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<b>let </b><font color="Maroon" title="c1">A</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> ;<br/><b>let </b><font color="Maroon" title="c2">L</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="lattice5.html#NM2" title="LATTICE5:NM.2">Sublattice</a> of  <a href="lattice5.html#K1" title="LATTICE5:func.1">EqRelLATT</a> <font color="Maroon" title="c1">A</font>;<br/>



<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:10_1"/><i><font color="Green" title="E6">A1</font></i>: 
for <font color="Olive" title="b1">e</font> being  <a href="eqrel_1.html#NM2" title="EQREL_1:NM.2">Equivalence_Relation</a> of <font color="Maroon" title="c1">A</font>  holds <br/>(  not <font color="Olive" title="b1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L</font> or  not <font color="Olive" title="b1">e</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <font color="Maroon" title="c1">A</font> )
 ;<br/><b>thus </b><a NAME="E2:10_1"/>
<font color="Maroon" title="c2">L</font> is <a href="realset2.html#V3" title="REALSET2:attr.3">trivial</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon" title="c3">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E2:10_1_1"/><i><font color="Green" title="E7">A2</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/>
<a NAME="E3:10_1_1"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="lattice5.html#K1" title="LATTICE5:func.1">EqRelLATT</a> <font color="Maroon" title="c1">A</font></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_0.html#D13" title="YELLOW_0:def.13">YELLOW_0:def 13</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c4">e</font> = <font color="Maroon" title="c3">x</font> as   <a href="eqrel_1.html#NM2" title="EQREL_1:NM.2">Equivalence_Relation</a> of <font color="Maroon" title="c1">A</font> <b>by </b><i><a class="txt" href="lattice8.html#E2:10_1_1"><i><font color="Green" title="E7">A2</font></i></a>, <a class="ref" href="lattice5.html#T4" title="LATTICE5:th.4">LATTICE5:4</a></i>;<br/>
<a NAME="E5:10_1_1"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:10_1_1_1"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:10_1_1_1_1"/><i><font color="Green" title="E8">A3</font></i>: 
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L</font>
 ;<br/>

<a NAME="E2:10_1_1_1_1"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="lattice5.html#K1" title="LATTICE5:func.1">EqRelLATT</a> <font color="Maroon" title="c1">A</font></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_0.html#D13" title="YELLOW_0:def.13">YELLOW_0:def 13</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c6">B</font> = <font color="Maroon" title="c5">a</font> as   <a href="eqrel_1.html#NM2" title="EQREL_1:NM.2">Equivalence_Relation</a> of <font color="Maroon" title="c1">A</font> <b>by </b><i><a class="txt" href="lattice8.html#E1:10_1_1_1_1"><i><font color="Green" title="E8">A3</font></i></a>, <a class="ref" href="lattice5.html#T4" title="LATTICE5:th.4">LATTICE5:4</a></i>;<br/>
<font color="Maroon" title="c6">B</font> = 
 <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <font color="Maroon" title="c1">A</font>
<b>by </b><i><a class="txt" href="lattice8.html#E1:10_1"><i><font color="Green" title="E6">A1</font></i></a>, <a class="txt" href="lattice8.html#E1:10_1_1_1_1"><i><font color="Green" title="E8">A3</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c4">e</font>
<b>by </b><i><a class="txt" href="lattice8.html#E1:10_1"><i><font color="Green" title="E6">A1</font></i></a>, <a class="txt" href="lattice8.html#E2:10_1_1"><i><font color="Green" title="E7">A2</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E5:10_1_1_1_1"/>
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon" title="c5">A</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:10_1_1_1"/>
<font color="Maroon" title="c5">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/>

<b>hence </b><a NAME="E3:10_1_1_1"/>
<font color="Maroon" title="c5">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L</font>
 <b>by </b><i><a class="txt" href="lattice8.html#E2:10_1_1"><i><font color="Green" title="E7">A2</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:10_1_1"/><b>then </b>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L</font> is <a href="realset1.html#V1" title="REALSET1:attr.1">trivial</a>
 
;<br/>
<b>hence </b><a NAME="E7:10_1_1"/>
<font color="Maroon" title="c2">L</font> is <a href="realset2.html#V3" title="REALSET2:attr.3">trivial</a>
 <b>by </b><i><a class="ref" href="realset2.html#D5" title="REALSET2:def.5">REALSET2:def 5</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:10"/>
( <font color="Maroon" title="c2">L</font> is <a href="realset2.html#V3" title="REALSET2:attr.3">trivial</a> or  ex <font color="Olive" title="b1">e</font> being  <a href="eqrel_1.html#NM2" title="EQREL_1:NM.2">Equivalence_Relation</a> of <font color="Maroon" title="c1">A</font> st <br/>( <font color="Olive" title="b1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L</font> &amp; <font color="Olive" title="b1">e</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <font color="Maroon" title="c1">A</font> ) )
 ;<br/>


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