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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> ; <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">R</font> being  <a href="eqrel_1.html#NM2" title="EQREL_1:NM.2">Equivalence_Relation</a> of <font color="Maroon" title="c1">Y</font> holds <br/> ( <span class="p1">(<span class="default"><a href="eqrel_1.html#K1" title="EQREL_1:func.1">nabla</a> <font color="Maroon" title="c1">Y</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">*</a> <font color="Olive" title="b1">R</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="eqrel_1.html#K1" title="EQREL_1:func.1">nabla</a> <font color="Maroon" title="c1">Y</font> &amp; <font color="Olive" title="b1">R</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">*</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K1" title="EQREL_1:func.1">nabla</a> <font color="Maroon" title="c1">Y</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="eqrel_1.html#K1" title="EQREL_1:func.1">nabla</a> <font color="Maroon" title="c1">Y</font> )</span><br/><b>let </b><font color="Maroon" title="c2">R</font> be   <a href="eqrel_1.html#NM2" title="EQREL_1:NM.2">Equivalence_Relation</a> of <font color="Maroon" title="c1">Y</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1">(<span class="default"><a href="eqrel_1.html#K1" title="EQREL_1:func.1">nabla</a> <font color="Maroon" title="c1">Y</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">*</a> <font color="Maroon" title="c2">R</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="eqrel_1.html#K1" title="EQREL_1:func.1">nabla</a> <font color="Maroon" title="c1">Y</font> &amp; <font color="Maroon" title="c2">R</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">*</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K1" title="EQREL_1:func.1">nabla</a> <font color="Maroon" title="c1">Y</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="eqrel_1.html#K1" title="EQREL_1:func.1">nabla</a> <font color="Maroon" title="c1">Y</font> )</span><br/>



<a NAME="E1:5"/><i><font color="Green" title="E5">A1</font></i>: 
<span class="p1">(<span class="default"><a href="eqrel_1.html#K1" title="EQREL_1:func.1">nabla</a> <font color="Maroon" title="c1">Y</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">*</a> <font color="Maroon" title="c2">R</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a>  <a href="eqrel_1.html#K1" title="EQREL_1:func.1">nabla</a> <font color="Maroon" title="c1">Y</font>
 
<b>by </b><i><a class="ref" href="partit_2.html#T3" target="_self" title="PARTIT_2:th.3">Th3</a></i>;<br/>
<a NAME="E2:5"/>
<span class="p1">(<span class="default"><a href="eqrel_1.html#K1" title="EQREL_1:func.1">nabla</a> <font color="Maroon" title="c1">Y</font></span>)</span> <a href="eqrel_1.html#K3" title="EQREL_1:func.3">\/</a> <font color="Maroon" title="c2">R</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K1" title="EQREL_1:func.1">nabla</a> <font color="Maroon" title="c1">Y</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">*</a> <font color="Maroon" title="c2">R</font>
 
<b>by </b><i><a class="ref" href="partit_2.html#T2" target="_self" title="PARTIT_2:th.2">Th2</a></i>;<br/>
<a NAME="E3:5"/><b>then </b>
 <a href="eqrel_1.html#K1" title="EQREL_1:func.1">nabla</a> <font color="Maroon" title="c1">Y</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K1" title="EQREL_1:func.1">nabla</a> <font color="Maroon" title="c1">Y</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">*</a> <font color="Maroon" title="c2">R</font>
 
<b>by </b><i><a class="ref" href="eqrel_1.html#T1" title="EQREL_1:th.1">EQREL_1:1</a></i>;<br/>
<b>hence </b><a NAME="E4:5"/>
<span class="p1">(<span class="default"><a href="eqrel_1.html#K1" title="EQREL_1:func.1">nabla</a> <font color="Maroon" title="c1">Y</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">*</a> <font color="Maroon" title="c2">R</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="eqrel_1.html#K1" title="EQREL_1:func.1">nabla</a> <font color="Maroon" title="c1">Y</font>
 <b>by </b><i><a class="txt" href="#E1:5"><i><font color="Green" title="E5">A1</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"> <font color="Maroon" title="c2">R</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">*</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K1" title="EQREL_1:func.1">nabla</a> <font color="Maroon" title="c1">Y</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="eqrel_1.html#K1" title="EQREL_1:func.1">nabla</a> <font color="Maroon" title="c1">Y</font></span><br/>

<a NAME="E5:5"/><i><font color="Green" title="E6">A2</font></i>: 
<font color="Maroon" title="c2">R</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">*</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K1" title="EQREL_1:func.1">nabla</a> <font color="Maroon" title="c1">Y</font></span>)</span> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a>  <a href="eqrel_1.html#K1" title="EQREL_1:func.1">nabla</a> <font color="Maroon" title="c1">Y</font>
 
<b>by </b><i><a class="ref" href="partit_2.html#T3" target="_self" title="PARTIT_2:th.3">Th3</a></i>;<br/>
<a NAME="E6:5"/>
<span class="p1">(<span class="default"><a href="eqrel_1.html#K1" title="EQREL_1:func.1">nabla</a> <font color="Maroon" title="c1">Y</font></span>)</span> <a href="eqrel_1.html#K3" title="EQREL_1:func.3">\/</a> <font color="Maroon" title="c2">R</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <font color="Maroon" title="c2">R</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">*</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K1" title="EQREL_1:func.1">nabla</a> <font color="Maroon" title="c1">Y</font></span>)</span>
 
<b>by </b><i><a class="ref" href="partit_2.html#T2" target="_self" title="PARTIT_2:th.2">Th2</a></i>;<br/>
<a NAME="E7:5"/><b>then </b>
 <a href="eqrel_1.html#K1" title="EQREL_1:func.1">nabla</a> <font color="Maroon" title="c1">Y</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <font color="Maroon" title="c2">R</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">*</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K1" title="EQREL_1:func.1">nabla</a> <font color="Maroon" title="c1">Y</font></span>)</span>
 
<b>by </b><i><a class="ref" href="eqrel_1.html#T1" title="EQREL_1:th.1">EQREL_1:1</a></i>;<br/>
<b>hence </b><a NAME="E8:5"/>
<font color="Maroon" title="c2">R</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">*</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K1" title="EQREL_1:func.1">nabla</a> <font color="Maroon" title="c1">Y</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="eqrel_1.html#K1" title="EQREL_1:func.1">nabla</a> <font color="Maroon" title="c1">Y</font>
 <b>by </b><i><a class="txt" href="#E5:5"><i><font color="Green" title="E6">A2</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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