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

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



<b>assume </b><a NAME="E1:42"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">L</font> is <a href="yellow11.html#V1">modular</a>
 ;<br/>

<b>let </b><font color="Maroon">d</font> be   <a href="lattice5.html#NM3">BiFunction</a> of <font color="Maroon">A</font>,<font color="Maroon">L</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E2:42"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">d</font> is <a href="lattice5.html#V1">symmetric</a>
 <b>and </b><br/><a NAME="E3:42"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">d</font> is <a href="lattice5.html#V3">u.t.i.</a>
 ;<br/>

<b>let </b><font color="Maroon">O</font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/><b>let </b><font color="Maroon">q</font> be    <a href="lattice5.html#M1">QuadrSeq</a> of <font color="Maroon">d</font>;<br/>



<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="ordinal1.html#NM2">Ordinal</a>] <b>means </b>( <font color="Maroon">a<sub>1</sub></font> <a href="ordinal1.html#R1">c=</a>  <a href="lattice5.html#K8">DistEsti</a> <font color="Maroon">d</font> implies  <a href="lattice8.html#K6" target="_self">ConsecutiveDelta2</a> <font color="Maroon">q</font>,<font color="Maroon">a<sub>1</sub></font> is <a href="lattice5.html#V3">u.t.i.</a> );<br/>
<a NAME="E4:42"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="xboole_0.html#K1">{}</a> ]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:42_1"/>
 <a href="xboole_0.html#K1">{}</a>  <a href="ordinal1.html#R1">c=</a>  <a href="lattice5.html#K8">DistEsti</a> <font color="Maroon">d</font>
 ;<br/>

<b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">A</font>,<a href="xboole_0.html#K1">{}</a> , <font color="Maroon">y</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">A</font>,<a href="xboole_0.html#K1">{}</a> , <font color="Maroon">z</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">A</font>,<a href="xboole_0.html#K1">{}</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice5.html#D8">LATTICE5:def 8</a><br/>

<b>reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font>, <font color="Maroon">y'</font> = <font color="Maroon">y</font>, <font color="Maroon">z'</font> = <font color="Maroon">z</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> <b>by </b><i/>;<br/>
<a NAME="E3:42_1"/><i><font color="Green">E60</font></i>: 
 <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> <font color="Maroon">d</font>
 
<b>by </b><i/>;<br/>
<a NAME="E4:42_1"/>
<font color="Maroon">d</font> <a href="lattice5.html#K4">.</a> <font color="Maroon">x'</font>,<font color="Maroon">z'</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">d</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> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice5.html#K4">.</a> <font color="Maroon">y'</font>,<font color="Maroon">z'</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="lattice5.html#D8">LATTICE5:def 8</a></i>;<br/>
<b>hence </b><a NAME="E5:42_1"/>
<span class="p1">(<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> <a href="lattice5.html#K4">.</a> <font color="Maroon">x</font>,<font color="Maroon">z</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><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> <a href="lattice5.html#K4">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><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> <a href="lattice5.html#K4">.</a> <font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span>
 <b>by </b><i><a class="txt" href="lattice8.html#E1:42"><i><font color="Green">E39</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:42"/><i><font color="Green">E61</font></i>: 
for <font color="Olive">O1</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">O1</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[ <a href="ordinal1.html#K1">succ</a> <font color="Olive">O1</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">O1</font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:42_2"/><i><font color="Green">E62</font></i>: 
( <font color="Maroon">O1</font> <a href="ordinal1.html#R1">c=</a>  <a href="lattice5.html#K8">DistEsti</a> <font color="Maroon">d</font> implies  <a href="lattice8.html#K6" target="_self">ConsecutiveDelta2</a> <font color="Maroon">q</font>,<font color="Maroon">O1</font> is <a href="lattice5.html#V3">u.t.i.</a> )
 <b>and </b><br/><a NAME="E2:42_2"/><i><font color="Green">E63</font></i>: 
 <a href="ordinal1.html#K1">succ</a> <font color="Maroon">O1</font> <a href="ordinal1.html#R1">c=</a>  <a href="lattice5.html#K8">DistEsti</a> <font color="Maroon">d</font>
 ;<br/>

<a NAME="E3:42_2"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">O1</font> <a href="hidden.html#R2">in</a>  <a href="lattice5.html#K8">DistEsti</a> <font color="Maroon">d</font>
 
<b>by </b><i>, <a class="ref" href="ordinal1.html#T33">ORDINAL1:33</a></i>;<br/>
<a NAME="E4:42_2"/><i><font color="Green">E66</font></i>: 
 <a href="lattice8.html#K6" target="_self">ConsecutiveDelta2</a> <font color="Maroon">q</font>,<font color="Maroon">O1</font> is <a href="lattice5.html#V1">symmetric</a>
 
<b>by </b><i>, </i>;<br/>
<b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">A</font>,<span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">O1</font></span>)</span>, <font color="Maroon">y</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">A</font>,<span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">O1</font></span>)</span>, <font color="Maroon">z</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">A</font>,<span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">O1</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice5.html#D8">LATTICE5:def 8</a><br/>

<b>set </b><font color="Maroon">f</font> =  <a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <span class="p1">(<span class="default"><a href="lattice8.html#K6" target="_self">ConsecutiveDelta2</a> <font color="Maroon">q</font>,<font color="Maroon">O1</font></span>)</span>,<span class="p1">(<span class="default"><a href="lattice8.html#K4" target="_self">Quadr2</a> <font color="Maroon">q</font>,<font color="Maroon">O1</font></span>)</span>;<br/>
<b>reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font>, <font color="Maroon">y'</font> = <font color="Maroon">y</font>, <font color="Maroon">z'</font> = <font color="Maroon">z</font> as    <a href="subset_1.html#M1">Element</a> of  <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> <font color="Maroon">A</font>,<font color="Maroon">O1</font></span>)</span> <b>by </b><i/>;<br/>
<a NAME="E6:42_2"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">O1</font> <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="txt" href="lattice8.html#E2:42"><i><font color="Green">E42</font></i></a>, <a class="ref" href="lattice5.html#T28">LATTICE5:28</a></i>;<br/>
<a NAME="E7:42_2"/><b>then </b>
<font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">O1</font> <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="E8:42_2"/><b>then </b>
<font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">O1</font> <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 <font color="Maroon">A</font>, <font color="Olive">v</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</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> : <font color="Maroon">d</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="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 <font color="Maroon">A</font>, <font color="Maroon">v</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</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="E10:42_2"/><i><font color="Green">E71</font></i>: 
( <font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">O1</font> <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; <font color="Maroon">d</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> )
 ;<br/>
<b>set </b><font color="Maroon">X</font> = <span class="p1">(<span class="default"><a href="lattice8.html#K4" target="_self">Quadr2</a> <font color="Maroon">q</font>,<font color="Maroon">O1</font></span>)</span> <a href="mcart_1.html#K8">`1</a> ;<br/>
<b>set </b><font color="Maroon">Y</font> = <span class="p1">(<span class="default"><a href="lattice8.html#K4" target="_self">Quadr2</a> <font color="Maroon">q</font>,<font color="Maroon">O1</font></span>)</span> <a href="mcart_1.html#K9">`2</a> ;<br/>
<b>reconsider </b><font color="Maroon">a</font> = <span class="p1">(<span class="default"><a href="lattice8.html#K4" target="_self">Quadr2</a> <font color="Maroon">q</font>,<font color="Maroon">O1</font></span>)</span> <a href="mcart_1.html#K10">`3</a> , <font color="Maroon">b</font> = <span class="p1">(<span class="default"><a href="lattice8.html#K4" target="_self">Quadr2</a> <font color="Maroon">q</font>,<font color="Maroon">O1</font></span>)</span> <a href="mcart_1.html#K11">`4</a>  as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> ;<br/>
<a NAME="E12:42_2"/><i><font color="Green">E74</font></i>: 
 <a href="lattice8.html#K4" target="_self">Quadr2</a> <font color="Maroon">q</font>,<font color="Maroon">O1</font> <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/>
<a NAME="E13:42_2"/><b>then </b><i><font color="Green">E75</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="lattice8.html#K4" target="_self">Quadr2</a> <font color="Maroon">q</font>,<font color="Maroon">O1</font></span>)</span> <a href="mcart_1.html#K8">`1</a> 
 
<b>by </b><i><a class="ref" href="mcart_1.html#D8">MCART_1:def 8</a></i>;<br/>
<a NAME="E14:42_2"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="lattice8.html#K4" target="_self">Quadr2</a> <font color="Maroon">q</font>,<font color="Maroon">O1</font></span>)</span> <a href="mcart_1.html#K9">`2</a> 
 
<b>by </b><i><a class="ref" href="lattice8.html#D2" target="_self">Def2</a>, <a class="ref" href="mcart_1.html#D9">MCART_1:def 9</a></i>;<br/>
<a NAME="E15:42_2"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">a'</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>, <a class="ref" href="mcart_1.html#D10">MCART_1:def 10</a></i>;<br/>
<a NAME="E16:42_2"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">b'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 
<b>by </b><i><a class="ref" href="lattice8.html#D2" target="_self">Def2</a>, <a class="ref" href="mcart_1.html#D11">MCART_1:def 11</a></i>;<br/>
<a NAME="E17:42_2"/><i><font color="Green">E80</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">A</font></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="E18:42_2"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">d</font> <a href="tarski.html#R1">c=</a>  <a href="lattice8.html#K6" target="_self">ConsecutiveDelta2</a> <font color="Maroon">q</font>,<font color="Maroon">O1</font>
 
<b>by </b><i/>;<br/>
<font color="Maroon">d</font> <a href="lattice5.html#K4">.</a> <font color="Maroon">u</font>,<font color="Maroon">v</font> = 
<font color="Maroon">d</font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">u</font>,<font color="Maroon">v</font></span><a href="domain_1.html#K1">]</a></span>

<br/>.= 
<span class="p1">(<span class="default"><a href="lattice8.html#K6" target="_self">ConsecutiveDelta2</a> <font color="Maroon">q</font>,<font color="Maroon">O1</font></span>)</span> <a href="lattice5.html#K4">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K4" target="_self">Quadr2</a> <font color="Maroon">q</font>,<font color="Maroon">O1</font></span>)</span> <a href="mcart_1.html#K8">`1</a> </span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K4" target="_self">Quadr2</a> <font color="Maroon">q</font>,<font color="Maroon">O1</font></span>)</span> <a href="mcart_1.html#K9">`2</a> </span>)</span>
<b>by </b><i>, , , , <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>
;<br/>
<a NAME="E20:42_2"/><b>then </b><i><font color="Green">E82</font></i>: 
 <a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <span class="p1">(<span class="default"><a href="lattice8.html#K6" target="_self">ConsecutiveDelta2</a> <font color="Maroon">q</font>,<font color="Maroon">O1</font></span>)</span>,<span class="p1">(<span class="default"><a href="lattice8.html#K4" target="_self">Quadr2</a> <font color="Maroon">q</font>,<font color="Maroon">O1</font></span>)</span> is <a href="lattice5.html#V3">u.t.i.</a>
 
<b>by </b><i>, , <a class="txt" href="lattice8.html#E2:42"><i><font color="Green">E42</font></i></a>, , , , , , <a class="ref" href="ordinal1.html#D2">ORDINAL1:def 2</a></i>;<br/>
<i><font color="Green">E83</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> <font color="Maroon">O1</font></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>,<font color="Maroon">O1</font></span>)</span>,<span class="p2">(<span class="default"><a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">A</font>,<font color="Maroon">O1</font></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>,<font color="Maroon">O1</font></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="lattice8.html#K6" target="_self">ConsecutiveDelta2</a> <font color="Maroon">q</font>,<font color="Maroon">O1</font></span>)</span>,<span class="p1">(<span class="default"><a href="lattice8.html#K4" target="_self">Quadr2</a> <font color="Maroon">q</font>,<font color="Maroon">O1</font></span>)</span>
<b>by </b><i><a class="ref" href="lattice5.html#D16">LATTICE5:def 16</a></i>
;<br/>
<a NAME="E22:42_2"/>
<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="lattice8.html#K6" target="_self">ConsecutiveDelta2</a> <font color="Maroon">q</font>,<font color="Maroon">O1</font></span>)</span>,<span class="p2">(<span class="default"><a href="lattice8.html#K4" target="_self">Quadr2</a> <font color="Maroon">q</font>,<font color="Maroon">O1</font></span>)</span></span>)</span> <a href="lattice5.html#K4">.</a> <font color="Maroon">x'</font>,<font color="Maroon">z'</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <span class="p3">(<span class="default"><a href="lattice8.html#K6" target="_self">ConsecutiveDelta2</a> <font color="Maroon">q</font>,<font color="Maroon">O1</font></span>)</span>,<span class="p3">(<span class="default"><a href="lattice8.html#K4" target="_self">Quadr2</a> <font color="Maroon">q</font>,<font color="Maroon">O1</font></span>)</span></span>)</span> <a href="lattice5.html#K4">.</a> <font color="Maroon">x'</font>,<font color="Maroon">y'</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" target="_self">new_bi_fun2</a> <span class="p3">(<span class="default"><a href="lattice8.html#K6" target="_self">ConsecutiveDelta2</a> <font color="Maroon">q</font>,<font color="Maroon">O1</font></span>)</span>,<span class="p3">(<span class="default"><a href="lattice8.html#K4" target="_self">Quadr2</a> <font color="Maroon">q</font>,<font color="Maroon">O1</font></span>)</span></span>)</span> <a href="lattice5.html#K4">.</a> <font color="Maroon">y'</font>,<font color="Maroon">z'</font></span>)</span>
 
<b>by </b><i><a class="ref" href="lattice8.html#D3" target="_self">Def3</a>, <a class="ref" href="lattice5.html#D8">LATTICE5:def 8</a></i>;<br/>
<b>hence </b><a NAME="E23:42_2"/>
<span class="p1">(<span class="default"><a href="lattice8.html#K6" target="_self">ConsecutiveDelta2</a> <font color="Maroon">q</font>,<span class="p2">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">O1</font></span>)</span></span>)</span> <a href="lattice5.html#K4">.</a> <font color="Maroon">x</font>,<font color="Maroon">z</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K6" target="_self">ConsecutiveDelta2</a> <font color="Maroon">q</font>,<span class="p3">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">O1</font></span>)</span></span>)</span> <a href="lattice5.html#K4">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K6" target="_self">ConsecutiveDelta2</a> <font color="Maroon">q</font>,<span class="p3">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">O1</font></span>)</span></span>)</span> <a href="lattice5.html#K4">.</a> <font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E6:42"/><i><font color="Green">E84</font></i>: 
for <font color="Olive">O1</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Olive">O1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Olive">O1</font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a>  &amp; ( for <font color="Olive">O2</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Olive">O2</font> <a href="hidden.html#R2">in</a> <font color="Olive">O1</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">O2</font>] ) holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">O1</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">O2</font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:42_3"/><i><font color="Green">E85</font></i>: 
( <font color="Maroon">O2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">O2</font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a>  &amp; ( for <font color="Olive">O1</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Olive">O1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">O2</font> &amp; <font color="Olive">O1</font> <a href="ordinal1.html#R1">c=</a>  <a href="lattice5.html#K8">DistEsti</a> <font color="Maroon">d</font> holds <br/> <a href="lattice8.html#K6" target="_self">ConsecutiveDelta2</a> <font color="Maroon">q</font>,<font color="Olive">O1</font> is <a href="lattice5.html#V3">u.t.i.</a> ) )
 <b>and </b><br/><a NAME="E2:42_3"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">O2</font> <a href="ordinal1.html#R1">c=</a>  <a href="lattice5.html#K8">DistEsti</a> <font color="Maroon">d</font>
 ;<br/>

<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="ordinal1.html#NM2">Ordinal</a>) <b>-&gt; </b>  <a href="lattice5.html#NM3">BiFunction</a> of <span class="p1">(<span class="default"><a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">A</font>,<font color="Maroon">a<sub>1</sub></font></span>)</span>,<font color="Maroon">L</font> =  <a href="lattice8.html#K6" target="_self">ConsecutiveDelta2</a> <font color="Maroon">q</font>,<font color="Maroon">a<sub>1</sub></font>;<br/>
<b>consider </b><font color="Maroon">Ls</font> being   <a href="ordinal1.html#NM3">T-Sequence</a><b> such that </b><br/><a NAME="E4:42_3"/><i><font color="Green">E87</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">Ls</font> <a href="hidden.html#R1">=</a> <font color="Maroon">O2</font> &amp; ( for <font color="Olive">O1</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Olive">O1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">O2</font> holds <br/><font color="Maroon">Ls</font> <a href="funct_1.html#K1">.</a> <font color="Olive">O1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">O1</font>) ) )
 <b>from </b><i><a class="ref" href="ordinal2.html#S2">ORDINAL2:sch 2</a>();<br/></i>
<a NAME="E5:42_3"/><i><font color="Green">E88</font></i>: 
 <a href="lattice8.html#K6" target="_self">ConsecutiveDelta2</a> <font color="Maroon">q</font>,<font color="Maroon">O2</font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">Ls</font></span>)</span>
 
<b>by </b><i>, , </i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(  <a href="ordinal1.html#NM2">Ordinal</a>) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  =  <a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">A</font>,<font color="Maroon">a<sub>1</sub></font>;<br/>
<b>consider </b><font color="Maroon">Ts</font> being   <a href="ordinal1.html#NM3">T-Sequence</a><b> such that </b><br/><a NAME="E7:42_3"/><i><font color="Green">E89</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">Ts</font> <a href="hidden.html#R1">=</a> <font color="Maroon">O2</font> &amp; ( for <font color="Olive">O1</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Olive">O1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">O2</font> holds <br/><font color="Maroon">Ts</font> <a href="funct_1.html#K1">.</a> <font color="Olive">O1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">O1</font>) ) )
 <b>from </b><i><a class="ref" href="ordinal2.html#S2">ORDINAL2:sch 2</a>();<br/></i>
<a NAME="E8:42_3"/><i><font color="Green">E90</font></i>: 
 <a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">A</font>,<font color="Maroon">O2</font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">Ts</font></span>)</span>
 
<b>by </b><i>, , </i>;<br/>
<b>set </b><font color="Maroon">CS</font> =  <a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">A</font>,<font color="Maroon">O2</font>;<br/>
<b>reconsider </b><font color="Maroon">f</font> =  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">Ls</font></span>)</span> as   <a href="lattice5.html#NM3">BiFunction</a> of <span class="p1">(<span class="default"><a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">A</font>,<font color="Maroon">O2</font></span>)</span>,<font color="Maroon">L</font> <b>by </b><i/>;<br/>
<a NAME="E10:42_3"/>
<font color="Maroon">f</font> is <a href="lattice5.html#V3">u.t.i.</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">A</font>,<font color="Maroon">O2</font>, <font color="Maroon">y</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">A</font>,<font color="Maroon">O2</font>, <font color="Maroon">z</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">A</font>,<font color="Maroon">O2</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice5.html#D8">LATTICE5:def 8</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="E2:42_3_1"/><i><font color="Green">E91</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">X</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">Ts</font> )
 <b>by </b><i><a class="ref" href="lattice8.html#D2" target="_self">Def2</a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">o1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:42_3_1"/><i><font color="Green">E92</font></i>: 
( <font color="Maroon">o1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">Ts</font> &amp; <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Ts</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o1</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">Y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:42_3_1"/><i><font color="Green">E93</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">Ts</font> )
 <b>by </b><i><a class="ref" href="lattice8.html#D2" target="_self">Def2</a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">o2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E8:42_3_1"/><i><font color="Green">E94</font></i>: 
( <font color="Maroon">o2</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">Ts</font> &amp; <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Ts</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o2</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">Z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E10:42_3_1"/><i><font color="Green">E95</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> &amp; <font color="Maroon">Z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">Ts</font> )
 <b>by </b><i><a class="ref" href="lattice8.html#D2" target="_self">Def2</a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">o3</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E12:42_3_1"/><i><font color="Green">E125</font></i>: 
( <font color="Maroon">o3</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">Ts</font> &amp; <font color="Maroon">Z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Ts</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o3</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">o1</font> = <font color="Maroon">o1</font>, <font color="Maroon">o2</font> = <font color="Maroon">o2</font>, <font color="Maroon">o3</font> = <font color="Maroon">o3</font> as   <a href="ordinal1.html#NM2">Ordinal</a> <b>by </b><i>, , , <a class="ref" href="ordinal1.html#T23">ORDINAL1:23</a></i>;<br/>
<a NAME="E14:42_3_1"/><i><font color="Green">E126</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">A</font>,<font color="Maroon">o1</font>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E15:42_3_1"/><i><font color="Green">E127</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">A</font>,<font color="Maroon">o2</font>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E16:42_3_1"/><i><font color="Green">E128</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">A</font>,<font color="Maroon">o3</font>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E17:42_3_1"/><i><font color="Green">E129</font></i>: 
<font color="Maroon">Ls</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o1</font> <a href="hidden.html#R1">=</a>  <a href="lattice8.html#K6" target="_self">ConsecutiveDelta2</a> <font color="Maroon">q</font>,<font color="Maroon">o1</font>
 
<b>by </b><i>, , </i>;<br/>
<b>then reconsider </b><font color="Maroon">h1</font> = <font color="Maroon">Ls</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o1</font> as   <a href="lattice5.html#NM3">BiFunction</a> of <span class="p1">(<span class="default"><a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">A</font>,<font color="Maroon">o1</font></span>)</span>,<font color="Maroon">L</font> ;<br/>
<a NAME="E19:42_3_1"/><i><font color="Green">E130</font></i>: 
<font color="Maroon">h1</font> is <a href="lattice5.html#V3">u.t.i.</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_3_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">A</font>,<font color="Maroon">o1</font>, <font color="Maroon">y</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">A</font>,<font color="Maroon">o1</font>, <font color="Maroon">z</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">A</font>,<font color="Maroon">o1</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice5.html#D8">LATTICE5:def 8</a><br/>

<a NAME="E1:42_3_1_1"/><i><font color="Green">E131</font></i>: 
 <a href="lattice8.html#K6" target="_self">ConsecutiveDelta2</a> <font color="Maroon">q</font>,<font color="Maroon">o1</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">h1</font>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E2:42_3_1_1"/>
<font color="Maroon">o1</font> <a href="ordinal1.html#R1">c=</a>  <a href="lattice5.html#K8">DistEsti</a> <font color="Maroon">d</font>
 
<b>by </b><i><a class="txt" href="lattice8.html#E2:42"><i><font color="Green">E42</font></i></a>, , , <a class="ref" href="ordinal1.html#D2">ORDINAL1:def 2</a></i>;<br/>
<a NAME="E3:42_3_1_1"/><b>then </b>
 <a href="lattice8.html#K6" target="_self">ConsecutiveDelta2</a> <font color="Maroon">q</font>,<font color="Maroon">o1</font> is <a href="lattice5.html#V3">u.t.i.</a>
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E4:42_3_1_1"/>
<font color="Maroon">h1</font> <a href="lattice5.html#K4">.</a> <font color="Maroon">x</font>,<font color="Maroon">z</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">h1</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> <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="lattice5.html#K4">.</a> <font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span>
 <b>by </b><i><a class="txt" href="lattice8.html#E3:42"><i><font color="Green">E58</font></i></a>, <a class="ref" href="lattice5.html#D8">LATTICE5:def 8</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E20:42_3_1"/><i><font color="Green">E132</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">h1</font> <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#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">A</font>,<font color="Maroon">o1</font></span>)</span>,<span class="p2">(<span class="default"><a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">A</font>,<font color="Maroon">o1</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="E21:42_3_1"/><i><font color="Green">E133</font></i>: 
<font color="Maroon">Ls</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o2</font> <a href="hidden.html#R1">=</a>  <a href="lattice8.html#K6" target="_self">ConsecutiveDelta2</a> <font color="Maroon">q</font>,<font color="Maroon">o2</font>
 
<b>by </b><i>, , </i>;<br/>
<b>then reconsider </b><font color="Maroon">h2</font> = <font color="Maroon">Ls</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o2</font> as   <a href="lattice5.html#NM3">BiFunction</a> of <span class="p1">(<span class="default"><a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">A</font>,<font color="Maroon">o2</font></span>)</span>,<font color="Maroon">L</font> ;<br/>
<a NAME="E23:42_3_1"/><i><font color="Green">E134</font></i>: 
<font color="Maroon">h2</font> is <a href="lattice5.html#V3">u.t.i.</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_3_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">A</font>,<font color="Maroon">o2</font>, <font color="Maroon">y</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">A</font>,<font color="Maroon">o2</font>, <font color="Maroon">z</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">A</font>,<font color="Maroon">o2</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice5.html#D8">LATTICE5:def 8</a><br/>

<a NAME="E1:42_3_1_2"/><i><font color="Green">E135</font></i>: 
 <a href="lattice8.html#K6" target="_self">ConsecutiveDelta2</a> <font color="Maroon">q</font>,<font color="Maroon">o2</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">h2</font>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E2:42_3_1_2"/>
<font color="Maroon">o2</font> <a href="ordinal1.html#R1">c=</a>  <a href="lattice5.html#K8">DistEsti</a> <font color="Maroon">d</font>
 
<b>by </b><i><a class="txt" href="lattice8.html#E2:42"><i><font color="Green">E42</font></i></a>, , , <a class="ref" href="ordinal1.html#D2">ORDINAL1:def 2</a></i>;<br/>
<a NAME="E3:42_3_1_2"/><b>then </b>
 <a href="lattice8.html#K6" target="_self">ConsecutiveDelta2</a> <font color="Maroon">q</font>,<font color="Maroon">o2</font> is <a href="lattice5.html#V3">u.t.i.</a>
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E4:42_3_1_2"/>
<font color="Maroon">h2</font> <a href="lattice5.html#K4">.</a> <font color="Maroon">x</font>,<font color="Maroon">z</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">h2</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> <span class="p1">(<span class="default"><font color="Maroon">h2</font> <a href="lattice5.html#K4">.</a> <font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span>
 <b>by </b><i><a class="txt" href="lattice8.html#E5:42"><i><font color="Green">E61</font></i></a>, <a class="ref" href="lattice5.html#D8">LATTICE5:def 8</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E24:42_3_1"/><i><font color="Green">E136</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">h2</font> <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#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">A</font>,<font color="Maroon">o2</font></span>)</span>,<span class="p2">(<span class="default"><a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">A</font>,<font color="Maroon">o2</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="E25:42_3_1"/><i><font color="Green">E137</font></i>: 
<font color="Maroon">Ls</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o3</font> <a href="hidden.html#R1">=</a>  <a href="lattice8.html#K6" target="_self">ConsecutiveDelta2</a> <font color="Maroon">q</font>,<font color="Maroon">o3</font>
 
<b>by </b><i>, , </i>;<br/>
<b>then reconsider </b><font color="Maroon">h3</font> = <font color="Maroon">Ls</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o3</font> as   <a href="lattice5.html#NM3">BiFunction</a> of <span class="p1">(<span class="default"><a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">A</font>,<font color="Maroon">o3</font></span>)</span>,<font color="Maroon">L</font> ;<br/>
<a NAME="E27:42_3_1"/><i><font color="Green">E138</font></i>: 
<font color="Maroon">h3</font> is <a href="lattice5.html#V3">u.t.i.</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_3_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">A</font>,<font color="Maroon">o3</font>, <font color="Maroon">y</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">A</font>,<font color="Maroon">o3</font>, <font color="Maroon">z</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">A</font>,<font color="Maroon">o3</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice5.html#D8">LATTICE5:def 8</a><br/>

<a NAME="E1:42_3_1_3"/><i><font color="Green">E139</font></i>: 
 <a href="lattice8.html#K6" target="_self">ConsecutiveDelta2</a> <font color="Maroon">q</font>,<font color="Maroon">o3</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">h3</font>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E2:42_3_1_3"/>
<font color="Maroon">o3</font> <a href="ordinal1.html#R1">c=</a>  <a href="lattice5.html#K8">DistEsti</a> <font color="Maroon">d</font>
 
<b>by </b><i><a class="txt" href="lattice8.html#E2:42"><i><font color="Green">E42</font></i></a>, , , <a class="ref" href="ordinal1.html#D2">ORDINAL1:def 2</a></i>;<br/>
<a NAME="E3:42_3_1_3"/><b>then </b>
 <a href="lattice8.html#K6" target="_self">ConsecutiveDelta2</a> <font color="Maroon">q</font>,<font color="Maroon">o3</font> is <a href="lattice5.html#V3">u.t.i.</a>
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E4:42_3_1_3"/>
<font color="Maroon">h3</font> <a href="lattice5.html#K4">.</a> <font color="Maroon">x</font>,<font color="Maroon">z</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">h3</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> <span class="p1">(<span class="default"><font color="Maroon">h3</font> <a href="lattice5.html#K4">.</a> <font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="lattice5.html#D8">LATTICE5:def 8</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E28:42_3_1"/><i><font color="Green">E140</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">h3</font> <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#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">A</font>,<font color="Maroon">o3</font></span>)</span>,<span class="p2">(<span class="default"><a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">A</font>,<font color="Maroon">o3</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/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_3_1_4"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:42_3_1_4"/>
( <font color="Maroon">o1</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">o3</font> or <font color="Maroon">o3</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">o1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_3_1_4_1"><b>suppose </b></a><a NAME="E1:42_3_1_4_1"/><i><font color="Green">E141</font></i>: 
<font color="Maroon">o1</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">o3</font>
 ;<br/><div class="add"><a NAME="E2:42_3_1_4_1"/><b>then </b><i><font color="Green">E161</font></i>: 
 <a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">A</font>,<font color="Maroon">o1</font> <a href="tarski.html#R1">c=</a>  <a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">A</font>,<font color="Maroon">o3</font>
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E3:42_3_1_4_1"/>
<span class="p1">(<span class="default"><font color="Maroon">f</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> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="lattice5.html#K4">.</a> <font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">f</font> <a href="lattice5.html#K4">.</a> <font color="Maroon">x</font>,<font color="Maroon">z</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_3_1_4_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_3_1_4_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:42_3_1_4_1_1_1"/>
( <font color="Maroon">o2</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">o3</font> or <font color="Maroon">o3</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">o2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_3_1_4_1_1_1_1"><b>suppose </b></a><a NAME="E1:42_3_1_4_1_1_1_1"/>
<font color="Maroon">o2</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">o3</font>
 ;<br/><div class="add"><a NAME="E2:42_3_1_4_1_1_1_1"/><b>then </b><i><font color="Green">E162</font></i>: 
 <a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">A</font>,<font color="Maroon">o2</font> <a href="tarski.html#R1">c=</a>  <a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">A</font>,<font color="Maroon">o3</font>
 <b>by </b><i/>;<br/><b>then reconsider </b><font color="Maroon">y'</font> = <font color="Maroon">y</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">A</font>,<font color="Maroon">o3</font> <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">A</font>,<font color="Maroon">o3</font> <b>by </b><i><a class="ref" href="lattice8.html#D3" target="_self">Def3</a>, </i>;<br/><b>reconsider </b><font color="Maroon">z'</font> = <font color="Maroon">z</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">A</font>,<font color="Maroon">o3</font> <b>by </b><i>, , </i>;<br/><a NAME="E6:42_3_1_4_1_1_1_1"/>
 <a href="lattice8.html#K6" target="_self">ConsecutiveDelta2</a> <font color="Maroon">q</font>,<font color="Maroon">o3</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">Ls</font>
 <b>by </b><i>, , , , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E7:42_3_1_4_1_1_1_1"/><b>then </b><i><font color="Green">E163</font></i>: 
<font color="Maroon">h3</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">f</font>
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/><a NAME="E8:42_3_1_4_1_1_1_1"/><i><font color="Green">E164</font></i>: 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h3</font>
 <b>by </b><i><a class="ref" href="lattice8.html#D3" target="_self">Def3</a>, , , , , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E9:42_3_1_4_1_1_1_1"/><i><font color="Green">E165</font></i>: 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h3</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#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E10:42_3_1_4_1_1_1_1"/><i><font color="Green">E166</font></i>: 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">z</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h3</font>
 <b>by </b><i><a class="ref" href="lattice8.html#D3" target="_self">Def3</a>, <a class="txt" href="lattice8.html#E4"><i><font color="Green">Lemma55</font></i></a>, , , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E11:42_3_1_4_1_1_1_1"/><i><font color="Green">E167</font></i>: 
<font color="Maroon">f</font> <a href="lattice5.html#K4">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h3</font> <a href="lattice5.html#K4">.</a> <font color="Maroon">x'</font>,<font color="Maroon">y'</font>
 <b>by </b><i>, , <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>;<br/><a NAME="E12:42_3_1_4_1_1_1_1"/><i><font color="Green">E168</font></i>: 
<font color="Maroon">f</font> <a href="lattice5.html#K4">.</a> <font color="Maroon">y</font>,<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h3</font> <a href="lattice5.html#K4">.</a> <font color="Maroon">y'</font>,<font color="Maroon">z'</font>
 <b>by </b><i>, , <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>;<br/><a NAME="E13:42_3_1_4_1_1_1_1"/>
<font color="Maroon">f</font> <a href="lattice5.html#K4">.</a> <font color="Maroon">x</font>,<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h3</font> <a href="lattice5.html#K4">.</a> <font color="Maroon">x'</font>,<font color="Maroon">z'</font>
 <b>by </b><i>, , <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>;<br/><b>hence </b><a NAME="E14:42_3_1_4_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">f</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> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="lattice5.html#K4">.</a> <font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">f</font> <a href="lattice5.html#K4">.</a> <font color="Maroon">x</font>,<font color="Maroon">z</font>
 <b>by </b><i>, , , <a class="ref" href="lattice5.html#D8">LATTICE5:def 8</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_3_1_4_1_1_1_2"><b>suppose </b></a><a NAME="E1:42_3_1_4_1_1_1_2"/>
<font color="Maroon">o3</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">o2</font>
 ;<br/><div class="add"><a NAME="E2:42_3_1_4_1_1_1_2"/><b>then </b><i><font color="Green">E169</font></i>: 
 <a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">A</font>,<font color="Maroon">o3</font> <a href="tarski.html#R1">c=</a>  <a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">A</font>,<font color="Maroon">o2</font>
 <b>by </b><i/>;<br/><b>then reconsider </b><font color="Maroon">z'</font> = <font color="Maroon">z</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">A</font>,<font color="Maroon">o2</font> <b>by </b><i><a class="txt" href="lattice8.html#E4"><i><font color="Green">Lemma55</font></i></a></i>;<br/><a NAME="E4:42_3_1_4_1_1_1_2"/>
 <a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">A</font>,<font color="Maroon">o1</font> <a href="tarski.html#R1">c=</a>  <a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">A</font>,<font color="Maroon">o3</font>
 <b>by </b><i>, </i>;<br/><a NAME="E5:42_3_1_4_1_1_1_2"/><b>then </b><i><font color="Green">E170</font></i>: 
 <a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">A</font>,<font color="Maroon">o1</font> <a href="tarski.html#R1">c=</a>  <a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">A</font>,<font color="Maroon">o2</font>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>then reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">A</font>,<font color="Maroon">o2</font> <b>by </b><i><a class="ref" href="lattice8.html#D3" target="_self">Def3</a></i>;<br/><b>reconsider </b><font color="Maroon">y'</font> = <font color="Maroon">y</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">A</font>,<font color="Maroon">o2</font> <b>by </b><i>, , </i>;<br/><a NAME="E8:42_3_1_4_1_1_1_2"/>
 <a href="lattice8.html#K6" target="_self">ConsecutiveDelta2</a> <font color="Maroon">q</font>,<font color="Maroon">o2</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">Ls</font>
 <b>by </b><i>, , , <a class="txt" href="lattice8.html#E4:42"><i><font color="Green">E59</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E9:42_3_1_4_1_1_1_2"/><b>then </b><i><font color="Green">E171</font></i>: 
<font color="Maroon">h2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="txt" href="lattice8.html#E4:42"><i><font color="Green">E59</font></i></a>, <a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/><a NAME="E10:42_3_1_4_1_1_1_2"/><i><font color="Green">E172</font></i>: 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h2</font>
 <b>by </b><i><a class="ref" href="lattice8.html#D3" target="_self">Def3</a>, , <a class="txt" href="lattice8.html#E5:42"><i><font color="Green">E61</font></i></a>, , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E11:42_3_1_4_1_1_1_2"/><i><font color="Green">E173</font></i>: 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h2</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#E5:42"><i><font color="Green">E61</font></i></a>, , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E12:42_3_1_4_1_1_1_2"/><i><font color="Green">E174</font></i>: 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">z</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h2</font>
 <b>by </b><i><a class="ref" href="lattice8.html#D3" target="_self">Def3</a>, <a class="txt" href="lattice8.html#E4"><i><font color="Green">Lemma55</font></i></a>, <a class="txt" href="lattice8.html#E5:42"><i><font color="Green">E61</font></i></a>, , , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E13:42_3_1_4_1_1_1_2"/><i><font color="Green">E175</font></i>: 
<font color="Maroon">f</font> <a href="lattice5.html#K4">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h2</font> <a href="lattice5.html#K4">.</a> <font color="Maroon">x'</font>,<font color="Maroon">y'</font>
 <b>by </b><i>, , <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>;<br/><a NAME="E14:42_3_1_4_1_1_1_2"/><i><font color="Green">E176</font></i>: 
<font color="Maroon">f</font> <a href="lattice5.html#K4">.</a> <font color="Maroon">y</font>,<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h2</font> <a href="lattice5.html#K4">.</a> <font color="Maroon">y'</font>,<font color="Maroon">z'</font>
 <b>by </b><i>, , <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>;<br/><a NAME="E15:42_3_1_4_1_1_1_2"/>
<font color="Maroon">f</font> <a href="lattice5.html#K4">.</a> <font color="Maroon">x</font>,<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h2</font> <a href="lattice5.html#K4">.</a> <font color="Maroon">x'</font>,<font color="Maroon">z'</font>
 <b>by </b><i>, , <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>;<br/><b>hence </b><a NAME="E16:42_3_1_4_1_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">f</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> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="lattice5.html#K4">.</a> <font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">f</font> <a href="lattice5.html#K4">.</a> <font color="Maroon">x</font>,<font color="Maroon">z</font>
 <b>by </b><i>, , , <a class="ref" href="lattice5.html#D8">LATTICE5:def 8</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_3_1_4_2"><b>suppose </b></a><a NAME="E1:42_3_1_4_2"/><i><font color="Green">E177</font></i>: 
<font color="Maroon">o3</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">o1</font>
 ;<br/><div class="add"><a NAME="E2:42_3_1_4_2"/><b>then </b><i><font color="Green">E178</font></i>: 
 <a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">A</font>,<font color="Maroon">o3</font> <a href="tarski.html#R1">c=</a>  <a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">A</font>,<font color="Maroon">o1</font>
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E3:42_3_1_4_2"/>
<span class="p1">(<span class="default"><font color="Maroon">f</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> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="lattice5.html#K4">.</a> <font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">f</font> <a href="lattice5.html#K4">.</a> <font color="Maroon">x</font>,<font color="Maroon">z</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_3_1_4_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_3_1_4_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:42_3_1_4_2_1_1"/>
( <font color="Maroon">o1</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">o2</font> or <font color="Maroon">o2</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">o1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_3_1_4_2_1_1_1"><b>suppose </b></a><a NAME="E1:42_3_1_4_2_1_1_1"/>
<font color="Maroon">o1</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">o2</font>
 ;<br/><div class="add"><a NAME="E2:42_3_1_4_2_1_1_1"/><b>then </b><i><font color="Green">E179</font></i>: 
 <a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">A</font>,<font color="Maroon">o1</font> <a href="tarski.html#R1">c=</a>  <a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">A</font>,<font color="Maroon">o2</font>
 <b>by </b><i/>;<br/><b>then reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">A</font>,<font color="Maroon">o2</font> <b>by </b><i><a class="ref" href="lattice8.html#D3" target="_self">Def3</a></i>;<br/><a NAME="E4:42_3_1_4_2_1_1_1"/>
 <a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">A</font>,<font color="Maroon">o3</font> <a href="tarski.html#R1">c=</a>  <a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">A</font>,<font color="Maroon">o1</font>
 <b>by </b><i>, </i>;<br/><a NAME="E5:42_3_1_4_2_1_1_1"/><b>then </b><i><font color="Green">E180</font></i>: 
 <a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">A</font>,<font color="Maroon">o3</font> <a href="tarski.html#R1">c=</a>  <a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">A</font>,<font color="Maroon">o2</font>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>then reconsider </b><font color="Maroon">z'</font> = <font color="Maroon">z</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">A</font>,<font color="Maroon">o2</font> <b>by </b><i><a class="txt" href="lattice8.html#E4"><i><font color="Green">Lemma55</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">y'</font> = <font color="Maroon">y</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">A</font>,<font color="Maroon">o2</font> <b>by </b><i>, , </i>;<br/><a NAME="E8:42_3_1_4_2_1_1_1"/>
 <a href="lattice8.html#K6" target="_self">ConsecutiveDelta2</a> <font color="Maroon">q</font>,<font color="Maroon">o2</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">Ls</font>
 <b>by </b><i>, , , <a class="txt" href="lattice8.html#E4:42"><i><font color="Green">E59</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E9:42_3_1_4_2_1_1_1"/><b>then </b><i><font color="Green">E181</font></i>: 
<font color="Maroon">h2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="txt" href="lattice8.html#E4:42"><i><font color="Green">E59</font></i></a>, <a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/><a NAME="E10:42_3_1_4_2_1_1_1"/><i><font color="Green">E182</font></i>: 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h2</font>
 <b>by </b><i><a class="ref" href="lattice8.html#D3" target="_self">Def3</a>, , <a class="txt" href="lattice8.html#E5:42"><i><font color="Green">E61</font></i></a>, , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E11:42_3_1_4_2_1_1_1"/><i><font color="Green">E183</font></i>: 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h2</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#E5:42"><i><font color="Green">E61</font></i></a>, , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E12:42_3_1_4_2_1_1_1"/><i><font color="Green">E184</font></i>: 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">z</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h2</font>
 <b>by </b><i><a class="ref" href="lattice8.html#D3" target="_self">Def3</a>, <a class="txt" href="lattice8.html#E4"><i><font color="Green">Lemma55</font></i></a>, <a class="txt" href="lattice8.html#E5:42"><i><font color="Green">E61</font></i></a>, , , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E13:42_3_1_4_2_1_1_1"/><i><font color="Green">E185</font></i>: 
<font color="Maroon">f</font> <a href="lattice5.html#K4">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h2</font> <a href="lattice5.html#K4">.</a> <font color="Maroon">x'</font>,<font color="Maroon">y'</font>
 <b>by </b><i>, , <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>;<br/><a NAME="E14:42_3_1_4_2_1_1_1"/><i><font color="Green">E186</font></i>: 
<font color="Maroon">f</font> <a href="lattice5.html#K4">.</a> <font color="Maroon">y</font>,<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h2</font> <a href="lattice5.html#K4">.</a> <font color="Maroon">y'</font>,<font color="Maroon">z'</font>
 <b>by </b><i>, , <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>;<br/><a NAME="E15:42_3_1_4_2_1_1_1"/>
<font color="Maroon">f</font> <a href="lattice5.html#K4">.</a> <font color="Maroon">x</font>,<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h2</font> <a href="lattice5.html#K4">.</a> <font color="Maroon">x'</font>,<font color="Maroon">z'</font>
 <b>by </b><i>, , <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>;<br/><b>hence </b><a NAME="E16:42_3_1_4_2_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">f</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> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="lattice5.html#K4">.</a> <font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">f</font> <a href="lattice5.html#K4">.</a> <font color="Maroon">x</font>,<font color="Maroon">z</font>
 <b>by </b><i>, , , <a class="ref" href="lattice5.html#D8">LATTICE5:def 8</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_3_1_4_2_1_1_2"><b>suppose </b></a><a NAME="E1:42_3_1_4_2_1_1_2"/>
<font color="Maroon">o2</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">o1</font>
 ;<br/><div class="add"><a NAME="E2:42_3_1_4_2_1_1_2"/><b>then </b><i><font color="Green">E187</font></i>: 
 <a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">A</font>,<font color="Maroon">o2</font> <a href="tarski.html#R1">c=</a>  <a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">A</font>,<font color="Maroon">o1</font>
 <b>by </b><i/>;<br/><b>then reconsider </b><font color="Maroon">y'</font> = <font color="Maroon">y</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">A</font>,<font color="Maroon">o1</font> <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">z'</font> = <font color="Maroon">z</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">A</font>,<font color="Maroon">o1</font> <b>by </b><i><a class="txt" href="lattice8.html#E4"><i><font color="Green">Lemma55</font></i></a>, </i>;<br/><b>reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="lattice8.html#K3" target="_self">ConsecutiveSet2</a> <font color="Maroon">A</font>,<font color="Maroon">o1</font> <b>by </b><i>, , </i>;<br/><a NAME="E6:42_3_1_4_2_1_1_2"/>
 <a href="lattice8.html#K6" target="_self">ConsecutiveDelta2</a> <font color="Maroon">q</font>,<font color="Maroon">o1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">Ls</font>
 <b>by </b><i>, , , <a class="txt" href="lattice8.html#E5"><i><font color="Green">Lemma56</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E7:42_3_1_4_2_1_1_2"/><b>then </b><i><font color="Green">E188</font></i>: 
<font color="Maroon">h1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">f</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="E8:42_3_1_4_2_1_1_2"/><i><font color="Green">E189</font></i>: 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h1</font>
 <b>by </b><i><a class="ref" href="lattice8.html#D3" target="_self">Def3</a>, , <a class="txt" href="lattice8.html#E3:42"><i><font color="Green">E58</font></i></a>, , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E9:42_3_1_4_2_1_1_2"/><i><font color="Green">E190</font></i>: 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h1</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#E3:42"><i><font color="Green">E58</font></i></a>, , , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E10:42_3_1_4_2_1_1_2"/><i><font color="Green">E191</font></i>: 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">z</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h1</font>
 <b>by </b><i><a class="ref" href="lattice8.html#D3" target="_self">Def3</a>, <a class="txt" href="lattice8.html#E4"><i><font color="Green">Lemma55</font></i></a>, <a class="txt" href="lattice8.html#E3:42"><i><font color="Green">E58</font></i></a>, , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E11:42_3_1_4_2_1_1_2"/><i><font color="Green">E192</font></i>: 
<font color="Maroon">f</font> <a href="lattice5.html#K4">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h1</font> <a href="lattice5.html#K4">.</a> <font color="Maroon">x'</font>,<font color="Maroon">y'</font>
 <b>by </b><i>, , <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>;<br/><a NAME="E12:42_3_1_4_2_1_1_2"/><i><font color="Green">E193</font></i>: 
<font color="Maroon">f</font> <a href="lattice5.html#K4">.</a> <font color="Maroon">y</font>,<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h1</font> <a href="lattice5.html#K4">.</a> <font color="Maroon">y'</font>,<font color="Maroon">z'</font>
 <b>by </b><i>, , <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>;<br/><a NAME="E13:42_3_1_4_2_1_1_2"/>
<font color="Maroon">f</font> <a href="lattice5.html#K4">.</a> <font color="Maroon">x</font>,<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h1</font> <a href="lattice5.html#K4">.</a> <font color="Maroon">x'</font>,<font color="Maroon">z'</font>
 <b>by </b><i>, , <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>;<br/><b>hence </b><a NAME="E14:42_3_1_4_2_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">f</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> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="lattice5.html#K4">.</a> <font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">f</font> <a href="lattice5.html#K4">.</a> <font color="Maroon">x</font>,<font color="Maroon">z</font>
 <b>by </b><i>, , , <a class="ref" href="lattice5.html#D8">LATTICE5:def 8</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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

</div><b>end;</b></div>
<b>hence </b><a NAME="E11:42_3"/>
 <a href="lattice8.html#K6" target="_self">ConsecutiveDelta2</a> <font color="Maroon">q</font>,<font color="Maroon">O2</font> is <a href="lattice5.html#V3">u.t.i.</a>
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:42"/>
for <font color="Olive">O</font> being  <a href="ordinal1.html#NM2">Ordinal</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">O</font>]
 
<b>from </b><i><a class="ref" href="ordinal2.html#S1">ORDINAL2:sch 1</a>(, <a class="txt" href="lattice8.html#E1:42"><i><font color="Green">E39</font></i></a>, );<br/></i>
<b>hence </b><a NAME="E8:42"/>
( <font color="Maroon">O</font> <a href="ordinal1.html#R1">c=</a>  <a href="lattice5.html#K8">DistEsti</a> <font color="Maroon">d</font> implies  <a href="lattice8.html#K6" target="_self">ConsecutiveDelta2</a> <font color="Maroon">q</font>,<font color="Maroon">O</font> is <a href="lattice5.html#V3">u.t.i.</a> )
 ;<br/>


</div>
