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<b>let </b><font color="Maroon">Y</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">R</font> be   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">Y</font>, <font color="Maroon">S</font> be   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">Y</font>;<br/>



<b>assume </b><a NAME="E1:12"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">R</font> <a href="relset_1.html#K7">*</a> <font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">S</font> <a href="relset_1.html#K7">*</a> <font color="Maroon">R</font>
 ;<br/>

<a NAME="E2:12"/><i><font color="Green">E19</font></i>: 
 <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</font>
 
<b>by </b><i><a class="ref" href="orders_1.html#T97">ORDERS_1:97</a></i>;<br/>
<a NAME="E3:12"/><i><font color="Green">E29</font></i>: 
 <a href="relat_1.html#K3">field</a> <font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</font>
 
<b>by </b><i><a class="ref" href="orders_1.html#T97">ORDERS_1:97</a></i>;<br/>
<a NAME="E4:12"/><b>then </b>
( <font color="Maroon">R</font> <a href="relat_2.html#R1">is_reflexive_in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">S</font> <a href="relat_2.html#R1">is_reflexive_in</a> <font color="Maroon">Y</font> )
 
<b>by </b><i>, <a class="ref" href="relat_2.html#D9">RELAT_2:def 9</a></i>;<br/>
<a NAME="E5:12"/><b>then </b>
<font color="Maroon">R</font> <a href="relset_1.html#K7">*</a> <font color="Maroon">S</font> <a href="relat_2.html#R1">is_reflexive_in</a> <font color="Maroon">Y</font>
 
<b>by </b><i/>;<br/>
<a NAME="E6:12"/><b>then </b><i><font color="Green">E30</font></i>: 
(  <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="relset_1.html#K7">*</a> <font color="Maroon">S</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</font> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="relset_1.html#K7">*</a> <font color="Maroon">S</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</font> )
 
<b>by </b><i><a class="ref" href="orders_1.html#T98">ORDERS_1:98</a></i>;<br/>
<a NAME="E7:12"/>
( <font color="Maroon">R</font> <a href="relset_1.html#K7">*</a> <font color="Maroon">S</font> is <a href="partfun1.html#V1">total</a> &amp; <font color="Maroon">R</font> <a href="relset_1.html#K7">*</a> <font color="Maroon">S</font> is <a href="relat_2.html#V3">symmetric</a> &amp; <font color="Maroon">R</font> <a href="relset_1.html#K7">*</a> <font color="Maroon">S</font> is <a href="relat_2.html#V8">transitive</a> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:12_1"/>
<font color="Maroon">R</font> <a href="relset_1.html#K7">*</a> <font color="Maroon">S</font> is <a href="partfun1.html#V1">total</a>
 <b>by </b><i>, <a class="ref" href="partfun1.html#D4">PARTFUN1:def 4</a></i>;<br/>

<a NAME="E2:12_1"/>
<font color="Maroon">R</font> <a href="relset_1.html#K7">*</a> <font color="Maroon">S</font> <a href="relat_2.html#R3">is_symmetric_in</a> <font color="Maroon">Y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:12_1_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">R</font> <a href="relat_2.html#R3">is_symmetric_in</a> <font color="Maroon">Y</font>
 
<b>by </b><i>, <a class="ref" href="relat_2.html#D11">RELAT_2:def 11</a></i>;<br/>
<a NAME="E2:12_1_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">S</font> <a href="relat_2.html#R3">is_symmetric_in</a> <font color="Maroon">Y</font>
 
<b>by </b><i>, <a class="ref" href="relat_2.html#D11">RELAT_2:def 11</a></i>;<br/>
<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="relat_2.html#D3">RELAT_2:def 3</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E3:12_1_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>and </b><br/><a NAME="E4:12_1_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 ;<br/>

<b>assume </b><a NAME="E5:12_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> <a href="relset_1.html#K7">*</a> <font color="Maroon">S</font>
 ;<br/>

<b>then consider </b><font color="Maroon">z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:12_1_1"/><i><font color="Green">E43</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">z</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 <b>and </b><br/><a NAME="E8:12_1_1"/><i><font color="Green">E44</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">z</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 <b>by </b><i><a class="ref" href="relat_1.html#D8">RELAT_1:def 8</a></i>;<br/>
<a NAME="E9:12_1_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font>
 
<b>by </b><i>, <a class="ref" href="relat_1.html#T30">RELAT_1:30</a></i>;<br/>
<a NAME="E10:12_1_1"/><b>then </b><i><font color="Green">E45</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">z</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 
<b>by </b><i>, , , , <a class="ref" href="relat_2.html#D3">RELAT_2:def 3</a></i>;<br/>
<a NAME="E11:12_1_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">S</font>
 
<b>by </b><i><a class="txt" href="partit_2.html#E1:12"><i><font color="Green">E17</font></i></a>, <a class="ref" href="relat_1.html#T30">RELAT_1:30</a></i>;<br/>
<a NAME="E12:12_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 
<b>by </b><i>, , , <a class="txt" href="partit_2.html#E1:12"><i><font color="Green">E17</font></i></a>, <a class="ref" href="relat_2.html#D3">RELAT_2:def 3</a></i>;<br/>
<b>hence </b><a NAME="E13:12_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> <a href="relset_1.html#K7">*</a> <font color="Maroon">S</font>
 <b>by </b><i>, , <a class="ref" href="relat_1.html#D8">RELAT_1:def 8</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:12_1"/>
<font color="Maroon">R</font> <a href="relset_1.html#K7">*</a> <font color="Maroon">S</font> is <a href="relat_2.html#V3">symmetric</a>
 <b>by </b><i>, <a class="ref" href="relat_2.html#D11">RELAT_2:def 11</a></i>;<br/>

<a NAME="E4:12_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">R</font> <a href="relset_1.html#K7">*</a> <font color="Maroon">R</font> <a href="partit_2.html#R1" target="_self">c=</a> <font color="Maroon">R</font>
 
<b>by </b><i><a class="ref" href="relat_2.html#T44">RELAT_2:44</a></i>;<br/>
<a NAME="E5:12_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">S</font> <a href="relset_1.html#K7">*</a> <font color="Maroon">S</font> <a href="partit_2.html#R1" target="_self">c=</a> <font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="relat_2.html#T44">RELAT_2:44</a></i>;<br/>
<span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="relset_1.html#K7">*</a> <font color="Maroon">S</font></span>)</span> <a href="relset_1.html#K7">*</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="relset_1.html#K7">*</a> <font color="Maroon">S</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="relset_1.html#K7">*</a> <font color="Maroon">S</font></span>)</span> <a href="relset_1.html#K7">*</a> <font color="Maroon">R</font></span>)</span> <a href="relset_1.html#K7">*</a> <font color="Maroon">S</font>
<b>by </b><i><a class="ref" href="relat_1.html#T55">RELAT_1:55</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="relset_1.html#K7">*</a> <span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="relset_1.html#K7">*</a> <font color="Maroon">S</font></span>)</span></span>)</span> <a href="relset_1.html#K7">*</a> <font color="Maroon">S</font>
<b>by </b><i>, <a class="ref" href="relat_1.html#T55">RELAT_1:55</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="relset_1.html#K7">*</a> <font color="Maroon">R</font></span>)</span> <a href="relset_1.html#K7">*</a> <font color="Maroon">S</font></span>)</span> <a href="relset_1.html#K7">*</a> <font color="Maroon">S</font>
<b>by </b><i><a class="ref" href="relat_1.html#T55">RELAT_1:55</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="relset_1.html#K7">*</a> <font color="Maroon">R</font></span>)</span> <a href="relset_1.html#K7">*</a> <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="relset_1.html#K7">*</a> <font color="Maroon">S</font></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T55">RELAT_1:55</a></i>
;<br/>
<a NAME="E7:12_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="relset_1.html#K7">*</a> <font color="Maroon">S</font></span>)</span> <a href="relset_1.html#K7">*</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="relset_1.html#K7">*</a> <font color="Maroon">S</font></span>)</span> <a href="partit_2.html#R1" target="_self">c=</a> <font color="Maroon">R</font> <a href="relset_1.html#K7">*</a> <font color="Maroon">S</font>
 
<b>by </b><i>, , <a class="ref" href="relat_1.html#T50">RELAT_1:50</a></i>;<br/>
<b>hence </b><a NAME="E8:12_1"/>
<font color="Maroon">R</font> <a href="relset_1.html#K7">*</a> <font color="Maroon">S</font> is <a href="relat_2.html#V8">transitive</a>
 <b>by </b><i><a class="ref" href="relat_2.html#T44">RELAT_2:44</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:12"/>
<font color="Maroon">R</font> <a href="relset_1.html#K7">*</a> <font color="Maroon">S</font> is   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">Y</font>
 ;<br/>


</div>
