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<div class="add">

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means 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">a<sub>1</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>set </b><font color="Maroon">c<sub>5</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span> ;<br/>
<a NAME="E1:25_1_1"/><i><font color="Green">E18</font></i>: 
 <a href="relat_1.html#K3">field</a> <font color="Maroon">c<sub>3</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="E2:25_1_1"/><b>then </b>
 <a href="cqc_sim1.html#K13">id</a> <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="relat_2.html#T17">RELAT_2:17</a></i>;<br/>
<a NAME="E3:25_1_1"/><b>then </b>
not <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a>  <a href="cqc_sim1.html#K13">id</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E2:25"><i><font color="Green">E16</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:25_1_1"/><i><font color="Green">E19</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>and </b><br/><a NAME="E6:25_1_1"/><i><font color="Green">E20</font></i>: 
not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="cqc_sim1.html#K13">id</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="relat_1.html#D3">RELAT_1:def 3</a></i>;<br/>
<a NAME="E7:25_1_1"/><i><font color="Green">E21</font></i>: 
(  not <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> or <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>7</sub></font> )
 
<b>by </b><i><a class="txt" href="lattice5.html#E6:25_1_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="relat_1.html#D10">RELAT_1:def 10</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></font> being   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E9:25_1_1"/><i><font color="Green">E22</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/>
<a NAME="E10:25_1_1"/><i><font color="Green">E23</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> <a href="eqrel_1.html#K5">"\/"</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E5:25_1_1"><i><font color="Green">E19</font></i></a>, <a class="txt" href="lattice5.html#E9:25_1_1"><i><font color="Green">E22</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>10</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="E12:25_1_1"/><i><font color="Green">E24</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>and </b><br/><a NAME="E13:25_1_1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="lattice5.html#R1" target="_self">are_joint_by</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="lattice5.html#E1:25"><i><font color="Green">E15</font></i></a>, <a class="txt" href="lattice5.html#E3:25"><i><font color="Green">E17</font></i></a>, <a class="txt" href="lattice5.html#E9:25_1_1"><i><font color="Green">E22</font></i></a></i>;<br/>
<a NAME="E14:25_1_1"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>10</sub></font> holds <br/>( ( not <font color="Olive">b<sub>1</sub></font> is <a href="abian.html#V1">even</a> implies <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> ) &amp; ( <font color="Olive">b<sub>1</sub></font> is <a href="abian.html#V1">even</a> implies <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> ) ) ) )
 
<b>by </b><i><a class="txt" href="lattice5.html#E13:25_1_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="lattice5.html#D3" target="_self">Def3</a></i>;<br/>
<a NAME="E15:25_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:25_1_1_1"/><i><font color="Green">E27</font></i>: 
not <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2
 ;<br/>

<a NAME="E2:25_1_1_1"/><i><font color="Green">E28</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T18">NAT_1:18</a></i>;<br/>
<a NAME="E3:25_1_1_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="txt" href="lattice5.html#E12:25_1_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="lattice5.html#E1:25_1_1_1"><i><font color="Green">E27</font></i></a></i>;<br/>
<a NAME="E4:25_1_1_1"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 0 <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/>
<a NAME="E5:25_1_1_1"/><b>then </b>
( 0 <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>10</sub></font> or  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> 1 )
 
<b>by </b><i><a class="txt" href="lattice5.html#E2:25_1_1_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="nat_1.html#T27">NAT_1:27</a></i>;<br/>
<b>hence </b><a NAME="E6:25_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="lattice5.html#E1:25_1_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="lattice5.html#E5:25_1_1"><i><font color="Green">E19</font></i></a>, <a class="txt" href="lattice5.html#E7:25_1_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="lattice5.html#E14:25_1_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a>, <a class="ref" href="relat_1.html#T30">RELAT_1:30</a></i>;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon">c<sub>11</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E17:25_1_1"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 2 <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>12</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E19:25_1_1"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; ( 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>12</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> ) ) )
 <b>by </b><i><a class="txt" href="lattice5.html#E3:25"><i><font color="Green">E17</font></i></a>, <a class="txt" href="lattice5.html#E17:25_1_1"><i><font color="Green">E27</font></i></a></i>;<br/>
<a NAME="E20:25_1_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span> 
 
<b>by </b><i><a class="txt" href="lattice5.html#E19:25_1_1"><i><font color="Green">E28</font></i></a></i>;<br/>
<a NAME="E21:25_1_1"/>
<span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span>  is   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K5">NAT</a> 
 
<b>from </b><i><a class="ref" href="domain_1.html#S7">DOMAIN_1:sch 7</a>();<br/></i>
<b>then reconsider </b><font color="Maroon">c<sub>13</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span>  as  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="txt" href="lattice5.html#E20:25_1_1"><i><font color="Green">E29</font></i></a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>14</sub></font> =  <a href="cqc_sim1.html#K10">min</a> <font color="Maroon">c<sub>13</sub></font>;<br/>
<a NAME="E23:25_1_1"/>
 <a href="cqc_sim1.html#K10">min</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>13</sub></font>
 
<b>by </b><i><a class="ref" href="cqc_sim1.html#D8">CQC_SIM1:def 8</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>15</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E25:25_1_1"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a>  <a href="cqc_sim1.html#K10">min</a> <font color="Maroon">c<sub>13</sub></font> &amp; ( 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>15</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/>
<a NAME="E26:25_1_1"/>
ex <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>ex <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 <br/>( <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> &amp; ( for <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> holds <br/>(  not  <a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="cqc_sim1.html#K10">min</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1 or  not <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> ) ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:25_1_1_2"/><i><font color="Green">E31</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>  holds <br/>(  not <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> or  not <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> or  not <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> or 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> <span class="p1">(<span class="default"><a href="cqc_sim1.html#K10">min</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1 &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>then 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="E3:25_1_1_2"/><i><font color="Green">E32</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="cqc_sim1.html#K10">min</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</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>8</sub></font>,<font color="Maroon">c<sub>9</sub></font> )
 <b>by </b><i><a class="txt" href="lattice5.html#E1:25"><i><font color="Green">E15</font></i></a>, <a class="txt" href="lattice5.html#E9:25_1_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="lattice5.html#E10:25_1_1"><i><font color="Green">E23</font></i></a></i>;<br/>
<a NAME="E4:25_1_1_2"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">c<sub>16</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> )
 
<b>by </b><i><a class="txt" href="lattice5.html#E3:25_1_1_2"><i><font color="Green">E32</font></i></a>, <a class="ref" href="lattice5.html#D3" target="_self">Def3</a></i>;<br/>
<a NAME="E5:25_1_1_2"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>16</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:25_1_1_2_1"/><i><font color="Green">E34</font></i>: 
not  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>16</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2
 ;<br/>

<a NAME="E2:25_1_1_2_1"/><i><font color="Green">E35</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>16</sub></font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T18">NAT_1:18</a></i>;<br/>
<a NAME="E3:25_1_1_2_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>16</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="txt" href="lattice5.html#E1:25_1_1_2_1"><i><font color="Green">E34</font></i></a></i>;<br/>
<a NAME="E4:25_1_1_2_1"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>16</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 0 <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/>
<a NAME="E5:25_1_1_2_1"/><b>then </b>
( 0 <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>16</sub></font> or  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> 1 )
 
<b>by </b><i><a class="txt" href="lattice5.html#E2:25_1_1_2_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="nat_1.html#T27">NAT_1:27</a></i>;<br/>
<b>hence </b><a NAME="E6:25_1_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="lattice5.html#E1:25_1_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="lattice5.html#E5:25_1_1"><i><font color="Green">E19</font></i></a>, <a class="txt" href="lattice5.html#E7:25_1_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="lattice5.html#E4:25_1_1_2"><i><font color="Green">E33</font></i></a>, <a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a>, <a class="ref" href="relat_1.html#T30">RELAT_1:30</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:25_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="cqc_sim1.html#K10">min</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="txt" href="lattice5.html#E3:25_1_1_2"><i><font color="Green">E32</font></i></a></i>;<br/>
<a NAME="E7:25_1_1_2"/><b>then </b><i><font color="Green">E34</font></i>: 
 <a href="cqc_sim1.html#K10">min</a> <font color="Maroon">c<sub>13</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E8:25_1_1_2"/><b>then </b><i><font color="Green">E35</font></i>: 
<span class="p1">(<span class="default"><a href="cqc_sim1.html#K10">min</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/>
<a NAME="E9:25_1_1_2"/>
 <a href="cqc_sim1.html#K10">min</a> <font color="Maroon">c<sub>13</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="cqc_sim1.html#K10">min</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/>
<a NAME="E10:25_1_1_2"/><b>then </b><i><font color="Green">E36</font></i>: 
<span class="p1">(<span class="default"><a href="cqc_sim1.html#K10">min</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cqc_sim1.html#K10">min</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<a NAME="E11:25_1_1_2"/><i><font color="Green">E37</font></i>: 
<span class="p1">(<span class="default"><a href="cqc_sim1.html#K10">min</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cqc_sim1.html#K10">min</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 2
 
;<br/>
<a NAME="E12:25_1_1_2"/>
<span class="p1">(<span class="default"><a href="cqc_sim1.html#K10">min</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="cqc_sim1.html#K10">min</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1
 
<b>by </b><i><a class="txt" href="lattice5.html#E7:25_1_1_2"><i><font color="Green">E34</font></i></a>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/>
<a NAME="E13:25_1_1_2"/><b>then </b><i><font color="Green">E38</font></i>: 
<span class="p1">(<span class="default"><a href="cqc_sim1.html#K10">min</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1 <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>13</sub></font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E1:25_1_1_2"><i><font color="Green">E31</font></i></a>, <a class="txt" href="lattice5.html#E11:25_1_1_2"><i><font color="Green">E37</font></i></a></i>;<br/>
<a NAME="E14:25_1_1_2"/>
<span class="p1">(<span class="default"><a href="cqc_sim1.html#K10">min</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="cqc_sim1.html#K10">min</a> <font color="Maroon">c<sub>13</sub></font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E8:25_1_1_2"><i><font color="Green">E35</font></i></a>, <a class="txt" href="lattice5.html#E10:25_1_1_2"><i><font color="Green">E36</font></i></a>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/>
<b>hence </b><a NAME="E15:25_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="lattice5.html#E13:25_1_1_2"><i><font color="Green">E38</font></i></a>, <a class="ref" href="cqc_sim1.html#D8">CQC_SIM1:def 8</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E27:25_1_1"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( ( for <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</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>4</sub></font>, <font color="Olive">b<sub>5</sub></font> being   <a href="hidden.html#M1">set</a>  st <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; <font color="Olive">b<sub>3</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>4</sub></font>,<font color="Olive">b<sub>5</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>2</sub></font> <a href="eqrel_1.html#K5">"\/"</a> <font color="Olive">b<sub>3</sub></font> holds <br/>ex <font color="Olive">b<sub>6</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>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 2 &amp; <font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>5</sub></font> <a href="lattice5.html#R1" target="_self">are_joint_by</a> <font color="Olive">b<sub>6</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font> ) ) &amp; ex <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being  <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">c<sub>1</sub></font>ex <font color="Olive">b<sub>4</sub></font>, <font color="Olive">b<sub>5</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <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; <font color="Olive">b<sub>3</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>4</sub></font>,<font color="Olive">b<sub>5</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>2</sub></font> <a href="eqrel_1.html#K5">"\/"</a> <font color="Olive">b<sub>3</sub></font> &amp; ( for <font color="Olive">b<sub>6</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>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1 or  not <font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>5</sub></font> <a href="lattice5.html#R1" target="_self">are_joint_by</a> <font color="Olive">b<sub>6</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font> ) ) ) )
 <b>by </b><i><a class="txt" href="lattice5.html#E25:25_1_1"><i><font color="Green">E30</font></i></a></i>;<br/>


</div>
