<?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>let </b><font color="Maroon">S</font> be    <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/><b>let </b><font color="Maroon">FS</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">FD</font> be   <a href="lattice5.html#NM4">distance_function</a> of <font color="Maroon">FS</font>,<font color="Maroon">L</font>;<br/><b>let </b><font color="Maroon">f</font> be   <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>let </b><font color="Maroon">e1</font> be   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">FS</font>, <font color="Maroon">e2</font> be   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">FS</font>;<br/><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><b>that </b><br/><a NAME="E1:56"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#R2">=</a>  <a href="lattice5.html#K5">alpha</a> <font color="Maroon">FD</font>
 <b>and </b><br/><a NAME="E2:56"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">FS</font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">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>  &amp; <font color="Maroon">FD</font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">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>  )
 <b>and </b><br/><a NAME="E3:56"/><i><font color="Green">E58</font></i>: 
( <font color="Maroon">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="Maroon">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> )
 <b>and </b><br/><a NAME="E4:56"/><i><font color="Green">E59</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> <a href="eqrel_1.html#K5">"\/"</a> <font color="Maroon">e2</font>
 ;<br/>

<a NAME="E5:56"/><i><font color="Green">E60</font></i>: 
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> <a href="hidden.html#R1">=</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">a</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:56"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">a</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="E8:56"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">e1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon">b</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E10:56"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">b</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="E11:56"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">e2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</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">a</font> = <font color="Maroon">a</font>, <font color="Maroon">b</font> = <font color="Maroon">b</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> <b>by </b><i>, </i>;<br/>
<a NAME="E13:56"/>
 <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><font color="Maroon">e1</font> <a href="eqrel_1.html#K5">"\/"</a> <font color="Maroon">e2</font></span>)</span> <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/>
<b>then reconsider </b><font color="Maroon">u</font> = <font color="Maroon">x</font>, <font color="Maroon">v</font> = <font color="Maroon">y</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">FS</font> <b>by </b><i>, <a class="ref" href="relat_1.html#T30">RELAT_1:30</a></i>;<br/>
<b>reconsider </b><font color="Maroon">a</font> = <font color="Maroon">a</font>, <font color="Maroon">b</font> = <font color="Maroon">b</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> ;<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="E17:56"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">e1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font>
 <b>and </b><br/><a NAME="E18:56"/><i><font color="Green">E69</font></i>: 
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">e1'</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">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="E20:56"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">e2'</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="E21:56"/><i><font color="Green">E74</font></i>: 
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">e2'</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">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">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="E23:56"/><i><font color="Green">E75</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> <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="E24:56"/><i><font color="Green">E76</font></i>: 
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">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/>
<font color="Maroon">e</font> = 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <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"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font></span>)</span>
<b>by </b><i><a class="txt" href="lattice8.html#E3:56"><i><font color="Green">E58</font></i></a>, <a class="ref" href="waybel_6.html#T2">WAYBEL_6:2</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="ref" href="lattice8.html#D3" target="_self">Def3</a>, <a class="ref" href="lattice5.html#T10">LATTICE5:10</a></i>
;<br/>
<a NAME="E26:56"/><b>then </b><i><font color="Green">E78</font></i>: 
<font color="Maroon">FD</font> <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>
 
<b>by </b><i>, <a class="txt" href="lattice8.html#E4:56"><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="E28:56"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">FD</font> <a href="lattice5.html#K4">.</a> <font color="Maroon">u</font>,<font color="Maroon">z1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 <b>and </b><br/><a NAME="E29:56"/><i><font color="Green">E80</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">u</font>,<font color="Maroon">v</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="E30:56"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">FD</font> <a href="lattice5.html#K4">.</a> <font color="Maroon">z2</font>,<font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="lattice8.html#D2" target="_self">Def2</a>, </i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b>( ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> 1 implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">u</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">v</font> ) );<br/>
<a NAME="E31:56"/><i><font color="Green">E82</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#E2:56"><i><font color="Green">E42</font></i></a></i>;<br/>
<a NAME="E32:56"/><i><font color="Green">E83</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="56_2"><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="E1:56_2"/><i><font color="Green">E84</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="56_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:56_2_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#E17:56"><i><font color="Green">E66</font></i></a>, <a class="txt" href="lattice8.html#E2:56"><i><font color="Green">E42</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_2_1_1"><b>suppose </b></a><a NAME="E1:56_2_1_1"/><i><font color="Green">E85</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="E2:56_2_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="56_2_1_2"><b>suppose </b></a><a NAME="E1:56_2_1_2"/><i><font color="Green">E86</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="E2:56_2_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="56_2_1_3"><b>suppose </b></a><a NAME="E1:56_2_1_3"/><i><font color="Green">E87</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="E2:56_2_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="56_2_1_4"><b>suppose </b></a><a NAME="E1:56_2_1_4"/><i><font color="Green">E88</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="E2:56_2_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="E33:56"/>
 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>(, <a class="txt" href="lattice8.html#E11:56"><i><font color="Green">E65</font></i></a>);<br/></i>
<b>then consider </b><font color="Maroon">h</font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E35:56"/><i><font color="Green">E89</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="E36:56"/><i><font color="Green">E90</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">u</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">v</font> ) )
 ;<br/>
<a NAME="E37:56"/>
 <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="E38:56"/><b>then </b><i><font color="Green">E91</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="E39:56"/><i><font color="Green">E92</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="txt" href="lattice8.html#E17:56"><i><font color="Green">E66</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E40:56"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">FS</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_3"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:56_3"/>
<font color="Maroon">w</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font>
 ;<br/>

<b>then consider </b><font color="Maroon">j</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:56_3"/><i><font color="Green">E93</font></i>: 
( <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">h</font> &amp; <font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:56_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 or <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 4 )
 </span><b>by </b><i><a class="txt" href="lattice8.html#E17:56"><i><font color="Green">E66</font></i></a>, , <a class="txt" href="lattice8.html#E2:56"><i><font color="Green">E42</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_3_1_1"><b>suppose </b></a><a NAME="E1:56_3_1_1"/>
<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:56_3_1_1"/><b>then </b>
<font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E3:56_3_1_1"/>
<font color="Maroon">w</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FS</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_3_1_2"><b>suppose </b></a><a NAME="E1:56_3_1_2"/>
<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 2
 ;<br/><div class="add"><a NAME="E2:56_3_1_2"/><b>then </b>
<font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z1</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E3:56_3_1_2"/>
<font color="Maroon">w</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FS</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_3_1_3"><b>suppose </b></a><a NAME="E1:56_3_1_3"/>
<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 3
 ;<br/><div class="add"><a NAME="E2:56_3_1_3"/><b>then </b>
<font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z2</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E3:56_3_1_3"/>
<font color="Maroon">w</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FS</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_3_1_4"><b>suppose </b></a><a NAME="E1:56_3_1_4"/>
<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 4
 ;<br/><div class="add"><a NAME="E2:56_3_1_4"/><b>then </b>
<font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E3:56_3_1_4"/>
<font color="Maroon">w</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FS</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">h</font> = <font color="Maroon">h</font> as    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">FS</font> <b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<a NAME="E42:56"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">h</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="txt" href="lattice8.html#E17:56"><i><font color="Green">E66</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">h</font> = <font color="Maroon">h</font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">FS</font> <b>by </b><i><a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/>
<b>take </b>
<font color="Maroon">h</font>
;<br/>

<b>thus </b><a NAME="E44:56"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> 2 <a href="nat_1.html#K1">+</a> 2
 <b>by </b><i><a class="txt" href="lattice8.html#E17:56"><i><font color="Green">E66</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>

<a NAME="E45:56"/><i><font color="Green">E94</font></i>: 
<font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 
<b>by </b><i>, </i>;<br/>
<i><font color="Green">E95</font></i>: <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> = 
<font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> 4
<b>by </b><i><a class="txt" href="lattice8.html#E17:56"><i><font color="Green">E66</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
<br/>.= 
<font color="Maroon">y</font>
<b>by </b><i>, </i>
;<br/>
<a NAME="E47:56"/>
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/>( ( not <font color="Olive">j</font> is <a href="abian.html#V1">even</a> implies <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">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> ) &amp; ( <font color="Olive">j</font> is <a href="abian.html#V1">even</a> implies <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">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">e2</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_5"><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="E1:56_5"/><i><font color="Green">E125</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="56_5_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:56_5_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#E18:56"><i><font color="Green">E69</font></i></a>, , </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_5_1_1"><b>suppose </b></a><a NAME="E1:56_5_1_1"/><i><font color="Green">E126</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:56_5_1_1"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">u</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>, <a class="txt" href="lattice8.html#E7:56"><i><font color="Green">E61</font></i></a></i>;<br/><a NAME="E3:56_5_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> 1</span>)</span>,<font color="Maroon">z1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">e1'</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E4:56_5_1_1"/>
( ( not <font color="Maroon">j</font> is <a href="abian.html#V1">even</a> implies <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">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> ) &amp; ( <font color="Maroon">j</font> is <a href="abian.html#V1">even</a> implies <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">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">e2</font> ) )
 <b>by </b><i>, , , , , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_5_1_2"><b>suppose </b></a><a NAME="E1:56_5_1_2"/><i><font color="Green">E127</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 2
 ;<br/><div class="add"><b>consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:56_5_1_2"/><i><font color="Green">E128</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><a NAME="E4:56_5_1_2"/><i><font color="Green">E129</font></i>: 
2 <a href="nat_1.html#K2">*</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font>
 <b>by </b><i>, <a class="txt" href="lattice8.html#E21:56"><i><font color="Green">E74</font></i></a></i>;<br/><a NAME="E5:56_5_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">FD</font> <a href="lattice5.html#K4">.</a> <font color="Maroon">u</font>,<font color="Maroon">v</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="txt" href="lattice8.html#E5:56"><i><font color="Green">E60</font></i></a>, <a class="ref" href="waybel_1.html#T3">WAYBEL_1:3</a></i>;<br/><a NAME="E6:56_5_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">u</font>,<font color="Maroon">v</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="E7:56_5_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">u</font>,<font color="Maroon">v</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="E8:56_5_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">u</font>,<font color="Maroon">v</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="E9:56_5_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">u</font>,<font color="Maroon">v</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="E10:56_5_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>, <a class="txt" href="lattice8.html#E8:56"><i><font color="Green">E62</font></i></a></i>;<br/><a NAME="E11:56_5_1_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> 2</span>)</span>,<font color="Maroon">z2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">e2'</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E12:56_5_1_2"/>
( ( not <font color="Maroon">j</font> is <a href="abian.html#V1">even</a> implies <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">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> ) &amp; ( <font color="Maroon">j</font> is <a href="abian.html#V1">even</a> implies <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">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">e2</font> ) )
 <b>by </b><i><a class="ref" href="lattice8.html#D3" target="_self">Def3</a>, <a class="txt" href="lattice8.html#E5"><i><font color="Green">Lemma56</font></i></a>, , , , <a class="txt" href="lattice8.html#E23:56"><i><font color="Green">E75</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_5_1_3"><b>suppose </b></a><a NAME="E1:56_5_1_3"/><i><font color="Green">E130</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> 3
 ;<br/><div class="add"><b>consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:56_5_1_3"/><i><font color="Green">E131</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><a NAME="E4:56_5_1_3"/><i><font color="Green">E132</font></i>: 
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">i</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j</font>
 <b>by </b><i>, <a class="txt" href="lattice8.html#E21:56"><i><font color="Green">E74</font></i></a></i>;<br/><a NAME="E5:56_5_1_3"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">z2</font>,<font color="Maroon">v</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>, <a class="txt" href="lattice8.html#E10:56"><i><font color="Green">E63</font></i></a></i>;<br/><a NAME="E6:56_5_1_3"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> 3</span>)</span>,<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>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E7:56_5_1_3"/>
( ( not <font color="Maroon">j</font> is <a href="abian.html#V1">even</a> implies <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">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> ) &amp; ( <font color="Maroon">j</font> is <a href="abian.html#V1">even</a> implies <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">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">e2</font> ) )
 <b>by </b><i>, , , , , <a class="txt" href="lattice8.html#E23:56"><i><font color="Green">E75</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E48:56"/>
<font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="lattice5.html#R1">are_joint_by</a> <font color="Maroon">h</font>,<font color="Maroon">e1</font>,<font color="Maroon">e2</font>
 <b>by </b><i>, <a class="txt" href="lattice8.html#E20:56"><i><font color="Green">E71</font></i></a>, <a class="ref" href="lattice5.html#D3">LATTICE5:def 3</a></i>;<br/>


</div>
