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

<b>let </b><font color="Maroon">L</font> be  <a href="yellow_0.html#V1">lower-bounded</a> <a href="waybel_0.html#NM8">LATTICE</a>;<br/>

<b>set </b><font color="Maroon">A</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>;<br/>
<b>set </b><font color="Maroon">D</font> =  <a href="lattice5.html#K17" target="_self">BasicDF</a> <font color="Maroon">L</font>;<br/>
<a NAME="E1:84"/><i><font color="Green">E29</font></i>: 
 <a href="lattice5.html#K17" target="_self">BasicDF</a> <font color="Maroon">L</font> is <a href="funct_2.html#V2">onto</a>
 
<b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">S</font> being    <a href="lattice5.html#M2" target="_self">ExtensionSeq</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>, <a href="lattice5.html#K17" target="_self">BasicDF</a> <font color="Maroon">L</font>;<br/>
<b>set </b><font color="Maroon">FS</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">S</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : verum </span>}</span> ;<br/>
<a NAME="E3:84"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">S</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">L</font>,<span class="p2">(<span class="default"><a href="lattice5.html#K17" target="_self">BasicDF</a> <font color="Maroon">L</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
 
<b>by </b><i/>;<br/>
<a NAME="E4:84"/><b>then </b><i><font color="Green">E39</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">S</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">L</font>
 
<b>by </b><i><a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>;<br/>
<a NAME="E5:84"/>
<span class="p1">(<span class="default"><font color="Maroon">S</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">S</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : verum </span>}</span> 
 
;<br/>
<a NAME="E6:84"/><b>then </b><i><font color="Green">E42</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</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">S</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : verum </span>}</span> 
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/>
<b>consider </b><font color="Maroon">xx</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E8:84"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">xx</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</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">FS</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">S</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">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>, </i>;<br/>
<b>reconsider </b><font color="Maroon">FD</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">S</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : verum </span>}</span>  as   <a href="lattice5.html#NM4">distance_function</a> of <font color="Maroon">FS</font>,<font color="Maroon">L</font> <b>by </b><i/>;<br/>
<a NAME="E11:84"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">FD</font> is <a href="funct_2.html#V2">onto</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_1"><b>proof </b></a><div class="add">

<a NAME="E1:84_1"/><i><font color="Green">E50</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="lattice5.html#K17" target="_self">BasicDF</a> <font color="Maroon">L</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 
<b>by </b><i>, <a class="ref" href="funct_2.html#D3">FUNCT_2:def 3</a></i>;<br/>
<a NAME="E2:84_1"/>
for <font color="Olive">w</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">w</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> holds <br/> ex <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">FS</font>,<font color="Maroon">FS</font></span><a href="zfmisc_1.html#K2">:]</a></span> &amp; <font color="Olive">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">FD</font> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_1_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:84_1_1"/>
<font color="Maroon">w</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 ;<br/>

<b>then consider </b><font color="Maroon">z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:84_1_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">z</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">L</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>and </b><br/><a NAME="E4:84_1_1"/><i><font color="Green">E52</font></i>: 
<span class="p1">(<span class="default"><a href="lattice5.html#K17" target="_self">BasicDF</a> <font color="Maroon">L</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w</font>
 <b>by </b><i>, <a class="ref" href="funct_2.html#T17">FUNCT_2:17</a></i>;<br/>
<b>take </b>
<font color="Maroon">z</font>
;<br/>

<a NAME="E5:84_1_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">S</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">L</font>,<span class="p2">(<span class="default"><a href="lattice5.html#K17" target="_self">BasicDF</a> <font color="Maroon">L</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
 
<b>by </b><i><a class="txt" href="lattice5.html#E5:84_1_1"><i><font color="Green">E53</font></i></a></i>;<br/>
<a NAME="E6:84_1_1"/><b>then </b><i><font color="Green">E54</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">S</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="E7:84_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">S</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">S</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : verum </span>}</span> 
 
;<br/>
<a NAME="E8:84_1_1"/><b>then </b>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">FS</font>
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/>
<a NAME="E9:84_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">L</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</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">FS</font>,<font color="Maroon">FS</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="E10:84_1_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">FS</font>,<font color="Maroon">FS</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i/>;<br/>

<a NAME="E11:84_1_1"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="lattice5.html#K17" target="_self">BasicDF</a> <font color="Maroon">L</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E12:84_1_1"/><i><font color="Green">E56</font></i>: 
 <a href="lattice5.html#K17" target="_self">BasicDF</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">S</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="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>;<br/>
<a NAME="E13:84_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">S</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">S</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : verum </span>}</span> 
 
;<br/>
<a NAME="E14:84_1_1"/><b>then </b>
 <a href="lattice5.html#K17" target="_self">BasicDF</a> <font color="Maroon">L</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">FD</font>
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/>
<b>hence </b><a NAME="E15:84_1_1"/>
<font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">FD</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font>
 <b>by </b><i>, , <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:84_1"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">FD</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T16">FUNCT_2:16</a></i>;<br/>
<b>hence </b><a NAME="E4:84_1"/>
<font color="Maroon">FD</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="E12:84"/>
 <a href="lattice5.html#K5" target="_self">alpha</a> <font color="Maroon">FD</font> is <a href="waybel_0.html#V20">join-preserving</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</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">f</font> =  <a href="lattice5.html#K5" target="_self">alpha</a> <font color="Maroon">FD</font>;<br/>
<a NAME="E1:84_2"/><i><font color="Green">E57</font></i>: 
 <a href="yellow_0.html#R1">ex_sup_of</a> <span class="p1">(<span class="default"><a href="lattice5.html#K5" target="_self">alpha</a> <font color="Maroon">FD</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">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K7">}</a></span>, <a href="lattice5.html#K1" target="_self">EqRelLATT</a> <font color="Maroon">FS</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T17">YELLOW_0:17</a></i>;<br/>
<a NAME="E2:84_2"/><i><font color="Green">E58</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="lattice5.html#K5" target="_self">alpha</a> <font color="Maroon">FD</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</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">e1</font> being   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">FS</font><b> such that </b><br/><a NAME="E4:84_2"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">e1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="lattice5.html#K5" target="_self">alpha</a> <font color="Maroon">FD</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font>
 <b>and </b><br/><a NAME="E5:84_2"/><i><font color="Green">E60</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">FS</font> holds <br/> ( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">e1</font> iff <font color="Maroon">FD</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Olive">x</font>,<font color="Olive">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">a</font> )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">e2</font> being   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">FS</font><b> such that </b><br/><a NAME="E7:84_2"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">e2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="lattice5.html#K5" target="_self">alpha</a> <font color="Maroon">FD</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font>
 <b>and </b><br/><a NAME="E8:84_2"/><i><font color="Green">E62</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">FS</font> holds <br/> ( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">e2</font> iff <font color="Maroon">FD</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Olive">x</font>,<font color="Olive">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">b</font> )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">e3</font> being   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">FS</font><b> such that </b><br/><a NAME="E10:84_2"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">e3</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="lattice5.html#K5" target="_self">alpha</a> <font color="Maroon">FD</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span>
 <b>and </b><br/><a NAME="E11:84_2"/><i><font color="Green">E64</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">FS</font> holds <br/> ( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">e3</font> iff <font color="Maroon">FD</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Olive">x</font>,<font color="Olive">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font> )
 <b>by </b><i/>;<br/>
<a NAME="E12:84_2"/><i><font color="Green">E65</font></i>: 
 <a href="relat_1.html#K3">field</a> <font color="Maroon">e1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">FS</font>
 
<b>by </b><i><a class="ref" href="orders_1.html#T97">ORDERS_1:97</a></i>;<br/>
<a NAME="E13:84_2"/><i><font color="Green">E66</font></i>: 
 <a href="relat_1.html#K3">field</a> <font color="Maroon">e2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">FS</font>
 
<b>by </b><i><a class="ref" href="orders_1.html#T97">ORDERS_1:97</a></i>;<br/>
<a NAME="E14:84_2"/><i><font color="Green">E69</font></i>: 
 <a href="relat_1.html#K3">field</a> <font color="Maroon">e3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">FS</font>
 
<b>by </b><i><a class="ref" href="orders_1.html#T97">ORDERS_1:97</a></i>;<br/>
<a NAME="E15:84_2"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">e1</font> <a href="eqrel_1.html#K5">"\/"</a> <font color="Maroon">e2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">e3</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_2_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:84_2_1_1"/><i><font color="Green">E72</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">e1</font>
 ;<br/><b>then reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font>, <font color="Maroon">y'</font> = <font color="Maroon">y</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">FS</font> <b>by </b><i>, <a class="ref" href="relat_1.html#T30">RELAT_1:30</a></i>;<br/><a NAME="E3:84_2_1_1"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">FD</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">x'</font>,<font color="Maroon">y'</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="txt" href="lattice5.html#E3:84_2_1_1"><i><font color="Green">E73</font></i></a>, </i>;<br/><a NAME="E4:84_2_1_1"/>
<font color="Maroon">a</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T22">YELLOW_0:22</a></i>;<br/><a NAME="E5:84_2_1_1"/><b>then </b>
<font color="Maroon">FD</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">x'</font>,<font color="Maroon">y'</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font>
 <b>by </b><i>, <a class="ref" href="orders_2.html#T26">ORDERS_2:26</a></i>;<br/><b>hence </b><a NAME="E6:84_2_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">e3</font>
 <b>by </b><i><a class="txt" href="lattice5.html#E16:84_2"><i><font color="Green">E77</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E2:84_2_1"/><b>then </b><i><font color="Green">E74</font></i>: 
<font color="Maroon">e1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">e3</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="84_2_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:84_2_1_2"/><i><font color="Green">E75</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">e2</font>
 ;<br/><b>then reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font>, <font color="Maroon">y'</font> = <font color="Maroon">y</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">FS</font> <b>by </b><i>, <a class="ref" href="relat_1.html#T30">RELAT_1:30</a></i>;<br/><a NAME="E3:84_2_1_2"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">FD</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">x'</font>,<font color="Maroon">y'</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="txt" href="lattice5.html#E1:84_2_1_2"><i><font color="Green">E75</font></i></a>, </i>;<br/><a NAME="E4:84_2_1_2"/>
<font color="Maroon">b</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">b</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T22">YELLOW_0:22</a></i>;<br/><a NAME="E5:84_2_1_2"/><b>then </b>
<font color="Maroon">FD</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">x'</font>,<font color="Maroon">y'</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">b</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font>
 <b>by </b><i>, <a class="ref" href="orders_2.html#T26">ORDERS_2:26</a></i>;<br/><b>hence </b><a NAME="E6:84_2_1_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">e3</font>
 <b>by </b><i><a class="txt" href="lattice5.html#E16:84_2"><i><font color="Green">E77</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E4:84_2_1"/><b>then </b>
<font color="Maroon">e2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">e3</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#D3">RELAT_1:def 3</a></i>;<br/>
<a NAME="E5:84_2_1"/><b>then </b>
<font color="Maroon">e1</font> <a href="eqrel_1.html#K3">\/</a> <font color="Maroon">e2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">e3</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/>
<b>hence </b><a NAME="E6:84_2_1"/>
<font color="Maroon">e1</font> <a href="eqrel_1.html#K5">"\/"</a> <font color="Maroon">e2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">e3</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="E16:84_2"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">e3</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">e1</font> <a href="eqrel_1.html#K5">"\/"</a> <font color="Maroon">e2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_2_2"><b>proof </b></a><div class="add">

<a NAME="E1:84_2_2"/>
for <font color="Olive">u</font>, <font color="Olive">v</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">u</font>,<font color="Olive">v</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">e3</font> holds <br/><span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">u</font>,<font color="Olive">v</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">e1</font> <a href="eqrel_1.html#K5">"\/"</a> <font color="Maroon">e2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_2_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">u</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">v</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:84_2_2_1"/><i><font color="Green">E78</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">u</font>,<font color="Maroon">v</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">e3</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">u</font>, <font color="Maroon">y</font> = <font color="Maroon">v</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">FS</font> <b>by </b><i><a class="txt" href="lattice5.html#E3:84_2_2_1"><i><font color="Green">E80</font></i></a>, <a class="ref" href="relat_1.html#T30">RELAT_1:30</a></i>;<br/>
<a NAME="E3:84_2_2_1"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FS</font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E3:84_2_2_1"><i><font color="Green">E80</font></i></a>, , <a class="ref" href="relat_1.html#T30">RELAT_1:30</a></i>;<br/>
<a NAME="E4:84_2_2_1"/>
<font color="Maroon">FD</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E16:84_2"><i><font color="Green">E77</font></i></a>, </i>;<br/>
<b>then consider </b><font color="Maroon">z1</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">FS</font>, <font color="Maroon">z2</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">FS</font>, <font color="Maroon">z3</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">FS</font><b> such that </b><br/><a NAME="E6:84_2_2_1"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">FD</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">x</font>,<font color="Maroon">z1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 <b>and </b><br/><a NAME="E7:84_2_2_1"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">FD</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">z2</font>,<font color="Maroon">z3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 <b>and </b><br/><a NAME="E8:84_2_2_1"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">FD</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">z1</font>,<font color="Maroon">z2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 <b>and </b><br/><a NAME="E9:84_2_2_1"/><i><font color="Green">E87</font></i>: 
<font color="Maroon">FD</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">z3</font>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="txt" href="lattice5.html#E8:84_2"><i><font color="Green">E62</font></i></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> ] <b>means </b>( ( <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">x</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">z1</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">z2</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">z3</font> ) &amp; ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> 5 implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> ) );<br/>
<a NAME="E10:84_2_2_1"/><i><font color="Green">E88</font></i>: 
for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">w1</font>, <font color="Olive">w2</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 5 &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">m</font>,<font color="Olive">w1</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">m</font>,<font color="Olive">w2</font>] holds <br/><font color="Olive">w1</font> <a href="hidden.html#R1">=</a> <font color="Olive">w2</font>
 
<b>by </b><i><a class="txt" href="lattice5.html#E4:84_2"><i><font color="Green">E59</font></i></a></i>;<br/>
<a NAME="E11:84_2_2_1"/><i><font color="Green">E89</font></i>: 
for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 5 holds <br/> ex <font color="Olive">w</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">m</font>,<font color="Olive">w</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_2_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:84_2_2_1_1"/><i><font color="Green">E91</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 5
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="84_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="E1:84_2_2_1_1_1"/>
( <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 2 or <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 3 or <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 4 or <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 5 )
 </span><b>by </b><i><a class="txt" href="lattice5.html#E1:84_2_2_1_1"><i><font color="Green">E91</font></i></a>, <a class="txt" href="lattice5.html#E4:84_2"><i><font color="Green">E59</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="84_2_2_1_1_1_1"><b>suppose </b></a><a NAME="E1:84_2_2_1_1_1_1"/><i><font color="Green">E92</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><b>take </b><font color="Maroon">w</font> = <font color="Maroon">x</font>;<br/><b>thus </b><a NAME="E2:84_2_2_1_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">m</font>,<font color="Maroon">w</font>]
 <b>by </b><i><a class="txt" href="lattice5.html#E1:84_2_2_1_1_1_1"><i><font color="Green">E92</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="84_2_2_1_1_1_2"><b>suppose </b></a><a NAME="E1:84_2_2_1_1_1_2"/><i><font color="Green">E93</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 2
 ;<br/><div class="add"><b>take </b><font color="Maroon">w</font> = <font color="Maroon">z1</font>;<br/><b>thus </b><a NAME="E2:84_2_2_1_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">m</font>,<font color="Maroon">w</font>]
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="84_2_2_1_1_1_3"><b>suppose </b></a><a NAME="E1:84_2_2_1_1_1_3"/><i><font color="Green">E95</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 3
 ;<br/><div class="add"><b>take </b><font color="Maroon">w</font> = <font color="Maroon">z2</font>;<br/><b>thus </b><a NAME="E2:84_2_2_1_1_1_3"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">m</font>,<font color="Maroon">w</font>]
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="84_2_2_1_1_1_4"><b>suppose </b></a><a NAME="E1:84_2_2_1_1_1_4"/><i><font color="Green">E96</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 4
 ;<br/><div class="add"><b>take </b><font color="Maroon">w</font> = <font color="Maroon">z3</font>;<br/><b>thus </b><a NAME="E2:84_2_2_1_1_1_4"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">m</font>,<font color="Maroon">w</font>]
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="84_2_2_1_1_1_5"><b>suppose </b></a><a NAME="E1:84_2_2_1_1_1_5"/><i><font color="Green">E97</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 5
 ;<br/><div class="add"><b>take </b><font color="Maroon">w</font> = <font color="Maroon">y</font>;<br/><b>thus </b><a NAME="E2:84_2_2_1_1_1_5"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">m</font>,<font color="Maroon">w</font>]
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E12:84_2_2_1"/>
 ex <font color="Olive">p</font> being  <a href="finseq_1.html#NM1">FinSequence</a> st <br/>(  <a href="finsop_1.html#K5">dom</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> 5 &amp; ( for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 5 holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>,<font color="Olive">p</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font>] ) )
 
<b>from </b><i><a class="ref" href="finseq_1.html#S1">FINSEQ_1:sch 1</a>(<a class="txt" href="lattice5.html#E11:84_2_2_1"><i><font color="Green">E89</font></i></a>, );<br/></i>
<b>then consider </b><font color="Maroon">h</font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E14:84_2_2_1"/><i><font color="Green">E98</font></i>: 
 <a href="finsop_1.html#K5">dom</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> 5
 <b>and </b><br/><a NAME="E15:84_2_2_1"/><i><font color="Green">E99</font></i>: 
for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 5 holds <br/>( ( <font color="Olive">m</font> <a href="hidden.html#R1">=</a> 1 implies <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Olive">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> ) &amp; ( <font color="Olive">m</font> <a href="hidden.html#R1">=</a> 2 implies <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Olive">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z1</font> ) &amp; ( <font color="Olive">m</font> <a href="hidden.html#R1">=</a> 3 implies <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Olive">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z2</font> ) &amp; ( <font color="Olive">m</font> <a href="hidden.html#R1">=</a> 4 implies <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Olive">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z3</font> ) &amp; ( <font color="Olive">m</font> <a href="hidden.html#R1">=</a> 5 implies <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Olive">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> ) )
 ;<br/>
<a NAME="E16:84_2_2_1"/><i><font color="Green">E100</font></i>: 
( 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 5 &amp; 2 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 5 &amp; 3 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 5 &amp; 4 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 5 &amp; 5 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 5 )
 
;<br/>
<a NAME="E17:84_2_2_1"/><i><font color="Green">E101</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> 5
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E18:84_2_2_1"/><i><font color="Green">E103</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> 1
 
<b>by </b><i>, </i>;<br/>
<i><font color="Green">E104</font></i>: <font color="Maroon">v</font> = 
<font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> 5
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">h</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">h</font></span>)</span>
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
;<br/>
<a NAME="E20:84_2_2_1"/>
for <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j</font> &amp; <font color="Olive">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h</font> holds <br/><span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Olive">j</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">e1</font> <a href="eqrel_1.html#K3">\/</a> <font color="Maroon">e2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_2_2_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">j</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:84_2_2_1_3"/><i><font color="Green">E105</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">h</font> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="84_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="E1:84_2_2_1_3_1"/>
( <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 3 or <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 2 or <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 4 )
 </span><b>by </b><i>, , <a class="txt" href="lattice5.html#E2:84_2"><i><font color="Green">E58</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="84_2_2_1_3_1_1"><b>suppose </b></a><a NAME="E1:84_2_2_1_3_1_1"/><i><font color="Green">E106</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:84_2_2_1_3_1_1"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">z1</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">e1</font>
 <b>by </b><i>, </i>;<br/><a NAME="E3:84_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">h</font> <a href="funct_1.html#K1">.</a> 1</span>)</span>,<font color="Maroon">z1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">e1</font>
 <b>by </b><i>, </i>;<br/><a NAME="E4:84_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">h</font> <a href="funct_1.html#K1">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">h</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">e1</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E5:84_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">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">j</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">e1</font> <a href="eqrel_1.html#K3">\/</a> <font color="Maroon">e2</font>
 <b>by </b><i><a class="txt" href="lattice5.html#E14:84_2_2_1"><i><font color="Green">E98</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="84_2_2_1_3_1_2"><b>suppose </b></a><a NAME="E1:84_2_2_1_3_1_2"/><i><font color="Green">E107</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 3
 ;<br/><div class="add"><a NAME="E2:84_2_2_1_3_1_2"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">z2</font>,<font color="Maroon">z3</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">e1</font>
 <b>by </b><i>, </i>;<br/><a NAME="E3:84_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">h</font> <a href="funct_1.html#K1">.</a> 3</span>)</span>,<font color="Maroon">z3</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">e1</font>
 <b>by </b><i>, </i>;<br/><a NAME="E4:84_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">h</font> <a href="funct_1.html#K1">.</a> 3</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">h</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">e1</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E5:84_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">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">j</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">e1</font> <a href="eqrel_1.html#K3">\/</a> <font color="Maroon">e2</font>
 <b>by </b><i><a class="txt" href="lattice5.html#E14:84_2_2_1"><i><font color="Green">E98</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="84_2_2_1_3_1_3"><b>suppose </b></a><a NAME="E1:84_2_2_1_3_1_3"/><i><font color="Green">E108</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 2
 ;<br/><div class="add"><a NAME="E2:84_2_2_1_3_1_3"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">z1</font>,<font color="Maroon">z2</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">e2</font>
 <b>by </b><i>, <a class="txt" href="lattice5.html#E9:84_2_2_1"><i><font color="Green">E87</font></i></a></i>;<br/><a NAME="E3:84_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">h</font> <a href="funct_1.html#K1">.</a> 2</span>)</span>,<font color="Maroon">z2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">e2</font>
 <b>by </b><i>, </i>;<br/><a NAME="E4:84_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">h</font> <a href="funct_1.html#K1">.</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">h</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">e2</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E5:84_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">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">j</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">e1</font> <a href="eqrel_1.html#K3">\/</a> <font color="Maroon">e2</font>
 <b>by </b><i><a class="txt" href="lattice5.html#E14:84_2_2_1"><i><font color="Green">E98</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="84_2_2_1_3_1_4"><b>suppose </b></a><a NAME="E1:84_2_2_1_3_1_4"/><i><font color="Green">E110</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 4
 ;<br/><div class="add"><a NAME="E2:84_2_2_1_3_1_4"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">z3</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">e2</font>
 <b>by </b><i>, <a class="txt" href="lattice5.html#E10:84_2_2_1"><i><font color="Green">E88</font></i></a></i>;<br/><a NAME="E3:84_2_2_1_3_1_4"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> 4</span>)</span>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">e2</font>
 <b>by </b><i>, </i>;<br/><a NAME="E4:84_2_2_1_3_1_4"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> 4</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> 5</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">e2</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E5:84_2_2_1_3_1_4"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">j</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">e1</font> <a href="eqrel_1.html#K3">\/</a> <font color="Maroon">e2</font>
 <b>by </b><i><a class="txt" href="lattice5.html#E14:84_2_2_1"><i><font color="Green">E98</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E21:84_2_2_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">u</font>,<font color="Maroon">v</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">e1</font> <a href="eqrel_1.html#K5">"\/"</a> <font color="Maroon">e2</font>
 <b>by </b><i><a class="txt" href="lattice5.html#E8:84_2_2_1"><i><font color="Green">E84</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="E2:84_2_2"/>
<font color="Maroon">e3</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">e1</font> <a href="eqrel_1.html#K5">"\/"</a> <font color="Maroon">e2</font>
 <b>by </b><i><a class="ref" href="relat_1.html#D3">RELAT_1:def 3</a></i>;<br/>


</div><b>end;</b></div>
 <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K5" target="_self">alpha</a> <font color="Maroon">FD</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">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K7">}</a></span></span>)</span> = 
 <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lattice5.html#K5" target="_self">alpha</a> <font color="Maroon">FD</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lattice5.html#K5" target="_self">alpha</a> <font color="Maroon">FD</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font></span>)</span></span><a href="domain_1.html#K7">}</a></span>
<b>by </b><i>, <a class="ref" href="funct_1.html#T118">FUNCT_1:118</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K5" target="_self">alpha</a> <font color="Maroon">FD</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K5" target="_self">alpha</a> <font color="Maroon">FD</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font></span>)</span>
<b>by </b><i><a class="ref" href="yellow_0.html#T41">YELLOW_0:41</a></i>
<br/>.= 
<font color="Maroon">e1</font> <a href="eqrel_1.html#K5">"\/"</a> <font color="Maroon">e2</font>
<b>by </b><i>, , </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="lattice5.html#K5" target="_self">alpha</a> <font color="Maroon">FD</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font></span>)</span>
<b>by </b><i>, , , <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="lattice5.html#K5" target="_self">alpha</a> <font color="Maroon">FD</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">a</font>,<font color="Maroon">b</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="E18:84_2"/>
 <a href="lattice5.html#K5" target="_self">alpha</a> <font color="Maroon">FD</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">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K7">}</a></span>
 <b>by </b><i><a class="txt" href="lattice5.html#E15:84_2"><i><font color="Green">E70</font></i></a>, <a class="ref" href="waybel_0.html#D31">WAYBEL_0:def 31</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">f</font> =  <a href="lattice5.html#K5" target="_self">alpha</a> <font color="Maroon">FD</font> as   <a href="lattice5.html#NM1">Homomorphism</a> of <font color="Maroon">L</font>,<span class="p1">(<span class="default"><a href="lattice5.html#K1" target="_self">EqRelLATT</a> <font color="Maroon">FS</font></span>)</span> <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">FS</font>
;<br/>

<b>take </b>
<font color="Maroon">f</font>
;<br/>

<b>thus </b><a NAME="E14:84"/>
<font color="Maroon">f</font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i>, </i>;<br/>

<a NAME="E15:84"/><i><font color="Green">E112</font></i>: 
 <a href="lattice5.html#K2" target="_self">Image</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K5">subrelstr</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_2.html#D2">YELLOW_2:def 2</a></i>;<br/>
<a NAME="E16:84"/><i><font color="Green">E113</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E17:84"/><i><font color="Green">E114</font></i>: 
 ex <font color="Olive">e</font> being  <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">FS</font> st <br/>( <font color="Olive">e</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="lattice5.html#K2" target="_self">Image</a> <font color="Maroon">f</font></span>)</span> &amp; <font color="Olive">e</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="cqc_sim1.html#K13">id</a> <font color="Maroon">FS</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_3"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">A'</font> being  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> , <font color="Maroon">d'</font> being   <a href="lattice5.html#NM4">distance_function</a> of <font color="Maroon">A'</font>,<font color="Maroon">L</font>, <font color="Maroon">Aq'</font> being  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> , <font color="Maroon">dq'</font> being   <a href="lattice5.html#NM4">distance_function</a> of <font color="Maroon">Aq'</font>,<font color="Maroon">L</font><b> such that </b><br/><a NAME="E2:84_3"/><i><font color="Green">E115</font></i>: 
<font color="Maroon">Aq'</font>,<font color="Maroon">dq'</font> <a href="lattice5.html#R2" target="_self">is_extension_of</a> <font color="Maroon">A'</font>,<font color="Maroon">d'</font>
 <b>and </b><br/><a NAME="E3:84_3"/><i><font color="Green">E116</font></i>: 
( <font color="Maroon">S</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">A'</font>,<font color="Maroon">d'</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">S</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">Aq'</font>,<font color="Maroon">dq'</font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i/>;<br/>
<a NAME="E4:84_3"/>
( <font color="Maroon">A'</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> &amp; <font color="Maroon">d'</font> <a href="funct_2.html#R1">=</a>  <a href="lattice5.html#K17" target="_self">BasicDF</a> <font color="Maroon">L</font> )
 
<b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<b>then consider </b><font color="Maroon">q</font> being    <a href="lattice5.html#M1" target="_self">QuadrSeq</a> of  <a href="lattice5.html#K17" target="_self">BasicDF</a> <font color="Maroon">L</font><b> such that </b><br/><a NAME="E6:84_3"/><i><font color="Green">E117</font></i>: 
<font color="Maroon">Aq'</font> <a href="hidden.html#R1">=</a>  <a href="lattice5.html#K14" target="_self">NextSet</a> <span class="p1">(<span class="default"><a href="lattice5.html#K17" target="_self">BasicDF</a> <font color="Maroon">L</font></span>)</span>
 <b>and </b><br/><a NAME="E7:84_3"/><i><font color="Green">E118</font></i>: 
<font color="Maroon">dq'</font> <a href="funct_2.html#R1">=</a>  <a href="lattice5.html#K16" target="_self">NextDelta</a> <font color="Maroon">q</font>
 <b>by </b><i>, </i>;<br/>
<a NAME="E8:84_3"/><i><font color="Green">E119</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> 1</span>)</span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a>  <a href="lattice5.html#K16" target="_self">NextDelta</a> <font color="Maroon">q</font>
 
<b>by </b><i>, , <a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>;<br/>
<a NAME="E9:84_3"/>
<span class="p1">(<span class="default"><font color="Maroon">S</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">S</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> <a href="mcart_1.html#K2">`2</a> </span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : verum </span>}</span> 
 
;<br/>
<a NAME="E10:84_3"/><b>then </b><i><font color="Green">E120</font></i>: 
 <a href="lattice5.html#K16" target="_self">NextDelta</a> <font color="Maroon">q</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">FD</font>
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/>
<a NAME="E11:84_3"/><i><font color="Green">E122</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> 1</span>)</span> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a>  <a href="lattice5.html#K14" target="_self">NextSet</a> <span class="p1">(<span class="default"><a href="lattice5.html#K17" target="_self">BasicDF</a> <font color="Maroon">L</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>;<br/>
<a NAME="E12:84_3"/>
<span class="p1">(<span class="default"><font color="Maroon">S</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">S</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> <a href="mcart_1.html#K1">`1</a> </span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : verum </span>}</span> 
 
;<br/>
<a NAME="E13:84_3"/><b>then </b><i><font color="Green">E123</font></i>: 
 <a href="lattice5.html#K14" target="_self">NextSet</a> <span class="p1">(<span class="default"><a href="lattice5.html#K17" target="_self">BasicDF</a> <font color="Maroon">L</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">FS</font>
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/>
<a NAME="E14:84_3"/>
 <a href="ordinal1.html#K1">succ</a> <a href="xboole_0.html#K1">{}</a>  <a href="ordinal1.html#R1">c=</a>  <a href="lattice5.html#K8" target="_self">DistEsti</a> <span class="p1">(<span class="default"><a href="lattice5.html#K17" target="_self">BasicDF</a> <font color="Maroon">L</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E15:84_3"/><b>then </b>
 <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a>  <a href="lattice5.html#K8" target="_self">DistEsti</a> <span class="p1">(<span class="default"><a href="lattice5.html#K17" target="_self">BasicDF</a> <font color="Maroon">L</font></span>)</span>
 
<b>by </b><i><a class="ref" href="ordinal1.html#T33">ORDINAL1:33</a></i>;<br/>
<a NAME="E16:84_3"/><b>then </b><i><font color="Green">E124</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">q</font>
 
<b>by </b><i/>;<br/>
<a NAME="E17:84_3"/><b>then </b>
<font color="Maroon">q</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">q</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E18:84_3"/><b>then </b>
<font color="Maroon">q</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">u</font>,<font color="Olive">v</font>,<font color="Olive">a'</font>,<font color="Olive">b'</font></span><a href="domain_1.html#K5">]</a></span> where <font color="Olive">u</font> is    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>, <font color="Olive">v</font> is    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>, <font color="Olive">a'</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Olive">b'</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : <span class="p1">(<span class="default"><a href="lattice5.html#K17" target="_self">BasicDF</a> <font color="Maroon">L</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Olive">u</font>,<font color="Olive">v</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">a'</font> <a href="lattice3.html#K13">"\/"</a> <font color="Olive">b'</font> </span>}</span> 
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">u</font> being    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>, <font color="Maroon">v</font> being    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>, <font color="Maroon">a</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">b</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E20:84_3"/><i><font color="Green">E125</font></i>: 
( <font color="Maroon">q</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">u</font>,<font color="Maroon">v</font>,<font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K5">]</a></span> &amp; <span class="p1">(<span class="default"><a href="lattice5.html#K17" target="_self">BasicDF</a> <font color="Maroon">L</font></span>)</span> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">u</font>,<font color="Maroon">v</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font> )
 ;<br/>
<a NAME="E21:84_3"/>
 <a href="lattice5.html#K9" target="_self">ConsecutiveSet</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</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">L</font>
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">Q</font> =  <a href="lattice5.html#K10" target="_self">Quadr</a> <font color="Maroon">q</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">L</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font></span><a href="zfmisc_1.html#K4">:]</a></span> ;<br/>
<a NAME="E23:84_3"/><i><font color="Green">E127</font></i>: 
 <a href="lattice5.html#K10" target="_self">Quadr</a> <font color="Maroon">q</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">u</font>,<font color="Maroon">v</font>,<font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K5">]</a></span>
 
<b>by </b><i>, , <a class="txt" href="lattice5.html#E3:84"><i><font color="Green">E37</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">e</font> being   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">FS</font><b> such that </b><br/><a NAME="E25:84_3"/><i><font color="Green">E128</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font>
 <b>and </b><br/><a NAME="E26:84_3"/><i><font color="Green">E130</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">FS</font> holds <br/> ( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">e</font> iff <font color="Maroon">FD</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Olive">x</font>,<font color="Olive">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">b</font> )
 <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">e</font>
;<br/>

<a NAME="E27:84_3"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font>
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E28:84_3"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="lattice5.html#K2" target="_self">Image</a> <font color="Maroon">f</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">E131</font></i>:  <a href="lattice5.html#K6" target="_self">new_set</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> = 
 <a href="lattice5.html#K6" target="_self">new_set</a> <span class="p1">(<span class="default"><a href="lattice5.html#K9" target="_self">ConsecutiveSet</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>,<a href="xboole_0.html#K1">{}</a> </span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="lattice5.html#K9" target="_self">ConsecutiveSet</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</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/>
;<br/>
<a NAME="E30:84_3"/><i><font color="Green">E132</font></i>: 
 <a href="ordinal1.html#K1">succ</a> <a href="xboole_0.html#K1">{}</a>  <a href="ordinal1.html#R1">c=</a>  <a href="lattice5.html#K8" target="_self">DistEsti</a> <span class="p1">(<span class="default"><a href="lattice5.html#K17" target="_self">BasicDF</a> <font color="Maroon">L</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E31:84_3"/><b>then </b><i><font color="Green">E133</font></i>: 
 <a href="lattice5.html#K6" target="_self">new_set</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> <a href="tarski.html#R1">c=</a>  <a href="lattice5.html#K14" target="_self">NextSet</a> <span class="p1">(<span class="default"><a href="lattice5.html#K17" target="_self">BasicDF</a> <font color="Maroon">L</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E32:84_3"/><i><font color="Green">E134</font></i>: 
(  <a href="lattice5.html#K9" target="_self">ConsecutiveSet</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</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">L</font> &amp;  <a href="lattice5.html#K13" target="_self">ConsecutiveDelta</a> <font color="Maroon">q</font>,<a href="xboole_0.html#K1">{}</a>  <a href="funct_2.html#R1">=</a>  <a href="lattice5.html#K17" target="_self">BasicDF</a> <font color="Maroon">L</font> )
 
<b>by </b><i>, </i>;<br/>
<i><font color="Green">E135</font></i>:  <a href="lattice5.html#K13" target="_self">ConsecutiveDelta</a> <font color="Maroon">q</font>,<span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <a href="xboole_0.html#K1">{}</a> </span>)</span> = 
 <a href="lattice5.html#K7" target="_self">new_bi_fun</a> <span class="p1">(<span class="default"><a href="lattice5.html#K11" target="_self">BiFun</a> <span class="p2">(<span class="default"><a href="lattice5.html#K13" target="_self">ConsecutiveDelta</a> <font color="Maroon">q</font>,<a href="xboole_0.html#K1">{}</a> </span>)</span>,<span class="p2">(<span class="default"><a href="lattice5.html#K9" target="_self">ConsecutiveSet</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>,<a href="xboole_0.html#K1">{}</a> </span>)</span>,<font color="Maroon">L</font></span>)</span>,<span class="p1">(<span class="default"><a href="lattice5.html#K10" target="_self">Quadr</a> <font color="Maroon">q</font>,<a href="xboole_0.html#K1">{}</a> </span>)</span>
<b>by </b><i><a class="txt" href="lattice5.html#E6:84"><i><font color="Green">E42</font></i></a></i>
<br/>.= 
 <a href="lattice5.html#K7" target="_self">new_bi_fun</a> <span class="p1">(<span class="default"><a href="lattice5.html#K17" target="_self">BasicDF</a> <font color="Maroon">L</font></span>)</span>,<font color="Maroon">Q</font>
<b>by </b><i>, <a class="txt" href="lattice5.html#E4:84"><i><font color="Green">E39</font></i></a></i>
;<br/>
<a NAME="E34:84_3"/>
 <a href="lattice5.html#K7" target="_self">new_bi_fun</a> <span class="p1">(<span class="default"><a href="lattice5.html#K17" target="_self">BasicDF</a> <font color="Maroon">L</font></span>)</span>,<font color="Maroon">Q</font> <a href="tarski.html#R1">c=</a>  <a href="lattice5.html#K16" target="_self">NextDelta</a> <font color="Maroon">q</font>
 
<b>by </b><i>, , <a class="txt" href="lattice5.html#E8:84"><i><font color="Green">E47</font></i></a></i>;<br/>
<a NAME="E35:84_3"/><b>then </b><i><font color="Green">E136</font></i>: 
 <a href="lattice5.html#K7" target="_self">new_bi_fun</a> <span class="p1">(<span class="default"><a href="lattice5.html#K17" target="_self">BasicDF</a> <font color="Maroon">L</font></span>)</span>,<font color="Maroon">Q</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">FD</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E36:84_3"/><i><font color="Green">E137</font></i>: 
 <a href="lattice5.html#K6" target="_self">new_set</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">FS</font>
 
<b>by </b><i>, , <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E37:84_3"/><i><font color="Green">E138</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <span class="p2">(<span class="default"><a href="lattice5.html#K17" target="_self">BasicDF</a> <font color="Maroon">L</font></span>)</span>,<font color="Maroon">Q</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K6" target="_self">new_set</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font></span>)</span>,<span class="p2">(<span class="default"><a href="lattice5.html#K6" target="_self">new_set</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</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="E38:84_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">L</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</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">L</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><span class="p4"><a href="tarski.html#K1">{</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="enumset1.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="enumset1.html#D1">ENUMSET1:def 1</a></i>;<br/>
<a NAME="E39:84_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">L</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">L</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</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">L</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><span class="p4"><a href="tarski.html#K1">{</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="enumset1.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E40:84_3"/><b>then </b><i><font color="Green">E139</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">L</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R2">in</a>  <a href="lattice5.html#K6" target="_self">new_set</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 
;<br/>
<a NAME="E41:84_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">L</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</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">L</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><span class="p4"><a href="tarski.html#K1">{</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="enumset1.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="enumset1.html#D1">ENUMSET1:def 1</a></i>;<br/>
<a NAME="E42:84_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">L</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">L</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</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">L</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><span class="p4"><a href="tarski.html#K1">{</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="enumset1.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E43:84_3"/><b>then </b><i><font color="Green">E140</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">L</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R2">in</a>  <a href="lattice5.html#K6" target="_self">new_set</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 
;<br/>
<a NAME="E44:84_3"/><b>then </b><i><font color="Green">E223</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">L</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">L</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <span class="p2">(<span class="default"><a href="lattice5.html#K17" target="_self">BasicDF</a> <font color="Maroon">L</font></span>)</span>,<font color="Maroon">Q</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>reconsider </b><font color="Maroon">W</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">L</font></span><a href="tarski.html#K1">}</a></span>, <font color="Maroon">V</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">L</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">FS</font> <b>by </b><i>, , </i>;<br/>
<font color="Maroon">FD</font> <a href="lattice5.html#K4" target="_self">.</a> <font color="Maroon">W</font>,<font color="Maroon">V</font> = 
<span class="p1">(<span class="default"><a href="lattice5.html#K7" target="_self">new_bi_fun</a> <span class="p2">(<span class="default"><a href="lattice5.html#K17" target="_self">BasicDF</a> <font color="Maroon">L</font></span>)</span>,<font color="Maroon">Q</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">L</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">L</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i>, , <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>
<br/>.= 
<font color="Maroon">Q</font> <a href="mcart_1.html#K11">`4</a> 
<b>by </b><i/>
<br/>.= 
<font color="Maroon">b</font>
<b>by </b><i>, <a class="ref" href="mcart_1.html#D11">MCART_1:def 11</a></i>
;<br/>
<a NAME="E47:84_3"/><b>then </b><i><font color="Green">E224</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">L</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">L</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">e</font>
 
<b>by </b><i/>;<br/>
<a NAME="E48:84_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">L</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">L</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="84_3_4"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:84_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">L</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">L</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<a NAME="E2:84_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">L</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">L</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="E3:84_3_4"/>
contradiction
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E49:84_3"/>
<font color="Maroon">e</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="cqc_sim1.html#K13">id</a> <font color="Maroon">FS</font>
 <b>by </b><i>, <a class="ref" href="relat_1.html#D10">RELAT_1:def 10</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E18:84"/>
for <font color="Olive">e1</font>, <font color="Olive">e2</font> being  <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">FS</font><br/> for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">e1</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="lattice5.html#K2" target="_self">Image</a> <font color="Maroon">f</font></span>)</span> &amp; <font color="Olive">e2</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="lattice5.html#K2" target="_self">Image</a> <font color="Maroon">f</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Olive">e1</font> <a href="eqrel_1.html#K5">"\/"</a> <font color="Olive">e2</font> holds <br/> ex <font color="Olive">F</font> being non <a href="xboole_0.html#V1">empty</a>  <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">FS</font> st <br/>(  <a href="finseq_1.html#K3">len</a> <font color="Olive">F</font> <a href="hidden.html#R1">=</a> 3 <a href="nat_1.html#K1">+</a> 2 &amp; <font color="Olive">x</font>,<font color="Olive">y</font> <a href="lattice5.html#R1" target="_self">are_joint_by</a> <font color="Olive">F</font>,<font color="Olive">e1</font>,<font color="Olive">e2</font> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E19:84"/>
 <a href="lattice5.html#K3" target="_self">type_of</a> <span class="p1">(<span class="default"><a href="lattice5.html#K2" target="_self">Image</a> <font color="Maroon">f</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 3
 <b>by </b><i>, </i>;<br/>


</div>
