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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="yellow_0.html#V1">lower-bounded</a> <a href="waybel_0.html#NM8">LATTICE</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="lattice5.html#M2" target="_self">ExtensionSeq</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>, <a href="lattice5.html#K17" target="_self">BasicDF</a> <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></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">c<sub>4</sub></font> be   <a href="lattice5.html#NM4">distance_function</a> of <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="lattice5.html#NM1">Homomorphism</a> of <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="lattice5.html#K1" target="_self">EqRelLATT</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>;<br/><b>let </b><font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>3</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></font> be   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">c<sub>3</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>10</sub></font>, <font color="Maroon">c<sub>11</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

















<b>assume </b><b>that </b><br/><a NAME="E1:82"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#R2">=</a>  <a href="lattice5.html#K5" target="_self">alpha</a> <font color="Maroon">c<sub>4</sub></font>
 <b>and </b><br/><a NAME="E2:82"/><i><font color="Green">E65</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> : verum </span>}</span>  &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> : verum </span>}</span>  )
 <b>and </b><br/><a NAME="E3:82"/><i><font color="Green">E66</font></i>: 
( <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 <span class="p1">(<span class="default"><a href="lattice5.html#K2" target="_self">Image</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="lattice5.html#K2" target="_self">Image</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> )
 <b>and </b><br/><a NAME="E4:82"/><i><font color="Green">E67</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</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>
 ;<br/>

<a NAME="E5:82"/>
 <a href="lattice5.html#K2" target="_self">Image</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K5">subrelstr</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_2.html#D2">YELLOW_2:def 2</a></i>;<br/>
<a NAME="E6:82"/><b>then </b><i><font color="Green">E68</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="lattice5.html#K2" target="_self">Image</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>12</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E8:82"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font>
 <b>and </b><br/><a NAME="E9:82"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="lattice5.html#E3:82"><i><font color="Green">E66</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>13</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E11:82"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font>
 <b>and </b><br/><a NAME="E12:82"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="lattice5.html#E3:82"><i><font color="Green">E66</font></i></a>, <a class="txt" href="lattice5.html#E6:82"><i><font color="Green">E68</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>12</sub></font>, <font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>13</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="lattice5.html#E8:82"><i><font color="Green">E69</font></i></a>, <a class="txt" href="lattice5.html#E11:82"><i><font color="Green">E71</font></i></a></i>;<br/>
<a NAME="E14:82"/>
 <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="eqrel_1.html#K5">"\/"</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="orders_1.html#T97">ORDERS_1:97</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>10</sub></font>, <font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>11</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="txt" href="lattice5.html#E4:82"><i><font color="Green">E67</font></i></a>, <a class="ref" href="relat_1.html#T30">RELAT_1:30</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>18</sub></font> = <font color="Maroon">c<sub>14</sub></font>, <font color="Maroon">c<sub>19</sub></font> = <font color="Maroon">c<sub>15</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<b>consider </b><font color="Maroon">c<sub>20</sub></font> being   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">c<sub>3</sub></font><b> such that </b><br/><a NAME="E18:82"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>18</sub></font>
 <b>and </b><br/><a NAME="E19:82"/><i><font color="Green">E74</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>3</sub></font> holds <br/>( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>20</sub></font> iff <font color="Maroon">c<sub>4</sub></font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>18</sub></font> )
 <b>by </b><i><a class="txt" href="lattice5.html#E1:82"><i><font color="Green">E64</font></i></a>, <a class="ref" href="lattice5.html#D9" target="_self">Def9</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>21</sub></font> being   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">c<sub>3</sub></font><b> such that </b><br/><a NAME="E21:82"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>19</sub></font>
 <b>and </b><br/><a NAME="E22:82"/><i><font color="Green">E76</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>3</sub></font> holds <br/>( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>21</sub></font> iff <font color="Maroon">c<sub>4</sub></font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>19</sub></font> )
 <b>by </b><i><a class="txt" href="lattice5.html#E1:82"><i><font color="Green">E64</font></i></a>, <a class="ref" href="lattice5.html#D9" target="_self">Def9</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>22</sub></font> being   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">c<sub>3</sub></font><b> such that </b><br/><a NAME="E24:82"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>19</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E25:82"/><i><font color="Green">E78</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>3</sub></font> holds <br/>( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>22</sub></font> iff <font color="Maroon">c<sub>4</sub></font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>18</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>19</sub></font> )
 <b>by </b><i><a class="txt" href="lattice5.html#E1:82"><i><font color="Green">E64</font></i></a>, <a class="ref" href="lattice5.html#D9" target="_self">Def9</a></i>;<br/>
<font color="Maroon">c<sub>22</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="lattice5.html#E24:82"><i><font color="Green">E77</font></i></a>, <a class="ref" href="waybel_6.html#T2">WAYBEL_6:2</a></i>
<br/>.= 
<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#E9:82"><i><font color="Green">E70</font></i></a>, <a class="txt" href="lattice5.html#E12:82"><i><font color="Green">E72</font></i></a>, <a class="ref" href="lattice5.html#T10" target="_self">Th10</a></i>
;<br/>
<a NAME="E27:82"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>18</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>19</sub></font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E4:82"><i><font color="Green">E67</font></i></a>, <a class="txt" href="lattice5.html#E25:82"><i><font color="Green">E78</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>23</sub></font>, <font color="Maroon">c<sub>24</sub></font>, <font color="Maroon">c<sub>25</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>3</sub></font><b> such that </b><br/><a NAME="E29:82"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font>
 <b>and </b><br/><a NAME="E30:82"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">c<sub>24</sub></font>,<font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font>
 <b>and </b><br/><a NAME="E31:82"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font>
 <b>and </b><br/><a NAME="E32:82"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">c<sub>25</sub></font>,<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font>
 <b>by </b><i><a class="txt" href="lattice5.html#E2:82"><i><font color="Green">E65</font></i></a>, <a class="ref" href="lattice5.html#T45" target="_self">Th45</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] means ( ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> 1 implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> ) &amp; ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> 2 implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> ) &amp; ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> 3 implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>24</sub></font> ) &amp; ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> 4 implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>25</sub></font> ) &amp; ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> 5 implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> ) );<br/>
<a NAME="E33:82"/><i><font color="Green">E83</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a><br/>for <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</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>  <a href="finseq_1.html#K2">Seg</a> 5 &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>3</sub></font>] holds <br/><font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E68"><i><font color="Green">Lemma59</font></i></a></i>;<br/>
<a NAME="E34:82"/><i><font color="Green">E84</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 5 holds <br/>ex <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</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="82_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>26</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:82_2"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 5
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="82_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:82_2_1"/>
( <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> 2 or <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> 3 or <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> 4 or <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> 5 )
 </span><b>by </b><i><a class="txt" href="lattice5.html#E1:82_2"><i><font color="Green">E85</font></i></a>, <a class="txt" href="lattice5.html#E68"><i><font color="Green">Lemma59</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="82_2_1_1"><b>suppose </b></a><a NAME="E1:82_2_1_1"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><b>take </b><font color="Maroon">c<sub>27</sub></font> = <font color="Maroon">c<sub>10</sub></font>;<br/><b>thus </b><a NAME="E2:82_2_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>26</sub></font>,<font color="Maroon">c<sub>27</sub></font>]
 <b>by </b><i><a class="txt" href="lattice5.html#E1:82_2_1_1"><i><font color="Green">E86</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="82_2_1_2"><b>suppose </b></a><a NAME="E1:82_2_1_2"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> 2
 ;<br/><div class="add"><b>take </b><font color="Maroon">c<sub>27</sub></font> = <font color="Maroon">c<sub>23</sub></font>;<br/><b>thus </b><a NAME="E2:82_2_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>26</sub></font>,<font color="Maroon">c<sub>27</sub></font>]
 <b>by </b><i><a class="txt" href="lattice5.html#E1:82_2_1_2"><i><font color="Green">E86</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="82_2_1_3"><b>suppose </b></a><a NAME="E1:82_2_1_3"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> 3
 ;<br/><div class="add"><b>take </b><font color="Maroon">c<sub>27</sub></font> = <font color="Maroon">c<sub>24</sub></font>;<br/><b>thus </b><a NAME="E2:82_2_1_3"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>26</sub></font>,<font color="Maroon">c<sub>27</sub></font>]
 <b>by </b><i><a class="txt" href="lattice5.html#E1:82_2_1_3"><i><font color="Green">E86</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="82_2_1_4"><b>suppose </b></a><a NAME="E1:82_2_1_4"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> 4
 ;<br/><div class="add"><b>take </b><font color="Maroon">c<sub>27</sub></font> = <font color="Maroon">c<sub>25</sub></font>;<br/><b>thus </b><a NAME="E2:82_2_1_4"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>26</sub></font>,<font color="Maroon">c<sub>27</sub></font>]
 <b>by </b><i><a class="txt" href="lattice5.html#E1:82_2_1_4"><i><font color="Green">E86</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="82_2_1_5"><b>suppose </b></a><a NAME="E1:82_2_1_5"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> 5
 ;<br/><div class="add"><b>take </b><font color="Maroon">c<sub>27</sub></font> = <font color="Maroon">c<sub>11</sub></font>;<br/><b>thus </b><a NAME="E2:82_2_1_5"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>26</sub></font>,<font color="Maroon">c<sub>27</sub></font>]
 <b>by </b><i><a class="txt" href="lattice5.html#E1:82_2_1_5"><i><font color="Green">E86</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E35:82"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="finseq_1.html#NM1">FinSequence</a> st <br/>(  <a href="finsop_1.html#K5">dom</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> 5 &amp; ( for <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 5 holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font>] ) )
 
<b>from </b><i><a class="ref" href="finseq_1.html#S1">FINSEQ_1:sch 1</a>(<a class="txt" href="lattice5.html#E33:82"><i><font color="Green">E83</font></i></a>, <a class="txt" href="lattice5.html#E34:82"><i><font color="Green">E84</font></i></a>);<br/></i>
<b>then consider </b><font color="Maroon">c<sub>26</sub></font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E37:82"/><i><font color="Green">E85</font></i>: 
 <a href="finsop_1.html#K5">dom</a> <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> 5
 <b>and </b><br/><a NAME="E38:82"/><i><font color="Green">E86</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 5 holds <br/>( ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 1 implies <font color="Maroon">c<sub>26</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> ) &amp; ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 2 implies <font color="Maroon">c<sub>26</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> ) &amp; ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 3 implies <font color="Maroon">c<sub>26</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>24</sub></font> ) &amp; ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 4 implies <font color="Maroon">c<sub>26</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>25</sub></font> ) &amp; ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 5 implies <font color="Maroon">c<sub>26</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> ) )
 ;<br/>
<a NAME="E39:82"/><i><font color="Green">E87</font></i>: 
( 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 5 &amp; 2 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 5 &amp; 3 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 5 &amp; 4 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 5 &amp; 5 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 5 )
 
;<br/>
<a NAME="E40:82"/><i><font color="Green">E88</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> 5
 
<b>by </b><i><a class="txt" href="lattice5.html#E37:82"><i><font color="Green">E85</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E41:82"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>26</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="82_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>27</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:82_3"/>
<font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>26</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>28</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:82_3"/><i><font color="Green">E89</font></i>: 
( <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">c<sub>26</sub></font> &amp; <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>26</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>28</sub></font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="82_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:82_3_1"/>
( <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R1">=</a> 2 or <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R1">=</a> 3 or <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R1">=</a> 4 or <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R1">=</a> 5 )
 </span><b>by </b><i><a class="txt" href="lattice5.html#E37:82"><i><font color="Green">E85</font></i></a>, <a class="txt" href="lattice5.html#E3:82_3"><i><font color="Green">E89</font></i></a>, <a class="txt" href="lattice5.html#E68"><i><font color="Green">Lemma59</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="82_3_1_1"><b>suppose </b></a><a NAME="E1:82_3_1_1"/>
<font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:82_3_1_1"/><b>then </b>
<font color="Maroon">c<sub>26</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="txt" href="lattice5.html#E38:82"><i><font color="Green">E86</font></i></a>, <a class="txt" href="lattice5.html#E39:82"><i><font color="Green">E87</font></i></a></i>;<br/><b>hence </b><a NAME="E3:82_3_1_1"/>
<font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="lattice5.html#E3:82_3"><i><font color="Green">E89</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="82_3_1_2"><b>suppose </b></a><a NAME="E1:82_3_1_2"/>
<font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R1">=</a> 2
 ;<br/><div class="add"><a NAME="E2:82_3_1_2"/><b>then </b>
<font color="Maroon">c<sub>26</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font>
 <b>by </b><i><a class="txt" href="lattice5.html#E38:82"><i><font color="Green">E86</font></i></a>, <a class="txt" href="lattice5.html#E39:82"><i><font color="Green">E87</font></i></a></i>;<br/><b>hence </b><a NAME="E3:82_3_1_2"/>
<font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="lattice5.html#E3:82_3"><i><font color="Green">E89</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="82_3_1_3"><b>suppose </b></a><a NAME="E1:82_3_1_3"/>
<font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R1">=</a> 3
 ;<br/><div class="add"><a NAME="E2:82_3_1_3"/><b>then </b>
<font color="Maroon">c<sub>26</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>24</sub></font>
 <b>by </b><i><a class="txt" href="lattice5.html#E38:82"><i><font color="Green">E86</font></i></a>, <a class="txt" href="lattice5.html#E39:82"><i><font color="Green">E87</font></i></a></i>;<br/><b>hence </b><a NAME="E3:82_3_1_3"/>
<font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="lattice5.html#E3:82_3"><i><font color="Green">E89</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="82_3_1_4"><b>suppose </b></a><a NAME="E1:82_3_1_4"/>
<font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R1">=</a> 4
 ;<br/><div class="add"><a NAME="E2:82_3_1_4"/><b>then </b>
<font color="Maroon">c<sub>26</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>25</sub></font>
 <b>by </b><i><a class="txt" href="lattice5.html#E38:82"><i><font color="Green">E86</font></i></a>, <a class="txt" href="lattice5.html#E39:82"><i><font color="Green">E87</font></i></a></i>;<br/><b>hence </b><a NAME="E3:82_3_1_4"/>
<font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="lattice5.html#E3:82_3"><i><font color="Green">E89</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="82_3_1_5"><b>suppose </b></a><a NAME="E1:82_3_1_5"/>
<font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R1">=</a> 5
 ;<br/><div class="add"><a NAME="E2:82_3_1_5"/><b>then </b>
<font color="Maroon">c<sub>26</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="txt" href="lattice5.html#E38:82"><i><font color="Green">E86</font></i></a>, <a class="txt" href="lattice5.html#E39:82"><i><font color="Green">E87</font></i></a></i>;<br/><b>hence </b><a NAME="E3:82_3_1_5"/>
<font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="lattice5.html#E3:82_3"><i><font color="Green">E89</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>27</sub></font> = <font color="Maroon">c<sub>26</sub></font> as    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<a NAME="E43:82"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="txt" href="lattice5.html#E37:82"><i><font color="Green">E85</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>28</sub></font> = <font color="Maroon">c<sub>27</sub></font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>28</sub></font>
;<br/>

<b>thus </b><a NAME="E45:82"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R1">=</a> 3 <a href="nat_1.html#K1">+</a> 2
 <b>by </b><i><a class="txt" href="lattice5.html#E37:82"><i><font color="Green">E85</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>

<a NAME="E46:82"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">c<sub>28</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E38:82"><i><font color="Green">E86</font></i></a>, <a class="txt" href="lattice5.html#E39:82"><i><font color="Green">E87</font></i></a></i>;<br/>
<i><font color="Green">E90</font></i>: <font color="Maroon">c<sub>28</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>28</sub></font></span>)</span> = 
<font color="Maroon">c<sub>28</sub></font> <a href="funct_1.html#K1">.</a> 5
<b>by </b><i><a class="txt" href="lattice5.html#E37:82"><i><font color="Green">E85</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
<br/>.= 
<font color="Maroon">c<sub>11</sub></font>
<b>by </b><i><a class="txt" href="lattice5.html#E38:82"><i><font color="Green">E86</font></i></a>, <a class="txt" href="lattice5.html#E39:82"><i><font color="Green">E87</font></i></a></i>
;<br/>
<a NAME="E48:82"/>
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>28</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>28</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>28</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>28</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>28</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> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="82_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>29</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:82_5"/><i><font color="Green">E91</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>29</sub></font> &amp; <font color="Maroon">c<sub>29</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>28</sub></font> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="82_5_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:82_5_1"/>
( <font color="Maroon">c<sub>29</sub></font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">c<sub>29</sub></font> <a href="hidden.html#R1">=</a> 3 or <font color="Maroon">c<sub>29</sub></font> <a href="hidden.html#R1">=</a> 2 or <font color="Maroon">c<sub>29</sub></font> <a href="hidden.html#R1">=</a> 4 )
 </span><b>by </b><i><a class="txt" href="lattice5.html#E40:82"><i><font color="Green">E88</font></i></a>, <a class="txt" href="lattice5.html#E1:82_5"><i><font color="Green">E91</font></i></a>, <a class="txt" href="lattice5.html#E67"><i><font color="Green">Lemma58</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="82_5_1_1"><b>suppose </b></a><a NAME="E1:82_5_1_1"/>
<font color="Maroon">c<sub>29</sub></font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:82_5_1_1"/><b>then </b><i><font color="Green">E92</font></i>: 
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> 0</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>29</sub></font>
 ;<br/><a NAME="E3:82_5_1_1"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>23</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>20</sub></font>
 <b>by </b><i><a class="txt" href="lattice5.html#E19:82"><i><font color="Green">E74</font></i></a>, <a class="txt" href="lattice5.html#E29:82"><i><font color="Green">E79</font></i></a></i>;<br/><a NAME="E4:82_5_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>28</sub></font> <a href="funct_1.html#K1">.</a> 1</span>)</span>,<font color="Maroon">c<sub>23</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>20</sub></font>
 <b>by </b><i><a class="txt" href="lattice5.html#E38:82"><i><font color="Green">E86</font></i></a>, <a class="txt" href="lattice5.html#E39:82"><i><font color="Green">E87</font></i></a></i>;<br/><b>hence </b><a NAME="E5:82_5_1_1"/>
( ( not <font color="Maroon">c<sub>29</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>28</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>29</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>28</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>29</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="Maroon">c<sub>29</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>28</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>29</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>28</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>29</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#E9:82"><i><font color="Green">E70</font></i></a>, <a class="txt" href="lattice5.html#E18:82"><i><font color="Green">E73</font></i></a>, <a class="txt" href="lattice5.html#E38:82"><i><font color="Green">E86</font></i></a>, <a class="txt" href="lattice5.html#E39:82"><i><font color="Green">E87</font></i></a>, <a class="txt" href="lattice5.html#E2:82_5_1_1"><i><font color="Green">E92</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="82_5_1_2"><b>suppose </b></a><a NAME="E1:82_5_1_2"/>
<font color="Maroon">c<sub>29</sub></font> <a href="hidden.html#R1">=</a> 3
 ;<br/><div class="add"><a NAME="E2:82_5_1_2"/><b>then </b><i><font color="Green">E92</font></i>: 
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>29</sub></font>
 ;<br/><a NAME="E3:82_5_1_2"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>24</sub></font>,<font color="Maroon">c<sub>25</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>20</sub></font>
 <b>by </b><i><a class="txt" href="lattice5.html#E19:82"><i><font color="Green">E74</font></i></a>, <a class="txt" href="lattice5.html#E30:82"><i><font color="Green">E80</font></i></a></i>;<br/><a NAME="E4:82_5_1_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>28</sub></font> <a href="funct_1.html#K1">.</a> 3</span>)</span>,<font color="Maroon">c<sub>25</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>20</sub></font>
 <b>by </b><i><a class="txt" href="lattice5.html#E38:82"><i><font color="Green">E86</font></i></a>, <a class="txt" href="lattice5.html#E39:82"><i><font color="Green">E87</font></i></a></i>;<br/><b>hence </b><a NAME="E5:82_5_1_2"/>
( ( not <font color="Maroon">c<sub>29</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>28</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>29</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>28</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>29</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="Maroon">c<sub>29</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>28</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>29</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>28</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>29</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#E9:82"><i><font color="Green">E70</font></i></a>, <a class="txt" href="lattice5.html#E18:82"><i><font color="Green">E73</font></i></a>, <a class="txt" href="lattice5.html#E38:82"><i><font color="Green">E86</font></i></a>, <a class="txt" href="lattice5.html#E39:82"><i><font color="Green">E87</font></i></a>, <a class="txt" href="lattice5.html#E2:82_5_1_2"><i><font color="Green">E92</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="82_5_1_3"><b>suppose </b></a><a NAME="E1:82_5_1_3"/>
<font color="Maroon">c<sub>29</sub></font> <a href="hidden.html#R1">=</a> 2
 ;<br/><div class="add"><a NAME="E2:82_5_1_3"/><b>then </b><i><font color="Green">E92</font></i>: 
2 <a href="nat_1.html#K2">*</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>29</sub></font>
 ;<br/><a NAME="E3:82_5_1_3"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>24</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>21</sub></font>
 <b>by </b><i><a class="txt" href="lattice5.html#E22:82"><i><font color="Green">E76</font></i></a>, <a class="txt" href="lattice5.html#E31:82"><i><font color="Green">E81</font></i></a></i>;<br/><a NAME="E4:82_5_1_3"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>28</sub></font> <a href="funct_1.html#K1">.</a> 2</span>)</span>,<font color="Maroon">c<sub>24</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>21</sub></font>
 <b>by </b><i><a class="txt" href="lattice5.html#E38:82"><i><font color="Green">E86</font></i></a>, <a class="txt" href="lattice5.html#E39:82"><i><font color="Green">E87</font></i></a></i>;<br/><b>hence </b><a NAME="E5:82_5_1_3"/>
( ( not <font color="Maroon">c<sub>29</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>28</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>29</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>28</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>29</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="Maroon">c<sub>29</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>28</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>29</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>28</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>29</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#E12:82"><i><font color="Green">E72</font></i></a>, <a class="txt" href="lattice5.html#E21:82"><i><font color="Green">E75</font></i></a>, <a class="txt" href="lattice5.html#E38:82"><i><font color="Green">E86</font></i></a>, <a class="txt" href="lattice5.html#E39:82"><i><font color="Green">E87</font></i></a>, <a class="txt" href="lattice5.html#E2:82_5_1_3"><i><font color="Green">E92</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="82_5_1_4"><b>suppose </b></a><a NAME="E1:82_5_1_4"/>
<font color="Maroon">c<sub>29</sub></font> <a href="hidden.html#R1">=</a> 4
 ;<br/><div class="add"><a NAME="E2:82_5_1_4"/><b>then </b><i><font color="Green">E92</font></i>: 
2 <a href="nat_1.html#K2">*</a> 2 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>29</sub></font>
 ;<br/><a NAME="E3:82_5_1_4"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>25</sub></font>,<font color="Maroon">c<sub>17</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>21</sub></font>
 <b>by </b><i><a class="txt" href="lattice5.html#E22:82"><i><font color="Green">E76</font></i></a>, <a class="txt" href="lattice5.html#E32:82"><i><font color="Green">E82</font></i></a></i>;<br/><a NAME="E4:82_5_1_4"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>28</sub></font> <a href="funct_1.html#K1">.</a> 4</span>)</span>,<font color="Maroon">c<sub>17</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>21</sub></font>
 <b>by </b><i><a class="txt" href="lattice5.html#E38:82"><i><font color="Green">E86</font></i></a>, <a class="txt" href="lattice5.html#E39:82"><i><font color="Green">E87</font></i></a></i>;<br/><b>hence </b><a NAME="E5:82_5_1_4"/>
( ( not <font color="Maroon">c<sub>29</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>28</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>29</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>28</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>29</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="Maroon">c<sub>29</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>28</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>29</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>28</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>29</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#E12:82"><i><font color="Green">E72</font></i></a>, <a class="txt" href="lattice5.html#E21:82"><i><font color="Green">E75</font></i></a>, <a class="txt" href="lattice5.html#E38:82"><i><font color="Green">E86</font></i></a>, <a class="txt" href="lattice5.html#E39:82"><i><font color="Green">E87</font></i></a>, <a class="txt" href="lattice5.html#E2:82_5_1_4"><i><font color="Green">E92</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E49:82"/>
<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font> <a href="lattice5.html#R1" target="_self">are_joint_by</a> <font color="Maroon">c<sub>28</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#E46:82"><i><font color="Green">E89</font></i></a>, <a class="txt" href="lattice5.html#E47:82"><i><font color="Green">E90</font></i></a>, <a class="ref" href="lattice5.html#D3" target="_self">Def3</a></i>;<br/>


</div>
