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<b>let </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:25_1_2"/><i><font color="Green">E18</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">c<sub>1</sub></font><br/>for <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>4</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> <a href="eqrel_1.html#K5">"\/"</a> <font color="Olive">b<sub>2</sub></font> holds <br/>ex <font color="Olive">b<sub>5</sub></font> being non <a href="xboole_0.html#V1">empty</a>  <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>(  <a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 2 &amp; <font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>4</sub></font> <a href="lattice5.html#R1" target="_self">are_joint_by</a> <font color="Olive">b<sub>5</sub></font>,<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> )
 ;<br/>

<b>given </b><font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font> being   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>10</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><a NAME="E2:25_1_2"/><i><font color="Green">E19</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> <a href="eqrel_1.html#K5">"\/"</a> <font color="Maroon">c<sub>8</sub></font> )
 <b>and </b><br/><a NAME="E3:25_1_2"/><i><font color="Green">E20</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being non <a href="xboole_0.html#V1">empty</a>  <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/>(  not  <a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1 or  not <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="lattice5.html#R1" target="_self">are_joint_by</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font> )
 ;<br/>

<b>assume </b><a NAME="E4:25_1_2"/><i><font color="Green">E21</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">c<sub>1</sub></font><br/>for <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>4</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> <a href="eqrel_1.html#K5">"\/"</a> <font color="Olive">b<sub>2</sub></font> holds <br/>ex <font color="Olive">b<sub>5</sub></font> being non <a href="xboole_0.html#V1">empty</a>  <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>(  <a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 2 &amp; <font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>4</sub></font> <a href="lattice5.html#R1" target="_self">are_joint_by</a> <font color="Olive">b<sub>5</sub></font>,<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> )
 ;<br/>

<b>given </b><font color="Maroon">c<sub>11</sub></font>, <font color="Maroon">c<sub>12</sub></font> being   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>13</sub></font>, <font color="Maroon">c<sub>14</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><a NAME="E5:25_1_2"/><i><font color="Green">E22</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font> <a href="eqrel_1.html#K5">"\/"</a> <font color="Maroon">c<sub>12</sub></font> )
 <b>and </b><br/><a NAME="E6:25_1_2"/><i><font color="Green">E23</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being non <a href="xboole_0.html#V1">empty</a>  <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/>(  not  <a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1 or  not <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font> <a href="lattice5.html#R1" target="_self">are_joint_by</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font> )
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>15</sub></font> being  non <a href="xboole_0.html#V1">empty</a>  <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E8:25_1_2"/><i><font color="Green">E24</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 2 &amp; <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font> <a href="lattice5.html#R1" target="_self">are_joint_by</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font> )
 <b>by </b><i><a class="txt" href="lattice5.html#E1:25_1_2"><i><font color="Green">E18</font></i></a>, <a class="txt" href="lattice5.html#E5:25_1_2"><i><font color="Green">E22</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>16</sub></font> being  non <a href="xboole_0.html#V1">empty</a>  <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E10:25_1_2"/><i><font color="Green">E25</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 2 &amp; <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="lattice5.html#R1" target="_self">are_joint_by</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font> )
 <b>by </b><i><a class="txt" href="lattice5.html#E2:25_1_2"><i><font color="Green">E19</font></i></a>, <a class="txt" href="lattice5.html#E4:25_1_2"><i><font color="Green">E21</font></i></a></i>;<br/>
<a NAME="E11:25_1_2"/>
(  <a href="relat_1.html#K3">field</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="relat_1.html#K3">field</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="ref" href="eqrel_1.html#T16">EQREL_1:16</a></i>;<br/>
<a NAME="E12:25_1_2"/><b>then </b><i><font color="Green">E26</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="relat_2.html#R1">is_reflexive_in</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="relat_2.html#R1">is_reflexive_in</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="ref" href="relat_2.html#D9">RELAT_2:def 9</a></i>;<br/>
<a NAME="E13:25_1_2"/>
(  <a href="relat_1.html#K3">field</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="relat_1.html#K3">field</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="ref" href="eqrel_1.html#T16">EQREL_1:16</a></i>;<br/>
<a NAME="E14:25_1_2"/><b>then </b><i><font color="Green">E27</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="relat_2.html#R1">is_reflexive_in</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="relat_2.html#R1">is_reflexive_in</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="ref" href="relat_2.html#D9">RELAT_2:def 9</a></i>;<br/>
<b>assume </b><a NAME="E15:25_1_2"/><i><font color="Green">E28</font></i>: 
not <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></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">c<sub>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font> or <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font> )
 </span><b>by </b><i><a class="txt" href="lattice5.html#E15:25_1_2"><i><font color="Green">E28</font></i></a>, <a class="ref" href="real_1.html#D5">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">c<sub>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><div class="add"><a NAME="E2:25_1_2_1_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E3:25_1_2_1_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1
 ;<br/><a NAME="E4:25_1_2_1_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</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#E6:25_1_2"><i><font color="Green">E23</font></i></a>, <a class="txt" href="lattice5.html#E8:25_1_2"><i><font color="Green">E24</font></i></a>, <a class="txt" href="lattice5.html#E12:25_1_2"><i><font color="Green">E26</font></i></a>, <a class="ref" href="lattice5.html#T14" target="_self">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">c<sub>6</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><div class="add"><a NAME="E2:25_1_2_1_2"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E3:25_1_2_1_2"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1
 ;<br/><a NAME="E4:25_1_2_1_2"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</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#E3:25_1_2"><i><font color="Green">E20</font></i></a>, <a class="txt" href="lattice5.html#E10:25_1_2"><i><font color="Green">E25</font></i></a>, <a class="txt" href="lattice5.html#E14:25_1_2"><i><font color="Green">E27</font></i></a>, <a class="ref" href="lattice5.html#T14" target="_self">Th14</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
