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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>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:7"/><i><font color="Green" title="E18">A1</font></i>: 
<font color="Maroon" title="c1">R</font> is <a href="relat_2.html#V3" title="RELAT_2:attr.3">symmetric</a>
 <b>and </b><br/><a NAME="E2:7"/><i><font color="Green" title="E19">A2</font></i>: 
<font color="Maroon" title="c1">R</font> is <a href="relat_2.html#V8" title="RELAT_2:attr.8">transitive</a>
 ;<br/>

<a NAME="E3:7"/><i><font color="Green" title="E20">A3</font></i>: 
<font color="Maroon" title="c1">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="c1">R</font>
 
<b>by </b><i><a class="txt" href="relat_2.html#E1:7"><i><font color="Green" title="E18">A1</font></i></a>, <a class="ref" href="relat_2.html#D11" target="_self" title="RELAT_2:def.11">Def11</a></i>;<br/>
<a NAME="E4:7"/><i><font color="Green" title="E21">A4</font></i>: 
<font color="Maroon" title="c1">R</font> <a href="relat_2.html#R8" title="RELAT_2:pred.8">is_transitive_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="relat_2.html#E2:7"><i><font color="Green" title="E19">A2</font></i></a>, <a class="ref" href="relat_2.html#D16" target="_self" title="RELAT_2:def.16">Def16</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c2">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:7_1"/>
<font color="Maroon" title="c2">a</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>
 ;<br/><a NAME="E2:7_1"/><b>then </b><i><font color="Green" title="E22">A5</font></i>: 
<font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">R</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK4" title="RELAT_1:NK.4">rng</a> <font color="Maroon" title="c1">R</font></span>)</span>
 <b>by </b><i><a class="ref" href="relat_1.html#D6" title="RELAT_1:def.6">RELAT_1:def 6</a></i>;<br/><div><i><font color="Green" title="E23">A6</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:7_1_1"/>
<font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">R</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c3">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:7_1_1"/><i><font color="Green" title="E24">A7</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">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">R</font>
 <b>by </b><i><a class="ref" href="relat_1.html#D4" title="RELAT_1:def.4">RELAT_1:def 4</a></i>;<br/><a NAME="E4:7_1_1"/><i><font color="Green" title="E25">A8</font></i>: 
( <font color="Maroon" title="c2">a</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">b</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="relat_2.html#E3:7_1_1"><i><font color="Green" title="E24">A7</font></i></a>, <a class="ref" href="relat_1.html#T30" title="RELAT_1:th.30">RELAT_1:30</a></i>;<br/><a NAME="E5:7_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">b</font>,<font color="Maroon" title="c2">a</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>
 <b>by </b><i><a class="txt" href="relat_2.html#E3:7"><i><font color="Green" title="E20">A3</font></i></a>, <a class="txt" href="relat_2.html#E3:7_1_1"><i><font color="Green" title="E24">A7</font></i></a>, <a class="ref" href="relat_2.html#D3" target="_self" title="RELAT_2:def.3">Def3</a></i>;<br/><b>hence </b><a NAME="E6:7_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c2">a</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>
 <b>by </b><i><a class="txt" href="relat_2.html#E4:7"><i><font color="Green" title="E21">A4</font></i></a>, <a class="txt" href="relat_2.html#E3:7_1_1"><i><font color="Green" title="E24">A7</font></i></a>, <a class="txt" href="relat_2.html#E4:7_1_1"><i><font color="Green" title="E25">A8</font></i></a>, <a class="ref" href="relat_2.html#D8" target="_self" title="RELAT_2:def.8">Def8</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:7_1_2"/>
<font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK4" title="RELAT_1:NK.4">rng</a> <font color="Maroon" title="c1">R</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c3">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:7_1_2"/><i><font color="Green" title="E24">A9</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c2">a</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>
 <b>by </b><i><a class="ref" href="relat_1.html#D5" title="RELAT_1:def.5">RELAT_1:def 5</a></i>;<br/><a NAME="E4:7_1_2"/><i><font color="Green" title="E25">A10</font></i>: 
( <font color="Maroon" title="c2">a</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">b</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="relat_2.html#E3:7_1_2"><i><font color="Green" title="E24">A9</font></i></a>, <a class="ref" href="relat_1.html#T30" title="RELAT_1:th.30">RELAT_1:30</a></i>;<br/><a NAME="E5:7_1_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">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">R</font>
 <b>by </b><i><a class="txt" href="relat_2.html#E3:7"><i><font color="Green" title="E20">A3</font></i></a>, <a class="txt" href="relat_2.html#E3:7_1_2"><i><font color="Green" title="E24">A9</font></i></a>, <a class="ref" href="relat_2.html#D3" target="_self" title="RELAT_2:def.3">Def3</a></i>;<br/><b>hence </b><a NAME="E6:7_1_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c2">a</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>
 <b>by </b><i><a class="txt" href="relat_2.html#E4:7"><i><font color="Green" title="E21">A4</font></i></a>, <a class="txt" href="relat_2.html#E3:7_1_2"><i><font color="Green" title="E24">A9</font></i></a>, <a class="txt" href="relat_2.html#E4:7_1_2"><i><font color="Green" title="E25">A10</font></i></a>, <a class="ref" href="relat_2.html#D8" target="_self" title="RELAT_2:def.8">Def8</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:7_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c2">a</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>
 <b>by </b><i><a class="txt" href="relat_2.html#E2:7_1"><i><font color="Green" title="E22">A5</font></i></a>, <a class="txt" href="relat_2.html#E3:7_1"><i><font color="Green" title="E23">A6</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div>
<a NAME="E6:7"/><b>then </b>
<font color="Maroon" title="c1">R</font> <a href="relat_2.html#R1" title="RELAT_2:pred.1">is_reflexive_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="ref" href="relat_2.html#D1" target="_self" title="RELAT_2:def.1">Def1</a></i>;<br/>
<b>hence </b><a NAME="E7:7"/>
<font color="Maroon" title="c1">R</font> is <a href="relat_2.html#V1" title="RELAT_2:attr.1">reflexive</a>
 <b>by </b><i><a class="ref" href="relat_2.html#D9" target="_self" title="RELAT_2:def.9">Def9</a></i>;<br/>


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