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<b>let </b><font color="Maroon" title="c5">n1</font>, <font color="Maroon" title="c6">n2</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:25_1_2"/><i><font color="Green" title="E18">A29</font></i>: 
for <font color="Olive" title="b1">e1</font>, <font color="Olive" title="b2">e2</font> being  <a href="eqrel_1.html#NM2" title="EQREL_1:NM.2">Equivalence_Relation</a> of <font color="Maroon" title="c1">X</font><br/> for <font color="Olive" title="b3">x</font>, <font color="Olive" title="b4">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">e1</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">Y</font> &amp; <font color="Olive" title="b2">e2</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">Y</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">x</font>,<font color="Olive" title="b4">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">e1</font> <a href="eqrel_1.html#K5" title="EQREL_1:func.5">"\/"</a> <font color="Olive" title="b2">e2</font> holds <br/> ex <font color="Olive" title="b5">F</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">X</font> st <br/>(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b5">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 &amp; <font color="Olive" title="b3">x</font>,<font color="Olive" title="b4">y</font> <a href="lattice5.html#R1" title="LATTICE5:pred.1">are_joint_by</a> <font color="Olive" title="b5">F</font>,<font color="Olive" title="b1">e1</font>,<font color="Olive" title="b2">e2</font> )
 ;<br/>

<b>given </b><font color="Maroon" title="c7">e1'</font>, <font color="Maroon" title="c8">e2'</font> being   <a href="eqrel_1.html#NM2" title="EQREL_1:NM.2">Equivalence_Relation</a> of <font color="Maroon" title="c1">X</font>, <font color="Maroon" title="c9">x'</font>, <font color="Maroon" title="c10">y'</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><a NAME="E3:25_1_2"/><i><font color="Green" title="E19">A30</font></i>: 
( <font color="Maroon" title="c7">e1'</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">Y</font> &amp; <font color="Maroon" title="c8">e2'</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">Y</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">x'</font>,<font color="Maroon" title="c10">y'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">e1'</font> <a href="eqrel_1.html#K5" title="EQREL_1:func.5">"\/"</a> <font color="Maroon" title="c8">e2'</font> )
 <b>and </b><br/><a NAME="E4:25_1_2"/><i><font color="Green" title="E20">A31</font></i>: 
for <font color="Olive" title="b1">F</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">X</font>  holds <br/>(  not  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 or  not <font color="Maroon" title="c9">x'</font>,<font color="Maroon" title="c10">y'</font> <a href="lattice5.html#R1" title="LATTICE5:pred.1">are_joint_by</a> <font color="Olive" title="b1">F</font>,<font color="Maroon" title="c7">e1'</font>,<font color="Maroon" title="c8">e2'</font> )
 ;<br/>

<b>assume </b><a NAME="E5:25_1_2"/><i><font color="Green" title="E21">A32</font></i>: 
for <font color="Olive" title="b1">e1</font>, <font color="Olive" title="b2">e2</font> being  <a href="eqrel_1.html#NM2" title="EQREL_1:NM.2">Equivalence_Relation</a> of <font color="Maroon" title="c1">X</font><br/> for <font color="Olive" title="b3">x</font>, <font color="Olive" title="b4">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">e1</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">Y</font> &amp; <font color="Olive" title="b2">e2</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">Y</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">x</font>,<font color="Olive" title="b4">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">e1</font> <a href="eqrel_1.html#K5" title="EQREL_1:func.5">"\/"</a> <font color="Olive" title="b2">e2</font> holds <br/> ex <font color="Olive" title="b5">F</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">X</font> st <br/>(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b5">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 &amp; <font color="Olive" title="b3">x</font>,<font color="Olive" title="b4">y</font> <a href="lattice5.html#R1" title="LATTICE5:pred.1">are_joint_by</a> <font color="Olive" title="b5">F</font>,<font color="Olive" title="b1">e1</font>,<font color="Olive" title="b2">e2</font> )
 ;<br/>

<b>given </b><font color="Maroon" title="c11">e1</font>, <font color="Maroon" title="c12">e2</font> being   <a href="eqrel_1.html#NM2" title="EQREL_1:NM.2">Equivalence_Relation</a> of <font color="Maroon" title="c1">X</font>, <font color="Maroon" title="c13">x</font>, <font color="Maroon" title="c14">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><a NAME="E7:25_1_2"/><i><font color="Green" title="E22">A33</font></i>: 
( <font color="Maroon" title="c11">e1</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">Y</font> &amp; <font color="Maroon" title="c12">e2</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">Y</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">x</font>,<font color="Maroon" title="c14">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">e1</font> <a href="eqrel_1.html#K5" title="EQREL_1:func.5">"\/"</a> <font color="Maroon" title="c12">e2</font> )
 <b>and </b><br/><a NAME="E8:25_1_2"/><i><font color="Green" title="E23">A34</font></i>: 
for <font color="Olive" title="b1">F</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">X</font>  holds <br/>(  not  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 or  not <font color="Maroon" title="c13">x</font>,<font color="Maroon" title="c14">y</font> <a href="lattice5.html#R1" title="LATTICE5:pred.1">are_joint_by</a> <font color="Olive" title="b1">F</font>,<font color="Maroon" title="c11">e1</font>,<font color="Maroon" title="c12">e2</font> )
 ;<br/>

<b>consider </b><font color="Maroon" title="c15">F1</font> being  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E10:25_1_2"/><i><font color="Green" title="E24">A35</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 &amp; <font color="Maroon" title="c13">x</font>,<font color="Maroon" title="c14">y</font> <a href="lattice5.html#R1" title="LATTICE5:pred.1">are_joint_by</a> <font color="Maroon" title="c15">F1</font>,<font color="Maroon" title="c11">e1</font>,<font color="Maroon" title="c12">e2</font> )
 <b>by </b><i><a class="txt" href="lattice5.html#E1:25_1_2"><i><font color="Green" title="E18">A29</font></i></a>, <a class="txt" href="lattice5.html#E7:25_1_2"><i><font color="Green" title="E22">A33</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c16">F2</font> being  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E12:25_1_2"/><i><font color="Green" title="E25">A36</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">F2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 &amp; <font color="Maroon" title="c9">x'</font>,<font color="Maroon" title="c10">y'</font> <a href="lattice5.html#R1" title="LATTICE5:pred.1">are_joint_by</a> <font color="Maroon" title="c16">F2</font>,<font color="Maroon" title="c7">e1'</font>,<font color="Maroon" title="c8">e2'</font> )
 <b>by </b><i><a class="txt" href="lattice5.html#E3:25_1_2"><i><font color="Green" title="E19">A30</font></i></a>, <a class="txt" href="lattice5.html#E5:25_1_2"><i><font color="Green" title="E21">A32</font></i></a></i>;<br/>
<a NAME="E13:25_1_2"/>
(  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c11">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> &amp;  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c12">e2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> )
 
<b>by </b><i><a class="ref" href="eqrel_1.html#T16" title="EQREL_1:th.16">EQREL_1:16</a></i>;<br/>
<a NAME="E14:25_1_2"/><b>then </b><i><font color="Green" title="E26">A37</font></i>: 
( <font color="Maroon" title="c11">e1</font> <a href="relat_2.html#R1" title="RELAT_2:pred.1">is_reflexive_in</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c12">e2</font> <a href="relat_2.html#R1" title="RELAT_2:pred.1">is_reflexive_in</a> <font color="Maroon" title="c1">X</font> )
 
<b>by </b><i><a class="ref" href="relat_2.html#D9" title="RELAT_2:def.9">RELAT_2:def 9</a></i>;<br/>
<a NAME="E15:25_1_2"/>
(  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c7">e1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> &amp;  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c8">e2'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> )
 
<b>by </b><i><a class="ref" href="eqrel_1.html#T16" title="EQREL_1:th.16">EQREL_1:16</a></i>;<br/>
<a NAME="E16:25_1_2"/><b>then </b><i><font color="Green" title="E27">A38</font></i>: 
( <font color="Maroon" title="c7">e1'</font> <a href="relat_2.html#R1" title="RELAT_2:pred.1">is_reflexive_in</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c8">e2'</font> <a href="relat_2.html#R1" title="RELAT_2:pred.1">is_reflexive_in</a> <font color="Maroon" title="c1">X</font> )
 
<b>by </b><i><a class="ref" href="relat_2.html#D9" title="RELAT_2:def.9">RELAT_2:def 9</a></i>;<br/>
<b>assume </b><a NAME="E17:25_1_2"/><i><font color="Green" title="E28">A39</font></i>: 
not <font color="Maroon" title="c5">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">n2</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:25_1_2_1"/>
( <font color="Maroon" title="c5">n1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">n2</font> or <font color="Maroon" title="c6">n2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">n1</font> )
 </span><b>by </b><i><a class="txt" href="lattice5.html#E17:25_1_2"><i><font color="Green" title="E28">A39</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_2_1_1"><b>suppose </b></a><a NAME="E1:25_1_2_1_1"/>
<font color="Maroon" title="c5">n1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">n2</font>
 ;<br/><div class="add"><a NAME="E2:25_1_2_1_1"/><b>then </b>
<font color="Maroon" title="c5">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E3:25_1_2_1_1"/><b>then </b>
<font color="Maroon" title="c5">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ;<br/><a NAME="E4:25_1_2_1_1"/><b>then </b>
<font color="Maroon" title="c5">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>hence </b><a NAME="E5:25_1_2_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="lattice5.html#E8:25_1_2"><i><font color="Green" title="E23">A34</font></i></a>, <a class="txt" href="lattice5.html#E10:25_1_2"><i><font color="Green" title="E24">A35</font></i></a>, <a class="txt" href="lattice5.html#E14:25_1_2"><i><font color="Green" title="E26">A37</font></i></a>, <a class="ref" href="lattice5.html#T14" target="_self" title="LATTICE5:th.14">Th14</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_2_1_2"><b>suppose </b></a><a NAME="E1:25_1_2_1_2"/>
<font color="Maroon" title="c6">n2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">n1</font>
 ;<br/><div class="add"><a NAME="E2:25_1_2_1_2"/><b>then </b>
<font color="Maroon" title="c6">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E3:25_1_2_1_2"/><b>then </b>
<font color="Maroon" title="c6">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ;<br/><a NAME="E4:25_1_2_1_2"/><b>then </b>
<font color="Maroon" title="c6">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">n1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>hence </b><a NAME="E5:25_1_2_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="lattice5.html#E4:25_1_2"><i><font color="Green" title="E20">A31</font></i></a>, <a class="txt" href="lattice5.html#E12:25_1_2"><i><font color="Green" title="E25">A36</font></i></a>, <a class="txt" href="lattice5.html#E16:25_1_2"><i><font color="Green" title="E27">A38</font></i></a>, <a class="ref" href="lattice5.html#T14" target="_self" title="LATTICE5:th.14">Th14</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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