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

<b>let </b><font color="Maroon" title="c1">A</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">L</font> be  <a href="yellow_0.html#V1" title="YELLOW_0:attr.1">lower-bounded</a> <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a>;<br/>



<b>assume </b><a NAME="E1:42"/><i><font color="Green" title="E35">A1</font></i>: 
<font color="Maroon" title="c2">L</font> is <a href="yellow11.html#V1" title="YELLOW11:attr.1">modular</a>
 ;<br/>

<b>let </b><font color="Maroon" title="c3">d</font> be   <a href="lattice5.html#NM3" title="LATTICE5:NM.3">BiFunction</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">L</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E2:42"/><i><font color="Green" title="E36">A2</font></i>: 
<font color="Maroon" title="c3">d</font> is <a href="lattice5.html#V1" title="LATTICE5:attr.1">symmetric</a>
 <b>and </b><br/><a NAME="E3:42"/><i><font color="Green" title="E37">A3</font></i>: 
<font color="Maroon" title="c3">d</font> is <a href="lattice5.html#V3" title="LATTICE5:attr.3">u.t.i.</a>
 ;<br/>

<b>let </b><font color="Maroon" title="c4">O</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>;<br/><b>let </b><font color="Maroon" title="c5">q</font> be    <a href="lattice5.html#M1" title="LATTICE5:mode.1">QuadrSeq</a> of <font color="Maroon" title="c3">d</font>;<br/>



<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>] <b>means </b>( $1 <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="lattice5.html#K8" title="LATTICE5:func.8">DistEsti</a> <font color="Maroon" title="c3">d</font> implies  <a href="lattice8.html#K6" title="LATTICE8:func.6">ConsecutiveDelta2</a> <font color="Maroon" title="c5">q</font>,$1 is <a href="lattice5.html#V3" title="LATTICE5:attr.3">u.t.i.</a> );<br/>
<a NAME="E4:42"/><i><font color="Green" title="E38">A4</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</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" title="XBOOLE_0:func.1">{}</a>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="lattice5.html#K8" title="LATTICE5:func.8">DistEsti</a> <font color="Maroon" title="c3">d</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c6">x</font>, <font color="Maroon" title="c7">y</font>, <font color="Maroon" title="c8">z</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> <font color="Maroon" title="c1">A</font>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice5.html#D8" title="LATTICE5:def.8">LATTICE5:def 8</a><br/>

<b>reconsider </b><font color="Maroon" title="c9">x'</font> = <font color="Maroon" title="c6">x</font>, <font color="Maroon" title="c10">y'</font> = <font color="Maroon" title="c7">y</font>, <font color="Maroon" title="c11">z'</font> = <font color="Maroon" title="c8">z</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font> <b>by </b><i><a class="ref" href="lattice8.html#T15" target="_self" title="LATTICE8:th.15">Th15</a></i>;<br/>
<a NAME="E3:42_1"/><i><font color="Green" title="E39">A5</font></i>: 
 <a href="lattice8.html#K6" title="LATTICE8:func.6">ConsecutiveDelta2</a> <font color="Maroon" title="c5">q</font>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c3">d</font>
 
<b>by </b><i><a class="ref" href="lattice8.html#T19" target="_self" title="LATTICE8:th.19">Th19</a></i>;<br/>
<a NAME="E4:42_1"/>
<font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c9">x'</font>,<font color="Maroon" title="c11">z'</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c9">x'</font>,<font color="Maroon" title="c10">y'</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c10">y'</font>,<font color="Maroon" title="c11">z'</font></span>)</span>
 
<b>by </b><i><a class="txt" href="lattice8.html#E3:42"><i><font color="Green" title="E37">A3</font></i></a>, <a class="ref" href="lattice5.html#D8" title="LATTICE5:def.8">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" title="LATTICE8:func.6">ConsecutiveDelta2</a> <font color="Maroon" title="c5">q</font>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c8">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K6" title="LATTICE8:func.6">ConsecutiveDelta2</a> <font color="Maroon" title="c5">q</font>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K6" title="LATTICE8:func.6">ConsecutiveDelta2</a> <font color="Maroon" title="c5">q</font>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c8">z</font></span>)</span>
 <b>by </b><i><a class="txt" href="lattice8.html#E3:42_1"><i><font color="Green" title="E39">A5</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:42"/><i><font color="Green" title="E39">A6</font></i>: 
for <font color="Olive" title="b1">O1</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">O1</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[ <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Olive" title="b1">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" title="c6">O1</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:42_2"/><i><font color="Green" title="E40">A7</font></i>: 
( <font color="Maroon" title="c6">O1</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="lattice5.html#K8" title="LATTICE5:func.8">DistEsti</a> <font color="Maroon" title="c3">d</font> implies  <a href="lattice8.html#K6" title="LATTICE8:func.6">ConsecutiveDelta2</a> <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c6">O1</font> is <a href="lattice5.html#V3" title="LATTICE5:attr.3">u.t.i.</a> )
 <b>and </b><br/><a NAME="E2:42_2"/><i><font color="Green" title="E41">A8</font></i>: 
 <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c6">O1</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="lattice5.html#K8" title="LATTICE5:func.8">DistEsti</a> <font color="Maroon" title="c3">d</font>
 ;<br/>

<a NAME="E3:42_2"/><i><font color="Green" title="E42">A9</font></i>: 
<font color="Maroon" title="c6">O1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lattice5.html#K8" title="LATTICE5:func.8">DistEsti</a> <font color="Maroon" title="c3">d</font>
 
<b>by </b><i><a class="txt" href="lattice8.html#E2:42_2"><i><font color="Green" title="E41">A8</font></i></a>, <a class="ref" href="ordinal1.html#T33" title="ORDINAL1:th.33">ORDINAL1:33</a></i>;<br/>
<a NAME="E4:42_2"/><i><font color="Green" title="E43">A10</font></i>: 
 <a href="lattice8.html#K6" title="LATTICE8:func.6">ConsecutiveDelta2</a> <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c6">O1</font> is <a href="lattice5.html#V1" title="LATTICE5:attr.1">symmetric</a>
 
<b>by </b><i><a class="txt" href="lattice8.html#E2:42"><i><font color="Green" title="E36">A2</font></i></a>, <a class="ref" href="lattice8.html#T27" target="_self" title="LATTICE8:th.27">Th27</a></i>;<br/>
<b>let </b><font color="Maroon" title="c7">x</font>, <font color="Maroon" title="c8">y</font>, <font color="Maroon" title="c9">z</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> <font color="Maroon" title="c1">A</font>,<span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c6">O1</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice5.html#D8" title="LATTICE5:def.8">LATTICE5:def 8</a><br/>

<b>set </b><font color="Maroon" title="c10">f</font> =  <a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <span class="p1">(<span class="default"><a href="lattice8.html#K6" title="LATTICE8:func.6">ConsecutiveDelta2</a> <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c6">O1</font></span>)</span>,<span class="p1">(<span class="default"><a href="lattice8.html#K4" title="LATTICE8:func.4">Quadr2</a> <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c6">O1</font></span>)</span>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">x'</font> = <font color="Maroon" title="c7">x</font>, <font color="Maroon" title="c12">y'</font> = <font color="Maroon" title="c8">y</font>, <font color="Maroon" title="c13">z'</font> = <font color="Maroon" title="c9">z</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="lattice8.html#K1" title="LATTICE8:func.1">new_set2</a> <span class="p1">(<span class="default"><a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c6">O1</font></span>)</span> <b>by </b><i><a class="ref" href="lattice8.html#T16" target="_self" title="LATTICE8:th.16">Th16</a></i>;<br/>
<a NAME="E6:42_2"/><i><font color="Green" title="E44">A11</font></i>: 
<font color="Maroon" title="c6">O1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c5">q</font>
 
<b>by </b><i><a class="txt" href="lattice8.html#E3:42_2"><i><font color="Green" title="E42">A9</font></i></a>, <a class="ref" href="lattice5.html#T28" title="LATTICE5:th.28">LATTICE5:28</a></i>;<br/>
<a NAME="E7:42_2"/><b>then </b>
<font color="Maroon" title="c5">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">O1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">q</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E8:42_2"/><b>then </b>
<font color="Maroon" title="c5">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">O1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K5" title="DOMAIN_1:func.5">[</a><span class="default"><font color="Olive" title="b1">u</font>,<font color="Olive" title="b2">v</font>,<font color="Olive" title="b3">a'</font>,<font color="Olive" title="b4">b'</font></span><a href="domain_1.html#K5" title="DOMAIN_1:func.5">]</a></span> where <font color="Olive" title="b1">u</font>, <font color="Olive" title="b2">v</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>, <font color="Olive" title="b3">a'</font>, <font color="Olive" title="b4">b'</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L</font> : <font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Olive" title="b1">u</font>,<font color="Olive" title="b2">v</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b3">a'</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Olive" title="b4">b'</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="lattice5.html#D14" title="LATTICE5:def.14">LATTICE5:def 14</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c14">u</font>, <font color="Maroon" title="c15">v</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>, <font color="Maroon" title="c16">a'</font>, <font color="Maroon" title="c17">b'</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L</font><b> such that </b><br/><a NAME="E10:42_2"/><i><font color="Green" title="E45">A12</font></i>: 
( <font color="Maroon" title="c5">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">O1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K5" title="DOMAIN_1:func.5">[</a><span class="default"><font color="Maroon" title="c14">u</font>,<font color="Maroon" title="c15">v</font>,<font color="Maroon" title="c16">a'</font>,<font color="Maroon" title="c17">b'</font></span><a href="domain_1.html#K5" title="DOMAIN_1:func.5">]</a></span> &amp; <font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">u</font>,<font color="Maroon" title="c15">v</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c16">a'</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c17">b'</font> )
 ;<br/>
<b>set </b><font color="Maroon" title="c18">X</font> = <span class="p1">(<span class="default"><a href="lattice8.html#K4" title="LATTICE8:func.4">Quadr2</a> <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c6">O1</font></span>)</span> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> ;<br/>
<b>set </b><font color="Maroon" title="c19">Y</font> = <span class="p1">(<span class="default"><a href="lattice8.html#K4" title="LATTICE8:func.4">Quadr2</a> <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c6">O1</font></span>)</span> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> ;<br/>
<b>reconsider </b><font color="Maroon" title="c20">a</font> = <span class="p1">(<span class="default"><a href="lattice8.html#K4" title="LATTICE8:func.4">Quadr2</a> <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c6">O1</font></span>)</span> <a href="mcart_1.html#K10" title="MCART_1:func.10">`3</a> , <font color="Maroon" title="c21">b</font> = <span class="p1">(<span class="default"><a href="lattice8.html#K4" title="LATTICE8:func.4">Quadr2</a> <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c6">O1</font></span>)</span> <a href="mcart_1.html#K11" title="MCART_1:func.11">`4</a>  as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L</font> ;<br/>
<a NAME="E12:42_2"/><i><font color="Green" title="E46">A13</font></i>: 
 <a href="lattice8.html#K4" title="LATTICE8:func.4">Quadr2</a> <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c6">O1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K5" title="DOMAIN_1:func.5">[</a><span class="default"><font color="Maroon" title="c14">u</font>,<font color="Maroon" title="c15">v</font>,<font color="Maroon" title="c16">a'</font>,<font color="Maroon" title="c17">b'</font></span><a href="domain_1.html#K5" title="DOMAIN_1:func.5">]</a></span>
 
<b>by </b><i><a class="txt" href="lattice8.html#E6:42_2"><i><font color="Green" title="E44">A11</font></i></a>, <a class="txt" href="lattice8.html#E10:42_2"><i><font color="Green" title="E45">A12</font></i></a>, <a class="ref" href="lattice8.html#D7" target="_self" title="LATTICE8:def.7">Def7</a></i>;<br/>
<a NAME="E13:42_2"/><b>then </b><i><font color="Green" title="E47">A14</font></i>: 
<font color="Maroon" title="c14">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="lattice8.html#K4" title="LATTICE8:func.4">Quadr2</a> <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c6">O1</font></span>)</span> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> 
 
<b>by </b><i><a class="ref" href="mcart_1.html#D8" title="MCART_1:def.8">MCART_1:def 8</a></i>;<br/>
<a NAME="E14:42_2"/><i><font color="Green" title="E48">A15</font></i>: 
<font color="Maroon" title="c15">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="lattice8.html#K4" title="LATTICE8:func.4">Quadr2</a> <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c6">O1</font></span>)</span> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> 
 
<b>by </b><i><a class="txt" href="lattice8.html#E12:42_2"><i><font color="Green" title="E46">A13</font></i></a>, <a class="ref" href="mcart_1.html#D9" title="MCART_1:def.9">MCART_1:def 9</a></i>;<br/>
<a NAME="E15:42_2"/><i><font color="Green" title="E49">A16</font></i>: 
<font color="Maroon" title="c16">a'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">a</font>
 
<b>by </b><i><a class="txt" href="lattice8.html#E12:42_2"><i><font color="Green" title="E46">A13</font></i></a>, <a class="ref" href="mcart_1.html#D10" title="MCART_1:def.10">MCART_1:def 10</a></i>;<br/>
<a NAME="E16:42_2"/><i><font color="Green" title="E50">A17</font></i>: 
<font color="Maroon" title="c17">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">b</font>
 
<b>by </b><i><a class="txt" href="lattice8.html#E12:42_2"><i><font color="Green" title="E46">A13</font></i></a>, <a class="ref" href="mcart_1.html#D11" title="MCART_1:def.11">MCART_1:def 11</a></i>;<br/>
<a NAME="E17:42_2"/><i><font color="Green" title="E51">A18</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E18:42_2"/><i><font color="Green" title="E52">A19</font></i>: 
<font color="Maroon" title="c3">d</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="lattice8.html#K6" title="LATTICE8:func.6">ConsecutiveDelta2</a> <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c6">O1</font>
 
<b>by </b><i><a class="ref" href="lattice8.html#T24" target="_self" title="LATTICE8:th.24">Th24</a></i>;<br/>
<font color="Maroon" title="c3">d</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c14">u</font>,<font color="Maroon" title="c15">v</font> = 
<font color="Maroon" title="c3">d</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c14">u</font>,<font color="Maroon" title="c15">v</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>

<br/>.= 
<span class="p1">(<span class="default"><a href="lattice8.html#K6" title="LATTICE8:func.6">ConsecutiveDelta2</a> <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c6">O1</font></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K4" title="LATTICE8:func.4">Quadr2</a> <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c6">O1</font></span>)</span> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> </span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K4" title="LATTICE8:func.4">Quadr2</a> <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c6">O1</font></span>)</span> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> </span>)</span>
<b>by </b><i><a class="txt" href="lattice8.html#E13:42_2"><i><font color="Green" title="E47">A14</font></i></a>, <a class="txt" href="lattice8.html#E14:42_2"><i><font color="Green" title="E48">A15</font></i></a>, <a class="txt" href="lattice8.html#E17:42_2"><i><font color="Green" title="E51">A18</font></i></a>, <a class="txt" href="lattice8.html#E18:42_2"><i><font color="Green" title="E52">A19</font></i></a>, <a class="ref" href="grfunc_1.html#T8" title="GRFUNC_1:th.8">GRFUNC_1:8</a></i>
;<br/>
<a NAME="E20:42_2"/><b>then </b><i><font color="Green" title="E53">A20</font></i>: 
 <a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <span class="p1">(<span class="default"><a href="lattice8.html#K6" title="LATTICE8:func.6">ConsecutiveDelta2</a> <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c6">O1</font></span>)</span>,<span class="p1">(<span class="default"><a href="lattice8.html#K4" title="LATTICE8:func.4">Quadr2</a> <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c6">O1</font></span>)</span> is <a href="lattice5.html#V3" title="LATTICE5:attr.3">u.t.i.</a>
 
<b>by </b><i><a class="txt" href="lattice8.html#E1:42"><i><font color="Green" title="E35">A1</font></i></a>, <a class="txt" href="lattice8.html#E1:42_2"><i><font color="Green" title="E40">A7</font></i></a>, <a class="txt" href="lattice8.html#E3:42_2"><i><font color="Green" title="E42">A9</font></i></a>, <a class="txt" href="lattice8.html#E4:42_2"><i><font color="Green" title="E43">A10</font></i></a>, <a class="txt" href="lattice8.html#E10:42_2"><i><font color="Green" title="E45">A12</font></i></a>, <a class="txt" href="lattice8.html#E15:42_2"><i><font color="Green" title="E49">A16</font></i></a>, <a class="txt" href="lattice8.html#E16:42_2"><i><font color="Green" title="E50">A17</font></i></a>, <a class="ref" href="lattice8.html#T12" target="_self" title="LATTICE8:th.12">Th12</a>, <a class="ref" href="ordinal1.html#D2" title="ORDINAL1:def.2">ORDINAL1:def 2</a></i>;<br/>
<i><font color="Green" title="E54">A21</font></i>:  <a href="lattice8.html#K6" title="LATTICE8:func.6">ConsecutiveDelta2</a> <font color="Maroon" title="c5">q</font>,<span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c6">O1</font></span>)</span> = 
 <a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <span class="p1">(<span class="default"><a href="lattice5.html#K11" title="LATTICE5:func.11">BiFun</a> <span class="p2">(<span class="default"><a href="lattice8.html#K6" title="LATTICE8:func.6">ConsecutiveDelta2</a> <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c6">O1</font></span>)</span>,<span class="p2">(<span class="default"><a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c6">O1</font></span>)</span>,<font color="Maroon" title="c2">L</font></span>)</span>,<span class="p1">(<span class="default"><a href="lattice8.html#K4" title="LATTICE8:func.4">Quadr2</a> <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c6">O1</font></span>)</span>
<b>by </b><i><a class="ref" href="lattice8.html#T20" target="_self" title="LATTICE8:th.20">Th20</a></i>
<br/>.= 
 <a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <span class="p1">(<span class="default"><a href="lattice8.html#K6" title="LATTICE8:func.6">ConsecutiveDelta2</a> <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c6">O1</font></span>)</span>,<span class="p1">(<span class="default"><a href="lattice8.html#K4" title="LATTICE8:func.4">Quadr2</a> <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c6">O1</font></span>)</span>
<b>by </b><i><a class="ref" href="lattice5.html#D16" title="LATTICE5:def.16">LATTICE5:def 16</a></i>
;<br/>
<a NAME="E22:42_2"/>
<span class="p1">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <span class="p2">(<span class="default"><a href="lattice8.html#K6" title="LATTICE8:func.6">ConsecutiveDelta2</a> <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c6">O1</font></span>)</span>,<span class="p2">(<span class="default"><a href="lattice8.html#K4" title="LATTICE8:func.4">Quadr2</a> <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c6">O1</font></span>)</span></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">x'</font>,<font color="Maroon" title="c13">z'</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <span class="p3">(<span class="default"><a href="lattice8.html#K6" title="LATTICE8:func.6">ConsecutiveDelta2</a> <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c6">O1</font></span>)</span>,<span class="p3">(<span class="default"><a href="lattice8.html#K4" title="LATTICE8:func.4">Quadr2</a> <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c6">O1</font></span>)</span></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">x'</font>,<font color="Maroon" title="c12">y'</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K2" title="LATTICE8:func.2">new_bi_fun2</a> <span class="p3">(<span class="default"><a href="lattice8.html#K6" title="LATTICE8:func.6">ConsecutiveDelta2</a> <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c6">O1</font></span>)</span>,<span class="p3">(<span class="default"><a href="lattice8.html#K4" title="LATTICE8:func.4">Quadr2</a> <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c6">O1</font></span>)</span></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">y'</font>,<font color="Maroon" title="c13">z'</font></span>)</span>
 
<b>by </b><i><a class="txt" href="lattice8.html#E20:42_2"><i><font color="Green" title="E53">A20</font></i></a>, <a class="ref" href="lattice5.html#D8" title="LATTICE5:def.8">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" title="LATTICE8:func.6">ConsecutiveDelta2</a> <font color="Maroon" title="c5">q</font>,<span class="p2">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c6">O1</font></span>)</span></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c9">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K6" title="LATTICE8:func.6">ConsecutiveDelta2</a> <font color="Maroon" title="c5">q</font>,<span class="p3">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c6">O1</font></span>)</span></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K6" title="LATTICE8:func.6">ConsecutiveDelta2</a> <font color="Maroon" title="c5">q</font>,<span class="p3">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c6">O1</font></span>)</span></span>)</span> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c8">y</font>,<font color="Maroon" title="c9">z</font></span>)</span>
 <b>by </b><i><a class="txt" href="lattice8.html#E21:42_2"><i><font color="Green" title="E54">A21</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:42"/><i><font color="Green" title="E40">A22</font></i>: 
for <font color="Olive" title="b1">O1</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b1">O1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Olive" title="b1">O1</font> <a href="ordinal1.html#NR2" title="ORDINAL1:NR.2">is_limit_ordinal</a>  &amp; ( for <font color="Olive" title="b2">O2</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b2">O2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">O1</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">O2</font>] ) holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">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" title="c6">O2</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:42_3"/><i><font color="Green" title="E41">A23</font></i>: 
( <font color="Maroon" title="c6">O2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c6">O2</font> <a href="ordinal1.html#NR2" title="ORDINAL1:NR.2">is_limit_ordinal</a>  &amp; ( for <font color="Olive" title="b1">O1</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b1">O1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">O2</font> &amp; <font color="Olive" title="b1">O1</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="lattice5.html#K8" title="LATTICE5:func.8">DistEsti</a> <font color="Maroon" title="c3">d</font> holds <br/> <a href="lattice8.html#K6" title="LATTICE8:func.6">ConsecutiveDelta2</a> <font color="Maroon" title="c5">q</font>,<font color="Olive" title="b1">O1</font> is <a href="lattice5.html#V3" title="LATTICE5:attr.3">u.t.i.</a> ) )
 <b>and </b><br/><a NAME="E2:42_3"/><i><font color="Green" title="E42">A24</font></i>: 
<font color="Maroon" title="c6">O2</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="lattice5.html#K8" title="LATTICE5:func.8">DistEsti</a> <font color="Maroon" title="c3">d</font>
 ;<br/>

<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>) <b>-&gt; </b>  <a href="lattice5.html#NM3" title="LATTICE5:NM.3">BiFunction</a> of <span class="p1">(<span class="default"><a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> <font color="Maroon" title="c1">A</font>,$1</span>)</span>,<font color="Maroon" title="c2">L</font> =  <a href="lattice8.html#K6" title="LATTICE8:func.6">ConsecutiveDelta2</a> <font color="Maroon" title="c5">q</font>,$1;<br/>
<b>consider </b><font color="Maroon" title="c7">Ls</font> being   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">T-Sequence</a><b> such that </b><br/><a NAME="E4:42_3"/><i><font color="Green" title="E43">A25</font></i>: 
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c7">Ls</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">O2</font> &amp; ( for <font color="Olive" title="b1">O1</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b1">O1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">O2</font> holds <br/><font color="Maroon" title="c7">Ls</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">O1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">O1</font>) ) )
 <b>from </b><i><a class="ref" href="ordinal2.html#S2" title="ORDINAL2:sch.2">ORDINAL2:sch 2</a>();<br/></i>
<a NAME="E5:42_3"/><i><font color="Green" title="E44">A26</font></i>: 
 <a href="lattice8.html#K6" title="LATTICE8:func.6">ConsecutiveDelta2</a> <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c6">O2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">Ls</font></span>)</span>
 
<b>by </b><i><a class="txt" href="lattice8.html#E1:42_3"><i><font color="Green" title="E41">A23</font></i></a>, <a class="txt" href="lattice8.html#E4:42_3"><i><font color="Green" title="E43">A25</font></i></a>, <a class="ref" href="lattice8.html#T21" target="_self" title="LATTICE8:th.21">Th21</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  =  <a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> <font color="Maroon" title="c1">A</font>,$1;<br/>
<b>consider </b><font color="Maroon" title="c8">Ts</font> being   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">T-Sequence</a><b> such that </b><br/><a NAME="E7:42_3"/><i><font color="Green" title="E45">A27</font></i>: 
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c8">Ts</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">O2</font> &amp; ( for <font color="Olive" title="b1">O1</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b1">O1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">O2</font> holds <br/><font color="Maroon" title="c8">Ts</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">O1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">O1</font>) ) )
 <b>from </b><i><a class="ref" href="ordinal2.html#S2" title="ORDINAL2:sch.2">ORDINAL2:sch 2</a>();<br/></i>
<a NAME="E8:42_3"/><i><font color="Green" title="E46">A28</font></i>: 
 <a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c6">O2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">Ts</font></span>)</span>
 
<b>by </b><i><a class="txt" href="lattice8.html#E1:42_3"><i><font color="Green" title="E41">A23</font></i></a>, <a class="txt" href="lattice8.html#E7:42_3"><i><font color="Green" title="E45">A27</font></i></a>, <a class="ref" href="lattice8.html#T17" target="_self" title="LATTICE8:th.17">Th17</a></i>;<br/>
<b>set </b><font color="Maroon" title="c9">CS</font> =  <a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c6">O2</font>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">f</font> =  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">Ls</font></span>)</span> as   <a href="lattice5.html#NM3" title="LATTICE5:NM.3">BiFunction</a> of <span class="p1">(<span class="default"><a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c6">O2</font></span>)</span>,<font color="Maroon" title="c2">L</font> <b>by </b><i><a class="txt" href="lattice8.html#E5:42_3"><i><font color="Green" title="E44">A26</font></i></a></i>;<br/>
<a NAME="E10:42_3"/>
<font color="Maroon" title="c10">f</font> is <a href="lattice5.html#V3" title="LATTICE5:attr.3">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" title="c11">x</font>, <font color="Maroon" title="c12">y</font>, <font color="Maroon" title="c13">z</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c6">O2</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice5.html#D8" title="LATTICE5:def.8">LATTICE5:def 8</a><br/>

<b>consider </b><font color="Maroon" title="c14">X</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E2:42_3_1"/><i><font color="Green" title="E47">A29</font></i>: 
( <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">X</font> &amp; <font color="Maroon" title="c14">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">Ts</font> )
 <b>by </b><i><a class="txt" href="lattice8.html#E8:42_3"><i><font color="Green" title="E46">A28</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c15">o1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:42_3_1"/><i><font color="Green" title="E48">A30</font></i>: 
( <font color="Maroon" title="c15">o1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c8">Ts</font> &amp; <font color="Maroon" title="c14">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">Ts</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">o1</font> )
 <b>by </b><i><a class="txt" href="lattice8.html#E2:42_3_1"><i><font color="Green" title="E47">A29</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c16">Y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E6:42_3_1"/><i><font color="Green" title="E49">A31</font></i>: 
( <font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">Y</font> &amp; <font color="Maroon" title="c16">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">Ts</font> )
 <b>by </b><i><a class="txt" href="lattice8.html#E8:42_3"><i><font color="Green" title="E46">A28</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c17">o2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E8:42_3_1"/><i><font color="Green" title="E50">A32</font></i>: 
( <font color="Maroon" title="c17">o2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c8">Ts</font> &amp; <font color="Maroon" title="c16">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">Ts</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">o2</font> )
 <b>by </b><i><a class="txt" href="lattice8.html#E6:42_3_1"><i><font color="Green" title="E49">A31</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c18">Z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E10:42_3_1"/><i><font color="Green" title="E51">A33</font></i>: 
( <font color="Maroon" title="c13">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">Z</font> &amp; <font color="Maroon" title="c18">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">Ts</font> )
 <b>by </b><i><a class="txt" href="lattice8.html#E8:42_3"><i><font color="Green" title="E46">A28</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c19">o3</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E12:42_3_1"/><i><font color="Green" title="E52">A34</font></i>: 
( <font color="Maroon" title="c19">o3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c8">Ts</font> &amp; <font color="Maroon" title="c18">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">Ts</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">o3</font> )
 <b>by </b><i><a class="txt" href="lattice8.html#E10:42_3_1"><i><font color="Green" title="E51">A33</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c20">o1</font> = <font color="Maroon" title="c15">o1</font>, <font color="Maroon" title="c21">o2</font> = <font color="Maroon" title="c17">o2</font>, <font color="Maroon" title="c22">o3</font> = <font color="Maroon" title="c19">o3</font> as   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> <b>by </b><i><a class="txt" href="lattice8.html#E4:42_3_1"><i><font color="Green" title="E48">A30</font></i></a>, <a class="txt" href="lattice8.html#E8:42_3_1"><i><font color="Green" title="E50">A32</font></i></a>, <a class="txt" href="lattice8.html#E12:42_3_1"><i><font color="Green" title="E52">A34</font></i></a>, <a class="ref" href="ordinal1.html#T23" title="ORDINAL1:th.23">ORDINAL1:23</a></i>;<br/>
<a NAME="E14:42_3_1"/><i><font color="Green" title="E53">A35</font></i>: 
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c20">o1</font>
 
<b>by </b><i><a class="txt" href="lattice8.html#E7:42_3"><i><font color="Green" title="E45">A27</font></i></a>, <a class="txt" href="lattice8.html#E2:42_3_1"><i><font color="Green" title="E47">A29</font></i></a>, <a class="txt" href="lattice8.html#E4:42_3_1"><i><font color="Green" title="E48">A30</font></i></a></i>;<br/>
<a NAME="E15:42_3_1"/><i><font color="Green" title="E54">A36</font></i>: 
<font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c21">o2</font>
 
<b>by </b><i><a class="txt" href="lattice8.html#E7:42_3"><i><font color="Green" title="E45">A27</font></i></a>, <a class="txt" href="lattice8.html#E6:42_3_1"><i><font color="Green" title="E49">A31</font></i></a>, <a class="txt" href="lattice8.html#E8:42_3_1"><i><font color="Green" title="E50">A32</font></i></a></i>;<br/>
<a NAME="E16:42_3_1"/><i><font color="Green" title="E55">A37</font></i>: 
<font color="Maroon" title="c13">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c22">o3</font>
 
<b>by </b><i><a class="txt" href="lattice8.html#E7:42_3"><i><font color="Green" title="E45">A27</font></i></a>, <a class="txt" href="lattice8.html#E10:42_3_1"><i><font color="Green" title="E51">A33</font></i></a>, <a class="txt" href="lattice8.html#E12:42_3_1"><i><font color="Green" title="E52">A34</font></i></a></i>;<br/>
<a NAME="E17:42_3_1"/><i><font color="Green" title="E56">A38</font></i>: 
<font color="Maroon" title="c7">Ls</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">o1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice8.html#K6" title="LATTICE8:func.6">ConsecutiveDelta2</a> <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c20">o1</font>
 
<b>by </b><i><a class="txt" href="lattice8.html#E4:42_3"><i><font color="Green" title="E43">A25</font></i></a>, <a class="txt" href="lattice8.html#E7:42_3"><i><font color="Green" title="E45">A27</font></i></a>, <a class="txt" href="lattice8.html#E4:42_3_1"><i><font color="Green" title="E48">A30</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c23">h1</font> = <font color="Maroon" title="c7">Ls</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">o1</font> as   <a href="lattice5.html#NM3" title="LATTICE5:NM.3">BiFunction</a> of <span class="p1">(<span class="default"><a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c20">o1</font></span>)</span>,<font color="Maroon" title="c2">L</font> ;<br/>
<a NAME="E19:42_3_1"/><i><font color="Green" title="E57">A39</font></i>: 
<font color="Maroon" title="c23">h1</font> is <a href="lattice5.html#V3" title="LATTICE5:attr.3">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" title="c24">x</font>, <font color="Maroon" title="c25">y</font>, <font color="Maroon" title="c26">z</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c20">o1</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice5.html#D8" title="LATTICE5:def.8">LATTICE5:def 8</a><br/>

<a NAME="E1:42_3_1_1"/><i><font color="Green" title="E58">A40</font></i>: 
 <a href="lattice8.html#K6" title="LATTICE8:func.6">ConsecutiveDelta2</a> <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c20">o1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c23">h1</font>
 
<b>by </b><i><a class="txt" href="lattice8.html#E4:42_3"><i><font color="Green" title="E43">A25</font></i></a>, <a class="txt" href="lattice8.html#E7:42_3"><i><font color="Green" title="E45">A27</font></i></a>, <a class="txt" href="lattice8.html#E4:42_3_1"><i><font color="Green" title="E48">A30</font></i></a></i>;<br/>
<a NAME="E2:42_3_1_1"/>
<font color="Maroon" title="c20">o1</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="lattice5.html#K8" title="LATTICE5:func.8">DistEsti</a> <font color="Maroon" title="c3">d</font>
 
<b>by </b><i><a class="txt" href="lattice8.html#E2:42_3"><i><font color="Green" title="E42">A24</font></i></a>, <a class="txt" href="lattice8.html#E7:42_3"><i><font color="Green" title="E45">A27</font></i></a>, <a class="txt" href="lattice8.html#E4:42_3_1"><i><font color="Green" title="E48">A30</font></i></a>, <a class="ref" href="ordinal1.html#D2" title="ORDINAL1:def.2">ORDINAL1:def 2</a></i>;<br/>
<a NAME="E3:42_3_1_1"/><b>then </b>
 <a href="lattice8.html#K6" title="LATTICE8:func.6">ConsecutiveDelta2</a> <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c20">o1</font> is <a href="lattice5.html#V3" title="LATTICE5:attr.3">u.t.i.</a>
 
<b>by </b><i><a class="txt" href="lattice8.html#E1:42_3"><i><font color="Green" title="E41">A23</font></i></a>, <a class="txt" href="lattice8.html#E7:42_3"><i><font color="Green" title="E45">A27</font></i></a>, <a class="txt" href="lattice8.html#E4:42_3_1"><i><font color="Green" title="E48">A30</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:42_3_1_1"/>
<font color="Maroon" title="c23">h1</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c24">x</font>,<font color="Maroon" title="c26">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">h1</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c24">x</font>,<font color="Maroon" title="c25">y</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">h1</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c25">y</font>,<font color="Maroon" title="c26">z</font></span>)</span>
 <b>by </b><i><a class="txt" href="lattice8.html#E1:42_3_1_1"><i><font color="Green" title="E58">A40</font></i></a>, <a class="ref" href="lattice5.html#D8" title="LATTICE5:def.8">LATTICE5:def 8</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E20:42_3_1"/><i><font color="Green" title="E58">A41</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c23">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c20">o1</font></span>)</span>,<span class="p2">(<span class="default"><a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c20">o1</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E21:42_3_1"/><i><font color="Green" title="E59">A42</font></i>: 
<font color="Maroon" title="c7">Ls</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">o2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice8.html#K6" title="LATTICE8:func.6">ConsecutiveDelta2</a> <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c21">o2</font>
 
<b>by </b><i><a class="txt" href="lattice8.html#E4:42_3"><i><font color="Green" title="E43">A25</font></i></a>, <a class="txt" href="lattice8.html#E7:42_3"><i><font color="Green" title="E45">A27</font></i></a>, <a class="txt" href="lattice8.html#E8:42_3_1"><i><font color="Green" title="E50">A32</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c24">h2</font> = <font color="Maroon" title="c7">Ls</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">o2</font> as   <a href="lattice5.html#NM3" title="LATTICE5:NM.3">BiFunction</a> of <span class="p1">(<span class="default"><a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c21">o2</font></span>)</span>,<font color="Maroon" title="c2">L</font> ;<br/>
<a NAME="E23:42_3_1"/><i><font color="Green" title="E60">A43</font></i>: 
<font color="Maroon" title="c24">h2</font> is <a href="lattice5.html#V3" title="LATTICE5:attr.3">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" title="c25">x</font>, <font color="Maroon" title="c26">y</font>, <font color="Maroon" title="c27">z</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c21">o2</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice5.html#D8" title="LATTICE5:def.8">LATTICE5:def 8</a><br/>

<a NAME="E1:42_3_1_2"/><i><font color="Green" title="E61">A44</font></i>: 
 <a href="lattice8.html#K6" title="LATTICE8:func.6">ConsecutiveDelta2</a> <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c21">o2</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c24">h2</font>
 
<b>by </b><i><a class="txt" href="lattice8.html#E4:42_3"><i><font color="Green" title="E43">A25</font></i></a>, <a class="txt" href="lattice8.html#E7:42_3"><i><font color="Green" title="E45">A27</font></i></a>, <a class="txt" href="lattice8.html#E8:42_3_1"><i><font color="Green" title="E50">A32</font></i></a></i>;<br/>
<a NAME="E2:42_3_1_2"/>
<font color="Maroon" title="c21">o2</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="lattice5.html#K8" title="LATTICE5:func.8">DistEsti</a> <font color="Maroon" title="c3">d</font>
 
<b>by </b><i><a class="txt" href="lattice8.html#E2:42_3"><i><font color="Green" title="E42">A24</font></i></a>, <a class="txt" href="lattice8.html#E7:42_3"><i><font color="Green" title="E45">A27</font></i></a>, <a class="txt" href="lattice8.html#E8:42_3_1"><i><font color="Green" title="E50">A32</font></i></a>, <a class="ref" href="ordinal1.html#D2" title="ORDINAL1:def.2">ORDINAL1:def 2</a></i>;<br/>
<a NAME="E3:42_3_1_2"/><b>then </b>
 <a href="lattice8.html#K6" title="LATTICE8:func.6">ConsecutiveDelta2</a> <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c21">o2</font> is <a href="lattice5.html#V3" title="LATTICE5:attr.3">u.t.i.</a>
 
<b>by </b><i><a class="txt" href="lattice8.html#E1:42_3"><i><font color="Green" title="E41">A23</font></i></a>, <a class="txt" href="lattice8.html#E7:42_3"><i><font color="Green" title="E45">A27</font></i></a>, <a class="txt" href="lattice8.html#E8:42_3_1"><i><font color="Green" title="E50">A32</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:42_3_1_2"/>
<font color="Maroon" title="c24">h2</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c25">x</font>,<font color="Maroon" title="c27">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">h2</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c25">x</font>,<font color="Maroon" title="c26">y</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">h2</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c26">y</font>,<font color="Maroon" title="c27">z</font></span>)</span>
 <b>by </b><i><a class="txt" href="lattice8.html#E1:42_3_1_2"><i><font color="Green" title="E61">A44</font></i></a>, <a class="ref" href="lattice5.html#D8" title="LATTICE5:def.8">LATTICE5:def 8</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E24:42_3_1"/><i><font color="Green" title="E61">A45</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c24">h2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c21">o2</font></span>)</span>,<span class="p2">(<span class="default"><a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c21">o2</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E25:42_3_1"/><i><font color="Green" title="E62">A46</font></i>: 
<font color="Maroon" title="c7">Ls</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">o3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice8.html#K6" title="LATTICE8:func.6">ConsecutiveDelta2</a> <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c22">o3</font>
 
<b>by </b><i><a class="txt" href="lattice8.html#E4:42_3"><i><font color="Green" title="E43">A25</font></i></a>, <a class="txt" href="lattice8.html#E7:42_3"><i><font color="Green" title="E45">A27</font></i></a>, <a class="txt" href="lattice8.html#E12:42_3_1"><i><font color="Green" title="E52">A34</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c25">h3</font> = <font color="Maroon" title="c7">Ls</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">o3</font> as   <a href="lattice5.html#NM3" title="LATTICE5:NM.3">BiFunction</a> of <span class="p1">(<span class="default"><a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c22">o3</font></span>)</span>,<font color="Maroon" title="c2">L</font> ;<br/>
<a NAME="E27:42_3_1"/><i><font color="Green" title="E63">A47</font></i>: 
<font color="Maroon" title="c25">h3</font> is <a href="lattice5.html#V3" title="LATTICE5:attr.3">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" title="c26">x</font>, <font color="Maroon" title="c27">y</font>, <font color="Maroon" title="c28">z</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c22">o3</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice5.html#D8" title="LATTICE5:def.8">LATTICE5:def 8</a><br/>

<a NAME="E1:42_3_1_3"/><i><font color="Green" title="E64">A48</font></i>: 
 <a href="lattice8.html#K6" title="LATTICE8:func.6">ConsecutiveDelta2</a> <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c22">o3</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c25">h3</font>
 
<b>by </b><i><a class="txt" href="lattice8.html#E4:42_3"><i><font color="Green" title="E43">A25</font></i></a>, <a class="txt" href="lattice8.html#E7:42_3"><i><font color="Green" title="E45">A27</font></i></a>, <a class="txt" href="lattice8.html#E12:42_3_1"><i><font color="Green" title="E52">A34</font></i></a></i>;<br/>
<a NAME="E2:42_3_1_3"/>
<font color="Maroon" title="c22">o3</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="lattice5.html#K8" title="LATTICE5:func.8">DistEsti</a> <font color="Maroon" title="c3">d</font>
 
<b>by </b><i><a class="txt" href="lattice8.html#E2:42_3"><i><font color="Green" title="E42">A24</font></i></a>, <a class="txt" href="lattice8.html#E7:42_3"><i><font color="Green" title="E45">A27</font></i></a>, <a class="txt" href="lattice8.html#E12:42_3_1"><i><font color="Green" title="E52">A34</font></i></a>, <a class="ref" href="ordinal1.html#D2" title="ORDINAL1:def.2">ORDINAL1:def 2</a></i>;<br/>
<a NAME="E3:42_3_1_3"/><b>then </b>
 <a href="lattice8.html#K6" title="LATTICE8:func.6">ConsecutiveDelta2</a> <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c22">o3</font> is <a href="lattice5.html#V3" title="LATTICE5:attr.3">u.t.i.</a>
 
<b>by </b><i><a class="txt" href="lattice8.html#E1:42_3"><i><font color="Green" title="E41">A23</font></i></a>, <a class="txt" href="lattice8.html#E7:42_3"><i><font color="Green" title="E45">A27</font></i></a>, <a class="txt" href="lattice8.html#E12:42_3_1"><i><font color="Green" title="E52">A34</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:42_3_1_3"/>
<font color="Maroon" title="c25">h3</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c26">x</font>,<font color="Maroon" title="c28">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">h3</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c26">x</font>,<font color="Maroon" title="c27">y</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">h3</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c27">y</font>,<font color="Maroon" title="c28">z</font></span>)</span>
 <b>by </b><i><a class="txt" href="lattice8.html#E1:42_3_1_3"><i><font color="Green" title="E64">A48</font></i></a>, <a class="ref" href="lattice5.html#D8" title="LATTICE5:def.8">LATTICE5:def 8</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E28:42_3_1"/><i><font color="Green" title="E64">A49</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c25">h3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c22">o3</font></span>)</span>,<span class="p2">(<span class="default"><a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c22">o3</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">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" title="c20">o1</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c22">o3</font> or <font color="Maroon" title="c22">o3</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c20">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" title="E65">A50</font></i>: 
<font color="Maroon" title="c20">o1</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c22">o3</font>
 ;<br/><div class="add"><a NAME="E2:42_3_1_4_1"/><b>then </b><i><font color="Green" title="E66">A51</font></i>: 
 <a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c20">o1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c22">o3</font>
 <b>by </b><i><a class="ref" href="lattice8.html#T22" target="_self" title="LATTICE8:th.22">Th22</a></i>;<br/><b>thus </b><a NAME="E3:42_3_1_4_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c10">f</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c12">y</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">f</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">y</font>,<font color="Maroon" title="c13">z</font></span>)</span> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c10">f</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c13">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" title="c21">o2</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c22">o3</font> or <font color="Maroon" title="c22">o3</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c21">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" title="c21">o2</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c22">o3</font>
 ;<br/><div class="add"><a NAME="E2:42_3_1_4_1_1_1_1"/><b>then </b><i><font color="Green" title="E67">A52</font></i>: 
 <a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c21">o2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c22">o3</font>
 <b>by </b><i><a class="ref" href="lattice8.html#T22" target="_self" title="LATTICE8:th.22">Th22</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c26">y'</font> = <font color="Maroon" title="c12">y</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c22">o3</font> <b>by </b><i><a class="txt" href="lattice8.html#E15:42_3_1"><i><font color="Green" title="E54">A36</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c27">x'</font> = <font color="Maroon" title="c11">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c22">o3</font> <b>by </b><i><a class="txt" href="lattice8.html#E14:42_3_1"><i><font color="Green" title="E53">A35</font></i></a>, <a class="txt" href="lattice8.html#E2:42_3_1_4_1"><i><font color="Green" title="E66">A51</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c28">z'</font> = <font color="Maroon" title="c13">z</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c22">o3</font> <b>by </b><i><a class="txt" href="lattice8.html#E7:42_3"><i><font color="Green" title="E45">A27</font></i></a>, <a class="txt" href="lattice8.html#E10:42_3_1"><i><font color="Green" title="E51">A33</font></i></a>, <a class="txt" href="lattice8.html#E12:42_3_1"><i><font color="Green" title="E52">A34</font></i></a></i>;<br/><a NAME="E6:42_3_1_4_1_1_1_1"/>
 <a href="lattice8.html#K6" title="LATTICE8:func.6">ConsecutiveDelta2</a> <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c22">o3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">Ls</font>
 <b>by </b><i><a class="txt" href="lattice8.html#E4:42_3"><i><font color="Green" title="E43">A25</font></i></a>, <a class="txt" href="lattice8.html#E7:42_3"><i><font color="Green" title="E45">A27</font></i></a>, <a class="txt" href="lattice8.html#E12:42_3_1"><i><font color="Green" title="E52">A34</font></i></a>, <a class="txt" href="lattice8.html#E25:42_3_1"><i><font color="Green" title="E62">A46</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">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" title="E68">A53</font></i>: 
<font color="Maroon" title="c25">h3</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">f</font>
 <b>by </b><i><a class="txt" href="lattice8.html#E25:42_3_1"><i><font color="Green" title="E62">A46</font></i></a>, <a class="ref" href="zfmisc_1.html#T92" title="ZFMISC_1:th.92">ZFMISC_1:92</a></i>;<br/><a NAME="E8:42_3_1_4_1_1_1_1"/><i><font color="Green" title="E69">A54</font></i>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c12">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c25">h3</font>
 <b>by </b><i><a class="txt" href="lattice8.html#E14:42_3_1"><i><font color="Green" title="E53">A35</font></i></a>, <a class="txt" href="lattice8.html#E15:42_3_1"><i><font color="Green" title="E54">A36</font></i></a>, <a class="txt" href="lattice8.html#E28:42_3_1"><i><font color="Green" title="E64">A49</font></i></a>, <a class="txt" href="lattice8.html#E2:42_3_1_4_1"><i><font color="Green" title="E66">A51</font></i></a>, <a class="txt" href="lattice8.html#E2:42_3_1_4_1_1_1_1"><i><font color="Green" title="E67">A52</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><a NAME="E9:42_3_1_4_1_1_1_1"/><i><font color="Green" title="E70">A55</font></i>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c12">y</font>,<font color="Maroon" title="c13">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c25">h3</font>
 <b>by </b><i><a class="txt" href="lattice8.html#E15:42_3_1"><i><font color="Green" title="E54">A36</font></i></a>, <a class="txt" href="lattice8.html#E16:42_3_1"><i><font color="Green" title="E55">A37</font></i></a>, <a class="txt" href="lattice8.html#E28:42_3_1"><i><font color="Green" title="E64">A49</font></i></a>, <a class="txt" href="lattice8.html#E2:42_3_1_4_1_1_1_1"><i><font color="Green" title="E67">A52</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><a NAME="E10:42_3_1_4_1_1_1_1"/><i><font color="Green" title="E71">A56</font></i>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c13">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c25">h3</font>
 <b>by </b><i><a class="txt" href="lattice8.html#E14:42_3_1"><i><font color="Green" title="E53">A35</font></i></a>, <a class="txt" href="lattice8.html#E16:42_3_1"><i><font color="Green" title="E55">A37</font></i></a>, <a class="txt" href="lattice8.html#E28:42_3_1"><i><font color="Green" title="E64">A49</font></i></a>, <a class="txt" href="lattice8.html#E2:42_3_1_4_1"><i><font color="Green" title="E66">A51</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><a NAME="E11:42_3_1_4_1_1_1_1"/><i><font color="Green" title="E72">A57</font></i>: 
<font color="Maroon" title="c10">f</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c12">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">h3</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c27">x'</font>,<font color="Maroon" title="c26">y'</font>
 <b>by </b><i><a class="txt" href="lattice8.html#E7:42_3_1_4_1_1_1_1"><i><font color="Green" title="E68">A53</font></i></a>, <a class="txt" href="lattice8.html#E8:42_3_1_4_1_1_1_1"><i><font color="Green" title="E69">A54</font></i></a>, <a class="ref" href="grfunc_1.html#T8" title="GRFUNC_1:th.8">GRFUNC_1:8</a></i>;<br/><a NAME="E12:42_3_1_4_1_1_1_1"/><i><font color="Green" title="E73">A58</font></i>: 
<font color="Maroon" title="c10">f</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">y</font>,<font color="Maroon" title="c13">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">h3</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c26">y'</font>,<font color="Maroon" title="c28">z'</font>
 <b>by </b><i><a class="txt" href="lattice8.html#E7:42_3_1_4_1_1_1_1"><i><font color="Green" title="E68">A53</font></i></a>, <a class="txt" href="lattice8.html#E9:42_3_1_4_1_1_1_1"><i><font color="Green" title="E70">A55</font></i></a>, <a class="ref" href="grfunc_1.html#T8" title="GRFUNC_1:th.8">GRFUNC_1:8</a></i>;<br/><a NAME="E13:42_3_1_4_1_1_1_1"/>
<font color="Maroon" title="c10">f</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c13">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">h3</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c27">x'</font>,<font color="Maroon" title="c28">z'</font>
 <b>by </b><i><a class="txt" href="lattice8.html#E7:42_3_1_4_1_1_1_1"><i><font color="Green" title="E68">A53</font></i></a>, <a class="txt" href="lattice8.html#E10:42_3_1_4_1_1_1_1"><i><font color="Green" title="E71">A56</font></i></a>, <a class="ref" href="grfunc_1.html#T8" title="GRFUNC_1:th.8">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" title="c10">f</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c12">y</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">f</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">y</font>,<font color="Maroon" title="c13">z</font></span>)</span> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c10">f</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c13">z</font>
 <b>by </b><i><a class="txt" href="lattice8.html#E27:42_3_1"><i><font color="Green" title="E63">A47</font></i></a>, <a class="txt" href="lattice8.html#E11:42_3_1_4_1_1_1_1"><i><font color="Green" title="E72">A57</font></i></a>, <a class="txt" href="lattice8.html#E12:42_3_1_4_1_1_1_1"><i><font color="Green" title="E73">A58</font></i></a>, <a class="ref" href="lattice5.html#D8" title="LATTICE5:def.8">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" title="c22">o3</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c21">o2</font>
 ;<br/><div class="add"><a NAME="E2:42_3_1_4_1_1_1_2"/><b>then </b><i><font color="Green" title="E67">A59</font></i>: 
 <a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c22">o3</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c21">o2</font>
 <b>by </b><i><a class="ref" href="lattice8.html#T22" target="_self" title="LATTICE8:th.22">Th22</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c26">z'</font> = <font color="Maroon" title="c13">z</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c21">o2</font> <b>by </b><i><a class="txt" href="lattice8.html#E16:42_3_1"><i><font color="Green" title="E55">A37</font></i></a></i>;<br/><a NAME="E4:42_3_1_4_1_1_1_2"/>
 <a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c20">o1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c22">o3</font>
 <b>by </b><i><a class="txt" href="lattice8.html#E1:42_3_1_4_1"><i><font color="Green" title="E65">A50</font></i></a>, <a class="ref" href="lattice8.html#T22" target="_self" title="LATTICE8:th.22">Th22</a></i>;<br/><a NAME="E5:42_3_1_4_1_1_1_2"/><b>then </b><i><font color="Green" title="E68">A60</font></i>: 
 <a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c20">o1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c21">o2</font>
 <b>by </b><i><a class="txt" href="lattice8.html#E2:42_3_1_4_1_1_1_2"><i><font color="Green" title="E67">A59</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c27">x'</font> = <font color="Maroon" title="c11">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c21">o2</font> <b>by </b><i><a class="txt" href="lattice8.html#E14:42_3_1"><i><font color="Green" title="E53">A35</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c28">y'</font> = <font color="Maroon" title="c12">y</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c21">o2</font> <b>by </b><i><a class="txt" href="lattice8.html#E7:42_3"><i><font color="Green" title="E45">A27</font></i></a>, <a class="txt" href="lattice8.html#E6:42_3_1"><i><font color="Green" title="E49">A31</font></i></a>, <a class="txt" href="lattice8.html#E8:42_3_1"><i><font color="Green" title="E50">A32</font></i></a></i>;<br/><a NAME="E8:42_3_1_4_1_1_1_2"/>
 <a href="lattice8.html#K6" title="LATTICE8:func.6">ConsecutiveDelta2</a> <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c21">o2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">Ls</font>
 <b>by </b><i><a class="txt" href="lattice8.html#E4:42_3"><i><font color="Green" title="E43">A25</font></i></a>, <a class="txt" href="lattice8.html#E7:42_3"><i><font color="Green" title="E45">A27</font></i></a>, <a class="txt" href="lattice8.html#E8:42_3_1"><i><font color="Green" title="E50">A32</font></i></a>, <a class="txt" href="lattice8.html#E21:42_3_1"><i><font color="Green" title="E59">A42</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">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" title="E69">A61</font></i>: 
<font color="Maroon" title="c24">h2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">f</font>
 <b>by </b><i><a class="txt" href="lattice8.html#E21:42_3_1"><i><font color="Green" title="E59">A42</font></i></a>, <a class="ref" href="zfmisc_1.html#T92" title="ZFMISC_1:th.92">ZFMISC_1:92</a></i>;<br/><a NAME="E10:42_3_1_4_1_1_1_2"/><i><font color="Green" title="E70">A62</font></i>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c12">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c24">h2</font>
 <b>by </b><i><a class="txt" href="lattice8.html#E14:42_3_1"><i><font color="Green" title="E53">A35</font></i></a>, <a class="txt" href="lattice8.html#E15:42_3_1"><i><font color="Green" title="E54">A36</font></i></a>, <a class="txt" href="lattice8.html#E24:42_3_1"><i><font color="Green" title="E61">A45</font></i></a>, <a class="txt" href="lattice8.html#E5:42_3_1_4_1_1_1_2"><i><font color="Green" title="E68">A60</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><a NAME="E11:42_3_1_4_1_1_1_2"/><i><font color="Green" title="E71">A63</font></i>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c12">y</font>,<font color="Maroon" title="c13">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c24">h2</font>
 <b>by </b><i><a class="txt" href="lattice8.html#E15:42_3_1"><i><font color="Green" title="E54">A36</font></i></a>, <a class="txt" href="lattice8.html#E16:42_3_1"><i><font color="Green" title="E55">A37</font></i></a>, <a class="txt" href="lattice8.html#E24:42_3_1"><i><font color="Green" title="E61">A45</font></i></a>, <a class="txt" href="lattice8.html#E2:42_3_1_4_1_1_1_2"><i><font color="Green" title="E67">A59</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><a NAME="E12:42_3_1_4_1_1_1_2"/><i><font color="Green" title="E72">A64</font></i>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c13">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c24">h2</font>
 <b>by </b><i><a class="txt" href="lattice8.html#E14:42_3_1"><i><font color="Green" title="E53">A35</font></i></a>, <a class="txt" href="lattice8.html#E16:42_3_1"><i><font color="Green" title="E55">A37</font></i></a>, <a class="txt" href="lattice8.html#E24:42_3_1"><i><font color="Green" title="E61">A45</font></i></a>, <a class="txt" href="lattice8.html#E2:42_3_1_4_1_1_1_2"><i><font color="Green" title="E67">A59</font></i></a>, <a class="txt" href="lattice8.html#E5:42_3_1_4_1_1_1_2"><i><font color="Green" title="E68">A60</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><a NAME="E13:42_3_1_4_1_1_1_2"/><i><font color="Green" title="E73">A65</font></i>: 
<font color="Maroon" title="c10">f</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c12">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">h2</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c27">x'</font>,<font color="Maroon" title="c28">y'</font>
 <b>by </b><i><a class="txt" href="lattice8.html#E9:42_3_1_4_1_1_1_2"><i><font color="Green" title="E69">A61</font></i></a>, <a class="txt" href="lattice8.html#E10:42_3_1_4_1_1_1_2"><i><font color="Green" title="E70">A62</font></i></a>, <a class="ref" href="grfunc_1.html#T8" title="GRFUNC_1:th.8">GRFUNC_1:8</a></i>;<br/><a NAME="E14:42_3_1_4_1_1_1_2"/><i><font color="Green" title="E74">A66</font></i>: 
<font color="Maroon" title="c10">f</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">y</font>,<font color="Maroon" title="c13">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">h2</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c28">y'</font>,<font color="Maroon" title="c26">z'</font>
 <b>by </b><i><a class="txt" href="lattice8.html#E9:42_3_1_4_1_1_1_2"><i><font color="Green" title="E69">A61</font></i></a>, <a class="txt" href="lattice8.html#E11:42_3_1_4_1_1_1_2"><i><font color="Green" title="E71">A63</font></i></a>, <a class="ref" href="grfunc_1.html#T8" title="GRFUNC_1:th.8">GRFUNC_1:8</a></i>;<br/><a NAME="E15:42_3_1_4_1_1_1_2"/>
<font color="Maroon" title="c10">f</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c13">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">h2</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c27">x'</font>,<font color="Maroon" title="c26">z'</font>
 <b>by </b><i><a class="txt" href="lattice8.html#E9:42_3_1_4_1_1_1_2"><i><font color="Green" title="E69">A61</font></i></a>, <a class="txt" href="lattice8.html#E12:42_3_1_4_1_1_1_2"><i><font color="Green" title="E72">A64</font></i></a>, <a class="ref" href="grfunc_1.html#T8" title="GRFUNC_1:th.8">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" title="c10">f</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c12">y</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">f</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">y</font>,<font color="Maroon" title="c13">z</font></span>)</span> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c10">f</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c13">z</font>
 <b>by </b><i><a class="txt" href="lattice8.html#E23:42_3_1"><i><font color="Green" title="E60">A43</font></i></a>, <a class="txt" href="lattice8.html#E13:42_3_1_4_1_1_1_2"><i><font color="Green" title="E73">A65</font></i></a>, <a class="txt" href="lattice8.html#E14:42_3_1_4_1_1_1_2"><i><font color="Green" title="E74">A66</font></i></a>, <a class="ref" href="lattice5.html#D8" title="LATTICE5:def.8">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" title="E65">A67</font></i>: 
<font color="Maroon" title="c22">o3</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c20">o1</font>
 ;<br/><div class="add"><a NAME="E2:42_3_1_4_2"/><b>then </b><i><font color="Green" title="E66">A68</font></i>: 
 <a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c22">o3</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c20">o1</font>
 <b>by </b><i><a class="ref" href="lattice8.html#T22" target="_self" title="LATTICE8:th.22">Th22</a></i>;<br/><b>thus </b><a NAME="E3:42_3_1_4_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c10">f</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c12">y</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">f</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">y</font>,<font color="Maroon" title="c13">z</font></span>)</span> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c10">f</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c13">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" title="c20">o1</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c21">o2</font> or <font color="Maroon" title="c21">o2</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c20">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" title="c20">o1</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c21">o2</font>
 ;<br/><div class="add"><a NAME="E2:42_3_1_4_2_1_1_1"/><b>then </b><i><font color="Green" title="E67">A69</font></i>: 
 <a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c20">o1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c21">o2</font>
 <b>by </b><i><a class="ref" href="lattice8.html#T22" target="_self" title="LATTICE8:th.22">Th22</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c26">x'</font> = <font color="Maroon" title="c11">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c21">o2</font> <b>by </b><i><a class="txt" href="lattice8.html#E14:42_3_1"><i><font color="Green" title="E53">A35</font></i></a></i>;<br/><a NAME="E4:42_3_1_4_2_1_1_1"/>
 <a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c22">o3</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c20">o1</font>
 <b>by </b><i><a class="txt" href="lattice8.html#E1:42_3_1_4_2"><i><font color="Green" title="E65">A67</font></i></a>, <a class="ref" href="lattice8.html#T22" target="_self" title="LATTICE8:th.22">Th22</a></i>;<br/><a NAME="E5:42_3_1_4_2_1_1_1"/><b>then </b><i><font color="Green" title="E68">A70</font></i>: 
 <a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c22">o3</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c21">o2</font>
 <b>by </b><i><a class="txt" href="lattice8.html#E2:42_3_1_4_2_1_1_1"><i><font color="Green" title="E67">A69</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c27">z'</font> = <font color="Maroon" title="c13">z</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c21">o2</font> <b>by </b><i><a class="txt" href="lattice8.html#E16:42_3_1"><i><font color="Green" title="E55">A37</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c28">y'</font> = <font color="Maroon" title="c12">y</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c21">o2</font> <b>by </b><i><a class="txt" href="lattice8.html#E7:42_3"><i><font color="Green" title="E45">A27</font></i></a>, <a class="txt" href="lattice8.html#E6:42_3_1"><i><font color="Green" title="E49">A31</font></i></a>, <a class="txt" href="lattice8.html#E8:42_3_1"><i><font color="Green" title="E50">A32</font></i></a></i>;<br/><a NAME="E8:42_3_1_4_2_1_1_1"/>
 <a href="lattice8.html#K6" title="LATTICE8:func.6">ConsecutiveDelta2</a> <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c21">o2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">Ls</font>
 <b>by </b><i><a class="txt" href="lattice8.html#E4:42_3"><i><font color="Green" title="E43">A25</font></i></a>, <a class="txt" href="lattice8.html#E7:42_3"><i><font color="Green" title="E45">A27</font></i></a>, <a class="txt" href="lattice8.html#E8:42_3_1"><i><font color="Green" title="E50">A32</font></i></a>, <a class="txt" href="lattice8.html#E21:42_3_1"><i><font color="Green" title="E59">A42</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">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" title="E69">A71</font></i>: 
<font color="Maroon" title="c24">h2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">f</font>
 <b>by </b><i><a class="txt" href="lattice8.html#E21:42_3_1"><i><font color="Green" title="E59">A42</font></i></a>, <a class="ref" href="zfmisc_1.html#T92" title="ZFMISC_1:th.92">ZFMISC_1:92</a></i>;<br/><a NAME="E10:42_3_1_4_2_1_1_1"/><i><font color="Green" title="E70">A72</font></i>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c12">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c24">h2</font>
 <b>by </b><i><a class="txt" href="lattice8.html#E14:42_3_1"><i><font color="Green" title="E53">A35</font></i></a>, <a class="txt" href="lattice8.html#E15:42_3_1"><i><font color="Green" title="E54">A36</font></i></a>, <a class="txt" href="lattice8.html#E24:42_3_1"><i><font color="Green" title="E61">A45</font></i></a>, <a class="txt" href="lattice8.html#E2:42_3_1_4_2_1_1_1"><i><font color="Green" title="E67">A69</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><a NAME="E11:42_3_1_4_2_1_1_1"/><i><font color="Green" title="E71">A73</font></i>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c12">y</font>,<font color="Maroon" title="c13">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c24">h2</font>
 <b>by </b><i><a class="txt" href="lattice8.html#E15:42_3_1"><i><font color="Green" title="E54">A36</font></i></a>, <a class="txt" href="lattice8.html#E16:42_3_1"><i><font color="Green" title="E55">A37</font></i></a>, <a class="txt" href="lattice8.html#E24:42_3_1"><i><font color="Green" title="E61">A45</font></i></a>, <a class="txt" href="lattice8.html#E5:42_3_1_4_2_1_1_1"><i><font color="Green" title="E68">A70</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><a NAME="E12:42_3_1_4_2_1_1_1"/><i><font color="Green" title="E72">A74</font></i>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c13">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c24">h2</font>
 <b>by </b><i><a class="txt" href="lattice8.html#E14:42_3_1"><i><font color="Green" title="E53">A35</font></i></a>, <a class="txt" href="lattice8.html#E16:42_3_1"><i><font color="Green" title="E55">A37</font></i></a>, <a class="txt" href="lattice8.html#E24:42_3_1"><i><font color="Green" title="E61">A45</font></i></a>, <a class="txt" href="lattice8.html#E2:42_3_1_4_2_1_1_1"><i><font color="Green" title="E67">A69</font></i></a>, <a class="txt" href="lattice8.html#E5:42_3_1_4_2_1_1_1"><i><font color="Green" title="E68">A70</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><a NAME="E13:42_3_1_4_2_1_1_1"/><i><font color="Green" title="E73">A75</font></i>: 
<font color="Maroon" title="c10">f</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c12">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">h2</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c26">x'</font>,<font color="Maroon" title="c28">y'</font>
 <b>by </b><i><a class="txt" href="lattice8.html#E9:42_3_1_4_2_1_1_1"><i><font color="Green" title="E69">A71</font></i></a>, <a class="txt" href="lattice8.html#E10:42_3_1_4_2_1_1_1"><i><font color="Green" title="E70">A72</font></i></a>, <a class="ref" href="grfunc_1.html#T8" title="GRFUNC_1:th.8">GRFUNC_1:8</a></i>;<br/><a NAME="E14:42_3_1_4_2_1_1_1"/><i><font color="Green" title="E74">A76</font></i>: 
<font color="Maroon" title="c10">f</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">y</font>,<font color="Maroon" title="c13">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">h2</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c28">y'</font>,<font color="Maroon" title="c27">z'</font>
 <b>by </b><i><a class="txt" href="lattice8.html#E9:42_3_1_4_2_1_1_1"><i><font color="Green" title="E69">A71</font></i></a>, <a class="txt" href="lattice8.html#E11:42_3_1_4_2_1_1_1"><i><font color="Green" title="E71">A73</font></i></a>, <a class="ref" href="grfunc_1.html#T8" title="GRFUNC_1:th.8">GRFUNC_1:8</a></i>;<br/><a NAME="E15:42_3_1_4_2_1_1_1"/>
<font color="Maroon" title="c10">f</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c13">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">h2</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c26">x'</font>,<font color="Maroon" title="c27">z'</font>
 <b>by </b><i><a class="txt" href="lattice8.html#E9:42_3_1_4_2_1_1_1"><i><font color="Green" title="E69">A71</font></i></a>, <a class="txt" href="lattice8.html#E12:42_3_1_4_2_1_1_1"><i><font color="Green" title="E72">A74</font></i></a>, <a class="ref" href="grfunc_1.html#T8" title="GRFUNC_1:th.8">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" title="c10">f</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c12">y</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">f</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">y</font>,<font color="Maroon" title="c13">z</font></span>)</span> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c10">f</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c13">z</font>
 <b>by </b><i><a class="txt" href="lattice8.html#E23:42_3_1"><i><font color="Green" title="E60">A43</font></i></a>, <a class="txt" href="lattice8.html#E13:42_3_1_4_2_1_1_1"><i><font color="Green" title="E73">A75</font></i></a>, <a class="txt" href="lattice8.html#E14:42_3_1_4_2_1_1_1"><i><font color="Green" title="E74">A76</font></i></a>, <a class="ref" href="lattice5.html#D8" title="LATTICE5:def.8">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" title="c21">o2</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c20">o1</font>
 ;<br/><div class="add"><a NAME="E2:42_3_1_4_2_1_1_2"/><b>then </b><i><font color="Green" title="E67">A77</font></i>: 
 <a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c21">o2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c20">o1</font>
 <b>by </b><i><a class="ref" href="lattice8.html#T22" target="_self" title="LATTICE8:th.22">Th22</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c26">y'</font> = <font color="Maroon" title="c12">y</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c20">o1</font> <b>by </b><i><a class="txt" href="lattice8.html#E15:42_3_1"><i><font color="Green" title="E54">A36</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c27">z'</font> = <font color="Maroon" title="c13">z</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c20">o1</font> <b>by </b><i><a class="txt" href="lattice8.html#E16:42_3_1"><i><font color="Green" title="E55">A37</font></i></a>, <a class="txt" href="lattice8.html#E2:42_3_1_4_2"><i><font color="Green" title="E66">A68</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c28">x'</font> = <font color="Maroon" title="c11">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c20">o1</font> <b>by </b><i><a class="txt" href="lattice8.html#E7:42_3"><i><font color="Green" title="E45">A27</font></i></a>, <a class="txt" href="lattice8.html#E2:42_3_1"><i><font color="Green" title="E47">A29</font></i></a>, <a class="txt" href="lattice8.html#E4:42_3_1"><i><font color="Green" title="E48">A30</font></i></a></i>;<br/><a NAME="E6:42_3_1_4_2_1_1_2"/>
 <a href="lattice8.html#K6" title="LATTICE8:func.6">ConsecutiveDelta2</a> <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c20">o1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">Ls</font>
 <b>by </b><i><a class="txt" href="lattice8.html#E4:42_3"><i><font color="Green" title="E43">A25</font></i></a>, <a class="txt" href="lattice8.html#E7:42_3"><i><font color="Green" title="E45">A27</font></i></a>, <a class="txt" href="lattice8.html#E4:42_3_1"><i><font color="Green" title="E48">A30</font></i></a>, <a class="txt" href="lattice8.html#E17:42_3_1"><i><font color="Green" title="E56">A38</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">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" title="E68">A78</font></i>: 
<font color="Maroon" title="c23">h1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">f</font>
 <b>by </b><i><a class="txt" href="lattice8.html#E17:42_3_1"><i><font color="Green" title="E56">A38</font></i></a>, <a class="ref" href="zfmisc_1.html#T92" title="ZFMISC_1:th.92">ZFMISC_1:92</a></i>;<br/><a NAME="E8:42_3_1_4_2_1_1_2"/><i><font color="Green" title="E69">A79</font></i>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c12">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c23">h1</font>
 <b>by </b><i><a class="txt" href="lattice8.html#E14:42_3_1"><i><font color="Green" title="E53">A35</font></i></a>, <a class="txt" href="lattice8.html#E15:42_3_1"><i><font color="Green" title="E54">A36</font></i></a>, <a class="txt" href="lattice8.html#E20:42_3_1"><i><font color="Green" title="E58">A41</font></i></a>, <a class="txt" href="lattice8.html#E2:42_3_1_4_2_1_1_2"><i><font color="Green" title="E67">A77</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><a NAME="E9:42_3_1_4_2_1_1_2"/><i><font color="Green" title="E70">A80</font></i>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c12">y</font>,<font color="Maroon" title="c13">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c23">h1</font>
 <b>by </b><i><a class="txt" href="lattice8.html#E15:42_3_1"><i><font color="Green" title="E54">A36</font></i></a>, <a class="txt" href="lattice8.html#E16:42_3_1"><i><font color="Green" title="E55">A37</font></i></a>, <a class="txt" href="lattice8.html#E20:42_3_1"><i><font color="Green" title="E58">A41</font></i></a>, <a class="txt" href="lattice8.html#E2:42_3_1_4_2"><i><font color="Green" title="E66">A68</font></i></a>, <a class="txt" href="lattice8.html#E2:42_3_1_4_2_1_1_2"><i><font color="Green" title="E67">A77</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><a NAME="E10:42_3_1_4_2_1_1_2"/><i><font color="Green" title="E71">A81</font></i>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c13">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c23">h1</font>
 <b>by </b><i><a class="txt" href="lattice8.html#E14:42_3_1"><i><font color="Green" title="E53">A35</font></i></a>, <a class="txt" href="lattice8.html#E16:42_3_1"><i><font color="Green" title="E55">A37</font></i></a>, <a class="txt" href="lattice8.html#E20:42_3_1"><i><font color="Green" title="E58">A41</font></i></a>, <a class="txt" href="lattice8.html#E2:42_3_1_4_2"><i><font color="Green" title="E66">A68</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><a NAME="E11:42_3_1_4_2_1_1_2"/><i><font color="Green" title="E72">A82</font></i>: 
<font color="Maroon" title="c10">f</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c12">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">h1</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c28">x'</font>,<font color="Maroon" title="c26">y'</font>
 <b>by </b><i><a class="txt" href="lattice8.html#E7:42_3_1_4_2_1_1_2"><i><font color="Green" title="E68">A78</font></i></a>, <a class="txt" href="lattice8.html#E8:42_3_1_4_2_1_1_2"><i><font color="Green" title="E69">A79</font></i></a>, <a class="ref" href="grfunc_1.html#T8" title="GRFUNC_1:th.8">GRFUNC_1:8</a></i>;<br/><a NAME="E12:42_3_1_4_2_1_1_2"/><i><font color="Green" title="E73">A83</font></i>: 
<font color="Maroon" title="c10">f</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">y</font>,<font color="Maroon" title="c13">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">h1</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c26">y'</font>,<font color="Maroon" title="c27">z'</font>
 <b>by </b><i><a class="txt" href="lattice8.html#E7:42_3_1_4_2_1_1_2"><i><font color="Green" title="E68">A78</font></i></a>, <a class="txt" href="lattice8.html#E9:42_3_1_4_2_1_1_2"><i><font color="Green" title="E70">A80</font></i></a>, <a class="ref" href="grfunc_1.html#T8" title="GRFUNC_1:th.8">GRFUNC_1:8</a></i>;<br/><a NAME="E13:42_3_1_4_2_1_1_2"/>
<font color="Maroon" title="c10">f</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c13">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">h1</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c28">x'</font>,<font color="Maroon" title="c27">z'</font>
 <b>by </b><i><a class="txt" href="lattice8.html#E7:42_3_1_4_2_1_1_2"><i><font color="Green" title="E68">A78</font></i></a>, <a class="txt" href="lattice8.html#E10:42_3_1_4_2_1_1_2"><i><font color="Green" title="E71">A81</font></i></a>, <a class="ref" href="grfunc_1.html#T8" title="GRFUNC_1:th.8">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" title="c10">f</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c12">y</font></span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">f</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c12">y</font>,<font color="Maroon" title="c13">z</font></span>)</span> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c10">f</font> <a href="lattice5.html#K4" title="LATTICE5:func.4">.</a> <font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c13">z</font>
 <b>by </b><i><a class="txt" href="lattice8.html#E19:42_3_1"><i><font color="Green" title="E57">A39</font></i></a>, <a class="txt" href="lattice8.html#E11:42_3_1_4_2_1_1_2"><i><font color="Green" title="E72">A82</font></i></a>, <a class="txt" href="lattice8.html#E12:42_3_1_4_2_1_1_2"><i><font color="Green" title="E73">A83</font></i></a>, <a class="ref" href="lattice5.html#D8" title="LATTICE5:def.8">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" title="LATTICE8:func.6">ConsecutiveDelta2</a> <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c6">O2</font> is <a href="lattice5.html#V3" title="LATTICE5:attr.3">u.t.i.</a>
 <b>by </b><i><a class="txt" href="lattice8.html#E1:42_3"><i><font color="Green" title="E41">A23</font></i></a>, <a class="txt" href="lattice8.html#E4:42_3"><i><font color="Green" title="E43">A25</font></i></a>, <a class="ref" href="lattice8.html#T21" target="_self" title="LATTICE8:th.21">Th21</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:42"/>
for <font color="Olive" title="b1">O</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">O</font>]
 
<b>from </b><i><a class="ref" href="ordinal2.html#S1" title="ORDINAL2:sch.1">ORDINAL2:sch 1</a>(<a class="txt" href="lattice8.html#E4:42"><i><font color="Green" title="E38">A4</font></i></a>, <a class="txt" href="lattice8.html#E5:42"><i><font color="Green" title="E39">A6</font></i></a>, <a class="txt" href="lattice8.html#E6:42"><i><font color="Green" title="E40">A22</font></i></a>);<br/></i>
<b>hence </b><a NAME="E8:42"/>
( <font color="Maroon" title="c4">O</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="lattice5.html#K8" title="LATTICE5:func.8">DistEsti</a> <font color="Maroon" title="c3">d</font> implies  <a href="lattice8.html#K6" title="LATTICE8:func.6">ConsecutiveDelta2</a> <font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c4">O</font> is <a href="lattice5.html#V3" title="LATTICE5:attr.3">u.t.i.</a> )
 ;<br/>


</div>
