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<a NAME="E1:13_1_1"/><i><font color="Green" title="E7">A1</font></i>: 
 <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">EqR1</font> <a href="eqrel_1.html#K2" title="EQREL_1:func.2">/\</a> <font color="Maroon" title="c3">EqR2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:13_1_1_1"/><i><font color="Green" title="E8">A2</font></i>: 
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="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">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">x</font>,<font color="Olive" title="b1">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">EqR1</font>
 
<b>by </b><i><a class="ref" href="eqrel_1.html#T11" target="_self" title="EQREL_1:th.11">Th11</a></i>;<br/>
<a NAME="E2:13_1_1_1"/>
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="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">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">x</font>,<font color="Olive" title="b1">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="c3">EqR2</font>
 
<b>by </b><i><a class="ref" href="eqrel_1.html#T11" target="_self" title="EQREL_1:th.11">Th11</a></i>;<br/>
<a NAME="E3:13_1_1_1"/><b>then </b>
(  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <font color="Maroon" title="c1">X</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <font color="Maroon" title="c2">EqR1</font> &amp;  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <font color="Maroon" title="c1">X</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <font color="Maroon" title="c3">EqR2</font> )
 
<b>by </b><i><a class="txt" href="eqrel_1.html#E1:13_1_1_1"><i><font color="Green" title="E8">A2</font></i></a>, <a class="ref" href="relat_1.html#T73" title="RELAT_1:th.73">RELAT_1:73</a></i>;<br/>
<a NAME="E4:13_1_1_1"/><b>then </b>
 <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <font color="Maroon" title="c1">X</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <font color="Maroon" title="c2">EqR1</font> <a href="eqrel_1.html#K2" title="EQREL_1:func.2">/\</a> <font color="Maroon" title="c3">EqR2</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T19" title="XBOOLE_1:th.19">XBOOLE_1:19</a></i>;<br/>
<a NAME="E5:13_1_1_1"/><b>then </b>
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">EqR1</font> <a href="eqrel_1.html#K2" title="EQREL_1:func.2">/\</a> <font color="Maroon" title="c3">EqR2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">EqR1</font> <a href="eqrel_1.html#K2" title="EQREL_1:func.2">/\</a> <font color="Maroon" title="c3">EqR2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> )
 
<b>by </b><i><a class="ref" href="relset_1.html#T31" title="RELSET_1:th.31">RELSET_1:31</a>, <a class="ref" href="relset_1.html#T32" title="RELSET_1:th.32">RELSET_1:32</a></i>;<br/>
<b>hence </b><a NAME="E6:13_1_1_1"/>
 <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">EqR1</font> <a href="eqrel_1.html#K2" title="EQREL_1:func.2">/\</a> <font color="Maroon" title="c3">EqR2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E2:13_1_1"/><i><font color="Green" title="E8">A3</font></i>: 
<font color="Maroon" title="c2">EqR1</font> <a href="eqrel_1.html#K2" title="EQREL_1:func.2">/\</a> <font color="Maroon" title="c3">EqR2</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#T31" title="RELAT_2:th.31">RELAT_2:31</a></i>;<br/>
<a NAME="E3:13_1_1"/><i><font color="Green" title="E9">A4</font></i>: 
<font color="Maroon" title="c2">EqR1</font> <a href="eqrel_1.html#K2" title="EQREL_1:func.2">/\</a> <font color="Maroon" title="c3">EqR2</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#T35" title="RELAT_2:th.35">RELAT_2:35</a></i>;<br/>
<a NAME="E4:13_1_1"/>
<font color="Maroon" title="c2">EqR1</font> <a href="eqrel_1.html#K2" title="EQREL_1:func.2">/\</a> <font color="Maroon" title="c3">EqR2</font> is <a href="relat_2.html#V8" title="RELAT_2:attr.8">transitive</a>
 
<b>by </b><i><a class="ref" href="relat_2.html#T43" title="RELAT_2:th.43">RELAT_2:43</a></i>;<br/>
<b>hence </b><a NAME="E5:13_1_1"/>
<font color="Maroon" title="c2">EqR1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c3">EqR2</font> is   <a href="eqrel_1.html#NM2" title="EQREL_1:NM.2">Equivalence_Relation</a> of <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="eqrel_1.html#E1:13_1_1"><i><font color="Green" title="E7">A1</font></i></a>, <a class="txt" href="eqrel_1.html#E2:13_1_1"><i><font color="Green" title="E8">A3</font></i></a>, <a class="txt" href="eqrel_1.html#E3:13_1_1"><i><font color="Green" title="E9">A4</font></i></a>, <a class="ref" href="eqrel_1.html#T16" target="_self" title="EQREL_1:th.16">Th16</a></i>;<br/>


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