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

<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>set </b><font color="Maroon">c<sub>2</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>3</sub></font> =  <a href="lattice5.html#K17" target="_self">BasicDF</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<a NAME="E1:83"/><i><font color="Green">E64</font></i>: 
 <a href="lattice5.html#K17" target="_self">BasicDF</a> <font color="Maroon">c<sub>1</sub></font> is <a href="funct_2.html#V2">onto</a>
 
<b>by </b><i><a class="ref" href="lattice5.html#T43" target="_self">Th43</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>4</sub></font> being    <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>set </b><font color="Maroon">c<sub>5</sub></font> =  <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>4</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> ;<br/>
<a NAME="E3:83"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,<span class="p2">(<span class="default"><a href="lattice5.html#K17" target="_self">BasicDF</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span>
 
<b>by </b><i><a class="ref" href="lattice5.html#D21" target="_self">Def21</a></i>;<br/>
<a NAME="E4:83"/><b>then </b><i><font color="Green">E66</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> 0</span>)</span> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>;<br/>
<a NAME="E5:83"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> 0</span>)</span> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</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> 
 
;<br/>
<a NAME="E6:83"/><b>then </b><i><font color="Green">E67</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</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>4</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> 
 
<b>by </b><i><a class="txt" href="lattice5.html#E4:83"><i><font color="Green">E66</font></i></a>, <a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>6</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E8:83"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> =  <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>4</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>  as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="txt" href="lattice5.html#E6:83"><i><font color="Green">E67</font></i></a>, <a class="txt" href="lattice5.html#E8:83"><i><font color="Green">E68</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> =  <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>4</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>  as   <a href="lattice5.html#NM4">distance_function</a> of <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="lattice5.html#T44" target="_self">Th44</a></i>;<br/>
<a NAME="E11:83"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">c<sub>8</sub></font> is <a href="funct_2.html#V2">onto</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1"><b>proof </b></a><div class="add">

<a NAME="E1:83_1"/><i><font color="Green">E70</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="lattice5.html#K17" target="_self">BasicDF</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E1:83"><i><font color="Green">E64</font></i></a>, <a class="ref" href="funct_2.html#D3">FUNCT_2:def 3</a></i>;<br/>
<a NAME="E2:83_1"/>
for <font color="Olive">b<sub>1</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>1</sub></font> holds <br/>ex <font color="Olive">b<sub>2</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> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:83_1_1"/>
<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 <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>10</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:83_1_1"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>and </b><br/><a NAME="E4:83_1_1"/><i><font color="Green">E72</font></i>: 
<span class="p1">(<span class="default"><a href="lattice5.html#K17" target="_self">BasicDF</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="lattice5.html#E1:83_1"><i><font color="Green">E70</font></i></a>, <a class="ref" href="funct_2.html#T17">FUNCT_2:17</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>10</sub></font>
;<br/>

<a NAME="E5:83_1_1"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,<span class="p2">(<span class="default"><a href="lattice5.html#K17" target="_self">BasicDF</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span>
 
<b>by </b><i><a class="ref" href="lattice5.html#D21" target="_self">Def21</a></i>;<br/>
<a NAME="E6:83_1_1"/><b>then </b><i><font color="Green">E74</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> 0</span>)</span> <a href="mcart_1.html#K1">`1</a> 
 
<b>by </b><i><a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>;<br/>
<a NAME="E7:83_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> 0</span>)</span> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</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> 
 
;<br/>
<a NAME="E8:83_1_1"/><b>then </b>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E6:83_1_1"><i><font color="Green">E74</font></i></a>, <a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/>
<a NAME="E9:83_1_1"/><b>then </b>
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T119">ZFMISC_1:119</a></i>;<br/>
<b>hence </b><a NAME="E10:83_1_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="txt" href="lattice5.html#E3:83_1_1"><i><font color="Green">E71</font></i></a></i>;<br/>

<a NAME="E11:83_1_1"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="lattice5.html#K17" target="_self">BasicDF</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="lattice5.html#E3:83_1_1"><i><font color="Green">E71</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E12:83_1_1"/><i><font color="Green">E76</font></i>: 
 <a href="lattice5.html#K17" target="_self">BasicDF</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> 0</span>)</span> <a href="mcart_1.html#K2">`2</a> 
 
<b>by </b><i><a class="txt" href="lattice5.html#E5:83_1_1"><i><font color="Green">E73</font></i></a>, <a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>;<br/>
<a NAME="E13:83_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> 0</span>)</span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</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> 
 
;<br/>
<a NAME="E14:83_1_1"/><b>then </b>
 <a href="lattice5.html#K17" target="_self">BasicDF</a> <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E12:83_1_1"><i><font color="Green">E76</font></i></a>, <a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/>
<b>hence </b><a NAME="E15:83_1_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="lattice5.html#E4:83_1_1"><i><font color="Green">E72</font></i></a>, <a class="txt" href="lattice5.html#E11:83_1_1"><i><font color="Green">E75</font></i></a>, <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:83_1"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T16">FUNCT_2:16</a></i>;<br/>
<b>hence </b><a NAME="E4:83_1"/>
<font color="Maroon">c<sub>8</sub></font> is <a href="funct_2.html#V2">onto</a>
 <b>by </b><i><a class="ref" href="funct_2.html#D3">FUNCT_2:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E12:83"/>
 <a href="lattice5.html#K5" target="_self">alpha</a> <font color="Maroon">c<sub>8</sub></font> is <a href="waybel_0.html#V20">join-preserving</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>10</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D35">WAYBEL_0:def 35</a><br/>

<b>set </b><font color="Maroon">c<sub>11</sub></font> =  <a href="lattice5.html#K5" target="_self">alpha</a> <font color="Maroon">c<sub>8</sub></font>;<br/>
<a NAME="E1:83_2"/><i><font color="Green">E70</font></i>: 
 <a href="yellow_0.html#R1">ex_sup_of</a> <span class="p1">(<span class="default"><a href="lattice5.html#K5" target="_self">alpha</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="domain_1.html#K7">}</a></span>, <a href="lattice5.html#K1" target="_self">EqRelLATT</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T17">YELLOW_0:17</a></i>;<br/>
<a NAME="E2:83_2"/><i><font color="Green">E71</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="lattice5.html#K5" target="_self">alpha</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>12</sub></font> being   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">c<sub>7</sub></font><b> such that </b><br/><a NAME="E4:83_2"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="lattice5.html#K5" target="_self">alpha</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font>
 <b>and </b><br/><a NAME="E5:83_2"/><i><font color="Green">E73</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>7</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>12</sub></font> iff <font color="Maroon">c<sub>8</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>9</sub></font> )
 <b>by </b><i><a class="ref" href="lattice5.html#D9" target="_self">Def9</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>13</sub></font> being   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">c<sub>7</sub></font><b> such that </b><br/><a NAME="E7:83_2"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="lattice5.html#K5" target="_self">alpha</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>10</sub></font>
 <b>and </b><br/><a NAME="E8:83_2"/><i><font color="Green">E75</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>7</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>13</sub></font> iff <font color="Maroon">c<sub>8</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>10</sub></font> )
 <b>by </b><i><a class="ref" href="lattice5.html#D9" target="_self">Def9</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>14</sub></font> being   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">c<sub>7</sub></font><b> such that </b><br/><a NAME="E10:83_2"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="lattice5.html#K5" target="_self">alpha</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E11:83_2"/><i><font color="Green">E77</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>7</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>14</sub></font> iff <font color="Maroon">c<sub>8</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>9</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>10</sub></font> )
 <b>by </b><i><a class="ref" href="lattice5.html#D9" target="_self">Def9</a></i>;<br/>
<a NAME="E12:83_2"/><i><font color="Green">E78</font></i>: 
 <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>7</sub></font>
 
<b>by </b><i><a class="ref" href="orders_1.html#T97">ORDERS_1:97</a></i>;<br/>
<a NAME="E13:83_2"/><i><font color="Green">E79</font></i>: 
 <a href="relat_1.html#K3">field</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="ref" href="orders_1.html#T97">ORDERS_1:97</a></i>;<br/>
<a NAME="E14:83_2"/><i><font color="Green">E80</font></i>: 
 <a href="relat_1.html#K3">field</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="ref" href="orders_1.html#T97">ORDERS_1:97</a></i>;<br/>
<a NAME="E15:83_2"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="eqrel_1.html#K5">"\/"</a> <font color="Maroon">c<sub>13</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>14</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_2_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>15</sub></font>, <font color="Maroon">c<sub>16</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:83_2_1_1"/><i><font color="Green">E82</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font>
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>15</sub></font>, <font color="Maroon">c<sub>18</sub></font> = <font color="Maroon">c<sub>16</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>7</sub></font> <b>by </b><i><a class="txt" href="lattice5.html#E12:83_2"><i><font color="Green">E78</font></i></a>, <a class="ref" href="relat_1.html#T30">RELAT_1:30</a></i>;<br/><a NAME="E3:83_2_1_1"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>18</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="lattice5.html#E5:83_2"><i><font color="Green">E73</font></i></a>, <a class="txt" href="lattice5.html#E1:83_2_1_1"><i><font color="Green">E82</font></i></a></i>;<br/><a NAME="E4:83_2_1_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>9</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T22">YELLOW_0:22</a></i>;<br/><a NAME="E5:83_2_1_1"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>18</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>9</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="lattice5.html#E3:83_2_1_1"><i><font color="Green">E83</font></i></a>, <a class="ref" href="orders_2.html#T26">ORDERS_2:26</a></i>;<br/><b>hence </b><a NAME="E6:83_2_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="lattice5.html#E11:83_2"><i><font color="Green">E77</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E2:83_2_1"/><b>then </b><i><font color="Green">E82</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>14</sub></font>
 
<b>by </b><i><a class="ref" href="relat_1.html#D3">RELAT_1:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_2_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>15</sub></font>, <font color="Maroon">c<sub>16</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:83_2_1_2"/><i><font color="Green">E83</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>13</sub></font>
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>15</sub></font>, <font color="Maroon">c<sub>18</sub></font> = <font color="Maroon">c<sub>16</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>7</sub></font> <b>by </b><i><a class="txt" href="lattice5.html#E13:83_2"><i><font color="Green">E79</font></i></a>, <a class="ref" href="relat_1.html#T30">RELAT_1:30</a></i>;<br/><a NAME="E3:83_2_1_2"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>18</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="lattice5.html#E8:83_2"><i><font color="Green">E75</font></i></a>, <a class="txt" href="lattice5.html#E1:83_2_1_2"><i><font color="Green">E83</font></i></a></i>;<br/><a NAME="E4:83_2_1_2"/>
<font color="Maroon">c<sub>10</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>10</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T22">YELLOW_0:22</a></i>;<br/><a NAME="E5:83_2_1_2"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>18</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>10</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="lattice5.html#E3:83_2_1_2"><i><font color="Green">E84</font></i></a>, <a class="ref" href="orders_2.html#T26">ORDERS_2:26</a></i>;<br/><b>hence </b><a NAME="E6:83_2_1_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="lattice5.html#E11:83_2"><i><font color="Green">E77</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E4:83_2_1"/><b>then </b>
<font color="Maroon">c<sub>13</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>14</sub></font>
 
<b>by </b><i><a class="ref" href="relat_1.html#D3">RELAT_1:def 3</a></i>;<br/>
<a NAME="E5:83_2_1"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="eqrel_1.html#K3">\/</a> <font color="Maroon">c<sub>13</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>14</sub></font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E2:83_2_1"><i><font color="Green">E82</font></i></a>, <a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/>
<b>hence </b><a NAME="E6:83_2_1"/>
<font color="Maroon">c<sub>12</sub></font> <a href="eqrel_1.html#K5">"\/"</a> <font color="Maroon">c<sub>13</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="ref" href="eqrel_1.html#D3">EQREL_1:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E16:83_2"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>12</sub></font> <a href="eqrel_1.html#K5">"\/"</a> <font color="Maroon">c<sub>13</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_2_2"><b>proof </b></a><div class="add">

<a NAME="E1:83_2_2"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>14</sub></font> holds <br/><span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font> <a href="eqrel_1.html#K5">"\/"</a> <font color="Maroon">c<sub>13</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_2_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>15</sub></font>, <font color="Maroon">c<sub>16</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:83_2_2_1"/><i><font color="Green">E83</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>14</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>15</sub></font>, <font color="Maroon">c<sub>18</sub></font> = <font color="Maroon">c<sub>16</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>7</sub></font> <b>by </b><i><a class="txt" href="lattice5.html#E14:83_2"><i><font color="Green">E80</font></i></a>, <a class="ref" href="relat_1.html#T30">RELAT_1:30</a></i>;<br/>
<a NAME="E3:83_2_2_1"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E14:83_2"><i><font color="Green">E80</font></i></a>, <a class="txt" href="lattice5.html#E1:83_2_2_1"><i><font color="Green">E83</font></i></a>, <a class="ref" href="relat_1.html#T30">RELAT_1:30</a></i>;<br/>
<a NAME="E4:83_2_2_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>18</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>9</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E11:83_2"><i><font color="Green">E77</font></i></a>, <a class="txt" href="lattice5.html#E1:83_2_2_1"><i><font color="Green">E83</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>19</sub></font>, <font color="Maroon">c<sub>20</sub></font>, <font color="Maroon">c<sub>21</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>7</sub></font><b> such that </b><br/><a NAME="E6:83_2_2_1"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font>
 <b>and </b><br/><a NAME="E7:83_2_2_1"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font>
 <b>and </b><br/><a NAME="E8:83_2_2_1"/><i><font color="Green">E87</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font>
 <b>and </b><br/><a NAME="E9:83_2_2_1"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><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>17</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>19</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>20</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>21</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>18</sub></font> ) );<br/>
<a NAME="E10:83_2_2_1"/><i><font color="Green">E89</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="E11:83_2_2_1"/><i><font color="Green">E90</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="83_2_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>22</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:83_2_2_1_1"/><i><font color="Green">E91</font></i>: 
<font color="Maroon">c<sub>22</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="83_2_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:83_2_2_1_1_1"/>
( <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> 2 or <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> 3 or <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> 4 or <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> 5 )
 </span><b>by </b><i><a class="txt" href="lattice5.html#E1:83_2_2_1_1"><i><font color="Green">E91</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="83_2_2_1_1_1_1"><b>suppose </b></a><a NAME="E1:83_2_2_1_1_1_1"/><i><font color="Green">E92</font></i>: 
<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><b>take </b><font color="Maroon">c<sub>23</sub></font> = <font color="Maroon">c<sub>17</sub></font>;<br/><b>thus </b><a NAME="E2:83_2_2_1_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>22</sub></font>,<font color="Maroon">c<sub>23</sub></font>]
 <b>by </b><i><a class="txt" href="lattice5.html#E1:83_2_2_1_1_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="83_2_2_1_1_1_2"><b>suppose </b></a><a NAME="E1:83_2_2_1_1_1_2"/><i><font color="Green">E92</font></i>: 
<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> 2
 ;<br/><div class="add"><b>take </b><font color="Maroon">c<sub>23</sub></font> = <font color="Maroon">c<sub>19</sub></font>;<br/><b>thus </b><a NAME="E2:83_2_2_1_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>22</sub></font>,<font color="Maroon">c<sub>23</sub></font>]
 <b>by </b><i><a class="txt" href="lattice5.html#E1:83_2_2_1_1_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="83_2_2_1_1_1_3"><b>suppose </b></a><a NAME="E1:83_2_2_1_1_1_3"/><i><font color="Green">E92</font></i>: 
<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> 3
 ;<br/><div class="add"><b>take </b><font color="Maroon">c<sub>23</sub></font> = <font color="Maroon">c<sub>20</sub></font>;<br/><b>thus </b><a NAME="E2:83_2_2_1_1_1_3"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>22</sub></font>,<font color="Maroon">c<sub>23</sub></font>]
 <b>by </b><i><a class="txt" href="lattice5.html#E1:83_2_2_1_1_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="83_2_2_1_1_1_4"><b>suppose </b></a><a NAME="E1:83_2_2_1_1_1_4"/><i><font color="Green">E92</font></i>: 
<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> 4
 ;<br/><div class="add"><b>take </b><font color="Maroon">c<sub>23</sub></font> = <font color="Maroon">c<sub>21</sub></font>;<br/><b>thus </b><a NAME="E2:83_2_2_1_1_1_4"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>22</sub></font>,<font color="Maroon">c<sub>23</sub></font>]
 <b>by </b><i><a class="txt" href="lattice5.html#E1:83_2_2_1_1_1_4"><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="83_2_2_1_1_1_5"><b>suppose </b></a><a NAME="E1:83_2_2_1_1_1_5"/><i><font color="Green">E92</font></i>: 
<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> 5
 ;<br/><div class="add"><b>take </b><font color="Maroon">c<sub>23</sub></font> = <font color="Maroon">c<sub>18</sub></font>;<br/><b>thus </b><a NAME="E2:83_2_2_1_1_1_5"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>22</sub></font>,<font color="Maroon">c<sub>23</sub></font>]
 <b>by </b><i><a class="txt" href="lattice5.html#E1:83_2_2_1_1_1_5"><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>
<a NAME="E12:83_2_2_1"/>
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#E10:83_2_2_1"><i><font color="Green">E89</font></i></a>, <a class="txt" href="lattice5.html#E11:83_2_2_1"><i><font color="Green">E90</font></i></a>);<br/></i>
<b>then consider </b><font color="Maroon">c<sub>22</sub></font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E14:83_2_2_1"/><i><font color="Green">E91</font></i>: 
 <a href="finsop_1.html#K5">dom</a> <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> 5
 <b>and </b><br/><a NAME="E15:83_2_2_1"/><i><font color="Green">E92</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>22</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> ) &amp; ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 2 implies <font color="Maroon">c<sub>22</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>19</sub></font> ) &amp; ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 3 implies <font color="Maroon">c<sub>22</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>20</sub></font> ) &amp; ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 4 implies <font color="Maroon">c<sub>22</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>21</sub></font> ) &amp; ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 5 implies <font color="Maroon">c<sub>22</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>18</sub></font> ) )
 ;<br/>
<a NAME="E16:83_2_2_1"/><i><font color="Green">E93</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="E17:83_2_2_1"/><i><font color="Green">E94</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> 5
 
<b>by </b><i><a class="txt" href="lattice5.html#E14:83_2_2_1"><i><font color="Green">E91</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E18:83_2_2_1"/><i><font color="Green">E95</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> <a href="funct_1.html#K1">.</a> 1
 
<b>by </b><i><a class="txt" href="lattice5.html#E15:83_2_2_1"><i><font color="Green">E92</font></i></a>, <a class="txt" href="lattice5.html#E16:83_2_2_1"><i><font color="Green">E93</font></i></a></i>;<br/>
<i><font color="Green">E96</font></i>: <font color="Maroon">c<sub>16</sub></font> = 
<font color="Maroon">c<sub>22</sub></font> <a href="funct_1.html#K1">.</a> 5
<b>by </b><i><a class="txt" href="lattice5.html#E15:83_2_2_1"><i><font color="Green">E92</font></i></a>, <a class="txt" href="lattice5.html#E16:83_2_2_1"><i><font color="Green">E93</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>22</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>22</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="lattice5.html#E14:83_2_2_1"><i><font color="Green">E91</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
;<br/>
<a NAME="E20:83_2_2_1"/>
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>22</sub></font> holds <br/><span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>22</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>22</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>12</sub></font> <a href="eqrel_1.html#K3">\/</a> <font color="Maroon">c<sub>13</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_2_2_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>23</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:83_2_2_1_3"/><i><font color="Green">E97</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>23</sub></font> &amp; <font color="Maroon">c<sub>23</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>22</sub></font> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_2_2_1_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:83_2_2_1_3_1"/>
( <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a> 3 or <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a> 2 or <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a> 4 )
 </span><b>by </b><i><a class="txt" href="lattice5.html#E17:83_2_2_1"><i><font color="Green">E94</font></i></a>, <a class="txt" href="lattice5.html#E1:83_2_2_1_3"><i><font color="Green">E97</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="83_2_2_1_3_1_1"><b>suppose </b></a><a NAME="E1:83_2_2_1_3_1_1"/><i><font color="Green">E98</font></i>: 
<font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:83_2_2_1_3_1_1"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>19</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="lattice5.html#E5:83_2"><i><font color="Green">E73</font></i></a>, <a class="txt" href="lattice5.html#E6:83_2_2_1"><i><font color="Green">E85</font></i></a></i>;<br/><a NAME="E3:83_2_2_1_3_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>22</sub></font> <a href="funct_1.html#K1">.</a> 1</span>)</span>,<font color="Maroon">c<sub>19</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="lattice5.html#E15:83_2_2_1"><i><font color="Green">E92</font></i></a>, <a class="txt" href="lattice5.html#E16:83_2_2_1"><i><font color="Green">E93</font></i></a></i>;<br/><a NAME="E4:83_2_2_1_3_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>22</sub></font> <a href="funct_1.html#K1">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="funct_1.html#K1">.</a> 2</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="lattice5.html#E15:83_2_2_1"><i><font color="Green">E92</font></i></a>, <a class="txt" href="lattice5.html#E16:83_2_2_1"><i><font color="Green">E93</font></i></a></i>;<br/><b>hence </b><a NAME="E5:83_2_2_1_3_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>23</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>12</sub></font> <a href="eqrel_1.html#K3">\/</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="lattice5.html#E1:83_2_2_1_3_1_1"><i><font color="Green">E98</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_2_2_1_3_1_2"><b>suppose </b></a><a NAME="E1:83_2_2_1_3_1_2"/><i><font color="Green">E98</font></i>: 
<font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a> 3
 ;<br/><div class="add"><a NAME="E2:83_2_2_1_3_1_2"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>21</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="lattice5.html#E5:83_2"><i><font color="Green">E73</font></i></a>, <a class="txt" href="lattice5.html#E7:83_2_2_1"><i><font color="Green">E86</font></i></a></i>;<br/><a NAME="E3:83_2_2_1_3_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>22</sub></font> <a href="funct_1.html#K1">.</a> 3</span>)</span>,<font color="Maroon">c<sub>21</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="lattice5.html#E15:83_2_2_1"><i><font color="Green">E92</font></i></a>, <a class="txt" href="lattice5.html#E16:83_2_2_1"><i><font color="Green">E93</font></i></a></i>;<br/><a NAME="E4:83_2_2_1_3_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>22</sub></font> <a href="funct_1.html#K1">.</a> 3</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="funct_1.html#K1">.</a> 4</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="lattice5.html#E15:83_2_2_1"><i><font color="Green">E92</font></i></a>, <a class="txt" href="lattice5.html#E16:83_2_2_1"><i><font color="Green">E93</font></i></a></i>;<br/><b>hence </b><a NAME="E5:83_2_2_1_3_1_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>23</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>12</sub></font> <a href="eqrel_1.html#K3">\/</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="lattice5.html#E1:83_2_2_1_3_1_2"><i><font color="Green">E98</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_2_2_1_3_1_3"><b>suppose </b></a><a NAME="E1:83_2_2_1_3_1_3"/><i><font color="Green">E98</font></i>: 
<font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a> 2
 ;<br/><div class="add"><a NAME="E2:83_2_2_1_3_1_3"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>20</sub></font></span><a href="domain_1.html#K1">]</a></span> <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#E8:83_2"><i><font color="Green">E75</font></i></a>, <a class="txt" href="lattice5.html#E8:83_2_2_1"><i><font color="Green">E87</font></i></a></i>;<br/><a NAME="E3:83_2_2_1_3_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>22</sub></font> <a href="funct_1.html#K1">.</a> 2</span>)</span>,<font color="Maroon">c<sub>20</sub></font></span><a href="tarski.html#K4">]</a></span> <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#E15:83_2_2_1"><i><font color="Green">E92</font></i></a>, <a class="txt" href="lattice5.html#E16:83_2_2_1"><i><font color="Green">E93</font></i></a></i>;<br/><a NAME="E4:83_2_2_1_3_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>22</sub></font> <a href="funct_1.html#K1">.</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="funct_1.html#K1">.</a> 3</span>)</span></span><a href="tarski.html#K4">]</a></span> <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#E15:83_2_2_1"><i><font color="Green">E92</font></i></a>, <a class="txt" href="lattice5.html#E16:83_2_2_1"><i><font color="Green">E93</font></i></a></i>;<br/><b>hence </b><a NAME="E5:83_2_2_1_3_1_3"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>23</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>12</sub></font> <a href="eqrel_1.html#K3">\/</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="lattice5.html#E1:83_2_2_1_3_1_3"><i><font color="Green">E98</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="83_2_2_1_3_1_4"><b>suppose </b></a><a NAME="E1:83_2_2_1_3_1_4"/><i><font color="Green">E98</font></i>: 
<font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a> 4
 ;<br/><div class="add"><a NAME="E2:83_2_2_1_3_1_4"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>18</sub></font></span><a href="domain_1.html#K1">]</a></span> <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#E8:83_2"><i><font color="Green">E75</font></i></a>, <a class="txt" href="lattice5.html#E9:83_2_2_1"><i><font color="Green">E88</font></i></a></i>;<br/><a NAME="E3:83_2_2_1_3_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>22</sub></font> <a href="funct_1.html#K1">.</a> 4</span>)</span>,<font color="Maroon">c<sub>18</sub></font></span><a href="tarski.html#K4">]</a></span> <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#E15:83_2_2_1"><i><font color="Green">E92</font></i></a>, <a class="txt" href="lattice5.html#E16:83_2_2_1"><i><font color="Green">E93</font></i></a></i>;<br/><a NAME="E4:83_2_2_1_3_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>22</sub></font> <a href="funct_1.html#K1">.</a> 4</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="funct_1.html#K1">.</a> 5</span>)</span></span><a href="tarski.html#K4">]</a></span> <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#E15:83_2_2_1"><i><font color="Green">E92</font></i></a>, <a class="txt" href="lattice5.html#E16:83_2_2_1"><i><font color="Green">E93</font></i></a></i>;<br/><b>hence </b><a NAME="E5:83_2_2_1_3_1_4"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>23</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>12</sub></font> <a href="eqrel_1.html#K3">\/</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="lattice5.html#E1:83_2_2_1_3_1_4"><i><font color="Green">E98</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E21:83_2_2_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font> <a href="eqrel_1.html#K5">"\/"</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="lattice5.html#E3:83_2_2_1"><i><font color="Green">E84</font></i></a>, <a class="txt" href="lattice5.html#E17:83_2_2_1"><i><font color="Green">E94</font></i></a>, <a class="txt" href="lattice5.html#E18:83_2_2_1"><i><font color="Green">E95</font></i></a>, <a class="txt" href="lattice5.html#E19:83_2_2_1"><i><font color="Green">E96</font></i></a>, <a class="ref" href="eqrel_1.html#T36">EQREL_1:36</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:83_2_2"/>
<font color="Maroon">c<sub>14</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>12</sub></font> <a href="eqrel_1.html#K5">"\/"</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="ref" href="relat_1.html#D3">RELAT_1:def 3</a></i>;<br/>


</div><b>end;</b></div>
 <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K5" target="_self">alpha</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p2"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="domain_1.html#K7">}</a></span></span>)</span> = 
 <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lattice5.html#K5" target="_self">alpha</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lattice5.html#K5" target="_self">alpha</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span><a href="domain_1.html#K7">}</a></span>
<b>by </b><i><a class="txt" href="lattice5.html#E2:83_2"><i><font color="Green">E71</font></i></a>, <a class="ref" href="funct_1.html#T118">FUNCT_1:118</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K5" target="_self">alpha</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K5" target="_self">alpha</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="yellow_0.html#T41">YELLOW_0:41</a></i>
<br/>.= 
<font color="Maroon">c<sub>12</sub></font> <a href="eqrel_1.html#K5">"\/"</a> <font color="Maroon">c<sub>13</sub></font>
<b>by </b><i><a class="txt" href="lattice5.html#E4:83_2"><i><font color="Green">E72</font></i></a>, <a class="txt" href="lattice5.html#E7:83_2"><i><font color="Green">E74</font></i></a>, <a class="ref" href="lattice5.html#T10" target="_self">Th10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="lattice5.html#K5" target="_self">alpha</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="lattice5.html#E10:83_2"><i><font color="Green">E76</font></i></a>, <a class="txt" href="lattice5.html#E15:83_2"><i><font color="Green">E81</font></i></a>, <a class="txt" href="lattice5.html#E16:83_2"><i><font color="Green">E82</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="lattice5.html#K5" target="_self">alpha</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p2"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="domain_1.html#K7">}</a></span></span>)</span>
<b>by </b><i><a class="ref" href="yellow_0.html#T41">YELLOW_0:41</a></i>
;<br/>
<b>hence </b><a NAME="E18:83_2"/>
 <a href="lattice5.html#K5" target="_self">alpha</a> <font color="Maroon">c<sub>8</sub></font> <a href="waybel_0.html#R4">preserves_sup_of</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="domain_1.html#K7">}</a></span>
 <b>by </b><i><a class="txt" href="lattice5.html#E1:83_2"><i><font color="Green">E70</font></i></a>, <a class="ref" href="waybel_0.html#D31">WAYBEL_0:def 31</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>9</sub></font> =  <a href="lattice5.html#K5" target="_self">alpha</a> <font color="Maroon">c<sub>8</sub></font> as   <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>7</sub></font></span>)</span> <b>by </b><i><a class="ref" href="lattice5.html#T16" target="_self">Th16</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>7</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>9</sub></font>
;<br/>

<b>thus </b><a NAME="E14:83"/>
<font color="Maroon">c<sub>9</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i><a class="txt" href="lattice5.html#E11:83"><i><font color="Green">E69</font></i></a>, <a class="ref" href="lattice5.html#T17" target="_self">Th17</a></i>;<br/>

<a NAME="E15:83"/><i><font color="Green">E70</font></i>: 
 <a href="lattice5.html#K2" target="_self">Image</a> <font color="Maroon">c<sub>9</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>9</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="E16:83"/><i><font color="Green">E71</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E17:83"/><i><font color="Green">E72</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being  <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">c<sub>7</sub></font> 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 <span class="p1">(<span class="default"><a href="lattice5.html#K2" target="_self">Image</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="cqc_sim1.html#K13">id</a> <font color="Maroon">c<sub>7</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_3"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">c<sub>10</sub></font> being  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> , <font color="Maroon">c<sub>11</sub></font> being   <a href="lattice5.html#NM4">distance_function</a> of <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>12</sub></font> being  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> , <font color="Maroon">c<sub>13</sub></font> being   <a href="lattice5.html#NM4">distance_function</a> of <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E2:83_3"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font> <a href="lattice5.html#R2" target="_self">is_extension_of</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font>
 <b>and </b><br/><a NAME="E3:83_3"/><i><font color="Green">E74</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <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> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i><a class="ref" href="lattice5.html#D21" target="_self">Def21</a></i>;<br/>
<a NAME="E4:83_3"/>
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="funct_2.html#R1">=</a>  <a href="lattice5.html#K17" target="_self">BasicDF</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="txt" href="lattice5.html#E3:83"><i><font color="Green">E65</font></i></a>, <a class="txt" href="lattice5.html#E3:83_3"><i><font color="Green">E74</font></i></a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>14</sub></font> being    <a href="lattice5.html#M1" target="_self">QuadrSeq</a> of  <a href="lattice5.html#K17" target="_self">BasicDF</a> <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E6:83_3"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a>  <a href="lattice5.html#K14" target="_self">NextSet</a> <span class="p1">(<span class="default"><a href="lattice5.html#K17" target="_self">BasicDF</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E7:83_3"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="funct_2.html#R1">=</a>  <a href="lattice5.html#K16" target="_self">NextDelta</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="lattice5.html#E2:83_3"><i><font color="Green">E73</font></i></a>, <a class="ref" href="lattice5.html#D20" target="_self">Def20</a></i>;<br/>
<a NAME="E8:83_3"/><i><font color="Green">E77</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> 1</span>)</span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a>  <a href="lattice5.html#K16" target="_self">NextDelta</a> <font color="Maroon">c<sub>14</sub></font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E3:83_3"><i><font color="Green">E74</font></i></a>, <a class="txt" href="lattice5.html#E7:83_3"><i><font color="Green">E76</font></i></a>, <a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>;<br/>
<a NAME="E9:83_3"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> 1</span>)</span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</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> 
 
;<br/>
<a NAME="E10:83_3"/><b>then </b><i><font color="Green">E78</font></i>: 
 <a href="lattice5.html#K16" target="_self">NextDelta</a> <font color="Maroon">c<sub>14</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E8:83_3"><i><font color="Green">E77</font></i></a>, <a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/>
<a NAME="E11:83_3"/><i><font color="Green">E79</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> 1</span>)</span> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a>  <a href="lattice5.html#K14" target="_self">NextSet</a> <span class="p1">(<span class="default"><a href="lattice5.html#K17" target="_self">BasicDF</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="lattice5.html#E3:83_3"><i><font color="Green">E74</font></i></a>, <a class="txt" href="lattice5.html#E6:83_3"><i><font color="Green">E75</font></i></a>, <a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>;<br/>
<a NAME="E12:83_3"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> 1</span>)</span> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</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> 
 
;<br/>
<a NAME="E13:83_3"/><b>then </b><i><font color="Green">E80</font></i>: 
 <a href="lattice5.html#K14" target="_self">NextSet</a> <span class="p1">(<span class="default"><a href="lattice5.html#K17" target="_self">BasicDF</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E11:83_3"><i><font color="Green">E79</font></i></a>, <a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/>
<a NAME="E14:83_3"/>
 <a href="ordinal1.html#K1">succ</a> <a href="xboole_0.html#K1">{}</a>  <a href="ordinal1.html#R1">c=</a>  <a href="lattice5.html#K8" target="_self">DistEsti</a> <span class="p1">(<span class="default"><a href="lattice5.html#K17" target="_self">BasicDF</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="lattice5.html#E69"><i><font color="Green">Lemma60</font></i></a></i>;<br/>
<a NAME="E15:83_3"/><b>then </b>
 <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a>  <a href="lattice5.html#K8" target="_self">DistEsti</a> <span class="p1">(<span class="default"><a href="lattice5.html#K17" target="_self">BasicDF</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="ordinal1.html#T33">ORDINAL1:33</a></i>;<br/>
<a NAME="E16:83_3"/><b>then </b><i><font color="Green">E81</font></i>: 
 <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>14</sub></font>
 
<b>by </b><i><a class="ref" href="lattice5.html#T28" target="_self">Th28</a></i>;<br/>
<a NAME="E17:83_3"/><b>then </b>
<font color="Maroon">c<sub>14</sub></font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>14</sub></font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E18:83_3"/><b>then </b>
<font color="Maroon">c<sub>14</sub></font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K5">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>4</sub></font></span><a href="domain_1.html#K5">]</a></span> where B is    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>, B is    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>, B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>, B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : <span class="p1">(<span class="default"><a href="lattice5.html#K17" target="_self">BasicDF</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <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="Olive">b<sub>3</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Olive">b<sub>4</sub></font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="lattice5.html#D14" target="_self">Def14</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>15</sub></font>, <font color="Maroon">c<sub>16</sub></font> being    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>17</sub></font>, <font color="Maroon">c<sub>18</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E20:83_3"/><i><font color="Green">E82</font></i>: 
( <font color="Maroon">c<sub>14</sub></font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K5">[</a><span class="default"><font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>18</sub></font></span><a href="domain_1.html#K5">]</a></span> &amp; <span class="p1">(<span class="default"><a href="lattice5.html#K17" target="_self">BasicDF</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>17</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>18</sub></font> )
 ;<br/>
<a NAME="E21:83_3"/>
 <a href="lattice5.html#K9" target="_self">ConsecutiveSet</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,<a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="lattice5.html#T24" target="_self">Th24</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>19</sub></font> =  <a href="lattice5.html#K10" target="_self">Quadr</a> <font color="Maroon">c<sub>14</sub></font>,<a href="xboole_0.html#K1">{}</a>  as    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K4">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="zfmisc_1.html#K4">:]</a></span> ;<br/>
<a NAME="E23:83_3"/><i><font color="Green">E83</font></i>: 
 <a href="lattice5.html#K10" target="_self">Quadr</a> <font color="Maroon">c<sub>14</sub></font>,<a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K5">[</a><span class="default"><font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>18</sub></font></span><a href="domain_1.html#K5">]</a></span>
 
<b>by </b><i><a class="txt" href="lattice5.html#E16:83_3"><i><font color="Green">E81</font></i></a>, <a class="txt" href="lattice5.html#E20:83_3"><i><font color="Green">E82</font></i></a>, <a class="ref" href="lattice5.html#D15" target="_self">Def15</a></i>;<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>7</sub></font><b> such that </b><br/><a NAME="E25:83_3"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>18</sub></font>
 <b>and </b><br/><a NAME="E26:83_3"/><i><font color="Green">E85</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>7</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>8</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="ref" href="lattice5.html#D9" target="_self">Def9</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>20</sub></font>
;<br/>

<a NAME="E27:83_3"/>
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E16:83"><i><font color="Green">E71</font></i></a>, <a class="txt" href="lattice5.html#E25:83_3"><i><font color="Green">E84</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E28:83_3"/>
<font color="Maroon">c<sub>20</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>9</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="lattice5.html#E15:83"><i><font color="Green">E70</font></i></a>, <a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/>

<i><font color="Green">E86</font></i>:  <a href="lattice5.html#K6" target="_self">new_set</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> = 
 <a href="lattice5.html#K6" target="_self">new_set</a> <span class="p1">(<span class="default"><a href="lattice5.html#K9" target="_self">ConsecutiveSet</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,<a href="xboole_0.html#K1">{}</a> </span>)</span>
<b>by </b><i><a class="ref" href="lattice5.html#T24" target="_self">Th24</a></i>
<br/>.= 
 <a href="lattice5.html#K9" target="_self">ConsecutiveSet</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <a href="xboole_0.html#K1">{}</a> </span>)</span>
<b>by </b><i><a class="ref" href="lattice5.html#T25" target="_self">Th25</a></i>
;<br/>
<a NAME="E30:83_3"/><i><font color="Green">E87</font></i>: 
 <a href="ordinal1.html#K1">succ</a> <a href="xboole_0.html#K1">{}</a>  <a href="ordinal1.html#R1">c=</a>  <a href="lattice5.html#K8" target="_self">DistEsti</a> <span class="p1">(<span class="default"><a href="lattice5.html#K17" target="_self">BasicDF</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="lattice5.html#E69"><i><font color="Green">Lemma60</font></i></a></i>;<br/>
<a NAME="E31:83_3"/><b>then </b><i><font color="Green">E88</font></i>: 
 <a href="lattice5.html#K6" target="_self">new_set</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="lattice5.html#K14" target="_self">NextSet</a> <span class="p1">(<span class="default"><a href="lattice5.html#K17" target="_self">BasicDF</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="lattice5.html#E29:83_3"><i><font color="Green">E86</font></i></a>, <a class="ref" href="lattice5.html#T32" target="_self">Th32</a></i>;<br/>
<a NAME="E32:83_3"/><i><font color="Green">E89</font></i>: 
(  <a href="lattice5.html#K9" target="_self">ConsecutiveSet</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,<a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="lattice5.html#K13" target="_self">ConsecutiveDelta</a> <font color="Maroon">c<sub>14</sub></font>,<a href="xboole_0.html#K1">{}</a>  <a href="funct_2.html#R1">=</a>  <a href="lattice5.html#K17" target="_self">BasicDF</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="ref" href="lattice5.html#T24" target="_self">Th24</a>, <a class="ref" href="lattice5.html#T29" target="_self">Th29</a></i>;<br/>
<i><font color="Green">E90</font></i>:  <a href="lattice5.html#K13" target="_self">ConsecutiveDelta</a> <font color="Maroon">c<sub>14</sub></font>,<span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <a href="xboole_0.html#K1">{}</a> </span>)</span> = 
 <a href="lattice5.html#K7" target="_self">new_bi_fun</a> <span class="p1">(<span class="default"><a href="lattice5.html#K11" target="_self">BiFun</a> <span class="p2">(<span class="default"><a href="lattice5.html#K13" target="_self">ConsecutiveDelta</a> <font color="Maroon">c<sub>14</sub></font>,<a href="xboole_0.html#K1">{}</a> </span>)</span>,<span class="p2">(<span class="default"><a href="lattice5.html#K9" target="_self">ConsecutiveSet</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,<a href="xboole_0.html#K1">{}</a> </span>)</span>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="lattice5.html#K10" target="_self">Quadr</a> <font color="Maroon">c<sub>14</sub></font>,<a href="xboole_0.html#K1">{}</a> </span>)</span>
<b>by </b><i><a class="ref" href="lattice5.html#T30" target="_self">Th30</a></i>
<br/>.= 
 <a href="lattice5.html#K7" target="_self">new_bi_fun</a> <span class="p1">(<span class="default"><a href="lattice5.html#K17" target="_self">BasicDF</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>19</sub></font>
<b>by </b><i><a class="txt" href="lattice5.html#E32:83_3"><i><font color="Green">E89</font></i></a>, <a class="ref" href="lattice5.html#D16" target="_self">Def16</a></i>
;<br/>
<a NAME="E34:83_3"/>
 <a href="lattice5.html#K7" target="_self">new_bi_fun</a> <span class="p1">(<span class="default"><a href="lattice5.html#K17" target="_self">BasicDF</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>19</sub></font> <a href="tarski.html#R1">c=</a>  <a href="lattice5.html#K16" target="_self">NextDelta</a> <font color="Maroon">c<sub>14</sub></font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E30:83_3"><i><font color="Green">E87</font></i></a>, <a class="txt" href="lattice5.html#E33:83_3"><i><font color="Green">E90</font></i></a>, <a class="ref" href="lattice5.html#T35" target="_self">Th35</a></i>;<br/>
<a NAME="E35:83_3"/><b>then </b><i><font color="Green">E91</font></i>: 
 <a href="lattice5.html#K7" target="_self">new_bi_fun</a> <span class="p1">(<span class="default"><a href="lattice5.html#K17" target="_self">BasicDF</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>19</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E10:83_3"><i><font color="Green">E78</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E36:83_3"/><i><font color="Green">E92</font></i>: 
 <a href="lattice5.html#K6" target="_self">new_set</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E13:83_3"><i><font color="Green">E80</font></i></a>, <a class="txt" href="lattice5.html#E31:83_3"><i><font color="Green">E88</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E37:83_3"/><i><font color="Green">E93</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <span class="p2">(<span class="default"><a href="lattice5.html#K17" target="_self">BasicDF</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K6" target="_self">new_set</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="lattice5.html#K6" target="_self">new_set</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E38:83_3"/>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><span class="p4"><a href="tarski.html#K1">{</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="enumset1.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="enumset1.html#D1">ENUMSET1:def 1</a></i>;<br/>
<a NAME="E39:83_3"/><b>then </b>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><span class="p4"><a href="tarski.html#K1">{</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="enumset1.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E40:83_3"/><b>then </b><i><font color="Green">E94</font></i>: 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R2">in</a>  <a href="lattice5.html#K6" target="_self">new_set</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 
;<br/>
<a NAME="E41:83_3"/>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><span class="p4"><a href="tarski.html#K1">{</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="enumset1.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="enumset1.html#D1">ENUMSET1:def 1</a></i>;<br/>
<a NAME="E42:83_3"/><b>then </b>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><span class="p4"><a href="tarski.html#K1">{</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="enumset1.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E43:83_3"/><b>then </b><i><font color="Green">E95</font></i>: 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R2">in</a>  <a href="lattice5.html#K6" target="_self">new_set</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 
;<br/>
<a NAME="E44:83_3"/><b>then </b><i><font color="Green">E96</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <span class="p2">(<span class="default"><a href="lattice5.html#K17" target="_self">BasicDF</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>19</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="lattice5.html#E37:83_3"><i><font color="Green">E93</font></i></a>, <a class="txt" href="lattice5.html#E40:83_3"><i><font color="Green">E94</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>21</sub></font> = <span class="p1"><a href="tarski.html#K1">{</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span>, <font color="Maroon">c<sub>22</sub></font> = <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>7</sub></font> <b>by </b><i><a class="txt" href="lattice5.html#E36:83_3"><i><font color="Green">E92</font></i></a>, <a class="txt" href="lattice5.html#E40:83_3"><i><font color="Green">E94</font></i></a>, <a class="txt" href="lattice5.html#E43:83_3"><i><font color="Green">E95</font></i></a></i>;<br/>
<font color="Maroon">c<sub>8</sub></font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>22</sub></font> = 
<font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K4">]</a></span>

<br/>.= 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <span class="p2">(<span class="default"><a href="lattice5.html#K17" target="_self">BasicDF</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="lattice5.html#E35:83_3"><i><font color="Green">E91</font></i></a>, <a class="txt" href="lattice5.html#E44:83_3"><i><font color="Green">E96</font></i></a>, <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <span class="p2">(<span class="default"><a href="lattice5.html#K17" target="_self">BasicDF</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span>,<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span>

<br/>.= 
<font color="Maroon">c<sub>19</sub></font> <a href="mcart_1.html#K11">`4</a> 
<b>by </b><i><a class="ref" href="lattice5.html#D11" target="_self">Def11</a></i>
<br/>.= 
<font color="Maroon">c<sub>18</sub></font>
<b>by </b><i><a class="txt" href="lattice5.html#E23:83_3"><i><font color="Green">E83</font></i></a>, <a class="ref" href="mcart_1.html#D11">MCART_1:def 11</a></i>
;<br/>
<a NAME="E47:83_3"/><b>then </b><i><font color="Green">E97</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></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#E26:83_3"><i><font color="Green">E85</font></i></a></i>;<br/>
<a NAME="E48:83_3"/>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_3_4"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:83_3_4"/>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<a NAME="E2:83_3_4"/><b>then </b>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:83_3_4"/>
contradiction
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E49:83_3"/>
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="cqc_sim1.html#K13">id</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="lattice5.html#E47:83_3"><i><font color="Green">E97</font></i></a>, <a class="ref" href="relat_1.html#D10">RELAT_1:def 10</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E18:83"/>
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>7</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 <span class="p1">(<span class="default"><a href="lattice5.html#K2" target="_self">Image</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> &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 <span class="p1">(<span class="default"><a href="lattice5.html#K2" target="_self">Image</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> &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>7</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> 3 <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="ref" href="lattice5.html#T46" target="_self">Th46</a></i>;<br/>
<b>hence </b><a NAME="E19:83"/>
 <a href="lattice5.html#K3" target="_self">type_of</a> <span class="p1">(<span class="default"><a href="lattice5.html#K2" target="_self">Image</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 3
 <b>by </b><i><a class="txt" href="lattice5.html#E17:83"><i><font color="Green">E72</font></i></a>, <a class="ref" href="lattice5.html#T15" target="_self">Th15</a></i>;<br/>


</div>
