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<b>let </b><font color="Maroon" title="c1">Y</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">P</font>, <font color="Maroon" title="c3">Q</font>, <font color="Maroon" title="c4">R</font> be    <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of <font color="Maroon" title="c1">Y</font>;<br/>



<b>assume </b><a NAME="E1:8"/><i><font color="Green" title="E6">A1</font></i>: 
 <a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <font color="Maroon" title="c4">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <font color="Maroon" title="c2">P</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">*</a> <span class="p1">(<span class="default"><a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <font color="Maroon" title="c3">Q</font></span>)</span>
 ;<br/>

<b>let </b><font color="Maroon" title="c5">x</font>, <font color="Maroon" title="c6">y</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">Y</font>;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:8_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bvfunc_1.html#K20" title="BVFUNC_1:func.20">EqClass</a> <font color="Maroon" title="c6">y</font>,<font color="Maroon" title="c4">R</font>
 ;<br/><a NAME="E2:8_1"/><b>then </b>
<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>  <a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <font color="Maroon" title="c4">R</font>
 <b>by </b><i><a class="ref" href="partit_2.html#T5" target="_self" title="PARTIT_2:th.5">Th5</a></i>;<br/><b>then consider </b><font color="Maroon" title="c7">z</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">Y</font><b> such that </b><br/><a NAME="E4:8_1"/><i><font color="Green" title="E7">A2</font></i>: 
<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="c7">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <font color="Maroon" title="c2">P</font>
 <b>and </b><br/><a NAME="E5:8_1"/><i><font color="Green" title="E8">A3</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">z</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>  <a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <font color="Maroon" title="c3">Q</font>
 <b>by </b><i><a class="txt" href="partit_2.html#E1:8"><i><font color="Green" title="E6">A1</font></i></a>, <a class="ref" href="relset_1.html#T51" title="RELSET_1:th.51">RELSET_1:51</a></i>;<br/><b>take </b><font color="Maroon" title="c8">z</font> = <font color="Maroon" title="c7">z</font>;<br/><b>thus </b><a NAME="E6:8_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bvfunc_1.html#K20" title="BVFUNC_1:func.20">EqClass</a> <font color="Maroon" title="c8">z</font>,<font color="Maroon" title="c2">P</font>
 <b>by </b><i><a class="txt" href="partit_2.html#E4:8_1"><i><font color="Green" title="E7">A2</font></i></a>, <a class="ref" href="partit_2.html#T5" target="_self" title="PARTIT_2:th.5">Th5</a></i>;<br/><b>thus </b><a NAME="E7:8_1"/>
<font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bvfunc_1.html#K20" title="BVFUNC_1:func.20">EqClass</a> <font color="Maroon" title="c6">y</font>,<font color="Maroon" title="c3">Q</font>
 <b>by </b><i><a class="txt" href="partit_2.html#E5:8_1"><i><font color="Green" title="E8">A3</font></i></a>, <a class="ref" href="partit_2.html#T5" target="_self" title="PARTIT_2:th.5">Th5</a></i>;<br/></div>
<b>end;</b></div>

<b>given </b><font color="Maroon" title="c7">z</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">Y</font><b> such that </b><a NAME="E3:8"/><i><font color="Green" title="E7">A4</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bvfunc_1.html#K20" title="BVFUNC_1:func.20">EqClass</a> <font color="Maroon" title="c7">z</font>,<font color="Maroon" title="c2">P</font>
 <b>and </b><br/><a NAME="E4:8"/><i><font color="Green" title="E8">A5</font></i>: 
<font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bvfunc_1.html#K20" title="BVFUNC_1:func.20">EqClass</a> <font color="Maroon" title="c6">y</font>,<font color="Maroon" title="c3">Q</font>
 ;<br/>

<a NAME="E5:8"/><i><font color="Green" title="E9">A6</font></i>: 
<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="c7">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <font color="Maroon" title="c2">P</font>
 
<b>by </b><i><a class="txt" href="partit_2.html#E3:8"><i><font color="Green" title="E7">A4</font></i></a>, <a class="ref" href="partit_2.html#T5" target="_self" title="PARTIT_2:th.5">Th5</a></i>;<br/>
<a NAME="E6:8"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">z</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>  <a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <font color="Maroon" title="c3">Q</font>
 
<b>by </b><i><a class="txt" href="partit_2.html#E4:8"><i><font color="Green" title="E8">A5</font></i></a>, <a class="ref" href="partit_2.html#T5" target="_self" title="PARTIT_2:th.5">Th5</a></i>;<br/>
<a NAME="E7:8"/><b>then </b>
<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>  <a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <font color="Maroon" title="c4">R</font>
 
<b>by </b><i><a class="txt" href="partit_2.html#E1:8"><i><font color="Green" title="E6">A1</font></i></a>, <a class="txt" href="partit_2.html#E5:8"><i><font color="Green" title="E9">A6</font></i></a>, <a class="ref" href="relat_1.html#D8" title="RELAT_1:def.8">RELAT_1:def 8</a></i>;<br/>
<b>hence </b><a NAME="E8:8"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bvfunc_1.html#K20" title="BVFUNC_1:func.20">EqClass</a> <font color="Maroon" title="c6">y</font>,<font color="Maroon" title="c4">R</font>
 <b>by </b><i><a class="ref" href="partit_2.html#T5" target="_self" title="PARTIT_2:th.5">Th5</a></i>;<br/>


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