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<span class="kw">let </span><font color="Maroon" title="c4">R1</font>, <font color="Maroon" title="c5">R2</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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">EqR1</font> <a href="eqrel_1.html#K3" title="EQREL_1:func.3">\/</a> <font color="Maroon" title="c3">EqR2</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <font color="Maroon" title="c4">R1</font> &amp; (  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="c1">X</font>  st <font color="Maroon" title="c2">EqR1</font> <a href="eqrel_1.html#K3" title="EQREL_1:func.3">\/</a> <font color="Maroon" title="c3">EqR2</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <font color="Olive" title="b1">EqR</font> holds <br/><font color="Maroon" title="c4">R1</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <font color="Olive" title="b1">EqR</font> ) &amp; <font color="Maroon" title="c2">EqR1</font> <a href="eqrel_1.html#K3" title="EQREL_1:func.3">\/</a> <font color="Maroon" title="c3">EqR2</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <font color="Maroon" title="c5">R2</font> &amp; (  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="c1">X</font>  st <font color="Maroon" title="c2">EqR1</font> <a href="eqrel_1.html#K3" title="EQREL_1:func.3">\/</a> <font color="Maroon" title="c3">EqR2</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <font color="Olive" title="b1">EqR</font> holds <br/><font color="Maroon" title="c5">R2</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <font color="Olive" title="b1">EqR</font> ) implies <font color="Maroon" title="c4">R1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">R2</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:17_1_1"/>
( <font color="Maroon" title="c2">EqR1</font> <a href="eqrel_1.html#K3" title="EQREL_1:func.3">\/</a> <font color="Maroon" title="c3">EqR2</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <font color="Maroon" title="c4">R1</font> &amp; (  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="c1">X</font>  st <font color="Maroon" title="c2">EqR1</font> <a href="eqrel_1.html#K3" title="EQREL_1:func.3">\/</a> <font color="Maroon" title="c3">EqR2</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <font color="Olive" title="b1">EqR</font> holds <br/><font color="Maroon" title="c4">R1</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <font color="Olive" title="b1">EqR</font> ) &amp; <font color="Maroon" title="c2">EqR1</font> <a href="eqrel_1.html#K3" title="EQREL_1:func.3">\/</a> <font color="Maroon" title="c3">EqR2</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <font color="Maroon" title="c5">R2</font> &amp; (  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="c1">X</font>  st <font color="Maroon" title="c2">EqR1</font> <a href="eqrel_1.html#K3" title="EQREL_1:func.3">\/</a> <font color="Maroon" title="c3">EqR2</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <font color="Olive" title="b1">EqR</font> holds <br/><font color="Maroon" title="c5">R2</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <font color="Olive" title="b1">EqR</font> ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">R1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">R2</font></span><br/>

<a NAME="E2:17_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c4">R1</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <font color="Maroon" title="c5">R2</font> &amp; <font color="Maroon" title="c5">R2</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <font color="Maroon" title="c4">R1</font> )
 
;<br/>
<span class="kw">hence </span><a NAME="E3:17_1_1"/>
<font color="Maroon" title="c4">R1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">R2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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