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<a NAME="E1:13_1_1"/><i><font color="Green">E38</font></i>: 
 <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><font color="Maroon">EqR1</font> <a href="eqrel_1.html#K2" target="_self">/\</a> <font color="Maroon">EqR2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">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">E39</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> holds <br/><span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">EqR1</font>
 
<b>by </b><i/>;<br/>
<a NAME="E2:13_1_1_1"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> holds <br/><span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">EqR2</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:13_1_1_1"/><b>then </b>
(  <a href="partfun1.html#K6">id</a> <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">EqR1</font> &amp;  <a href="partfun1.html#K6">id</a> <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">EqR2</font> )
 
<b>by </b><i>, <a class="ref" href="relat_1.html#T73">RELAT_1:73</a></i>;<br/>
<a NAME="E4:13_1_1_1"/><b>then </b>
 <a href="partfun1.html#K6">id</a> <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">EqR1</font> <a href="eqrel_1.html#K2" target="_self">/\</a> <font color="Maroon">EqR2</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T19">XBOOLE_1:19</a></i>;<br/>
<a NAME="E5:13_1_1_1"/><b>then </b>
(  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">EqR1</font> <a href="eqrel_1.html#K2" target="_self">/\</a> <font color="Maroon">EqR2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> &amp;  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">EqR1</font> <a href="eqrel_1.html#K2" target="_self">/\</a> <font color="Maroon">EqR2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> )
 
<b>by </b><i><a class="ref" href="relset_1.html#T31">RELSET_1:31</a>, <a class="ref" href="relset_1.html#T32">RELSET_1:32</a></i>;<br/>
<a NAME="E6:13_1_1_1"/><b>then </b>
 <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><font color="Maroon">EqR1</font> <a href="eqrel_1.html#K2" target="_self">/\</a> <font color="Maroon">EqR2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#D6">RELAT_1:def 6</a></i>;<br/>
<b>hence </b><a NAME="E7:13_1_1_1"/>
 <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><font color="Maroon">EqR1</font> <a href="eqrel_1.html#K2" target="_self">/\</a> <font color="Maroon">EqR2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E2:13_1_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">EqR1</font> <a href="eqrel_1.html#K2" target="_self">/\</a> <font color="Maroon">EqR2</font> is <a href="relat_2.html#V1">reflexive</a>
 
<b>by </b><i><a class="ref" href="relat_2.html#T31">RELAT_2:31</a></i>;<br/>
<a NAME="E3:13_1_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">EqR1</font> <a href="eqrel_1.html#K2" target="_self">/\</a> <font color="Maroon">EqR2</font> is <a href="relat_2.html#V3">symmetric</a>
 
<b>by </b><i><a class="ref" href="relat_2.html#T35">RELAT_2:35</a></i>;<br/>
<a NAME="E4:13_1_1"/>
<font color="Maroon">EqR1</font> <a href="eqrel_1.html#K2" target="_self">/\</a> <font color="Maroon">EqR2</font> is <a href="relat_2.html#V8">transitive</a>
 
<b>by </b><i><a class="ref" href="relat_2.html#T43">RELAT_2:43</a></i>;<br/>
<b>hence </b><a NAME="E5:13_1_1"/>
<font color="Maroon">EqR1</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">EqR2</font> is   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">X</font>
 <b>by </b><i>, , , </i>;<br/>


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