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<b>let </b><font color="Maroon" title="c3">C1</font>, <font color="Maroon" title="c4">C2</font> be   <a href="eqrel_1.html#NM2" title="EQREL_1:NM.2">Equivalence_Relation</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">R</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <font color="Maroon" title="c3">C1</font> &amp; ( for <font color="Olive" title="b1">EqR2</font> being  <a href="eqrel_1.html#NM2" title="EQREL_1:NM.2">Equivalence_Relation</a> of <font color="Maroon" title="c1">X</font>  st <font color="Maroon" title="c2">R</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <font color="Olive" title="b1">EqR2</font> holds <br/><font color="Maroon" title="c3">C1</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <font color="Olive" title="b1">EqR2</font> ) &amp; <font color="Maroon" title="c2">R</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <font color="Maroon" title="c4">C2</font> &amp; ( for <font color="Olive" title="b1">EqR2</font> being  <a href="eqrel_1.html#NM2" title="EQREL_1:NM.2">Equivalence_Relation</a> of <font color="Maroon" title="c1">X</font>  st <font color="Maroon" title="c2">R</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <font color="Olive" title="b1">EqR2</font> holds <br/><font color="Maroon" title="c4">C2</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <font color="Olive" title="b1">EqR2</font> ) implies <font color="Maroon" title="c3">C1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">C2</font> )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:1_1_1"/><i><font color="Green" title="E1">A1</font></i>: 
<font color="Maroon" title="c2">R</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <font color="Maroon" title="c3">C1</font>
 <b>and </b><br/><a NAME="E2:1_1_1"/><i><font color="Green" title="E2">A2</font></i>: 
for <font color="Olive" title="b1">EqR2</font> being  <a href="eqrel_1.html#NM2" title="EQREL_1:NM.2">Equivalence_Relation</a> of <font color="Maroon" title="c1">X</font>  st <font color="Maroon" title="c2">R</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <font color="Olive" title="b1">EqR2</font> holds <br/><font color="Maroon" title="c3">C1</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <font color="Olive" title="b1">EqR2</font>
 <b>and </b><br/><a NAME="E3:1_1_1"/><i><font color="Green" title="E3">A3</font></i>: 
<font color="Maroon" title="c2">R</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <font color="Maroon" title="c4">C2</font>
 <b>and </b><br/><a NAME="E4:1_1_1"/><i><font color="Green" title="E4">A4</font></i>: 
for <font color="Olive" title="b1">EqR2</font> being  <a href="eqrel_1.html#NM2" title="EQREL_1:NM.2">Equivalence_Relation</a> of <font color="Maroon" title="c1">X</font>  st <font color="Maroon" title="c2">R</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <font color="Olive" title="b1">EqR2</font> holds <br/><font color="Maroon" title="c4">C2</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <font color="Olive" title="b1">EqR2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">C1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">C2</font></span><br/>

<a NAME="E5:1_1_1"/><i><font color="Green" title="E5">A5</font></i>: 
<font color="Maroon" title="c3">C1</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <font color="Maroon" title="c4">C2</font>
 
<b>by </b><i><a class="txt" href="#E2:1_1_1"><i><font color="Green" title="E2">A2</font></i></a>, <a class="txt" href="#E3:1_1_1"><i><font color="Green" title="E3">A3</font></i></a></i>;<br/>
<a NAME="E6:1_1_1"/>
<font color="Maroon" title="c4">C2</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <font color="Maroon" title="c3">C1</font>
 
<b>by </b><i><a class="txt" href="#E1:1_1_1"><i><font color="Green" title="E1">A1</font></i></a>, <a class="txt" href="#E4:1_1_1"><i><font color="Green" title="E4">A4</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:1_1_1"/>
<font color="Maroon" title="c3">C1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c4">C2</font>
 <b>by </b><i><a class="txt" href="#E5:1_1_1"><i><font color="Green" title="E5">A5</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</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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