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<span class="kw">let </span><font color="Maroon" title="c1">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">R</font> being   <a href="relset_1.html#NM2" title="RELSET_1:NM.2">Relation</a> of <font color="Maroon" title="c1">X</font> holds <br/> ( <font color="Olive" title="b1">R</font> is   <a href="eqrel_1.html#NM2" title="EQREL_1:NM.2">Equivalence_Relation</a> of <font color="Maroon" title="c1">X</font> iff ( <font color="Olive" title="b1">R</font> is  <a href="relat_2.html#V1" title="RELAT_2:attr.1">reflexive</a>  &amp; <font color="Olive" title="b1">R</font> is  <a href="relat_2.html#V3" title="RELAT_2:attr.3">symmetric</a>  &amp; <font color="Olive" title="b1">R</font> is  <a href="relat_2.html#V8" title="RELAT_2:attr.8">transitive</a>  &amp;  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Olive" title="b1">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">R</font> be   <a href="relset_1.html#NM2" title="RELSET_1:NM.2">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">R</font> is   <a href="eqrel_1.html#NM2" title="EQREL_1:NM.2">Equivalence_Relation</a> of <font color="Maroon" title="c1">X</font> iff ( <font color="Maroon" title="c2">R</font> is  <a href="relat_2.html#V1" title="RELAT_2:attr.1">reflexive</a>  &amp; <font color="Maroon" title="c2">R</font> is  <a href="relat_2.html#V3" title="RELAT_2:attr.3">symmetric</a>  &amp; <font color="Maroon" title="c2">R</font> is  <a href="relat_2.html#V8" title="RELAT_2:attr.8">transitive</a>  &amp;  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c2">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> ) )</span><br/>



<span class="kw">thus </span><a NAME="E1:12"/>
( <font color="Maroon" title="c2">R</font> is   <a href="eqrel_1.html#NM2" title="EQREL_1:NM.2">Equivalence_Relation</a> of <font color="Maroon" title="c1">X</font> implies ( <font color="Maroon" title="c2">R</font> is  <a href="relat_2.html#V1" title="RELAT_2:attr.1">reflexive</a>  &amp; <font color="Maroon" title="c2">R</font> is  <a href="relat_2.html#V3" title="RELAT_2:attr.3">symmetric</a>  &amp; <font color="Maroon" title="c2">R</font> is  <a href="relat_2.html#V8" title="RELAT_2:attr.8">transitive</a>  &amp;  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c2">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="orders_1.html#T12" title="ORDERS_1:th.12">ORDERS_1:12</a></span>; <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">R</font> is  <a href="relat_2.html#V1" title="RELAT_2:attr.1">reflexive</a>  &amp; <font color="Maroon" title="c2">R</font> is  <a href="relat_2.html#V3" title="RELAT_2:attr.3">symmetric</a>  &amp; <font color="Maroon" title="c2">R</font> is  <a href="relat_2.html#V8" title="RELAT_2:attr.8">transitive</a>  &amp;  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c2">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> implies <font color="Maroon" title="c2">R</font> is   <a href="eqrel_1.html#NM2" title="EQREL_1:NM.2">Equivalence_Relation</a> of <font color="Maroon" title="c1">X</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:12"/><span class="lab"><font color="Green" title="E7">A1</font></span>: 
<font color="Maroon" title="c2">R</font> is  <a href="relat_2.html#V1" title="RELAT_2:attr.1">reflexive</a> 
 <span class="kw">and </span><br/><a NAME="E3:12"/><span class="lab"><font color="Green" title="E8">A2</font></span>: 
( <font color="Maroon" title="c2">R</font> is  <a href="relat_2.html#V3" title="RELAT_2:attr.3">symmetric</a>  &amp; <font color="Maroon" title="c2">R</font> is  <a href="relat_2.html#V8" title="RELAT_2:attr.8">transitive</a>  )
 <span class="kw">and </span><br/><a NAME="E4:12"/><span class="lab"><font color="Green" title="E9">A3</font></span>: 
 <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c2">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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">R</font> is   <a href="eqrel_1.html#NM2" title="EQREL_1:NM.2">Equivalence_Relation</a> of <font color="Maroon" title="c1">X</font></span><br/>

<a NAME="E5:12"/>
<font color="Maroon" title="c2">R</font> <a href="relat_2.html#R1" title="RELAT_2:pred.1">is_reflexive_in</a> <font color="Maroon" title="c1">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:12"><span class="lab"><font color="Green" title="E7">A1</font></span></a>, <a class="txt" href="#E4:12"><span class="lab"><font color="Green" title="E9">A3</font></span></a>, <a class="ref" href="relat_2.html#D9" title="RELAT_2:def.9">RELAT_2:def 9</a></span>;<br/>
<a NAME="E6:12"/><span class="kw">then </span>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="orders_1.html#T13" title="ORDERS_1:th.13">ORDERS_1:13</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:12"/>
<font color="Maroon" title="c2">R</font> is   <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">by</span> <span class="lab"><a class="txt" href="#E3:12"><span class="lab"><font color="Green" title="E8">A2</font></span></a>, <a class="ref" href="partfun1.html#D2" title="PARTFUN1:def.2">PARTFUN1:def 2</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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