT_2:def.11">RELAT_2:def 11</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">x</font>, <font color="Maroon" title="c6">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="c5">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="c4">R</font> <a href="toler_1.html#K1" title="TOLER_1:func.1">|_2</a> <font color="Maroon" title="c3">Z</font></span>)</span> &amp; <font color="Maroon" title="c6">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="c4">R</font> <a href="toler_1.html#K1" title="TOLER_1:func.1">|_2</a> <font color="Maroon" title="c3">Z</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">x</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="c4">R</font> <a href="toler_1.html#K1" title="TOLER_1:func.1">|_2</a> <font color="Maroon" title="c3">Z</font> implies <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">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="c4">R</font> <a href="toler_1.html#K1" title="TOLER_1:func.1">|_2</a> <font color="Maroon" title="c3">Z</font> )</span><br/><b>assume </b><a NAME="E1:21_1"/><i><font color="Green" title="E13">A2</font></i>: 
( <font color="Maroon" title="c5">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="c4">R</font> <a href="toler_1.html#K1" title="TOLER_1:func.1">|_2</a> <font color="Maroon" title="c3">Z</font></span>)</span> &amp; <font color="Maroon" title="c6">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="c4">R</font> <a href="toler_1.html#K1" title="TOLER_1:func.1">|_2</a> <font color="Maroon" title="c3">Z</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">x</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="c4">R</font> <a href="toler_1.html#K1" title="TOLER_1:func.1">|_2</a> <font color="Maroon" title="c3">Z</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <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">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="c4">R</font> <a href="toler_1.html#K1" title="TOLER_1:func.1">|_2</a> <font color="Maroon" title="c3">Z</font></span><br/><a NAME="E2:21_1"/><b>then </b><i><font color="Green" title="E14">A3</font></i>: 
( <font color="Maroon" title="c5">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="c4">R</font> &amp; <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z</font> )
 <b>by </b><i><a class="ref" href="wellord1.html#T19" title="WELLORD1:th.19">WELLORD1:19</a></i>;<br/><a NAME="E3:21_1"/><i><font color="Green" title="E15">A4</font></i>: 
( <font color="Maroon" title="c6">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="c4">R</font> &amp; <font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z</font> )
 <b>by </b><i><a class="txt" href="toler_1.html#E1:21_1"><i><font color="Green" title="E13">A2</font></i></a>, <a class="ref" href="wellord1.html#T19" title="WELLORD1:th.19">WELLORD1:19</a></i>;<br/><a NAME="E4:21_1"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">x</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="c4">R</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">x</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> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c3">Z</font>,<font color="Maroon" title="c3">Z</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> )
 <b>by </b><i><a class="txt" href="toler_1.html#E1:21_1"><i><font color="Green" title="E13">A2</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E5:21_1"/><b>then </b><i><font color="Green" title="E16">A5</font></i>: 
<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">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="c4">R</font>
 <b>by </b><i><a class="txt" href="toler_1.html#E2:21"><i><font color="Green" title="E12">A1</font></i></a>, <a class="txt" href="toler_1.html#E2:21_1"><i><font color="Green" title="E14">A3</font></i></a>, <a class="txt" href="toler_1.html#E3:21_1"><i><font color="Green" title="E15">A4</font></i></a>, <a class="ref" href="relat_2.html#D3" title="RELAT_2:def.3">RELAT_2:def 3</a></i>;<br/><a NAME="E6:21_1"/>
<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">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> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c3">Z</font>,<font color="Maroon" title="c3">Z</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="txt" href="toler_1.html#E1:21_1"><i><font color="Green" title="E13">A2</font></i></a>, <a class="ref" href="zfmisc_1.html#T107" title="ZFMISC_1:th.107">ZFMISC_1:107</a></i>;<br/><b>hence </b><a NAME="E7:21_1"/>
<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">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="c4">R</font> <a href="toler_1.html#K1" title="TOLER_1:func.1">|_2</a> <font color="Maroon" title="c3">Z</font>
 <b>by </b><i><a class="txt" href="toler_1.html#E5:21_1"><i><font color="Green" title="E16">A5</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</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>
<a NAME="E4:21"/><b>then </b>
<font color="Maroon" title="c4">R</font> <a href="toler_1.html#K1" title="TOLER_1:func.1">|_2</a> <font color="Maroon" title="c3">Z</font> <a href="relat_2.html#R3" title="RELAT_2:pred.3">is_symmetric_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="c4">R</font> <a href="toler_1.html#K1" title="TOLER_1:func.1">|_2</a> <font color="Maroon" title="c3">Z</font></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_2.html#D3" title="RELAT_2:def.3">RELAT_2:def 3</a></i>;<br/>
<b>hence </b><a NAME="E5:21"/>
<font color="Maroon" title="c4">R</font> <a href="toler_1.html#K1" title="TOLER_1:func.1">|_2</a> <font color="Maroon" title="c3">Z</font> is <a href="relat_2.html#V3" title="RELAT_2:attr.3">symmetric</a>
 <b>by </b><i><a class="ref" href="relat_2.html#D11" title="RELAT_2:def.11">RELAT_2:def 11</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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