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<b>let </b><font color="Maroon" title="c1">X</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">S1</font>, <font color="Maroon" title="c3">S2</font> be    <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of <font color="Maroon" title="c1">X</font>;<br/>



<b>assume </b><a NAME="E1:54"/><i><font color="Green" title="E26">A1</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> holds   <a href="eqrel_1.html#K11" title="EQREL_1:func.11">EqClass</a> <font color="Olive" title="b1">x</font>,<font color="Maroon" title="c2">S1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K11" title="EQREL_1:func.11">EqClass</a> <font color="Olive" title="b1">x</font>,<font color="Maroon" title="c3">S2</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">P</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>;<br/><b>thus </b><a NAME="E1:54_1"/>
( <font color="Maroon" title="c4">P</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S1</font> implies <font color="Maroon" title="c4">P</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">S2</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:54_1_1"/><i><font color="Green" title="E27">A2</font></i>: 
<font color="Maroon" title="c4">P</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S1</font>
 ;<br/>

<a NAME="E2:54_1_1"/><b>then </b><i><font color="Green" title="E28">A3</font></i>: 
not <font color="Maroon" title="c4">P</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 
<b>by </b><i><a class="ref" href="eqrel_1.html#D6" target="_self" title="EQREL_1:def.6">Def6</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c5">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">P</font>;<br/>
<a NAME="E4:54_1_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">P</font>
 
<b>by </b><i><a class="txt" href="eqrel_1.html#E2:54_1_1"><i><font color="Green" title="E28">A3</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c6">x</font> = <font color="Maroon" title="c5">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> ;<br/>
<a NAME="E6:54_1_1"/>
<font color="Maroon" title="c4">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K11" title="EQREL_1:func.11">EqClass</a> <font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c2">S1</font>
 
<b>by </b><i><a class="txt" href="eqrel_1.html#E1:54_1_1"><i><font color="Green" title="E27">A2</font></i></a>, <a class="txt" href="eqrel_1.html#E2:54_1_1"><i><font color="Green" title="E28">A3</font></i></a>, <a class="ref" href="eqrel_1.html#D8" target="_self" title="EQREL_1:def.8">Def8</a></i>;<br/>
<a NAME="E7:54_1_1"/><b>then </b>
<font color="Maroon" title="c4">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K11" title="EQREL_1:func.11">EqClass</a> <font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c3">S2</font>
 
<b>by </b><i><a class="txt" href="eqrel_1.html#E1:54"><i><font color="Green" title="E26">A1</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:54_1_1"/>
<font color="Maroon" title="c4">P</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">S2</font>
 <b>by </b><i><a class="ref" href="eqrel_1.html#D8" target="_self" title="EQREL_1:def.8">Def8</a></i>;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E2:54_1"/>
( <font color="Maroon" title="c4">P</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">S2</font> implies <font color="Maroon" title="c4">P</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S1</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:54_1_2"/><i><font color="Green" title="E27">A4</font></i>: 
<font color="Maroon" title="c4">P</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">S2</font>
 ;<br/>

<a NAME="E2:54_1_2"/><b>then </b><i><font color="Green" title="E28">A5</font></i>: 
<font color="Maroon" title="c4">P</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="ref" href="eqrel_1.html#D6" target="_self" title="EQREL_1:def.6">Def6</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c5">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">P</font>;<br/>
<a NAME="E4:54_1_2"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">P</font>
 
<b>by </b><i><a class="txt" href="eqrel_1.html#E2:54_1_2"><i><font color="Green" title="E28">A5</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c6">x</font> = <font color="Maroon" title="c5">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> ;<br/>
<a NAME="E6:54_1_2"/>
<font color="Maroon" title="c4">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K11" title="EQREL_1:func.11">EqClass</a> <font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c3">S2</font>
 
<b>by </b><i><a class="txt" href="eqrel_1.html#E1:54_1_2"><i><font color="Green" title="E27">A4</font></i></a>, <a class="txt" href="eqrel_1.html#E2:54_1_2"><i><font color="Green" title="E28">A5</font></i></a>, <a class="ref" href="eqrel_1.html#D8" target="_self" title="EQREL_1:def.8">Def8</a></i>;<br/>
<a NAME="E7:54_1_2"/><b>then </b>
<font color="Maroon" title="c4">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K11" title="EQREL_1:func.11">EqClass</a> <font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c2">S1</font>
 
<b>by </b><i><a class="txt" href="eqrel_1.html#E1:54"><i><font color="Green" title="E26">A1</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:54_1_2"/>
<font color="Maroon" title="c4">P</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S1</font>
 <b>by </b><i><a class="ref" href="eqrel_1.html#D8" target="_self" title="EQREL_1:def.8">Def8</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:54"/>
<font color="Maroon" title="c2">S1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">S2</font>
 <b>by </b><i><a class="ref" href="setfam_1.html#T44" title="SETFAM_1:th.44">SETFAM_1:44</a></i>;<br/>


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