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<b>let </b><font color="Maroon" title="c1">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">I</font> being 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/> for <font color="Olive" title="b2">EqR</font> being  <a href="eqrel_1.html#NM2" title="EQREL_1:NM.2">Equivalence_Relation</a> of <font color="Olive" title="b1">I</font><br/> for <font color="Olive" title="b3">c1</font>, <font color="Olive" title="b4">c2</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <font color="Olive" title="b2">EqR</font>  st <font color="Maroon" title="c1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">c1</font> &amp; <font color="Maroon" title="c1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b4">c2</font> holds <br/><font color="Olive" title="b3">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">c2</font></span><br/><b>let </b><font color="Maroon" title="c2">I</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">EqR</font> being  <a href="eqrel_1.html#NM2" title="EQREL_1:NM.2">Equivalence_Relation</a> of <font color="Maroon" title="c2">I</font><br/> for <font color="Olive" title="b2">c1</font>, <font color="Olive" title="b3">c2</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <font color="Olive" title="b1">EqR</font>  st <font color="Maroon" title="c1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">c1</font> &amp; <font color="Maroon" title="c1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">c2</font> holds <br/><font color="Olive" title="b2">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">c2</font></span><br/><b>let </b><font color="Maroon" title="c3">EqR</font> be   <a href="eqrel_1.html#NM2" title="EQREL_1:NM.2">Equivalence_Relation</a> of <font color="Maroon" title="c2">I</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">c1</font>, <font color="Olive" title="b2">c2</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <font color="Maroon" title="c3">EqR</font>  st <font color="Maroon" title="c1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">c1</font> &amp; <font color="Maroon" title="c1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">c2</font> holds <br/><font color="Olive" title="b1">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">c2</font></span><br/><b>let </b><font color="Maroon" title="c4">c1</font>, <font color="Maroon" title="c5">c2</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <font color="Maroon" title="c3">EqR</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">c1</font> &amp; <font color="Maroon" title="c1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">c2</font> implies <font color="Maroon" title="c4">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c2</font> )</span><br/>







<b>assume </b><b>that </b><br/><a NAME="E1:6"/><i><font color="Green" title="E4">A1</font></i>: 
<font color="Maroon" title="c1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">c1</font>
 <b>and </b><br/><a NAME="E2:6"/><i><font color="Green" title="E5">A2</font></i>: 
<font color="Maroon" title="c1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">c2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c2</font></span><br/>

<b>consider </b><font color="Maroon" title="c6">x1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:6"/>
<font color="Maroon" title="c6">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font>
 <b>and </b><br/><a NAME="E5:6"/><i><font color="Green" title="E6">A3</font></i>: 
<font color="Maroon" title="c4">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <font color="Maroon" title="c3">EqR</font>,<font color="Maroon" title="c6">x1</font>
 <b>by </b><i><a class="ref" href="eqrel_1.html#D5" title="EQREL_1:def.5">EQREL_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c7">x2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E7:6"/>
<font color="Maroon" title="c7">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font>
 <b>and </b><br/><a NAME="E8:6"/><i><font color="Green" title="E7">A4</font></i>: 
<font color="Maroon" title="c5">c2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <font color="Maroon" title="c3">EqR</font>,<font color="Maroon" title="c7">x2</font>
 <b>by </b><i><a class="ref" href="eqrel_1.html#D5" title="EQREL_1:def.5">EQREL_1:def 5</a></i>;<br/>
<a NAME="E9:6"/><i><font color="Green" title="E8">A5</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c6">x1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">EqR</font>
 
<b>by </b><i><a class="txt" href="pralg_3.html#E1:6"><i><font color="Green" title="E4">A1</font></i></a>, <a class="txt" href="pralg_3.html#E5:6"><i><font color="Green" title="E6">A3</font></i></a>, <a class="ref" href="eqrel_1.html#T27" title="EQREL_1:th.27">EQREL_1:27</a></i>;<br/>
<a NAME="E10:6"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c7">x2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">EqR</font>
 
<b>by </b><i><a class="txt" href="pralg_3.html#E2:6"><i><font color="Green" title="E5">A2</font></i></a>, <a class="txt" href="pralg_3.html#E8:6"><i><font color="Green" title="E7">A4</font></i></a>, <a class="ref" href="eqrel_1.html#T27" title="EQREL_1:th.27">EQREL_1:27</a></i>;<br/>
<b>then </b> <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <font color="Maroon" title="c3">EqR</font>,<font color="Maroon" title="c7">x2</font> = 
 <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <font color="Maroon" title="c3">EqR</font>,<font color="Maroon" title="c1">i</font>
<b>by </b><i><a class="txt" href="pralg_3.html#E1:6"><i><font color="Green" title="E4">A1</font></i></a>, <a class="ref" href="eqrel_1.html#T44" title="EQREL_1:th.44">EQREL_1:44</a></i>
<br/>.= 
<font color="Maroon" title="c4">c1</font>
<b>by </b><i><a class="txt" href="pralg_3.html#E1:6"><i><font color="Green" title="E4">A1</font></i></a>, <a class="txt" href="pralg_3.html#E5:6"><i><font color="Green" title="E6">A3</font></i></a>, <a class="txt" href="pralg_3.html#E9:6"><i><font color="Green" title="E8">A5</font></i></a>, <a class="ref" href="eqrel_1.html#T44" title="EQREL_1:th.44">EQREL_1:44</a></i>
;<br/>
<b>hence </b><a NAME="E12:6"/>
<font color="Maroon" title="c4">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c2</font>
 <b>by </b><i><a class="txt" href="pralg_3.html#E8:6"><i><font color="Green" title="E7">A4</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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