<?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="yellow11.html#V1">modular</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">BasicDF</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<a NAME="E1:57"/><i><font color="Green">E46</font></i>: 
 <a href="lattice5.html#K17">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">LATTICE5:43</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>4</sub></font> being    <a href="lattice8.html#M1" target="_self">ExtensionSeq2</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">BasicDF</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<a NAME="E3:57"/><i><font color="Green">E47</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">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="lattice8.html#D12" target="_self">Def12</a></i>;<br/>
<a NAME="E4:57"/><b>then </b><i><font color="Green">E48</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/>
<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="E5:57"/>
<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="E6:57"/><b>then </b><i><font color="Green">E49</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="lattice8.html#E4:57"><i><font color="Green">E48</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="E8:57"/><i><font color="Green">E50</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="lattice8.html#E6:57"><i><font color="Green">E49</font></i></a>, <a class="txt" href="lattice8.html#E8:57"><i><font color="Green">E50</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="lattice8.html#T33" target="_self">Th33</a></i>;<br/>
<a NAME="E11:57"/><i><font color="Green">E51</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="57_1"><b>proof </b></a><div class="add">

<a NAME="E1:57_1"/><i><font color="Green">E52</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="lattice5.html#K17">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="lattice8.html#E1:57"><i><font color="Green">E46</font></i></a>, <a class="ref" href="funct_2.html#D3">FUNCT_2:def 3</a></i>;<br/>
<a NAME="E2:57_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="57_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="E1:57_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="E3:57_1_1"/><i><font color="Green">E53</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="E4:57_1_1"/><i><font color="Green">E54</font></i>: 
<span class="p1">(<span class="default"><a href="lattice5.html#K17">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="lattice8.html#E1:57_1"><i><font color="Green">E52</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="E5:57_1_1"/><i><font color="Green">E55</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">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="lattice8.html#D12" target="_self">Def12</a></i>;<br/>
<a NAME="E6:57_1_1"/><b>then </b><i><font color="Green">E56</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="E7:57_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="E8:57_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="lattice8.html#E6:57_1_1"><i><font color="Green">E56</font></i></a>, <a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/>
<a NAME="E9:57_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="E10:57_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="lattice8.html#E3:57_1_1"><i><font color="Green">E53</font></i></a></i>;<br/>

<a NAME="E11:57_1_1"/><i><font color="Green">E57</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">BasicDF</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="lattice8.html#E3:57_1_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E12:57_1_1"/><i><font color="Green">E58</font></i>: 
 <a href="lattice5.html#K17">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="lattice8.html#E5:57_1_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>;<br/>
<a NAME="E13:57_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="E14:57_1_1"/><b>then </b>
 <a href="lattice5.html#K17">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="lattice8.html#E12:57_1_1"><i><font color="Green">E58</font></i></a>, <a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/>
<b>hence </b><a NAME="E15:57_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="lattice8.html#E4:57_1_1"><i><font color="Green">E54</font></i></a>, <a class="txt" href="lattice8.html#E11:57_1_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:57_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="E4:57_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="E12:57"/>
 <a href="lattice5.html#K5">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="57_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">alpha</a> <font color="Maroon">c<sub>8</sub></font>;<br/>
<a NAME="E1:57_2"/><i><font color="Green">E52</font></i>: 
 <a href="yellow_0.html#R1">ex_sup_of</a> <span class="p1">(<span class="default"><a href="lattice5.html#K5">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">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="E2:57_2"/><i><font color="Green">E53</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="lattice5.html#K5">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="E4:57_2"/><i><font color="Green">E54</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">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="E5:57_2"/><i><font color="Green">E55</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">.</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">LATTICE5:def 9</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="E7:57_2"/><i><font color="Green">E56</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">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="E8:57_2"/><i><font color="Green">E57</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">.</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">LATTICE5:def 9</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="E10:57_2"/><i><font color="Green">E58</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">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="E11:57_2"/><i><font color="Green">E59</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">.</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">LATTICE5:def 9</a></i>;<br/>
<a NAME="E12:57_2"/><i><font color="Green">E60</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="E13:57_2"/><i><font color="Green">E61</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="E14:57_2"/><i><font color="Green">E62</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="E15:57_2"/><i><font color="Green">E63</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="lattice8.html#R1" target="_self">c=</a> <font color="Maroon">c<sub>14</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_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="E1:57_2_1_1"/><i><font color="Green">E64</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="lattice8.html#E12:57_2"><i><font color="Green">E60</font></i></a>, <a class="ref" href="relat_1.html#T30">RELAT_1:30</a></i>;<br/><a NAME="E3:57_2_1_1"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="lattice5.html#K4">.</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="lattice8.html#E5:57_2"><i><font color="Green">E55</font></i></a>, <a class="txt" href="lattice8.html#E1:57_2_1_1"><i><font color="Green">E64</font></i></a></i>;<br/><a NAME="E4:57_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="E5:57_2_1_1"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="lattice5.html#K4">.</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="lattice8.html#E3:57_2_1_1"><i><font color="Green">E65</font></i></a>, <a class="ref" href="orders_2.html#T26">ORDERS_2:26</a></i>;<br/><b>hence </b><a NAME="E6:57_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="lattice8.html#E11:57_2"><i><font color="Green">E59</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E2:57_2_1"/><b>then </b><i><font color="Green">E64</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="lattice8.html#R1" target="_self">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="57_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="E1:57_2_1_2"/><i><font color="Green">E65</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="lattice8.html#E13:57_2"><i><font color="Green">E61</font></i></a>, <a class="ref" href="relat_1.html#T30">RELAT_1:30</a></i>;<br/><a NAME="E3:57_2_1_2"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="lattice5.html#K4">.</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="lattice8.html#E8:57_2"><i><font color="Green">E57</font></i></a>, <a class="txt" href="lattice8.html#E1:57_2_1_2"><i><font color="Green">E65</font></i></a></i>;<br/><a NAME="E4:57_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="E5:57_2_1_2"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="lattice5.html#K4">.</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="lattice8.html#E3:57_2_1_2"><i><font color="Green">E66</font></i></a>, <a class="ref" href="orders_2.html#T26">ORDERS_2:26</a></i>;<br/><b>hence </b><a NAME="E6:57_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="lattice8.html#E11:57_2"><i><font color="Green">E59</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E4:57_2_1"/><b>then </b>
<font color="Maroon">c<sub>13</sub></font> <a href="lattice8.html#R1" target="_self">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="E5:57_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="lattice8.html#R1" target="_self">c=</a> <font color="Maroon">c<sub>14</sub></font>
 
<b>by </b><i><a class="txt" href="lattice8.html#E2:57_2_1"><i><font color="Green">E64</font></i></a>, <a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/>
<b>hence </b><a NAME="E6:57_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="lattice8.html#R1" target="_self">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="E16:57_2"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="lattice8.html#R1" target="_self">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="57_2_2"><b>proof </b></a><div class="add">

<a NAME="E1:57_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="57_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="E1:57_2_2_1"/><i><font color="Green">E65</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="lattice8.html#E14:57_2"><i><font color="Green">E62</font></i></a>, <a class="ref" href="relat_1.html#T30">RELAT_1:30</a></i>;<br/>
<a NAME="E3:57_2_2_1"/><i><font color="Green">E66</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="lattice8.html#E14:57_2"><i><font color="Green">E62</font></i></a>, <a class="txt" href="lattice8.html#E1:57_2_2_1"><i><font color="Green">E65</font></i></a>, <a class="ref" href="relat_1.html#T30">RELAT_1:30</a></i>;<br/>
<a NAME="E4:57_2_2_1"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="lattice5.html#K4">.</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="lattice8.html#E11:57_2"><i><font color="Green">E59</font></i></a>, <a class="txt" href="lattice8.html#E1:57_2_2_1"><i><font color="Green">E65</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> 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="E6:57_2_2_1"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="lattice5.html#K4">.</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="E7:57_2_2_1"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="lattice5.html#K4">.</a> <font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="lattice5.html#K4">.</a> <font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>10</sub></font>
 <b>and </b><br/><a NAME="E8:57_2_2_1"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="lattice5.html#K4">.</a> <font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="ref" href="lattice8.html#T34" target="_self">Th34</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>18</sub></font> ) );<br/>
<a NAME="E9:57_2_2_1"/><i><font color="Green">E71</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> 4 &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="lattice8.html#E49"><i><font color="Green">Lemma42</font></i></a></i>;<br/>
<a NAME="E10:57_2_2_1"/><i><font color="Green">E72</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> 4 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="57_2_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>21</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:57_2_2_1_1"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 4
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_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="E1:57_2_2_1_1_1"/>
( <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> 2 or <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> 3 or <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> 4 )
 </span><b>by </b><i><a class="txt" href="lattice8.html#E1:57_2_2_1_1"><i><font color="Green">E73</font></i></a>, <a class="txt" href="lattice8.html#E49"><i><font color="Green">Lemma42</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_2_2_1_1_1_1"><b>suppose </b></a><a NAME="E1:57_2_2_1_1_1_1"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><b>take </b><font color="Maroon">c<sub>22</sub></font> = <font color="Maroon">c<sub>17</sub></font>;<br/><b>thus </b><a NAME="E2:57_2_2_1_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>22</sub></font>]
 <b>by </b><i><a class="txt" href="lattice8.html#E1:57_2_2_1_1_1_1"><i><font color="Green">E74</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_2_2_1_1_1_2"><b>suppose </b></a><a NAME="E1:57_2_2_1_1_1_2"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> 2
 ;<br/><div class="add"><b>take </b><font color="Maroon">c<sub>22</sub></font> = <font color="Maroon">c<sub>19</sub></font>;<br/><b>thus </b><a NAME="E2:57_2_2_1_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>22</sub></font>]
 <b>by </b><i><a class="txt" href="lattice8.html#E1:57_2_2_1_1_1_2"><i><font color="Green">E74</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_2_2_1_1_1_3"><b>suppose </b></a><a NAME="E1:57_2_2_1_1_1_3"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> 3
 ;<br/><div class="add"><b>take </b><font color="Maroon">c<sub>22</sub></font> = <font color="Maroon">c<sub>20</sub></font>;<br/><b>thus </b><a NAME="E2:57_2_2_1_1_1_3"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>22</sub></font>]
 <b>by </b><i><a class="txt" href="lattice8.html#E1:57_2_2_1_1_1_3"><i><font color="Green">E74</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_2_2_1_1_1_4"><b>suppose </b></a><a NAME="E1:57_2_2_1_1_1_4"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> 4
 ;<br/><div class="add"><b>take </b><font color="Maroon">c<sub>22</sub></font> = <font color="Maroon">c<sub>18</sub></font>;<br/><b>thus </b><a NAME="E2:57_2_2_1_1_1_4"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>22</sub></font>]
 <b>by </b><i><a class="txt" href="lattice8.html#E1:57_2_2_1_1_1_4"><i><font color="Green">E74</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E11:57_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="finseq_1.html#K4">dom</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> 4 &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> 4 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="lattice8.html#E9:57_2_2_1"><i><font color="Green">E71</font></i></a>, <a class="txt" href="lattice8.html#E10:57_2_2_1"><i><font color="Green">E72</font></i></a>);<br/></i>
<b>then consider </b><font color="Maroon">c<sub>21</sub></font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E13:57_2_2_1"/><i><font color="Green">E73</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> 4
 <b>and </b><br/><a NAME="E14:57_2_2_1"/><i><font color="Green">E74</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> 4 holds <br/>( ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 1 implies <font color="Maroon">c<sub>21</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>21</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>21</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>21</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="E15:57_2_2_1"/>
 <a href="finseq_1.html#K2">Seg</a> 4 <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( 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#R1">&lt;=</a> 4 ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#D1">FINSEQ_1:def 1</a></i>;<br/>
<a NAME="E16:57_2_2_1"/><b>then </b><i><font color="Green">E75</font></i>: 
( 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 4 &amp; 2 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 4 &amp; 3 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 4 &amp; 4 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 4 )
 
;<br/>
<a NAME="E17:57_2_2_1"/><i><font color="Green">E76</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> 4
 
<b>by </b><i><a class="txt" href="lattice8.html#E13:57_2_2_1"><i><font color="Green">E73</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E18:57_2_2_1"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> <a href="funct_1.html#K1">.</a> 1
 
<b>by </b><i><a class="txt" href="lattice8.html#E14:57_2_2_1"><i><font color="Green">E74</font></i></a>, <a class="txt" href="lattice8.html#E16:57_2_2_1"><i><font color="Green">E75</font></i></a></i>;<br/>
<i><font color="Green">E78</font></i>: <font color="Maroon">c<sub>16</sub></font> = 
<font color="Maroon">c<sub>21</sub></font> <a href="funct_1.html#K1">.</a> 4
<b>by </b><i><a class="txt" href="lattice8.html#E14:57_2_2_1"><i><font color="Green">E74</font></i></a>, <a class="txt" href="lattice8.html#E16:57_2_2_1"><i><font color="Green">E75</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>21</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>21</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="lattice8.html#E13:57_2_2_1"><i><font color="Green">E73</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
;<br/>
<a NAME="E20:57_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>21</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>21</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>21</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="57_2_2_1_3"><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="E1:57_2_2_1_3"/><i><font color="Green">E79</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>22</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>21</sub></font> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_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="E1:57_2_2_1_3_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 )
 </span><b>by </b><i><a class="txt" href="lattice8.html#E17:57_2_2_1"><i><font color="Green">E76</font></i></a>, <a class="txt" href="lattice8.html#E1:57_2_2_1_3"><i><font color="Green">E79</font></i></a>, <a class="txt" href="lattice8.html#E50"><i><font color="Green">Lemma43</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_2_2_1_3_1_1"><b>suppose </b></a><a NAME="E1:57_2_2_1_3_1_1"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:57_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="lattice8.html#E5:57_2"><i><font color="Green">E55</font></i></a>, <a class="txt" href="lattice8.html#E6:57_2_2_1"><i><font color="Green">E68</font></i></a></i>;<br/><a NAME="E3:57_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>21</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="lattice8.html#E14:57_2_2_1"><i><font color="Green">E74</font></i></a>, <a class="txt" href="lattice8.html#E16:57_2_2_1"><i><font color="Green">E75</font></i></a></i>;<br/><a NAME="E4:57_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>21</sub></font> <a href="funct_1.html#K1">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>21</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="lattice8.html#E14:57_2_2_1"><i><font color="Green">E74</font></i></a>, <a class="txt" href="lattice8.html#E16:57_2_2_1"><i><font color="Green">E75</font></i></a></i>;<br/><b>hence </b><a NAME="E5:57_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>21</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>22</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="lattice8.html#E1:57_2_2_1_3_1_1"><i><font color="Green">E80</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="57_2_2_1_3_1_2"><b>suppose </b></a><a NAME="E1:57_2_2_1_3_1_2"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> 2
 ;<br/><div class="add"><a NAME="E2:57_2_2_1_3_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="lattice5.html#K4">.</a> <font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>9</sub></font> <a href="orders_2.html#R3">&lt;=</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> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="lattice8.html#E4:57_2_2_1"><i><font color="Green">E67</font></i></a>, <a class="ref" href="waybel_1.html#T3">WAYBEL_1:3</a></i>;<br/><a NAME="E3:57_2_2_1_3_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="lattice5.html#K4">.</a> <font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>9</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>10</sub></font> <a href="lattice3.html#K13">"\/"</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>9</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="lattice3.html#T14">LATTICE3:14</a></i>;<br/><a NAME="E4:57_2_2_1_3_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="lattice5.html#K4">.</a> <font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>9</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_5.html#T1">YELLOW_5:1</a></i>;<br/><a NAME="E5:57_2_2_1_3_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="lattice5.html#K4">.</a> <font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="lattice3.html#K12">"/\"</a> <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#K12">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="waybel_1.html#T2">WAYBEL_1:2</a></i>;<br/><a NAME="E6:57_2_2_1_3_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="lattice5.html#K4">.</a> <font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>10</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="lattice3.html#T18">LATTICE3:18</a></i>;<br/><a NAME="E7:57_2_2_1_3_1_2"/><b>then </b>
<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="lattice8.html#E8:57_2"><i><font color="Green">E57</font></i></a>, <a class="txt" href="lattice8.html#E7:57_2_2_1"><i><font color="Green">E69</font></i></a></i>;<br/><a NAME="E8:57_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>21</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="lattice8.html#E14:57_2_2_1"><i><font color="Green">E74</font></i></a>, <a class="txt" href="lattice8.html#E16:57_2_2_1"><i><font color="Green">E75</font></i></a></i>;<br/><a NAME="E9:57_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>21</sub></font> <a href="funct_1.html#K1">.</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>21</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="lattice8.html#E14:57_2_2_1"><i><font color="Green">E74</font></i></a>, <a class="txt" href="lattice8.html#E16:57_2_2_1"><i><font color="Green">E75</font></i></a></i>;<br/><b>hence </b><a NAME="E10:57_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>21</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>22</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="lattice8.html#E1:57_2_2_1_3_1_2"><i><font color="Green">E80</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="57_2_2_1_3_1_3"><b>suppose </b></a><a NAME="E1:57_2_2_1_3_1_3"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> 3
 ;<br/><div class="add"><a NAME="E2:57_2_2_1_3_1_3"/>
<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>18</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="lattice8.html#E5:57_2"><i><font color="Green">E55</font></i></a>, <a class="txt" href="lattice8.html#E8:57_2_2_1"><i><font color="Green">E70</font></i></a></i>;<br/><a NAME="E3:57_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>21</sub></font> <a href="funct_1.html#K1">.</a> 3</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>12</sub></font>
 <b>by </b><i><a class="txt" href="lattice8.html#E14:57_2_2_1"><i><font color="Green">E74</font></i></a>, <a class="txt" href="lattice8.html#E16:57_2_2_1"><i><font color="Green">E75</font></i></a></i>;<br/><a NAME="E4:57_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>21</sub></font> <a href="funct_1.html#K1">.</a> 3</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>21</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="lattice8.html#E14:57_2_2_1"><i><font color="Green">E74</font></i></a>, <a class="txt" href="lattice8.html#E16:57_2_2_1"><i><font color="Green">E75</font></i></a></i>;<br/><b>hence </b><a NAME="E5:57_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>21</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>22</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>22</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="lattice8.html#E1:57_2_2_1_3_1_3"><i><font color="Green">E80</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="E21:57_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="lattice8.html#E3:57_2_2_1"><i><font color="Green">E66</font></i></a>, <a class="txt" href="lattice8.html#E17:57_2_2_1"><i><font color="Green">E76</font></i></a>, <a class="txt" href="lattice8.html#E18:57_2_2_1"><i><font color="Green">E77</font></i></a>, <a class="txt" href="lattice8.html#E19:57_2_2_1"><i><font color="Green">E78</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="E2:57_2_2"/>
<font color="Maroon">c<sub>14</sub></font> <a href="lattice8.html#R1" target="_self">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">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">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">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="lattice8.html#E2:57_2"><i><font color="Green">E53</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">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">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="lattice8.html#E4:57_2"><i><font color="Green">E54</font></i></a>, <a class="txt" href="lattice8.html#E7:57_2"><i><font color="Green">E56</font></i></a>, <a class="ref" href="lattice5.html#T10">LATTICE5:10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="lattice5.html#K5">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="lattice8.html#E10:57_2"><i><font color="Green">E58</font></i></a>, <a class="txt" href="lattice8.html#E15:57_2"><i><font color="Green">E63</font></i></a>, <a class="txt" href="lattice8.html#E16:57_2"><i><font color="Green">E64</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">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="E18:57_2"/>
 <a href="lattice5.html#K5">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="lattice8.html#E1:57_2"><i><font color="Green">E52</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">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">EqRelLATT</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <b>by </b><i><a class="ref" href="lattice5.html#T16">LATTICE5:16</a></i>;<br/>
<a NAME="E14:57"/><i><font color="Green">E52</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="E15:57"/><i><font color="Green">E53</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">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="partfun1.html#K6">id</a> <font color="Maroon">c<sub>7</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_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="E2:57_3"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font> <a href="lattice8.html#R3" target="_self">is_extension2_of</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font>
 <b>and </b><br/><a NAME="E3:57_3"/><i><font color="Green">E55</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="lattice8.html#D12" target="_self">Def12</a></i>;<br/>
<a NAME="E4:57_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">BasicDF</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="txt" href="lattice8.html#E3:57"><i><font color="Green">E47</font></i></a>, <a class="txt" href="lattice8.html#E3:57_3"><i><font color="Green">E55</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">QuadrSeq</a> of  <a href="lattice5.html#K17">BasicDF</a> <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E6:57_3"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a>  <a href="lattice8.html#K7" target="_self">NextSet2</a> <span class="p1">(<span class="default"><a href="lattice5.html#K17">BasicDF</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E7:57_3"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="funct_2.html#R1">=</a>  <a href="lattice8.html#K9" target="_self">NextDelta2</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="lattice8.html#E2:57_3"><i><font color="Green">E54</font></i></a>, <a class="ref" href="lattice8.html#D11" target="_self">Def11</a></i>;<br/>
<a NAME="E8:57_3"/><i><font color="Green">E58</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="lattice8.html#K9" target="_self">NextDelta2</a> <font color="Maroon">c<sub>14</sub></font>
 
<b>by </b><i><a class="txt" href="lattice8.html#E3:57_3"><i><font color="Green">E55</font></i></a>, <a class="txt" href="lattice8.html#E7:57_3"><i><font color="Green">E57</font></i></a>, <a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>;<br/>
<a NAME="E9:57_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="E10:57_3"/><b>then </b><i><font color="Green">E59</font></i>: 
 <a href="lattice8.html#K9" target="_self">NextDelta2</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="lattice8.html#E8:57_3"><i><font color="Green">E58</font></i></a>, <a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/>
<a NAME="E11:57_3"/><i><font color="Green">E60</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="lattice8.html#K7" target="_self">NextSet2</a> <span class="p1">(<span class="default"><a href="lattice5.html#K17">BasicDF</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="lattice8.html#E3:57_3"><i><font color="Green">E55</font></i></a>, <a class="txt" href="lattice8.html#E6:57_3"><i><font color="Green">E56</font></i></a>, <a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>;<br/>
<a NAME="E12:57_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="E13:57_3"/><b>then </b><i><font color="Green">E61</font></i>: 
 <a href="lattice8.html#K7" target="_self">NextSet2</a> <span class="p1">(<span class="default"><a href="lattice5.html#K17">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="lattice8.html#E11:57_3"><i><font color="Green">E60</font></i></a>, <a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/>
<a NAME="E14:57_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">DistEsti</a> <span class="p1">(<span class="default"><a href="lattice5.html#K17">BasicDF</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="lattice8.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E15:57_3"/><b>then </b>
 <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a>  <a href="lattice5.html#K8">DistEsti</a> <span class="p1">(<span class="default"><a href="lattice5.html#K17">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="E16:57_3"/><b>then </b><i><font color="Green">E62</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">LATTICE5:28</a></i>;<br/>
<a NAME="E17:57_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="E18:57_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">BasicDF</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="lattice5.html#K4">.</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">LATTICE5:def 14</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="E20:57_3"/><i><font color="Green">E63</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">BasicDF</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="lattice5.html#K4">.</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="E21:57_3"/>
 <a href="lattice8.html#K3" target="_self">ConsecutiveSet2</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="lattice8.html#T15" target="_self">Th15</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>19</sub></font> =  <a href="lattice8.html#K4" target="_self">Quadr2</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="E23:57_3"/><i><font color="Green">E64</font></i>: 
 <a href="lattice8.html#K4" target="_self">Quadr2</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="lattice8.html#E16:57_3"><i><font color="Green">E62</font></i></a>, <a class="txt" href="lattice8.html#E20:57_3"><i><font color="Green">E63</font></i></a>, <a class="ref" href="lattice8.html#D7" target="_self">Def7</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="E25:57_3"/><i><font color="Green">E65</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="E26:57_3"/><i><font color="Green">E66</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">.</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">LATTICE5:def 9</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>20</sub></font>
;<br/>

<a NAME="E27:57_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="lattice8.html#E14:57"><i><font color="Green">E52</font></i></a>, <a class="txt" href="lattice8.html#E25:57_3"><i><font color="Green">E65</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E28:57_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">Image</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/>

<i><font color="Green">E67</font></i>:  <a href="lattice8.html#K1" target="_self">new_set2</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> = 
 <a href="lattice8.html#K1" target="_self">new_set2</a> <span class="p1">(<span class="default"><a href="lattice8.html#K3" target="_self">ConsecutiveSet2</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="lattice8.html#T15" target="_self">Th15</a></i>
<br/>.= 
 <a href="lattice8.html#K3" target="_self">ConsecutiveSet2</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="lattice8.html#T16" target="_self">Th16</a></i>
;<br/>
<a NAME="E30:57_3"/><i><font color="Green">E68</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">DistEsti</a> <span class="p1">(<span class="default"><a href="lattice5.html#K17">BasicDF</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="lattice8.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E31:57_3"/><b>then </b><i><font color="Green">E69</font></i>: 
 <a href="lattice8.html#K1" target="_self">new_set2</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="lattice8.html#K7" target="_self">NextSet2</a> <span class="p1">(<span class="default"><a href="lattice5.html#K17">BasicDF</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="lattice8.html#E29:57_3"><i><font color="Green">E67</font></i></a>, <a class="ref" href="lattice8.html#T22" target="_self">Th22</a></i>;<br/>
<a NAME="E32:57_3"/><i><font color="Green">E70</font></i>: 
(  <a href="lattice8.html#K3" target="_self">ConsecutiveSet2</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="lattice8.html#K6" target="_self">ConsecutiveDelta2</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">BasicDF</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="ref" href="lattice8.html#T15" target="_self">Th15</a>, <a class="ref" href="lattice8.html#T19" target="_self">Th19</a></i>;<br/>
<i><font color="Green">E71</font></i>:  <a href="lattice8.html#K6" target="_self">ConsecutiveDelta2</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <span class="p1">(<span class="default"><a href="lattice5.html#K11">BiFun</a> <span class="p2">(<span class="default"><a href="lattice8.html#K6" target="_self">ConsecutiveDelta2</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="lattice8.html#K3" target="_self">ConsecutiveSet2</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="lattice8.html#K4" target="_self">Quadr2</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="lattice8.html#T20" target="_self">Th20</a></i>
<br/>.= 
 <a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <span class="p1">(<span class="default"><a href="lattice5.html#K17">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="lattice8.html#E32:57_3"><i><font color="Green">E70</font></i></a>, <a class="ref" href="lattice5.html#D16">LATTICE5:def 16</a></i>
;<br/>
<a NAME="E34:57_3"/>
 <a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <span class="p1">(<span class="default"><a href="lattice5.html#K17">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="lattice8.html#K9" target="_self">NextDelta2</a> <font color="Maroon">c<sub>14</sub></font>
 
<b>by </b><i><a class="txt" href="lattice8.html#E30:57_3"><i><font color="Green">E68</font></i></a>, <a class="txt" href="lattice8.html#E33:57_3"><i><font color="Green">E71</font></i></a>, <a class="ref" href="lattice8.html#T25" target="_self">Th25</a></i>;<br/>
<a NAME="E35:57_3"/><b>then </b><i><font color="Green">E72</font></i>: 
 <a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <span class="p1">(<span class="default"><a href="lattice5.html#K17">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="lattice8.html#E10:57_3"><i><font color="Green">E59</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E36:57_3"/><i><font color="Green">E73</font></i>: 
 <a href="lattice8.html#K1" target="_self">new_set2</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="lattice8.html#E13:57_3"><i><font color="Green">E61</font></i></a>, <a class="txt" href="lattice8.html#E31:57_3"><i><font color="Green">E69</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E37:57_3"/><i><font color="Green">E74</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <span class="p2">(<span class="default"><a href="lattice5.html#K17">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="lattice8.html#K1" target="_self">new_set2</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="lattice8.html#K1" target="_self">new_set2</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="E38:57_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="tarski.html#K2">{</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#K2">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<a NAME="E39:57_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="tarski.html#K2">{</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#K2">}</a></span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E40:57_3"/><b>then </b><i><font color="Green">E75</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="lattice8.html#K1" target="_self">new_set2</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 
;<br/>
<a NAME="E41:57_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="tarski.html#K2">{</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#K2">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<a NAME="E42:57_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="tarski.html#K2">{</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#K2">}</a></span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E43:57_3"/><b>then </b><i><font color="Green">E76</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="lattice8.html#K1" target="_self">new_set2</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 
;<br/>
<a NAME="E44:57_3"/><b>then </b><i><font color="Green">E77</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <span class="p2">(<span class="default"><a href="lattice5.html#K17">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="lattice8.html#E37:57_3"><i><font color="Green">E74</font></i></a>, <a class="txt" href="lattice8.html#E40:57_3"><i><font color="Green">E75</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E45:57_3"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> <a href="mcart_1.html#K11">`4</a> 
 
<b>by </b><i><a class="txt" href="lattice8.html#E23:57_3"><i><font color="Green">E64</font></i></a>, <a class="ref" href="mcart_1.html#D11">MCART_1:def 11</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="lattice8.html#E36:57_3"><i><font color="Green">E73</font></i></a>, <a class="txt" href="lattice8.html#E40:57_3"><i><font color="Green">E75</font></i></a>, <a class="txt" href="lattice8.html#E43:57_3"><i><font color="Green">E76</font></i></a></i>;<br/>
<font color="Maroon">c<sub>8</sub></font> <a href="lattice5.html#K4">.</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <span class="p2">(<span class="default"><a href="lattice5.html#K17">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="lattice8.html#E35:57_3"><i><font color="Green">E72</font></i></a>, <a class="txt" href="lattice8.html#E44:57_3"><i><font color="Green">E77</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <span class="p2">(<span class="default"><a href="lattice5.html#K17">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/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lattice5.html#K17">BasicDF</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="lattice5.html#K4">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="mcart_1.html#K8">`1</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="mcart_1.html#K10">`3</a> </span>)</span></span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="mcart_1.html#K11">`4</a> </span>)</span>
<b>by </b><i><a class="ref" href="lattice8.html#D5" target="_self">Def5</a></i>
;<br/>
<a NAME="E48:57_3"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="lattice5.html#K4">.</a> <font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>22</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>18</sub></font>
 
<b>by </b><i><a class="txt" href="lattice8.html#E45:57_3"><i><font color="Green">E78</font></i></a>, <a class="ref" href="yellow_0.html#T23">YELLOW_0:23</a></i>;<br/>
<a NAME="E49:57_3"/><b>then </b><i><font color="Green">E79</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="lattice8.html#E26:57_3"><i><font color="Green">E66</font></i></a></i>;<br/>
<a NAME="E50:57_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="57_3_4"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:57_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="E2:57_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="E3:57_3_4"/>
contradiction
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E51:57_3"/>
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="partfun1.html#K6">id</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="lattice8.html#E49:57_3"><i><font color="Green">E79</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="E16:57"/>
<font color="Maroon">c<sub>7</sub></font> is  non <a href="realset1.html#V1">trivial</a>  <a href="hidden.html#M1">set</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_4"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_4_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:57_4_1"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">c<sub>7</sub></font> is <a href="realset1.html#V1">trivial</a>
 ;<br/><b>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="E3:57_4_1"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="lattice8.html#E1:57_4_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="realset1.html#D4">REALSET1:def 4</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>11</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="E5:57_4_1"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="lattice5.html#K2">Image</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="partfun1.html#K6">id</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="txt" href="lattice8.html#E15:57"><i><font color="Green">E53</font></i></a></i>;<br/><a NAME="E6:57_4_1"/>
 <a href="relat_1.html#K3">field</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="lattice8.html#E3:57_4_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="eqrel_1.html#T16">EQREL_1:16</a></i>;<br/><a NAME="E7:57_4_1"/><b>then </b><i><font color="Green">E57</font></i>: 
 <a href="partfun1.html#K6">id</a> <font color="Maroon">c<sub>7</sub></font> <a href="lattice8.html#R1" target="_self">c=</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="lattice8.html#E3:57_4_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="relat_2.html#T17">RELAT_2:17</a></i>;<br/><a NAME="E8:57_4_1"/><i><font color="Green">E58</font></i>: 
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K1">}</a></span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K1">}</a></span></span><a href="zfmisc_1.html#K2">:]</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#K4">[</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T35">ZFMISC_1:35</a></i>;<br/><a NAME="E9:57_4_1"/>
 <a href="partfun1.html#K6">id</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>10</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#K4">[</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="sysrel.html#T30">SYSREL:30</a></i>;<br/><b>hence </b><a NAME="E10:57_4_1"/>
contradiction
 <b>by </b><i><a class="txt" href="lattice8.html#E3:57_4_1"><i><font color="Green">E55</font></i></a>, <a class="txt" href="lattice8.html#E5:57_4_1"><i><font color="Green">E56</font></i></a>, <a class="txt" href="lattice8.html#E7:57_4_1"><i><font color="Green">E57</font></i></a>, <a class="txt" href="lattice8.html#E8:57_4_1"><i><font color="Green">E58</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:57_4"/>
<font color="Maroon">c<sub>7</sub></font> is  non <a href="realset1.html#V1">trivial</a>  <a href="hidden.html#M1">set</a> 
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>7</sub></font> as  non <a href="realset1.html#V1">trivial</a>  <a href="hidden.html#M1">set</a>  ;<br/>
<b>take </b>
<font color="Maroon">c<sub>10</sub></font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice8.html#D3" target="_self">LATTICE8:def 3</a><br/>

<b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>9</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">EqRelLATT</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> ;<br/>
<b>take </b>
<font color="Maroon">c<sub>11</sub></font>
;<br/>

<b>thus </b><a NAME="E19:57"/>
<font color="Maroon">c<sub>11</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i><a class="txt" href="lattice8.html#E11:57"><i><font color="Green">E51</font></i></a>, <a class="ref" href="lattice5.html#T17">LATTICE5:17</a></i>;<br/>

<b>reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>8</sub></font> as   <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> ;<br/>
<b>thus </b><a NAME="E21:57"/>
 <a href="lattice5.html#K2">Image</a> <font color="Maroon">c<sub>11</sub></font> is <a href="lattice8.html#V1" target="_self">finitely_typed</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_5"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon">c<sub>10</sub></font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice8.html#D2" target="_self">LATTICE8:def 2</a><br/>

<b>thus </b><a NAME="E1:57_5"/>
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 <span class="p1">(<span class="default"><a href="lattice5.html#K2">Image</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> holds <br/><font color="Olive">b<sub>1</sub></font> is   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">c<sub>10</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_5_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>13</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:57_5_1"/>
<font color="Maroon">c<sub>13</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">Image</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>
 ;<br/>

<a NAME="E2:57_5_1"/><b>then </b>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>14</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:57_5_1"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>11</sub></font>
 <b>and </b><br/><a NAME="E5:57_5_1"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</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/>
<b>reconsider </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>14</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="lattice8.html#E4:57_5_1"><i><font color="Green">E54</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>16</sub></font> being   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">c<sub>10</sub></font><b> such that </b><br/><a NAME="E8:57_5_1"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>15</sub></font>
 <b>and </b><br/><a NAME="E9:57_5_1"/>
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>10</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>16</sub></font> iff <font color="Maroon">c<sub>12</sub></font> <a href="lattice5.html#K4">.</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>15</sub></font> )
 <b>by </b><i><a class="ref" href="lattice5.html#D9">LATTICE5:def 9</a></i>;<br/>
<b>thus </b><a NAME="E10:57_5_1"/>
<font color="Maroon">c<sub>13</sub></font> is   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="lattice8.html#E5:57_5_1"><i><font color="Green">E55</font></i></a>, <a class="txt" href="lattice8.html#E8:57_5_1"><i><font color="Green">E56</font></i></a></i>;<br/>


</div><b>end;</b></div>

<b>take </b>
2 <a href="nat_1.html#K1">+</a> 2
;<br/>

<b>thus </b><a NAME="E2:57_5"/>
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>10</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">Image</a> <font color="Maroon">c<sub>11</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">Image</a> <font color="Maroon">c<sub>11</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>10</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> 2 <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">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="lattice8.html#T35" target="_self">Th35</a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E22:57"/>
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>10</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">Image</a> <font color="Maroon">c<sub>11</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="partfun1.html#K6">id</a> <font color="Maroon">c<sub>10</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_6"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">c<sub>13</sub></font> being  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> , <font color="Maroon">c<sub>14</sub></font> being   <a href="lattice5.html#NM4">distance_function</a> of <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>15</sub></font> being  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> , <font color="Maroon">c<sub>16</sub></font> being   <a href="lattice5.html#NM4">distance_function</a> of <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E2:57_6"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font> <a href="lattice8.html#R3" target="_self">is_extension2_of</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font>
 <b>and </b><br/><a NAME="E3:57_6"/><i><font color="Green">E55</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>13</sub></font>,<font color="Maroon">c<sub>14</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>15</sub></font>,<font color="Maroon">c<sub>16</sub></font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i><a class="ref" href="lattice8.html#D12" target="_self">Def12</a></i>;<br/>
<a NAME="E4:57_6"/>
( <font color="Maroon">c<sub>13</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>14</sub></font> <a href="funct_2.html#R1">=</a>  <a href="lattice5.html#K17">BasicDF</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="txt" href="lattice8.html#E3:57"><i><font color="Green">E47</font></i></a>, <a class="txt" href="lattice8.html#E3:57_6"><i><font color="Green">E55</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>17</sub></font> being    <a href="lattice5.html#M1">QuadrSeq</a> of  <a href="lattice5.html#K17">BasicDF</a> <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E6:57_6"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a>  <a href="lattice8.html#K7" target="_self">NextSet2</a> <span class="p1">(<span class="default"><a href="lattice5.html#K17">BasicDF</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E7:57_6"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="funct_2.html#R1">=</a>  <a href="lattice8.html#K9" target="_self">NextDelta2</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="txt" href="lattice8.html#E2:57_6"><i><font color="Green">E54</font></i></a>, <a class="ref" href="lattice8.html#D11" target="_self">Def11</a></i>;<br/>
<a NAME="E8:57_6"/><i><font color="Green">E58</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="lattice8.html#K9" target="_self">NextDelta2</a> <font color="Maroon">c<sub>17</sub></font>
 
<b>by </b><i><a class="txt" href="lattice8.html#E3:57_6"><i><font color="Green">E55</font></i></a>, <a class="txt" href="lattice8.html#E7:57_6"><i><font color="Green">E57</font></i></a>, <a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>;<br/>
<a NAME="E9:57_6"/>
<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="E10:57_6"/><b>then </b><i><font color="Green">E59</font></i>: 
 <a href="lattice8.html#K9" target="_self">NextDelta2</a> <font color="Maroon">c<sub>17</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>12</sub></font>
 
<b>by </b><i><a class="txt" href="lattice8.html#E8:57_6"><i><font color="Green">E58</font></i></a>, <a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/>
<a NAME="E11:57_6"/><i><font color="Green">E60</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="lattice8.html#K7" target="_self">NextSet2</a> <span class="p1">(<span class="default"><a href="lattice5.html#K17">BasicDF</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="lattice8.html#E3:57_6"><i><font color="Green">E55</font></i></a>, <a class="txt" href="lattice8.html#E6:57_6"><i><font color="Green">E56</font></i></a>, <a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>;<br/>
<a NAME="E12:57_6"/>
<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="E13:57_6"/><b>then </b><i><font color="Green">E61</font></i>: 
 <a href="lattice8.html#K7" target="_self">NextSet2</a> <span class="p1">(<span class="default"><a href="lattice5.html#K17">BasicDF</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="txt" href="lattice8.html#E11:57_6"><i><font color="Green">E60</font></i></a>, <a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/>
<a NAME="E14:57_6"/>
 <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">DistEsti</a> <span class="p1">(<span class="default"><a href="lattice5.html#K17">BasicDF</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="lattice8.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E15:57_6"/><b>then </b>
 <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a>  <a href="lattice5.html#K8">DistEsti</a> <span class="p1">(<span class="default"><a href="lattice5.html#K17">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="E16:57_6"/><b>then </b><i><font color="Green">E62</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>17</sub></font>
 
<b>by </b><i><a class="ref" href="lattice5.html#T28">LATTICE5:28</a></i>;<br/>
<a NAME="E17:57_6"/><b>then </b>
<font color="Maroon">c<sub>17</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>17</sub></font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E18:57_6"/><b>then </b>
<font color="Maroon">c<sub>17</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">BasicDF</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="lattice5.html#K4">.</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">LATTICE5:def 14</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>18</sub></font>, <font color="Maroon">c<sub>19</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>20</sub></font>, <font color="Maroon">c<sub>21</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="E20:57_6"/><i><font color="Green">E63</font></i>: 
( <font color="Maroon">c<sub>17</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>18</sub></font>,<font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>21</sub></font></span><a href="domain_1.html#K5">]</a></span> &amp; <span class="p1">(<span class="default"><a href="lattice5.html#K17">BasicDF</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="lattice5.html#K4">.</a> <font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>20</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>21</sub></font> )
 ;<br/>
<a NAME="E21:57_6"/>
 <a href="lattice8.html#K3" target="_self">ConsecutiveSet2</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="lattice8.html#T15" target="_self">Th15</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>22</sub></font> =  <a href="lattice8.html#K4" target="_self">Quadr2</a> <font color="Maroon">c<sub>17</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="E23:57_6"/><i><font color="Green">E64</font></i>: 
 <a href="lattice8.html#K4" target="_self">Quadr2</a> <font color="Maroon">c<sub>17</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>18</sub></font>,<font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>21</sub></font></span><a href="domain_1.html#K5">]</a></span>
 
<b>by </b><i><a class="txt" href="lattice8.html#E16:57_6"><i><font color="Green">E62</font></i></a>, <a class="txt" href="lattice8.html#E20:57_6"><i><font color="Green">E63</font></i></a>, <a class="ref" href="lattice8.html#D7" target="_self">Def7</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>23</sub></font> being   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">c<sub>10</sub></font><b> such that </b><br/><a NAME="E25:57_6"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>21</sub></font>
 <b>and </b><br/><a NAME="E26:57_6"/><i><font color="Green">E66</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>10</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>23</sub></font> iff <font color="Maroon">c<sub>12</sub></font> <a href="lattice5.html#K4">.</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>21</sub></font> )
 <b>by </b><i><a class="ref" href="lattice5.html#D9">LATTICE5:def 9</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>23</sub></font>
;<br/>

<a NAME="E27:57_6"/>
<font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="txt" href="lattice8.html#E14:57"><i><font color="Green">E52</font></i></a>, <a class="txt" href="lattice8.html#E25:57_6"><i><font color="Green">E65</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E28:57_6"/>
<font color="Maroon">c<sub>23</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">Image</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/>

<i><font color="Green">E67</font></i>:  <a href="lattice8.html#K1" target="_self">new_set2</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> = 
 <a href="lattice8.html#K1" target="_self">new_set2</a> <span class="p1">(<span class="default"><a href="lattice8.html#K3" target="_self">ConsecutiveSet2</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="lattice8.html#T15" target="_self">Th15</a></i>
<br/>.= 
 <a href="lattice8.html#K3" target="_self">ConsecutiveSet2</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="lattice8.html#T16" target="_self">Th16</a></i>
;<br/>
<a NAME="E30:57_6"/><i><font color="Green">E68</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">DistEsti</a> <span class="p1">(<span class="default"><a href="lattice5.html#K17">BasicDF</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="lattice8.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E31:57_6"/><b>then </b><i><font color="Green">E69</font></i>: 
 <a href="lattice8.html#K1" target="_self">new_set2</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="lattice8.html#K7" target="_self">NextSet2</a> <span class="p1">(<span class="default"><a href="lattice5.html#K17">BasicDF</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="lattice8.html#E29:57_6"><i><font color="Green">E67</font></i></a>, <a class="ref" href="lattice8.html#T22" target="_self">Th22</a></i>;<br/>
<a NAME="E32:57_6"/><i><font color="Green">E70</font></i>: 
(  <a href="lattice8.html#K3" target="_self">ConsecutiveSet2</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="lattice8.html#K6" target="_self">ConsecutiveDelta2</a> <font color="Maroon">c<sub>17</sub></font>,<a href="xboole_0.html#K1">{}</a>  <a href="funct_2.html#R1">=</a>  <a href="lattice5.html#K17">BasicDF</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="ref" href="lattice8.html#T15" target="_self">Th15</a>, <a class="ref" href="lattice8.html#T19" target="_self">Th19</a></i>;<br/>
<i><font color="Green">E71</font></i>:  <a href="lattice8.html#K6" target="_self">ConsecutiveDelta2</a> <font color="Maroon">c<sub>17</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <span class="p1">(<span class="default"><a href="lattice5.html#K11">BiFun</a> <span class="p2">(<span class="default"><a href="lattice8.html#K6" target="_self">ConsecutiveDelta2</a> <font color="Maroon">c<sub>17</sub></font>,<a href="xboole_0.html#K1">{}</a> </span>)</span>,<span class="p2">(<span class="default"><a href="lattice8.html#K3" target="_self">ConsecutiveSet2</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="lattice8.html#K4" target="_self">Quadr2</a> <font color="Maroon">c<sub>17</sub></font>,<a href="xboole_0.html#K1">{}</a> </span>)</span>
<b>by </b><i><a class="ref" href="lattice8.html#T20" target="_self">Th20</a></i>
<br/>.= 
 <a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <span class="p1">(<span class="default"><a href="lattice5.html#K17">BasicDF</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>22</sub></font>
<b>by </b><i><a class="txt" href="lattice8.html#E32:57_6"><i><font color="Green">E70</font></i></a>, <a class="ref" href="lattice5.html#D16">LATTICE5:def 16</a></i>
;<br/>
<a NAME="E34:57_6"/>
 <a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <span class="p1">(<span class="default"><a href="lattice5.html#K17">BasicDF</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>22</sub></font> <a href="tarski.html#R1">c=</a>  <a href="lattice8.html#K9" target="_self">NextDelta2</a> <font color="Maroon">c<sub>17</sub></font>
 
<b>by </b><i><a class="txt" href="lattice8.html#E30:57_6"><i><font color="Green">E68</font></i></a>, <a class="txt" href="lattice8.html#E33:57_6"><i><font color="Green">E71</font></i></a>, <a class="ref" href="lattice8.html#T25" target="_self">Th25</a></i>;<br/>
<a NAME="E35:57_6"/><b>then </b><i><font color="Green">E72</font></i>: 
 <a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <span class="p1">(<span class="default"><a href="lattice5.html#K17">BasicDF</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>22</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>12</sub></font>
 
<b>by </b><i><a class="txt" href="lattice8.html#E10:57_6"><i><font color="Green">E59</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E36:57_6"/><i><font color="Green">E73</font></i>: 
 <a href="lattice8.html#K1" target="_self">new_set2</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>10</sub></font>
 
<b>by </b><i><a class="txt" href="lattice8.html#E13:57_6"><i><font color="Green">E61</font></i></a>, <a class="txt" href="lattice8.html#E31:57_6"><i><font color="Green">E69</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E37:57_6"/><i><font color="Green">E74</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <span class="p2">(<span class="default"><a href="lattice5.html#K17">BasicDF</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>22</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="lattice8.html#K1" target="_self">new_set2</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="lattice8.html#K1" target="_self">new_set2</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="E38:57_6"/>
<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#K2">{</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#K2">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<a NAME="E39:57_6"/><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="tarski.html#K2">{</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#K2">}</a></span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E40:57_6"/><b>then </b><i><font color="Green">E75</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="lattice8.html#K1" target="_self">new_set2</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 
;<br/>
<a NAME="E41:57_6"/>
<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="tarski.html#K2">{</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#K2">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<a NAME="E42:57_6"/><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="tarski.html#K2">{</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#K2">}</a></span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E43:57_6"/><b>then </b><i><font color="Green">E76</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="lattice8.html#K1" target="_self">new_set2</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 
;<br/>
<a NAME="E44:57_6"/><b>then </b><i><font color="Green">E77</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <span class="p2">(<span class="default"><a href="lattice5.html#K17">BasicDF</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>22</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="lattice8.html#E37:57_6"><i><font color="Green">E74</font></i></a>, <a class="txt" href="lattice8.html#E40:57_6"><i><font color="Green">E75</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E45:57_6"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> <a href="mcart_1.html#K11">`4</a> 
 
<b>by </b><i><a class="txt" href="lattice8.html#E23:57_6"><i><font color="Green">E64</font></i></a>, <a class="ref" href="mcart_1.html#D11">MCART_1:def 11</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>24</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>25</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>10</sub></font> <b>by </b><i><a class="txt" href="lattice8.html#E36:57_6"><i><font color="Green">E73</font></i></a>, <a class="txt" href="lattice8.html#E40:57_6"><i><font color="Green">E75</font></i></a>, <a class="txt" href="lattice8.html#E43:57_6"><i><font color="Green">E76</font></i></a></i>;<br/>
<font color="Maroon">c<sub>12</sub></font> <a href="lattice5.html#K4">.</a> <font color="Maroon">c<sub>24</sub></font>,<font color="Maroon">c<sub>25</sub></font> = 
<font color="Maroon">c<sub>12</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <span class="p2">(<span class="default"><a href="lattice5.html#K17">BasicDF</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>22</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="lattice8.html#E35:57_6"><i><font color="Green">E72</font></i></a>, <a class="txt" href="lattice8.html#E44:57_6"><i><font color="Green">E77</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <span class="p2">(<span class="default"><a href="lattice5.html#K17">BasicDF</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>22</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/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lattice5.html#K17">BasicDF</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="lattice5.html#K4">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="mcart_1.html#K8">`1</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="mcart_1.html#K10">`3</a> </span>)</span></span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="mcart_1.html#K11">`4</a> </span>)</span>
<b>by </b><i><a class="ref" href="lattice8.html#D5" target="_self">Def5</a></i>
;<br/>
<a NAME="E48:57_6"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="lattice5.html#K4">.</a> <font color="Maroon">c<sub>24</sub></font>,<font color="Maroon">c<sub>25</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>21</sub></font>
 
<b>by </b><i><a class="txt" href="lattice8.html#E45:57_6"><i><font color="Green">E78</font></i></a>, <a class="ref" href="yellow_0.html#T23">YELLOW_0:23</a></i>;<br/>
<a NAME="E49:57_6"/><b>then </b><i><font color="Green">E79</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>23</sub></font>
 
<b>by </b><i><a class="txt" href="lattice8.html#E26:57_6"><i><font color="Green">E66</font></i></a></i>;<br/>
<a NAME="E50:57_6"/>
<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="57_6_4"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:57_6_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="E2:57_6_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="E3:57_6_4"/>
contradiction
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E51:57_6"/>
<font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="partfun1.html#K6">id</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="lattice8.html#E49:57_6"><i><font color="Green">E79</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="E23:57"/>
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>10</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">Image</a> <font color="Maroon">c<sub>11</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">Image</a> <font color="Maroon">c<sub>11</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>10</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> 2 <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">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="lattice8.html#T35" target="_self">Th35</a></i>;<br/>
<b>hence </b><a NAME="E24:57"/>
 <a href="lattice5.html#K3">type_of</a> <span class="p1">(<span class="default"><a href="lattice5.html#K2">Image</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 2
 <b>by </b><i><a class="txt" href="lattice8.html#E15:57"><i><font color="Green">E53</font></i></a>, <a class="ref" href="lattice5.html#T15">LATTICE5:15</a></i>;<br/>


</div>
