<?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="yellow11.html#V1">modular</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">BasicDF</a> <font color="Maroon">L</font>;<br/>
<a NAME="E1:57"/><i><font color="Green">E39</font></i>: 
 <a href="lattice5.html#K17">BasicDF</a> <font color="Maroon">L</font> is <a href="funct_2.html#V2">onto</a>
 
<b>by </b><i><a class="ref" href="lattice5.html#T43">LATTICE5:43</a></i>;<br/>
<b>consider </b><font color="Maroon">S</font> being    <a href="lattice8.html#M1" target="_self">ExtensionSeq2</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>, <a href="lattice5.html#K17">BasicDF</a> <font color="Maroon">L</font>;<br/>
<a NAME="E3:57"/><i><font color="Green">E42</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">BasicDF</a> <font color="Maroon">L</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
 
<b>by </b><i/>;<br/>
<a NAME="E4:57"/><b>then </b><i><font color="Green">E58</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/>
<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#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : verum </span>}</span> ;<br/>
<a NAME="E5:57"/>
<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#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : verum </span>}</span> 
 
;<br/>
<a NAME="E6:57"/><b>then </b><i><font color="Green">E59</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#M1">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="E8:57"/><i><font color="Green">E60</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#M1">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#M1">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="E11:57"/><i><font color="Green">E61</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="57_1"><b>proof </b></a><div class="add">

<a NAME="E1:57_1"/><i><font color="Green">E62</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="lattice5.html#K17">BasicDF</a> <font color="Maroon">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="lattice8.html#D2" target="_self">Def2</a>, <a class="ref" href="funct_2.html#D3">FUNCT_2:def 3</a></i>;<br/>
<a NAME="E2:57_1"/>
for <font color="Olive">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="57_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="E1:57_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="E3:57_1_1"/><i><font color="Green">E63</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="E4:57_1_1"/><i><font color="Green">E65</font></i>: 
<span class="p1">(<span class="default"><a href="lattice5.html#K17">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="E5:57_1_1"/><i><font color="Green">E66</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">BasicDF</a> <font color="Maroon">L</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
 
<b>by </b><i/>;<br/>
<a NAME="E6:57_1_1"/><b>then </b><i><font color="Green">E69</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="E7:57_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#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : verum </span>}</span> 
 
;<br/>
<a NAME="E8:57_1_1"/><b>then </b>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">FS</font>
 
<b>by </b><i><a class="txt" href="lattice8.html#E5"><i><font color="Green">Lemma56</font></i></a>, <a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/>
<a NAME="E9:57_1_1"/><b>then </b>
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">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="E10:57_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><a class="ref" href="lattice8.html#D3" target="_self">Def3</a></i>;<br/>

<a NAME="E11:57_1_1"/><i><font color="Green">E71</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">BasicDF</a> <font color="Maroon">L</font></span>)</span>
 
<b>by </b><i><a class="ref" href="lattice8.html#D3" target="_self">Def3</a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E12:57_1_1"/><i><font color="Green">E74</font></i>: 
 <a href="lattice5.html#K17">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="txt" href="lattice8.html#E4"><i><font color="Green">Lemma55</font></i></a>, <a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>;<br/>
<a NAME="E13:57_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">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#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : verum </span>}</span> 
 
;<br/>
<a NAME="E14:57_1_1"/><b>then </b>
 <a href="lattice5.html#K17">BasicDF</a> <font color="Maroon">L</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">FD</font>
 
<b>by </b><i><a class="txt" href="lattice8.html#E4:57"><i><font color="Green">E58</font></i></a>, <a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/>
<b>hence </b><a NAME="E15:57_1_1"/>
<font color="Maroon">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="E3:57_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="E4:57_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="E12:57"/>
 <a href="lattice5.html#K5">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="57_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">alpha</a> <font color="Maroon">FD</font>;<br/>
<a NAME="E1:57_2"/><i><font color="Green">E75</font></i>: 
 <a href="yellow_0.html#R1">ex_sup_of</a> <span class="p1">(<span class="default"><a href="lattice5.html#K5">alpha</a> <font color="Maroon">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">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="E2:57_2"/><i><font color="Green">E76</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="lattice5.html#K5">alpha</a> <font color="Maroon">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="E4:57_2"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">e1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="lattice5.html#K5">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="E5:57_2"/><i><font color="Green">E79</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">.</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><a class="ref" href="lattice5.html#D9">LATTICE5:def 9</a></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="E7:57_2"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">e2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="lattice5.html#K5">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="E8:57_2"/><i><font color="Green">E81</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">.</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><a class="ref" href="lattice5.html#D9">LATTICE5:def 9</a></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="E10:57_2"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">e3</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="lattice5.html#K5">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="E11:57_2"/><i><font color="Green">E83</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">.</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><a class="ref" href="lattice5.html#D9">LATTICE5:def 9</a></i>;<br/>
<a NAME="E12:57_2"/><i><font color="Green">E84</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="E13:57_2"/><i><font color="Green">E85</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="E14:57_2"/><i><font color="Green">E86</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="E15:57_2"/><i><font color="Green">E87</font></i>: 
<font color="Maroon">e1</font> <a href="eqrel_1.html#K5">"\/"</a> <font color="Maroon">e2</font> <a href="lattice8.html#R1" target="_self">c=</a> <font color="Maroon">e3</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_2_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">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="E1:57_2_1_1"/><i><font color="Green">E88</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="txt" href="lattice8.html#E8:57"><i><font color="Green">E60</font></i></a>, <a class="ref" href="relat_1.html#T30">RELAT_1:30</a></i>;<br/><a NAME="E3:57_2_1_1"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">FD</font> <a href="lattice5.html#K4">.</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="lattice8.html#E4"><i><font color="Green">Lemma55</font></i></a>, </i>;<br/><a NAME="E4:57_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="E5:57_2_1_1"/><b>then </b>
<font color="Maroon">FD</font> <a href="lattice5.html#K4">.</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="E6:57_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="lattice8.html#E6:57"><i><font color="Green">E59</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E2:57_2_1"/><b>then </b><i><font color="Green">E90</font></i>: 
<font color="Maroon">e1</font> <a href="lattice8.html#R1" target="_self">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="57_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="E1:57_2_1_2"/><i><font color="Green">E91</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="txt" href="lattice8.html#E11:57"><i><font color="Green">E61</font></i></a>, <a class="ref" href="relat_1.html#T30">RELAT_1:30</a></i>;<br/><a NAME="E3:57_2_1_2"/><i><font color="Green">E92</font></i>: 
<font color="Maroon">FD</font> <a href="lattice5.html#K4">.</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>, </i>;<br/><a NAME="E4:57_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="E5:57_2_1_2"/><b>then </b>
<font color="Maroon">FD</font> <a href="lattice5.html#K4">.</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="E6:57_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="lattice8.html#E6:57"><i><font color="Green">E59</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E4:57_2_1"/><b>then </b>
<font color="Maroon">e2</font> <a href="lattice8.html#R1" target="_self">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="E5:57_2_1"/><b>then </b>
<font color="Maroon">e1</font> <a href="eqrel_1.html#K3">\/</a> <font color="Maroon">e2</font> <a href="lattice8.html#R1" target="_self">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="E6:57_2_1"/>
<font color="Maroon">e1</font> <a href="eqrel_1.html#K5">"\/"</a> <font color="Maroon">e2</font> <a href="lattice8.html#R1" target="_self">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="E16:57_2"/><i><font color="Green">E93</font></i>: 
<font color="Maroon">e3</font> <a href="lattice8.html#R1" target="_self">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="57_2_2"><b>proof </b></a><div class="add">

<a NAME="E1:57_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="57_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="E1:57_2_2_1"/><i><font color="Green">E94</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="ref" href="relat_1.html#T30">RELAT_1:30</a></i>;<br/>
<a NAME="E3:57_2_2_1"/><i><font color="Green">E95</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="ref" href="relat_1.html#T30">RELAT_1:30</a></i>;<br/>
<a NAME="E4:57_2_2_1"/><i><font color="Green">E125</font></i>: 
<font color="Maroon">FD</font> <a href="lattice5.html#K4">.</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="lattice8.html#E6:57"><i><font color="Green">E59</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><b> such that </b><br/><a NAME="E6:57_2_2_1"/><i><font color="Green">E126</font></i>: 
<font color="Maroon">FD</font> <a href="lattice5.html#K4">.</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="E7:57_2_2_1"/><i><font color="Green">E127</font></i>: 
<font color="Maroon">FD</font> <a href="lattice5.html#K4">.</a> <font color="Maroon">z1</font>,<font color="Maroon">z2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">FD</font> <a href="lattice5.html#K4">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font></span>)</span> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">b</font>
 <b>and </b><br/><a NAME="E8:57_2_2_1"/><i><font color="Green">E128</font></i>: 
<font color="Maroon">FD</font> <a href="lattice5.html#K4">.</a> <font color="Maroon">z2</font>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 <b>by </b><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">y</font> ) );<br/>
<a NAME="E9:57_2_2_1"/><i><font color="Green">E129</font></i>: 
for <font color="Olive">m</font> being   <a href="subset_1.html#M1">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> 4 &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="lattice8.html#E3:57"><i><font color="Green">E42</font></i></a></i>;<br/>
<a NAME="E10:57_2_2_1"/><i><font color="Green">E130</font></i>: 
for <font color="Olive">m</font> being   <a href="subset_1.html#M1">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> 4 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="57_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#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:57_2_2_1_1"/><i><font color="Green">E131</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 4
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_2_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:57_2_2_1_1_1"/>
( <font color="Maroon">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 )
 </span><b>by </b><i>, <a class="txt" href="lattice8.html#E3:57"><i><font color="Green">E42</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_2_2_1_1_1_1"><b>suppose </b></a><a NAME="E1:57_2_2_1_1_1_1"/><i><font color="Green">E132</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="E2:57_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/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_2_2_1_1_1_2"><b>suppose </b></a><a NAME="E1:57_2_2_1_1_1_2"/><i><font color="Green">E133</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="E2:57_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="57_2_2_1_1_1_3"><b>suppose </b></a><a NAME="E1:57_2_2_1_1_1_3"/><i><font color="Green">E134</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="E2:57_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="57_2_2_1_1_1_4"><b>suppose </b></a><a NAME="E1:57_2_2_1_1_1_4"/><i><font color="Green">E135</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">y</font>;<br/><b>thus </b><a NAME="E2:57_2_2_1_1_1_4"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">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="E11:57_2_2_1"/>
 ex <font color="Olive">p</font> being  <a href="finseq_1.html#NM1">FinSequence</a> st <br/>(  <a href="finseq_1.html#K4">dom</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> 4 &amp; ( for <font color="Olive">k</font> being   <a href="subset_1.html#M1">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> 4 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>(, );<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="E13:57_2_2_1"/><i><font color="Green">E136</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> 4
 <b>and </b><br/><a NAME="E14:57_2_2_1"/><i><font color="Green">E137</font></i>: 
for <font color="Olive">m</font> being   <a href="subset_1.html#M1">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> 4 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">y</font> ) )
 ;<br/>
<a NAME="E15:57_2_2_1"/>
 <a href="finseq_1.html#K2">Seg</a> 4 <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">n</font> where <font color="Olive">n</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> &amp; <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> 4 ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#D1">FINSEQ_1:def 1</a></i>;<br/>
<a NAME="E16:57_2_2_1"/><b>then </b><i><font color="Green">E138</font></i>: 
( 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 4 &amp; 2 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 4 &amp; 3 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 4 &amp; 4 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 4 )
 
;<br/>
<a NAME="E17:57_2_2_1"/><i><font color="Green">E139</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> 4
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E18:57_2_2_1"/><i><font color="Green">E140</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>, <a class="txt" href="lattice8.html#E1:57_2"><i><font color="Green">E75</font></i></a></i>;<br/>
<i><font color="Green">E141</font></i>: <font color="Maroon">v</font> = 
<font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> 4
<b>by </b><i>, <a class="txt" href="lattice8.html#E1:57_2"><i><font color="Green">E75</font></i></a></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="E20:57_2_2_1"/>
for <font color="Olive">j</font> being   <a href="subset_1.html#M1">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="57_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#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:57_2_2_1_3"/><i><font color="Green">E161</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="57_2_2_1_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:57_2_2_1_3_1"/>
( <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 1 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> 3 )
 </span><b>by </b><i><a class="txt" href="lattice8.html#E2:57_2"><i><font color="Green">E76</font></i></a>, <a class="txt" href="lattice8.html#E5:57_2"><i><font color="Green">E79</font></i></a>, </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_2_2_1_3_1_1"><b>suppose </b></a><a NAME="E1:57_2_2_1_3_1_1"/><i><font color="Green">E162</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:57_2_2_1_3_1_1"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">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><a class="txt" href="lattice8.html#E4"><i><font color="Green">Lemma55</font></i></a>, </i>;<br/><a NAME="E3:57_2_2_1_3_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">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>, <a class="txt" href="lattice8.html#E1:57_2"><i><font color="Green">E75</font></i></a></i>;<br/><a NAME="E4:57_2_2_1_3_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">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>, <a class="txt" href="lattice8.html#E1:57_2"><i><font color="Green">E75</font></i></a></i>;<br/><b>hence </b><a NAME="E5:57_2_2_1_3_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">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="lattice8.html#E7:57_2"><i><font color="Green">E80</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_2_2_1_3_1_2"><b>suppose </b></a><a NAME="E1:57_2_2_1_3_1_2"/><i><font color="Green">E163</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 2
 ;<br/><div class="add"><a NAME="E2:57_2_2_1_3_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">FD</font> <a href="lattice5.html#K4">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font> <a href="orders_2.html#R3">&lt;=</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> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font>
 <b>by </b><i>, <a class="ref" href="waybel_1.html#T3">WAYBEL_1:3</a></i>;<br/><a NAME="E3:57_2_2_1_3_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">FD</font> <a href="lattice5.html#K4">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">b</font> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font></span>)</span>
 <b>by </b><i><a class="ref" href="lattice3.html#T14">LATTICE3:14</a></i>;<br/><a NAME="E4:57_2_2_1_3_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">FD</font> <a href="lattice5.html#K4">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</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_5.html#T1">YELLOW_5:1</a></i>;<br/><a NAME="E5:57_2_2_1_3_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">FD</font> <a href="lattice5.html#K4">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font></span>)</span> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">b</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">b</font> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font></span>)</span>
 <b>by </b><i><a class="ref" href="waybel_1.html#T2">WAYBEL_1:2</a></i>;<br/><a NAME="E6:57_2_2_1_3_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">FD</font> <a href="lattice5.html#K4">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">a</font></span>)</span> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">b</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="lattice3.html#T18">LATTICE3:18</a></i>;<br/><a NAME="E7:57_2_2_1_3_1_2"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">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>, </i>;<br/><a NAME="E8:57_2_2_1_3_1_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">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>, <a class="txt" href="lattice8.html#E1:57_2"><i><font color="Green">E75</font></i></a></i>;<br/><a NAME="E9:57_2_2_1_3_1_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">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>, <a class="txt" href="lattice8.html#E1:57_2"><i><font color="Green">E75</font></i></a></i>;<br/><b>hence </b><a NAME="E10:57_2_2_1_3_1_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">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="lattice8.html#E7:57_2"><i><font color="Green">E80</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_2_2_1_3_1_3"><b>suppose </b></a><a NAME="E1:57_2_2_1_3_1_3"/><i><font color="Green">E164</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 3
 ;<br/><div class="add"><a NAME="E2:57_2_2_1_3_1_3"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">z2</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">e1</font>
 <b>by </b><i><a class="txt" href="lattice8.html#E4"><i><font color="Green">Lemma55</font></i></a>, </i>;<br/><a NAME="E3:57_2_2_1_3_1_3"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> 3</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">e1</font>
 <b>by </b><i>, <a class="txt" href="lattice8.html#E1:57_2"><i><font color="Green">E75</font></i></a></i>;<br/><a NAME="E4:57_2_2_1_3_1_3"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">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>, <a class="txt" href="lattice8.html#E1:57_2"><i><font color="Green">E75</font></i></a></i>;<br/><b>hence </b><a NAME="E5:57_2_2_1_3_1_3"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">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="lattice8.html#E7:57_2"><i><font color="Green">E80</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E21:57_2_2_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">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="lattice8.html#E2:57_2"><i><font color="Green">E76</font></i></a>, , <a class="txt" href="lattice8.html#E4:57_2"><i><font color="Green">E78</font></i></a>, <a class="ref" href="eqrel_1.html#T36">EQREL_1:36</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:57_2_2"/>
<font color="Maroon">e3</font> <a href="lattice8.html#R1" target="_self">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">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">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">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="lattice8.html#D3" target="_self">Def3</a>, <a class="ref" href="funct_1.html#T118">FUNCT_1:118</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice5.html#K5">alpha</a> <font color="Maroon">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">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>, <a class="txt" href="lattice8.html#E5"><i><font color="Green">Lemma56</font></i></a>, <a class="ref" href="lattice5.html#T10">LATTICE5:10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="lattice5.html#K5">alpha</a> <font color="Maroon">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="txt" href="lattice8.html#E4:57"><i><font color="Green">E58</font></i></a>, , , <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="lattice5.html#K5">alpha</a> <font color="Maroon">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="E18:57_2"/>
 <a href="lattice5.html#K5">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="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">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">EqRelLATT</a> <font color="Maroon">FS</font></span>)</span> <b>by </b><i><a class="ref" href="lattice5.html#T16">LATTICE5:16</a></i>;<br/>
<a NAME="E14:57"/><i><font color="Green">E165</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="E15:57"/><i><font color="Green">E166</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">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="partfun1.html#K6">id</a> <font color="Maroon">FS</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_3"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">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="E2:57_3"/><i><font color="Green">E167</font></i>: 
<font color="Maroon">Aq'</font>,<font color="Maroon">dq'</font> <a href="lattice8.html#R3" target="_self">is_extension2_of</a> <font color="Maroon">A'</font>,<font color="Maroon">d'</font>
 <b>and </b><br/><a NAME="E3:57_3"/><i><font color="Green">E168</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="E4:57_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">BasicDF</a> <font color="Maroon">L</font> )
 
<b>by </b><i>, <a class="txt" href="lattice8.html#E4"><i><font color="Green">Lemma55</font></i></a>, <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">QuadrSeq</a> of  <a href="lattice5.html#K17">BasicDF</a> <font color="Maroon">L</font><b> such that </b><br/><a NAME="E6:57_3"/><i><font color="Green">E169</font></i>: 
<font color="Maroon">Aq'</font> <a href="hidden.html#R1">=</a>  <a href="lattice8.html#K7" target="_self">NextSet2</a> <span class="p1">(<span class="default"><a href="lattice5.html#K17">BasicDF</a> <font color="Maroon">L</font></span>)</span>
 <b>and </b><br/><a NAME="E7:57_3"/><i><font color="Green">E170</font></i>: 
<font color="Maroon">dq'</font> <a href="funct_2.html#R1">=</a>  <a href="lattice8.html#K9" target="_self">NextDelta2</a> <font color="Maroon">q</font>
 <b>by </b><i>, </i>;<br/>
<a NAME="E8:57_3"/><i><font color="Green">E171</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="lattice8.html#K9" target="_self">NextDelta2</a> <font color="Maroon">q</font>
 
<b>by </b><i><a class="txt" href="lattice8.html#E4"><i><font color="Green">Lemma55</font></i></a>, , <a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>;<br/>
<a NAME="E9:57_3"/>
<span class="p1">(<span class="default"><font color="Maroon">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#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : verum </span>}</span> 
 
;<br/>
<a NAME="E10:57_3"/><b>then </b><i><font color="Green">E172</font></i>: 
 <a href="lattice8.html#K9" target="_self">NextDelta2</a> <font color="Maroon">q</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">FD</font>
 
<b>by </b><i><a class="txt" href="lattice8.html#E4:57"><i><font color="Green">E58</font></i></a>, <a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/>
<a NAME="E11:57_3"/><i><font color="Green">E173</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="lattice8.html#K7" target="_self">NextSet2</a> <span class="p1">(<span class="default"><a href="lattice5.html#K17">BasicDF</a> <font color="Maroon">L</font></span>)</span>
 
<b>by </b><i><a class="txt" href="lattice8.html#E4"><i><font color="Green">Lemma55</font></i></a>, <a class="txt" href="lattice8.html#E5"><i><font color="Green">Lemma56</font></i></a>, <a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>;<br/>
<a NAME="E12:57_3"/>
<span class="p1">(<span class="default"><font color="Maroon">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#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : verum </span>}</span> 
 
;<br/>
<a NAME="E13:57_3"/><b>then </b><i><font color="Green">E174</font></i>: 
 <a href="lattice8.html#K7" target="_self">NextSet2</a> <span class="p1">(<span class="default"><a href="lattice5.html#K17">BasicDF</a> <font color="Maroon">L</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">FS</font>
 
<b>by </b><i><a class="txt" href="lattice8.html#E8:57"><i><font color="Green">E60</font></i></a>, <a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/>
<a NAME="E14:57_3"/>
 <a href="ordinal1.html#K1">succ</a> <a href="xboole_0.html#K1">{}</a>  <a href="ordinal1.html#R1">c=</a>  <a href="lattice5.html#K8">DistEsti</a> <span class="p1">(<span class="default"><a href="lattice5.html#K17">BasicDF</a> <font color="Maroon">L</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E15:57_3"/><b>then </b>
 <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a>  <a href="lattice5.html#K8">DistEsti</a> <span class="p1">(<span class="default"><a href="lattice5.html#K17">BasicDF</a> <font color="Maroon">L</font></span>)</span>
 
<b>by </b><i><a class="ref" href="ordinal1.html#T33">ORDINAL1:33</a></i>;<br/>
<a NAME="E16:57_3"/><b>then </b><i><font color="Green">E175</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><a class="ref" href="lattice5.html#T28">LATTICE5:28</a></i>;<br/>
<a NAME="E17:57_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="E18:57_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">BasicDF</a> <font color="Maroon">L</font></span>)</span> <a href="lattice5.html#K4">.</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><a class="ref" href="lattice5.html#D14">LATTICE5:def 14</a></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="E20:57_3"/><i><font color="Green">E176</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">BasicDF</a> <font color="Maroon">L</font></span>)</span> <a href="lattice5.html#K4">.</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="E21:57_3"/>
 <a href="lattice8.html#K3" target="_self">ConsecutiveSet2</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="lattice8.html#K4" target="_self">Quadr2</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="E23:57_3"/><i><font color="Green">E177</font></i>: 
 <a href="lattice8.html#K4" target="_self">Quadr2</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>, , </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="E25:57_3"/><i><font color="Green">E178</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="E26:57_3"/><i><font color="Green">E179</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">.</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><a class="ref" href="lattice5.html#D9">LATTICE5:def 9</a></i>;<br/>
<b>take </b>
<font color="Maroon">e</font>
;<br/>

<a NAME="E27:57_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="E28:57_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">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">E180</font></i>:  <a href="lattice8.html#K1" target="_self">new_set2</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> = 
 <a href="lattice8.html#K1" target="_self">new_set2</a> <span class="p1">(<span class="default"><a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>,<a href="xboole_0.html#K1">{}</a> </span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="lattice8.html#K3" target="_self">ConsecutiveSet2</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="E30:57_3"/><i><font color="Green">E181</font></i>: 
 <a href="ordinal1.html#K1">succ</a> <a href="xboole_0.html#K1">{}</a>  <a href="ordinal1.html#R1">c=</a>  <a href="lattice5.html#K8">DistEsti</a> <span class="p1">(<span class="default"><a href="lattice5.html#K17">BasicDF</a> <font color="Maroon">L</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E31:57_3"/><b>then </b><i><font color="Green">E182</font></i>: 
 <a href="lattice8.html#K1" target="_self">new_set2</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> <a href="tarski.html#R1">c=</a>  <a href="lattice8.html#K7" target="_self">NextSet2</a> <span class="p1">(<span class="default"><a href="lattice5.html#K17">BasicDF</a> <font color="Maroon">L</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E32:57_3"/><i><font color="Green">E183</font></i>: 
(  <a href="lattice8.html#K3" target="_self">ConsecutiveSet2</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="lattice8.html#K6" target="_self">ConsecutiveDelta2</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">BasicDF</a> <font color="Maroon">L</font> )
 
<b>by </b><i>, </i>;<br/>
<i><font color="Green">E184</font></i>:  <a href="lattice8.html#K6" target="_self">ConsecutiveDelta2</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <span class="p1">(<span class="default"><a href="lattice5.html#K11">BiFun</a> <span class="p2">(<span class="default"><a href="lattice8.html#K6" target="_self">ConsecutiveDelta2</a> <font color="Maroon">q</font>,<a href="xboole_0.html#K1">{}</a> </span>)</span>,<span class="p2">(<span class="default"><a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">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="lattice8.html#K4" target="_self">Quadr2</a> <font color="Maroon">q</font>,<a href="xboole_0.html#K1">{}</a> </span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <span class="p1">(<span class="default"><a href="lattice5.html#K17">BasicDF</a> <font color="Maroon">L</font></span>)</span>,<font color="Maroon">Q</font>
<b>by </b><i>, <a class="ref" href="lattice5.html#D16">LATTICE5:def 16</a></i>
;<br/>
<a NAME="E34:57_3"/>
 <a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <span class="p1">(<span class="default"><a href="lattice5.html#K17">BasicDF</a> <font color="Maroon">L</font></span>)</span>,<font color="Maroon">Q</font> <a href="tarski.html#R1">c=</a>  <a href="lattice8.html#K9" target="_self">NextDelta2</a> <font color="Maroon">q</font>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E35:57_3"/><b>then </b><i><font color="Green">E185</font></i>: 
 <a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <span class="p1">(<span class="default"><a href="lattice5.html#K17">BasicDF</a> <font color="Maroon">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="txt" href="lattice8.html#E6:57"><i><font color="Green">E59</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E36:57_3"/><i><font color="Green">E186</font></i>: 
 <a href="lattice8.html#K1" target="_self">new_set2</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">FS</font>
 
<b>by </b><i><a class="txt" href="lattice8.html#E11:57"><i><font color="Green">E61</font></i></a>, , <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E37:57_3"/><i><font color="Green">E187</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <span class="p2">(<span class="default"><a href="lattice5.html#K17">BasicDF</a> <font color="Maroon">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="lattice8.html#K1" target="_self">new_set2</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="lattice8.html#K1" target="_self">new_set2</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="E38:57_3"/>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">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#K2">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<a NAME="E39:57_3"/><b>then </b>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">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="tarski.html#K2">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">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#K2">}</a></span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E40:57_3"/><b>then </b><i><font color="Green">E188</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="lattice8.html#K1" target="_self">new_set2</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 
;<br/>
<a NAME="E41:57_3"/>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">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="tarski.html#K2">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">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#K2">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<a NAME="E42:57_3"/><b>then </b>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">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="tarski.html#K2">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">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#K2">}</a></span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E43:57_3"/><b>then </b><i><font color="Green">E189</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="lattice8.html#K1" target="_self">new_set2</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 
;<br/>
<a NAME="E44:57_3"/><b>then </b><i><font color="Green">E190</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <span class="p2">(<span class="default"><a href="lattice5.html#K17">BasicDF</a> <font color="Maroon">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/>
<a NAME="E45:57_3"/><i><font color="Green">E191</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Q</font> <a href="mcart_1.html#K11">`4</a> 
 
<b>by </b><i>, <a class="ref" href="mcart_1.html#D11">MCART_1:def 11</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">.</a> <font color="Maroon">W</font>,<font color="Maroon">V</font> = 
<span class="p1">(<span class="default"><a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <span class="p2">(<span class="default"><a href="lattice5.html#K17">BasicDF</a> <font color="Maroon">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/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lattice5.html#K17">BasicDF</a> <font color="Maroon">L</font></span>)</span> <a href="lattice5.html#K4">.</a> <span class="p3">(<span class="default"><font color="Maroon">Q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">Q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">Q</font> <a href="mcart_1.html#K10">`3</a> </span>)</span></span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">Q</font> <a href="mcart_1.html#K11">`4</a> </span>)</span>
<b>by </b><i/>
;<br/>
<a NAME="E48:57_3"/><b>then </b>
<font color="Maroon">FD</font> <a href="lattice5.html#K4">.</a> <font color="Maroon">W</font>,<font color="Maroon">V</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">b</font>
 
<b>by </b><i>, <a class="ref" href="yellow_0.html#T23">YELLOW_0:23</a></i>;<br/>
<a NAME="E49:57_3"/><b>then </b><i><font color="Green">E192</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="E50:57_3"/>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">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="57_3_4"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:57_3_4"/>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">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="E2:57_3_4"/><b>then </b>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">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="E3:57_3_4"/>
contradiction
 ;<br/>


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

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_4_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:57_4_1"/><i><font color="Green">E193</font></i>: 
<font color="Maroon">FS</font> is <a href="realset1.html#V1">trivial</a>
 ;<br/><b>consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:57_4_1"/><i><font color="Green">E194</font></i>: 
<font color="Maroon">FS</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, <a class="ref" href="realset1.html#D4">REALSET1:def 4</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="E5:57_4_1"/><i><font color="Green">E195</font></i>: 
( <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">Image</a> <font color="Maroon">f</font></span>)</span> &amp; <font color="Maroon">e</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="partfun1.html#K6">id</a> <font color="Maroon">FS</font> )
 <b>by </b><i><a class="ref" href="lattice8.html#D3" target="_self">Def3</a></i>;<br/><a NAME="E6:57_4_1"/>
 <a href="relat_1.html#K3">field</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="lattice8.html#E4"><i><font color="Green">Lemma55</font></i></a>, <a class="ref" href="eqrel_1.html#T16">EQREL_1:16</a></i>;<br/><a NAME="E7:57_4_1"/><b>then </b><i><font color="Green">E196</font></i>: 
 <a href="partfun1.html#K6">id</a> <font color="Maroon">FS</font> <a href="lattice8.html#R1" target="_self">c=</a> <font color="Maroon">e</font>
 <b>by </b><i><a class="txt" href="lattice8.html#E4"><i><font color="Green">Lemma55</font></i></a>, <a class="ref" href="relat_2.html#T17">RELAT_2:17</a></i>;<br/><a NAME="E8:57_4_1"/><i><font color="Green">E197</font></i>: 
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T35">ZFMISC_1:35</a></i>;<br/><a NAME="E9:57_4_1"/>
 <a href="partfun1.html#K6">id</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="sysrel.html#T30">SYSREL:30</a></i>;<br/><b>hence </b><a NAME="E10:57_4_1"/>
contradiction
 <b>by </b><i><a class="txt" href="lattice8.html#E4"><i><font color="Green">Lemma55</font></i></a>, <a class="txt" href="lattice8.html#E5"><i><font color="Green">Lemma56</font></i></a>, , <a class="txt" href="lattice8.html#E4:57"><i><font color="Green">E58</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:57_4"/>
<font color="Maroon">FS</font> is  non <a href="realset1.html#V1">trivial</a>  <a href="hidden.html#M1">set</a> 
 ;<br/>


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

<b>reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</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">EqRelLATT</a> <font color="Maroon">FS</font></span>)</span> ;<br/>
<b>take </b>
<font color="Maroon">f</font>
;<br/>

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

<b>reconsider </b><font color="Maroon">FD</font> = <font color="Maroon">FD</font> as   <a href="lattice5.html#NM4">distance_function</a> of <font color="Maroon">FS</font>,<font color="Maroon">L</font> ;<br/>
<b>thus </b><a NAME="E21:57"/>
 <a href="lattice5.html#K2">Image</a> <font color="Maroon">f</font> is <a href="lattice8.html#V1" target="_self">finitely_typed</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_5"><b>proof </b></a><div class="add">

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

<b>thus </b><a NAME="E1:57_5"/>
for <font color="Olive">e</font> being   <a href="hidden.html#M1">set</a>   st <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">Image</a> <font color="Maroon">f</font></span>)</span> holds <br/><font color="Olive">e</font> is   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">FS</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_5_1"><b>proof </b></a><div class="add">

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

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

<a NAME="E2:57_5_1"/><b>then </b>
<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="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/>
<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:57_5_1"/><i><font color="Green">E198</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E5:57_5_1"/><i><font color="Green">E199</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> <b>by </b><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="E8:57_5_1"/><i><font color="Green">E200</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">x</font>
 <b>and </b><br/><a NAME="E9:57_5_1"/>
for <font color="Olive">u</font>, <font color="Olive">v</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">u</font>,<font color="Olive">v</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">.</a> <font color="Olive">u</font>,<font color="Olive">v</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="lattice5.html#D9">LATTICE5:def 9</a></i>;<br/>
<b>thus </b><a NAME="E10:57_5_1"/>
<font color="Maroon">e</font> is   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">FS</font>
 <b>by </b><i><a class="txt" href="lattice8.html#E4"><i><font color="Green">Lemma55</font></i></a>, <a class="txt" href="lattice8.html#E5"><i><font color="Green">Lemma56</font></i></a></i>;<br/>


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

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

<b>thus </b><a NAME="E2:57_5"/>
for <font color="Olive">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">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">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> 2 <a href="nat_1.html#K1">+</a> 2 &amp; <font color="Olive">x</font>,<font color="Olive">y</font> <a href="lattice5.html#R1">are_joint_by</a> <font color="Olive">F</font>,<font color="Olive">e1</font>,<font color="Olive">e2</font> )
 <b>by </b><i/>;<br/>


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

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

<b>consider </b><font color="Maroon">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="E2:57_6"/><i><font color="Green">E201</font></i>: 
<font color="Maroon">Aq'</font>,<font color="Maroon">dq'</font> <a href="lattice8.html#R3" target="_self">is_extension2_of</a> <font color="Maroon">A'</font>,<font color="Maroon">d'</font>
 <b>and </b><br/><a NAME="E3:57_6"/><i><font color="Green">E202</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="E4:57_6"/>
( <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">BasicDF</a> <font color="Maroon">L</font> )
 
<b>by </b><i>, <a class="txt" href="lattice8.html#E4"><i><font color="Green">Lemma55</font></i></a>, <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">QuadrSeq</a> of  <a href="lattice5.html#K17">BasicDF</a> <font color="Maroon">L</font><b> such that </b><br/><a NAME="E6:57_6"/><i><font color="Green">E203</font></i>: 
<font color="Maroon">Aq'</font> <a href="hidden.html#R1">=</a>  <a href="lattice8.html#K7" target="_self">NextSet2</a> <span class="p1">(<span class="default"><a href="lattice5.html#K17">BasicDF</a> <font color="Maroon">L</font></span>)</span>
 <b>and </b><br/><a NAME="E7:57_6"/><i><font color="Green">E289</font></i>: 
<font color="Maroon">dq'</font> <a href="funct_2.html#R1">=</a>  <a href="lattice8.html#K9" target="_self">NextDelta2</a> <font color="Maroon">q</font>
 <b>by </b><i>, </i>;<br/>
<a NAME="E8:57_6"/><i><font color="Green">E290</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="lattice8.html#K9" target="_self">NextDelta2</a> <font color="Maroon">q</font>
 
<b>by </b><i><a class="txt" href="lattice8.html#E4"><i><font color="Green">Lemma55</font></i></a>, , <a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>;<br/>
<a NAME="E9:57_6"/>
<span class="p1">(<span class="default"><font color="Maroon">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#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : verum </span>}</span> 
 
;<br/>
<a NAME="E10:57_6"/><b>then </b><i><font color="Green">E291</font></i>: 
 <a href="lattice8.html#K9" target="_self">NextDelta2</a> <font color="Maroon">q</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">FD</font>
 
<b>by </b><i><a class="txt" href="lattice8.html#E4:57"><i><font color="Green">E58</font></i></a>, <a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/>
<a NAME="E11:57_6"/><i><font color="Green">E292</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="lattice8.html#K7" target="_self">NextSet2</a> <span class="p1">(<span class="default"><a href="lattice5.html#K17">BasicDF</a> <font color="Maroon">L</font></span>)</span>
 
<b>by </b><i><a class="txt" href="lattice8.html#E4"><i><font color="Green">Lemma55</font></i></a>, <a class="txt" href="lattice8.html#E5"><i><font color="Green">Lemma56</font></i></a>, <a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>;<br/>
<a NAME="E12:57_6"/>
<span class="p1">(<span class="default"><font color="Maroon">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#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : verum </span>}</span> 
 
;<br/>
<a NAME="E13:57_6"/><b>then </b><i><font color="Green">E293</font></i>: 
 <a href="lattice8.html#K7" target="_self">NextSet2</a> <span class="p1">(<span class="default"><a href="lattice5.html#K17">BasicDF</a> <font color="Maroon">L</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">FS</font>
 
<b>by </b><i><a class="txt" href="lattice8.html#E8:57"><i><font color="Green">E60</font></i></a>, <a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/>
<a NAME="E14:57_6"/>
 <a href="ordinal1.html#K1">succ</a> <a href="xboole_0.html#K1">{}</a>  <a href="ordinal1.html#R1">c=</a>  <a href="lattice5.html#K8">DistEsti</a> <span class="p1">(<span class="default"><a href="lattice5.html#K17">BasicDF</a> <font color="Maroon">L</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E15:57_6"/><b>then </b>
 <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a>  <a href="lattice5.html#K8">DistEsti</a> <span class="p1">(<span class="default"><a href="lattice5.html#K17">BasicDF</a> <font color="Maroon">L</font></span>)</span>
 
<b>by </b><i><a class="ref" href="ordinal1.html#T33">ORDINAL1:33</a></i>;<br/>
<a NAME="E16:57_6"/><b>then </b><i><font color="Green">E294</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><a class="ref" href="lattice5.html#T28">LATTICE5:28</a></i>;<br/>
<a NAME="E17:57_6"/><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="E18:57_6"/><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">BasicDF</a> <font color="Maroon">L</font></span>)</span> <a href="lattice5.html#K4">.</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><a class="ref" href="lattice5.html#D14">LATTICE5:def 14</a></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="E20:57_6"/><i><font color="Green">E295</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">BasicDF</a> <font color="Maroon">L</font></span>)</span> <a href="lattice5.html#K4">.</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="E21:57_6"/>
 <a href="lattice8.html#K3" target="_self">ConsecutiveSet2</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="lattice8.html#K4" target="_self">Quadr2</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="E23:57_6"/><i><font color="Green">E296</font></i>: 
 <a href="lattice8.html#K4" target="_self">Quadr2</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>, , </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="E25:57_6"/><i><font color="Green">E319</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="E26:57_6"/><i><font color="Green">E320</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">.</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><a class="ref" href="lattice5.html#D9">LATTICE5:def 9</a></i>;<br/>
<b>take </b>
<font color="Maroon">e</font>
;<br/>

<a NAME="E27:57_6"/>
<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="E28:57_6"/>
<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">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">E321</font></i>:  <a href="lattice8.html#K1" target="_self">new_set2</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> = 
 <a href="lattice8.html#K1" target="_self">new_set2</a> <span class="p1">(<span class="default"><a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>,<a href="xboole_0.html#K1">{}</a> </span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="lattice8.html#K3" target="_self">ConsecutiveSet2</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="E30:57_6"/><i><font color="Green">E322</font></i>: 
 <a href="ordinal1.html#K1">succ</a> <a href="xboole_0.html#K1">{}</a>  <a href="ordinal1.html#R1">c=</a>  <a href="lattice5.html#K8">DistEsti</a> <span class="p1">(<span class="default"><a href="lattice5.html#K17">BasicDF</a> <font color="Maroon">L</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E31:57_6"/><b>then </b><i><font color="Green">E323</font></i>: 
 <a href="lattice8.html#K1" target="_self">new_set2</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> <a href="tarski.html#R1">c=</a>  <a href="lattice8.html#K7" target="_self">NextSet2</a> <span class="p1">(<span class="default"><a href="lattice5.html#K17">BasicDF</a> <font color="Maroon">L</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E32:57_6"/><i><font color="Green">E324</font></i>: 
(  <a href="lattice8.html#K3" target="_self">ConsecutiveSet2</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="lattice8.html#K6" target="_self">ConsecutiveDelta2</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">BasicDF</a> <font color="Maroon">L</font> )
 
<b>by </b><i>, </i>;<br/>
<i><font color="Green">E325</font></i>:  <a href="lattice8.html#K6" target="_self">ConsecutiveDelta2</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <span class="p1">(<span class="default"><a href="lattice5.html#K11">BiFun</a> <span class="p2">(<span class="default"><a href="lattice8.html#K6" target="_self">ConsecutiveDelta2</a> <font color="Maroon">q</font>,<a href="xboole_0.html#K1">{}</a> </span>)</span>,<span class="p2">(<span class="default"><a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">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="lattice8.html#K4" target="_self">Quadr2</a> <font color="Maroon">q</font>,<a href="xboole_0.html#K1">{}</a> </span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <span class="p1">(<span class="default"><a href="lattice5.html#K17">BasicDF</a> <font color="Maroon">L</font></span>)</span>,<font color="Maroon">Q</font>
<b>by </b><i>, <a class="ref" href="lattice5.html#D16">LATTICE5:def 16</a></i>
;<br/>
<a NAME="E34:57_6"/>
 <a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <span class="p1">(<span class="default"><a href="lattice5.html#K17">BasicDF</a> <font color="Maroon">L</font></span>)</span>,<font color="Maroon">Q</font> <a href="tarski.html#R1">c=</a>  <a href="lattice8.html#K9" target="_self">NextDelta2</a> <font color="Maroon">q</font>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E35:57_6"/><b>then </b><i><font color="Green">E326</font></i>: 
 <a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <span class="p1">(<span class="default"><a href="lattice5.html#K17">BasicDF</a> <font color="Maroon">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="txt" href="lattice8.html#E6:57"><i><font color="Green">E59</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E36:57_6"/><i><font color="Green">E327</font></i>: 
 <a href="lattice8.html#K1" target="_self">new_set2</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">FS</font>
 
<b>by </b><i><a class="txt" href="lattice8.html#E11:57"><i><font color="Green">E61</font></i></a>, , <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E37:57_6"/><i><font color="Green">E328</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <span class="p2">(<span class="default"><a href="lattice5.html#K17">BasicDF</a> <font color="Maroon">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="lattice8.html#K1" target="_self">new_set2</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="lattice8.html#K1" target="_self">new_set2</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="E38:57_6"/>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">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#K2">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<a NAME="E39:57_6"/><b>then </b>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">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="tarski.html#K2">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">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#K2">}</a></span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E40:57_6"/><b>then </b><i><font color="Green">E329</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="lattice8.html#K1" target="_self">new_set2</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 
;<br/>
<a NAME="E41:57_6"/>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">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="tarski.html#K2">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">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#K2">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<a NAME="E42:57_6"/><b>then </b>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">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="tarski.html#K2">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">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#K2">}</a></span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E43:57_6"/><b>then </b><i><font color="Green">E330</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="lattice8.html#K1" target="_self">new_set2</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 
;<br/>
<a NAME="E44:57_6"/><b>then </b><i><font color="Green">E331</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="lattice8.html#K2" target="_self">new_bi_fun2</a> <span class="p2">(<span class="default"><a href="lattice5.html#K17">BasicDF</a> <font color="Maroon">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/>
<a NAME="E45:57_6"/><i><font color="Green">E332</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Q</font> <a href="mcart_1.html#K11">`4</a> 
 
<b>by </b><i>, <a class="ref" href="mcart_1.html#D11">MCART_1:def 11</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">.</a> <font color="Maroon">W</font>,<font color="Maroon">V</font> = 
<span class="p1">(<span class="default"><a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <span class="p2">(<span class="default"><a href="lattice5.html#K17">BasicDF</a> <font color="Maroon">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/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="lattice5.html#K17">BasicDF</a> <font color="Maroon">L</font></span>)</span> <a href="lattice5.html#K4">.</a> <span class="p3">(<span class="default"><font color="Maroon">Q</font> <a href="mcart_1.html#K8">`1</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">Q</font> <a href="mcart_1.html#K9">`2</a> </span>)</span></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">Q</font> <a href="mcart_1.html#K10">`3</a> </span>)</span></span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">Q</font> <a href="mcart_1.html#K11">`4</a> </span>)</span>
<b>by </b><i/>
;<br/>
<a NAME="E48:57_6"/><b>then </b>
<font color="Maroon">FD</font> <a href="lattice5.html#K4">.</a> <font color="Maroon">W</font>,<font color="Maroon">V</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">b</font>
 
<b>by </b><i>, <a class="ref" href="yellow_0.html#T23">YELLOW_0:23</a></i>;<br/>
<a NAME="E49:57_6"/><b>then </b><i><font color="Green">E333</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="E50:57_6"/>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">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="57_6_4"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:57_6_4"/>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">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="E2:57_6_4"/><b>then </b>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">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="E3:57_6_4"/>
contradiction
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E51:57_6"/>
<font color="Maroon">e</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="partfun1.html#K6">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="E23:57"/>
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">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">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> 2 <a href="nat_1.html#K1">+</a> 2 &amp; <font color="Olive">x</font>,<font color="Olive">y</font> <a href="lattice5.html#R1">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="E24:57"/>
 <a href="lattice5.html#K3">type_of</a> <span class="p1">(<span class="default"><a href="lattice5.html#K2">Image</a> <font color="Maroon">f</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 2
 <b>by </b><i><a class="ref" href="lattice8.html#D3" target="_self">Def3</a>, <a class="ref" href="lattice5.html#T15">LATTICE5:15</a></i>;<br/>


</div>
