<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">X</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">Y</font> be   <a href="lattice5.html#NM2">Sublattice</a> of  <a href="lattice5.html#K1" target="_self">EqRelLATT</a> <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:26"/><i><font color="Green">E29</font></i>: 
 ex <font color="Olive">e</font> being  <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">X</font> st <br/>( <font color="Olive">e</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font> &amp; <font color="Olive">e</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="cqc_sim1.html#K13">id</a> <font color="Maroon">X</font> )
 <b>and </b><br/><a NAME="E2:26"/><i><font color="Green">E37</font></i>: 
for <font color="Olive">e1</font>, <font color="Olive">e2</font> being  <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">X</font><br/> for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">e1</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font> &amp; <font color="Olive">e2</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Olive">e1</font> <a href="eqrel_1.html#K5">"\/"</a> <font color="Olive">e2</font> holds <br/> ex <font color="Olive">F</font> being non <a href="xboole_0.html#V1">empty</a>  <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">X</font> st <br/>(  <a href="finseq_1.html#K3">len</a> <font color="Olive">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2 &amp; <font color="Olive">x</font>,<font color="Olive">y</font> <a href="lattice5.html#R1" target="_self">are_joint_by</a> <font color="Olive">F</font>,<font color="Olive">e1</font>,<font color="Olive">e2</font> )
 <b>and </b><br/><a NAME="E3:26"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="lattice5.html#K3" target="_self">type_of</a> <font color="Maroon">Y</font>
 ;<br/>

<a NAME="E4:26"/>
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="lattice5.html#K3" target="_self">type_of</a> <font color="Maroon">Y</font>
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<b>then consider </b><font color="Maroon">m</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E6:26"/><i><font color="Green">E42</font></i>: 
 <a href="lattice5.html#K3" target="_self">type_of</a> <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/>
<b>reconsider </b><font color="Maroon">m</font> = <font color="Maroon">m</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E8:26"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font></span>)</span> <a href="nat_1.html#K1">+</a> 2
 
;<br/>
<b>then consider </b><font color="Maroon">e1</font> being   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">X</font>, <font color="Maroon">e2</font> being   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">X</font>, <font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E10:26"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">e1</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font> &amp; <font color="Maroon">e2</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">e1</font> <a href="eqrel_1.html#K5">"\/"</a> <font color="Maroon">e2</font> )
 <b>and </b><br/><a NAME="E11:26"/><i><font color="Green">E49</font></i>: 
for <font color="Olive">F</font> being non <a href="xboole_0.html#V1">empty</a>  <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">X</font>  holds <br/>(  not  <a href="finseq_1.html#K3">len</a> <font color="Olive">F</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font></span>)</span> <a href="nat_1.html#K1">+</a> 2 or  not <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="lattice5.html#R1" target="_self">are_joint_by</a> <font color="Olive">F</font>,<font color="Maroon">e1</font>,<font color="Maroon">e2</font> )
 <b>by </b><i>, , </i>;<br/>
<b>consider </b><font color="Maroon">F1</font> being  non <a href="xboole_0.html#V1">empty</a>  <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E13:26"/><i><font color="Green">E50</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2 &amp; <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="lattice5.html#R1" target="_self">are_joint_by</a> <font color="Maroon">F1</font>,<font color="Maroon">e1</font>,<font color="Maroon">e2</font> )
 <b>by </b><i>, </i>;<br/>
<a NAME="E14:26"/>
(  <a href="relat_1.html#K3">field</a> <font color="Maroon">e1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> &amp;  <a href="relat_1.html#K3">field</a> <font color="Maroon">e2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> )
 
<b>by </b><i><a class="ref" href="eqrel_1.html#T16">EQREL_1:16</a></i>;<br/>
<a NAME="E15:26"/><b>then </b><i><font color="Green">E51</font></i>: 
( <font color="Maroon">e1</font> <a href="relat_2.html#R1">is_reflexive_in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">e2</font> <a href="relat_2.html#R1">is_reflexive_in</a> <font color="Maroon">X</font> )
 
<b>by </b><i><a class="ref" href="relat_2.html#D9">RELAT_2:def 9</a></i>;<br/>
<a NAME="E16:26"/>
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font></span>)</span> <a href="nat_1.html#K1">+</a> 2
 
;<br/>
<a NAME="E17:26"/><b>then </b>
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font></span>)</span> <a href="nat_1.html#K1">+</a> 2
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<b>hence </b><a NAME="E18:26"/>
contradiction
 <b>by </b><i>, , , </i>;<br/>


</div>
