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<span class="kw">let </span><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">S1</font> being   <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of <font color="Maroon" title="c1">X</font><br/> for <font color="Olive" title="b2">x</font>, <font color="Olive" title="b3">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>  st  <a href="eqrel_1.html#K11" title="EQREL_1:func.11">EqClass</a> <font color="Olive" title="b2">x</font>,<font color="Olive" title="b1">S1</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="eqrel_1.html#K11" title="EQREL_1:func.11">EqClass</a> <font color="Olive" title="b3">y</font>,<font color="Olive" title="b1">S1</font> holds <br/> <a href="eqrel_1.html#K11" title="EQREL_1:func.11">EqClass</a> <font color="Olive" title="b2">x</font>,<font color="Olive" title="b1">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="b3">y</font>,<font color="Olive" title="b1">S1</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">S1</font> be    <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>  st  <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="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="eqrel_1.html#K11" title="EQREL_1:func.11">EqClass</a> <font color="Olive" title="b2">y</font>,<font color="Maroon" title="c2">S1</font> holds <br/> <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="b2">y</font>,<font color="Maroon" title="c2">S1</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c4">y</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="eqrel_1.html#K11" title="EQREL_1:func.11">EqClass</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c2">S1</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="eqrel_1.html#K11" title="EQREL_1:func.11">EqClass</a> <font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c2">S1</font> implies  <a href="eqrel_1.html#K11" title="EQREL_1:func.11">EqClass</a> <font color="Maroon" title="c3">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="Maroon" title="c4">y</font>,<font color="Maroon" title="c2">S1</font> )</span><br/>





<span class="kw">consider </span><font color="Maroon" title="c5">EQR</font> being   <a href="eqrel_1.html#NM2" title="EQREL_1:NM.2">Equivalence_Relation</a> of <font color="Maroon" title="c1">X</font><span class="kw"> such that </span><br/><a NAME="E2:73"/><span class="lab"><font color="Green" title="E31">A1</font></span>: 
<font color="Maroon" title="c2">S1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <font color="Maroon" title="c5">EQR</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="eqrel_1.html#T43" target="_self" title="EQREL_1:th.43">Th43</a></span>;<br/>
<a NAME="E3:73"/><span class="lab"><font color="Green" title="E32">A2</font></span>: 
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <font color="Maroon" title="c5">EQR</font>,<font color="Maroon" title="c4">y</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="eqrel_1.html#T28" target="_self" title="EQREL_1:th.28">Th28</a></span>;<br/>
<a NAME="E4:73"/>
 <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <font color="Maroon" title="c5">EQR</font>,<font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:73"><span class="lab"><font color="Green" title="E31">A1</font></span></a>, <a class="ref" href="eqrel_1.html#D5" target="_self" title="EQREL_1:def.5">Def5</a></span>;<br/>
<a NAME="E5:73"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A3</font></span>: 
 <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <font color="Maroon" title="c5">EQR</font>,<font color="Maroon" title="c4">y</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="c4">y</font>,<font color="Maroon" title="c2">S1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:73"><span class="lab"><font color="Green" title="E32">A2</font></span></a>, <a class="ref" href="eqrel_1.html#D8" target="_self" title="EQREL_1:def.8">Def8</a></span>;<br/>
<a NAME="E6:73"/><span class="lab"><font color="Green" title="E34">A4</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <font color="Maroon" title="c5">EQR</font>,<font color="Maroon" title="c3">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="eqrel_1.html#T28" target="_self" title="EQREL_1:th.28">Th28</a></span>;<br/>
<a NAME="E7:73"/>
 <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <font color="Maroon" title="c5">EQR</font>,<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:73"><span class="lab"><font color="Green" title="E31">A1</font></span></a>, <a class="ref" href="eqrel_1.html#D5" target="_self" title="EQREL_1:def.5">Def5</a></span>;<br/>
<a NAME="E8:73"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A5</font></span>: 
 <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <font color="Maroon" title="c5">EQR</font>,<font color="Maroon" title="c3">x</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="c3">x</font>,<font color="Maroon" title="c2">S1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:73"><span class="lab"><font color="Green" title="E34">A4</font></span></a>, <a class="ref" href="eqrel_1.html#D8" target="_self" title="EQREL_1:def.8">Def8</a></span>;<br/>
<span class="kw">assume </span><a NAME="E9:73"/>
 <a href="eqrel_1.html#K11" title="EQREL_1:func.11">EqClass</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c2">S1</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="eqrel_1.html#K11" title="EQREL_1:func.11">EqClass</a> <font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c2">S1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="eqrel_1.html#K11" title="EQREL_1:func.11">EqClass</a> <font color="Maroon" title="c3">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="Maroon" title="c4">y</font>,<font color="Maroon" title="c2">S1</font></span><br/>

<span class="kw">hence </span><a NAME="E10:73"/>
 <a href="eqrel_1.html#K11" title="EQREL_1:func.11">EqClass</a> <font color="Maroon" title="c3">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="Maroon" title="c4">y</font>,<font color="Maroon" title="c2">S1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:73"><span class="lab"><font color="Green" title="E35">A5</font></span></a>, <a class="txt" href="#E5:73"><span class="lab"><font color="Green" title="E33">A3</font></span></a>, <a class="ref" href="eqrel_1.html#T32" target="_self" title="EQREL_1:th.32">Th32</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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