<?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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">L</font> being <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/> for <font color="Olive" title="b2">d</font> being  <a href="lattice5.html#NM3" title="LATTICE5:NM.3">BiFunction</a> of <font color="Maroon" title="c1">A</font>,<font color="Olive" title="b1">L</font><br/> for <font color="Olive" title="b3">q</font> being   <a href="lattice5.html#M1" title="LATTICE5:mode.1">QuadrSeq</a> of <font color="Olive" title="b2">d</font><br/> for <font color="Olive" title="b4">O</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> holds   <a href="lattice8.html#K5" title="LATTICE8:func.5">ConsecutiveDelta2</a> <font color="Olive" title="b3">q</font>,<font color="Olive" title="b4">O</font> is   <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="Olive" title="b4">O</font></span>)</span>,<font color="Olive" title="b1">L</font></span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">d</font> being  <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/> for <font color="Olive" title="b2">q</font> being   <a href="lattice5.html#M1" title="LATTICE5:mode.1">QuadrSeq</a> of <font color="Olive" title="b1">d</font><br/> for <font color="Olive" title="b3">O</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> holds   <a href="lattice8.html#K5" title="LATTICE8:func.5">ConsecutiveDelta2</a> <font color="Olive" title="b2">q</font>,<font color="Olive" title="b3">O</font> is   <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="Olive" title="b3">O</font></span>)</span>,<font color="Maroon" title="c2">L</font></span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">q</font> being   <a href="lattice5.html#M1" title="LATTICE5:mode.1">QuadrSeq</a> of <font color="Maroon" title="c3">d</font><br/> for <font color="Olive" title="b2">O</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> holds   <a href="lattice8.html#K5" title="LATTICE8:func.5">ConsecutiveDelta2</a> <font color="Olive" title="b1">q</font>,<font color="Olive" title="b2">O</font> is   <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="Olive" title="b2">O</font></span>)</span>,<font color="Maroon" title="c2">L</font></span><br/><b>let </b><font color="Maroon" title="c4">q</font> be    <a href="lattice5.html#M1" title="LATTICE5:mode.1">QuadrSeq</a> of <font color="Maroon" title="c3">d</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">O</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> holds   <a href="lattice8.html#K5" title="LATTICE8:func.5">ConsecutiveDelta2</a> <font color="Maroon" title="c4">q</font>,<font color="Olive" title="b1">O</font> is   <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="Olive" title="b1">O</font></span>)</span>,<font color="Maroon" title="c2">L</font></span><br/><b>let </b><font color="Maroon" title="c5">O</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="lattice8.html#K5" title="LATTICE8:func.5">ConsecutiveDelta2</a> <font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c5">O</font> is   <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="c5">O</font></span>)</span>,<font color="Maroon" title="c2">L</font></span><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> <a href="lattice8.html#K5" title="LATTICE8:func.5">ConsecutiveDelta2</a> <font color="Maroon" title="c4">q</font>,$1 is   <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>;<br/>
<a NAME="E1:35"/><i><font color="Green" title="E29">A1</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="35_1"><b>proof </b></a><div class="add">

<a NAME="E1:35_1"/>
(  <a href="lattice8.html#K5" title="LATTICE8:func.5">ConsecutiveDelta2</a> <font color="Maroon" title="c4">q</font>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">d</font> &amp;  <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>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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>, <a class="ref" href="lattice8.html#T19" target="_self" title="LATTICE8:th.19">Th19</a></i>;<br/>
<b>hence </b><a NAME="E2:35_1"/>
 <a href="lattice8.html#K5" title="LATTICE8:func.5">ConsecutiveDelta2</a> <font color="Maroon" title="c4">q</font>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  is   <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>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span>)</span>,<font color="Maroon" title="c2">L</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E2:35"/><i><font color="Green" title="E30">A2</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="35_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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">O1</font>] implies <font color="Maroon">S<sub>1</sub></font>[ <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c6">O1</font>] )</span><br/>

<b>assume </b><a NAME="E1:35_2"/>
 <a href="lattice8.html#K5" title="LATTICE8:func.5">ConsecutiveDelta2</a> <font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c6">O1</font> is   <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">O1</font></span>)</span>,<font color="Maroon" title="c2">L</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[ <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c6">O1</font>]</span><br/>

<b>then reconsider </b><font color="Maroon" title="c7">CD</font> =  <a href="lattice8.html#K5" title="LATTICE8:func.5">ConsecutiveDelta2</a> <font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c6">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="c6">O1</font></span>)</span>,<font color="Maroon" title="c2">L</font> ;<br/>
<a NAME="E3:35_2"/><i><font color="Green" title="E31">A3</font></i>: 
 <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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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 href="lattice8.html#K5" title="LATTICE8:func.5">ConsecutiveDelta2</a> <font color="Maroon" title="c4">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#K5" title="LATTICE8:func.5">ConsecutiveDelta2</a> <font color="Maroon" title="c4">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="c4">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> <font color="Maroon" title="c7">CD</font>,<span class="p1">(<span class="default"><a href="lattice8.html#K4" title="LATTICE8:func.4">Quadr2</a> <font color="Maroon" title="c4">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/>
<b>hence </b><a NAME="E5:35_2"/>
 <a href="lattice8.html#K5" title="LATTICE8:func.5">ConsecutiveDelta2</a> <font color="Maroon" title="c4">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> is   <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>,<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>,<font color="Maroon" title="c2">L</font>
 <b>by </b><i><a class="txt" href="#E3:35_2"><i><font color="Green" title="E31">A3</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E3:35"/><i><font color="Green" title="E31">A4</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> is <a href="ordinal1.html#V4" title="ORDINAL1:attr.4">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="35_3"><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c6">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="Maroon" title="c6">O1</font> is <a href="ordinal1.html#V4" title="ORDINAL1:attr.4">limit_ordinal</a> &amp; ( for <font color="Olive" title="b1">O2</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b1">O2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">O1</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">O2</font>] ) implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">O1</font>] )</span><br/>

<b>assume </b><a NAME="E1:35_3"/><i><font color="Green" title="E32">A5</font></i>: 
( <font color="Maroon" title="c6">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="Maroon" title="c6">O1</font> is <a href="ordinal1.html#V4" title="ORDINAL1:attr.4">limit_ordinal</a> &amp; ( for <font color="Olive" title="b1">O2</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b1">O2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">O1</font> holds <br/> <a href="lattice8.html#K5" title="LATTICE8:func.5">ConsecutiveDelta2</a> <font color="Maroon" title="c4">q</font>,<font color="Olive" title="b1">O2</font> is   <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="Olive" title="b1">O2</font></span>)</span>,<font color="Maroon" title="c2">L</font> ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">O1</font>]</span><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="hidden.html#M1" title="HIDDEN:mode.1">set</a>  =  <a href="lattice8.html#K5" title="LATTICE8:func.5">ConsecutiveDelta2</a> <font color="Maroon" title="c4">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="E3:35_3"/><i><font color="Green" title="E33">A6</font></i>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c7">Ls</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">O1</font> &amp; ( for <font color="Olive" title="b1">O2</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b1">O2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">O1</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">O2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">O2</font>) ) )
 <b>from </b><i><a class="ref" href="ordinal2.html#S2" title="ORDINAL2:sch.2">ORDINAL2:sch 2</a>();<br/></i>
<a NAME="E4:35_3"/><i><font color="Green" title="E34">A7</font></i>: 
 <a href="lattice8.html#K5" title="LATTICE8:func.5">ConsecutiveDelta2</a> <font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c6">O1</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="#E1:35_3"><i><font color="Green" title="E32">A5</font></i></a>, <a class="txt" href="#E3:35_3"><i><font color="Green" title="E33">A6</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="E6:35_3"/><i><font color="Green" title="E35">A8</font></i>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c8">Ts</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">O1</font> &amp; ( for <font color="Olive" title="b1">O2</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b1">O2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">O1</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">O2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">O2</font>) ) )
 <b>from </b><i><a class="ref" href="ordinal2.html#S2" title="ORDINAL2:sch.2">ORDINAL2:sch 2</a>();<br/></i>
<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">O1</font>;<br/>
<b>set </b><font color="Maroon" title="c10">Y</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L</font>;<br/>
<b>set </b><font color="Maroon" title="c11">X</font> = <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="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></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>;<br/>
<b>set </b><font color="Maroon" title="c12">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>;<br/>
<a NAME="E7:35_3"/><i><font color="Green" title="E36">A9</font></i>: 
for <font color="Olive" title="b1">O</font>, <font color="Olive" title="b2">O2</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b1">O</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Olive" title="b2">O2</font> &amp; <font color="Olive" title="b2">O2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c7">Ls</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">O</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">Ls</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">O2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">O</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">O2</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Maroon" title="c13">O</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Olive" title="b1">O2</font> &amp; <font color="Olive" title="b1">O2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c7">Ls</font> holds <br/><font color="Maroon" title="c7">Ls</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">O</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">Ls</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">O2</font></span><br/>

<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>] <b>means </b>( <font color="Maroon" title="c13">O</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> $1 &amp; $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c7">Ls</font> implies <font color="Maroon" title="c7">Ls</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">O</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">Ls</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 );<br/>
<a NAME="E1:35_3_1"/><i><font color="Green" title="E37">A10</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[ <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ]
 
;<br/>
<a NAME="E2:35_3_1"/><i><font color="Green" title="E38">A11</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>2</sub></font>[<font color="Olive" title="b1">O1</font>] holds <br/><font color="Maroon">S<sub>2</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="35_3_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c14">O2</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c14">O2</font>] implies <font color="Maroon">S<sub>2</sub></font>[ <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c14">O2</font>] )</span><br/>

<b>assume </b><a NAME="E1:35_3_1_1"/><i><font color="Green" title="E39">A12</font></i>: 
( <font color="Maroon" title="c13">O</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c14">O2</font> &amp; <font color="Maroon" title="c14">O2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c7">Ls</font> implies <font color="Maroon" title="c7">Ls</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">O</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">Ls</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">O2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>2</sub></font>[ <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c14">O2</font>]</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E2:35_3_1_1"/><i><font color="Green" title="E40">A13</font></i>: 
<font color="Maroon" title="c13">O</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c14">O2</font>
 <b>and </b><br/><a NAME="E3:35_3_1_1"/><i><font color="Green" title="E41">A14</font></i>: 
 <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c14">O2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c7">Ls</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">Ls</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">O</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">Ls</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c14">O2</font></span>)</span></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_3_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:35_3_1_1_1"/>
( <font color="Maroon" title="c13">O</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c14">O2</font> or <font color="Maroon" title="c13">O</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c14">O2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_3_1_1_1_1"><b>suppose </b></a><a NAME="E1:35_3_1_1_1_1"/>
<font color="Maroon" title="c13">O</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c14">O2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">Ls</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">O</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">Ls</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c14">O2</font></span>)</span></span><br/><div class="add"><b>hence </b><a NAME="E2:35_3_1_1_1_1"/>
<font color="Maroon" title="c7">Ls</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">O</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">Ls</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c14">O2</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_3_1_1_1_2"><b>suppose </b></a><a NAME="E1:35_3_1_1_1_2"/>
<font color="Maroon" title="c13">O</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c14">O2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">Ls</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">O</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">Ls</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c14">O2</font></span>)</span></span><br/><div class="add"><a NAME="E2:35_3_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c13">O</font> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c14">O2</font>
 <b>by </b><i><a class="txt" href="#E2:35_3_1_1"><i><font color="Green" title="E40">A13</font></i></a>, <a class="ref" href="xboole_0.html#D8" title="XBOOLE_0:def.8">XBOOLE_0:def 8</a></i>;<br/><a NAME="E3:35_3_1_1_1_2"/><b>then </b><i><font color="Green" title="E42">A15</font></i>: 
<font color="Maroon" title="c13">O</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c14">O2</font>
 <b>by </b><i><a class="ref" href="ordinal1.html#T21" title="ORDINAL1:th.21">ORDINAL1:21</a></i>;<br/><a NAME="E4:35_3_1_1_1_2"/><i><font color="Green" title="E43">A16</font></i>: 
<font color="Maroon" title="c14">O2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c14">O2</font>
 <b>by </b><i><a class="ref" href="ordinal1.html#T10" title="ORDINAL1:th.10">ORDINAL1:10</a></i>;<br/><a NAME="E5:35_3_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c14">O2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c7">Ls</font>
 <b>by </b><i><a class="txt" href="#E3:35_3_1_1"><i><font color="Green" title="E41">A14</font></i></a>, <a class="ref" href="ordinal1.html#T19" title="ORDINAL1:th.19">ORDINAL1:19</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c15">Def8</font> =  <a href="lattice8.html#K5" title="LATTICE8:func.5">ConsecutiveDelta2</a> <font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c14">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="c14">O2</font></span>)</span>,<font color="Maroon" title="c2">L</font> <b>by </b><i><a class="txt" href="#E1:35_3"><i><font color="Green" title="E32">A5</font></i></a>, <a class="txt" href="#E3:35_3"><i><font color="Green" title="E33">A6</font></i></a></i>;<br/><font color="Maroon" title="c7">Ls</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c14">O2</font></span>)</span> = 
 <a href="lattice8.html#K5" title="LATTICE8:func.5">ConsecutiveDelta2</a> <font color="Maroon" title="c4">q</font>,<span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c14">O2</font></span>)</span>
<b>by </b><i><a class="txt" href="#E3:35_3"><i><font color="Green" title="E33">A6</font></i></a>, <a class="txt" href="#E3:35_3_1_1"><i><font color="Green" title="E41">A14</font></i></a></i>
<br/>.= 
 <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#K5" title="LATTICE8:func.5">ConsecutiveDelta2</a> <font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c14">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="c14">O2</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="c4">q</font>,<font color="Maroon" title="c14">O2</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> <font color="Maroon" title="c15">Def8</font>,<span class="p1">(<span class="default"><a href="lattice8.html#K4" title="LATTICE8:func.4">Quadr2</a> <font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c14">O2</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="E8:35_3_1_1_1_2"/><b>then </b>
 <a href="lattice8.html#K5" title="LATTICE8:func.5">ConsecutiveDelta2</a> <font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c14">O2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">Ls</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c14">O2</font></span>)</span>
 <b>by </b><i><a class="ref" href="lattice8.html#T14" target="_self" title="LATTICE8:th.14">Th14</a></i>;<br/><a NAME="E9:35_3_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c7">Ls</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">O2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">Ls</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c14">O2</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E3:35_3"><i><font color="Green" title="E33">A6</font></i></a>, <a class="txt" href="#E3:35_3_1_1"><i><font color="Green" title="E41">A14</font></i></a>, <a class="txt" href="#E4:35_3_1_1_1_2"><i><font color="Green" title="E43">A16</font></i></a>, <a class="ref" href="ordinal1.html#T19" title="ORDINAL1:th.19">ORDINAL1:19</a></i>;<br/><b>hence </b><a NAME="E10:35_3_1_1_1_2"/>
<font color="Maroon" title="c7">Ls</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">O</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">Ls</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c14">O2</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:35_3_1_1"><i><font color="Green" title="E39">A12</font></i></a>, <a class="txt" href="#E3:35_3_1_1"><i><font color="Green" title="E41">A14</font></i></a>, <a class="txt" href="#E3:35_3_1_1_1_2"><i><font color="Green" title="E42">A15</font></i></a>, <a class="txt" href="#E4:35_3_1_1_1_2"><i><font color="Green" title="E43">A16</font></i></a>, <a class="ref" href="ordinal1.html#T19" title="ORDINAL1:th.19">ORDINAL1:19</a>, <a class="ref" href="ordinal1.html#T34" title="ORDINAL1:th.34">ORDINAL1:34</a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E3:35_3_1"/><i><font color="Green" title="E39">A17</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> is <a href="ordinal1.html#V4" title="ORDINAL1:attr.4">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>2</sub></font>[<font color="Olive" title="b2">O2</font>] ) holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">O1</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_3_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c14">O2</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c14">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="c14">O2</font> is <a href="ordinal1.html#V4" title="ORDINAL1:attr.4">limit_ordinal</a> &amp; ( for <font color="Olive" title="b1">O2</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b1">O2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">O2</font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">O2</font>] ) implies <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c14">O2</font>] )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:35_3_1_2"/><i><font color="Green" title="E40">A18</font></i>: 
( <font color="Maroon" title="c14">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="c14">O2</font> is <a href="ordinal1.html#V4" title="ORDINAL1:attr.4">limit_ordinal</a> )
 <b>and </b><br/><a NAME="E2:35_3_1_2"/>
for <font color="Olive" title="b1">O3</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b1">O3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">O2</font> &amp; <font color="Maroon" title="c13">O</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Olive" title="b1">O3</font> &amp; <font color="Olive" title="b1">O3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c7">Ls</font> holds <br/><font color="Maroon" title="c7">Ls</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">O</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">Ls</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">O3</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c14">O2</font>]</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E3:35_3_1_2"/><i><font color="Green" title="E41">A19</font></i>: 
<font color="Maroon" title="c13">O</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c14">O2</font>
 <b>and </b><br/><a NAME="E4:35_3_1_2"/><i><font color="Green" title="E42">A20</font></i>: 
<font color="Maroon" title="c14">O2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c7">Ls</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">Ls</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">O</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">Ls</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">O2</font></span><br/>

<b>deffunc </b><font color="Maroon">H<sub>3</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#K5" title="LATTICE8:func.5">ConsecutiveDelta2</a> <font color="Maroon" title="c4">q</font>,$1;<br/>
<b>consider </b><font color="Maroon" title="c15">Lt</font> being   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">T-Sequence</a><b> such that </b><br/><a NAME="E6:35_3_1_2"/><i><font color="Green" title="E43">A21</font></i>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c15">Lt</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">O2</font> &amp; ( for <font color="Olive" title="b1">O3</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b1">O3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">O2</font> holds <br/><font color="Maroon" title="c15">Lt</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">O3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive" title="b1">O3</font>) ) )
 <b>from </b><i><a class="ref" href="ordinal2.html#S2" title="ORDINAL2:sch.2">ORDINAL2:sch 2</a>();<br/></i>
<i><font color="Green" title="E44">A22</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="c14">O2</font> = 
 <a href="lattice8.html#K5" title="LATTICE8:func.5">ConsecutiveDelta2</a> <font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c14">O2</font>
<b>by </b><i><a class="txt" href="#E3:35_3"><i><font color="Green" title="E33">A6</font></i></a>, <a class="txt" href="#E4:35_3_1_2"><i><font color="Green" title="E42">A20</font></i></a></i>
<br/>.= 
 <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="c15">Lt</font></span>)</span>
<b>by </b><i><a class="txt" href="#E1:35_3_1_2"><i><font color="Green" title="E40">A18</font></i></a>, <a class="txt" href="#E6:35_3_1_2"><i><font color="Green" title="E43">A21</font></i></a>, <a class="ref" href="lattice8.html#T21" target="_self" title="LATTICE8:th.21">Th21</a></i>
;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_3_1_2_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:35_3_1_2_2"/>
( <font color="Maroon" title="c13">O</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">O2</font> or <font color="Maroon" title="c13">O</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c14">O2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_3_1_2_2_1"><b>suppose </b></a><a NAME="E1:35_3_1_2_2_1"/>
<font color="Maroon" title="c13">O</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">O2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">Ls</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">O</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">Ls</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">O2</font></span><br/><div class="add"><b>hence </b><a NAME="E2:35_3_1_2_2_1"/>
<font color="Maroon" title="c7">Ls</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">O</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">Ls</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">O2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_3_1_2_2_2"><b>suppose </b></a><a NAME="E1:35_3_1_2_2_2"/>
<font color="Maroon" title="c13">O</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c14">O2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">Ls</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">O</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">Ls</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">O2</font></span><br/><div class="add"><a NAME="E2:35_3_1_2_2_2"/><b>then </b><i><font color="Green" title="E45">A23</font></i>: 
<font color="Maroon" title="c13">O</font> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <font color="Maroon" title="c14">O2</font>
 <b>by </b><i><a class="txt" href="#E3:35_3_1_2"><i><font color="Green" title="E41">A19</font></i></a>, <a class="ref" href="xboole_0.html#D8" title="XBOOLE_0:def.8">XBOOLE_0:def 8</a></i>;<br/><a NAME="E3:35_3_1_2_2_2"/><b>then </b><i><font color="Green" title="E46">A24</font></i>: 
<font color="Maroon" title="c13">O</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">O2</font>
 <b>by </b><i><a class="ref" href="ordinal1.html#T21" title="ORDINAL1:th.21">ORDINAL1:21</a></i>;<br/><b>then </b><font color="Maroon" title="c7">Ls</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">O</font> = 
 <a href="lattice8.html#K5" title="LATTICE8:func.5">ConsecutiveDelta2</a> <font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c13">O</font>
<b>by </b><i><a class="txt" href="#E3:35_3"><i><font color="Green" title="E33">A6</font></i></a>, <a class="txt" href="#E4:35_3_1_2"><i><font color="Green" title="E42">A20</font></i></a>, <a class="ref" href="ordinal1.html#T19" title="ORDINAL1:th.19">ORDINAL1:19</a></i>
<br/>.= 
<font color="Maroon" title="c15">Lt</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">O</font>
<b>by </b><i><a class="txt" href="#E6:35_3_1_2"><i><font color="Green" title="E43">A21</font></i></a>, <a class="txt" href="#E2:35_3_1_2_2_2"><i><font color="Green" title="E45">A23</font></i></a>, <a class="ref" href="ordinal1.html#T21" title="ORDINAL1:th.21">ORDINAL1:21</a></i>
;<br/><a NAME="E5:35_3_1_2_2_2"/><b>then </b>
<font color="Maroon" title="c7">Ls</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">O</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="c15">Lt</font>
 <b>by </b><i><a class="txt" href="#E6:35_3_1_2"><i><font color="Green" title="E43">A21</font></i></a>, <a class="txt" href="#E3:35_3_1_2_2_2"><i><font color="Green" title="E46">A24</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>hence </b><a NAME="E6:35_3_1_2_2_2"/>
<font color="Maroon" title="c7">Ls</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">O</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">Ls</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">O2</font>
 <b>by </b><i><a class="txt" href="#E7:35_3_1_2"><i><font color="Green" title="E44">A22</font></i></a>, <a class="ref" href="zfmisc_1.html#T92" title="ZFMISC_1:th.92">ZFMISC_1:92</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>thus </b><a NAME="E4:35_3_1"/>
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>2</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="#E1:35_3_1"><i><font color="Green" title="E37">A10</font></i></a>, <a class="txt" href="#E2:35_3_1"><i><font color="Green" title="E38">A11</font></i></a>, <a class="txt" href="#E3:35_3_1"><i><font color="Green" title="E39">A17</font></i></a>);</i> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E8:35_3"/>
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">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="c7">Ls</font> &amp; <font color="Olive" title="b2">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="c7">Ls</font> holds <br/><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font> <a href="xboole_0.html#R3" title="XBOOLE_0:pred.3">are_c=-comparable</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_3_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">x</font>, <font color="Maroon" title="c14">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c13">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="c7">Ls</font> &amp; <font color="Maroon" title="c14">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="c7">Ls</font> implies <font color="Maroon" title="c13">x</font>,<font color="Maroon" title="c14">y</font> <a href="xboole_0.html#R3" title="XBOOLE_0:pred.3">are_c=-comparable</a>  )</span><br/>

<b>assume </b><a NAME="E1:35_3_2"/><i><font color="Green" title="E37">A25</font></i>: 
( <font color="Maroon" title="c13">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="c7">Ls</font> &amp; <font color="Maroon" title="c14">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="c7">Ls</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c13">x</font>,<font color="Maroon" title="c14">y</font> <a href="xboole_0.html#R3" title="XBOOLE_0:pred.3">are_c=-comparable</a> </span><br/>

<b>then 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="E3:35_3_2"/><i><font color="Green" title="E38">A26</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#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c7">Ls</font> &amp; <font color="Maroon" title="c7">Ls</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">o1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">x</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/>
<b>consider </b><font color="Maroon" title="c16">o2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:35_3_2"/><i><font color="Green" title="E39">A27</font></i>: 
( <font color="Maroon" title="c16">o2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c7">Ls</font> &amp; <font color="Maroon" title="c7">Ls</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">o2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">y</font> )
 <b>by </b><i><a class="txt" href="#E1:35_3_2"><i><font color="Green" title="E37">A25</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="c17">o1'</font> = <font color="Maroon" title="c15">o1</font>, <font color="Maroon" title="c18">o2'</font> = <font color="Maroon" title="c16">o2</font> as   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> <b>by </b><i><a class="txt" href="#E3:35_3_2"><i><font color="Green" title="E38">A26</font></i></a>, <a class="txt" href="#E5:35_3_2"><i><font color="Green" title="E39">A27</font></i></a></i>;<br/>
<a NAME="E7:35_3_2"/>
( <font color="Maroon" title="c17">o1'</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c18">o2'</font> or <font color="Maroon" title="c18">o2'</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c17">o1'</font> )
 
;<br/>
<a NAME="E8:35_3_2"/><b>then </b>
( <font color="Maroon" title="c13">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c14">y</font> or <font color="Maroon" title="c14">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c13">x</font> )
 
<b>by </b><i><a class="txt" href="#E7:35_3"><i><font color="Green" title="E36">A9</font></i></a>, <a class="txt" href="#E3:35_3_2"><i><font color="Green" title="E38">A26</font></i></a>, <a class="txt" href="#E5:35_3_2"><i><font color="Green" title="E39">A27</font></i></a></i>;<br/>
<b>hence </b><a NAME="E9:35_3_2"/>
<font color="Maroon" title="c13">x</font>,<font color="Maroon" title="c14">y</font> <a href="xboole_0.html#R3" title="XBOOLE_0:pred.3">are_c=-comparable</a> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D9" title="XBOOLE_0:def.9">XBOOLE_0:def 9</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E9:35_3"/><b>then </b><i><font color="Green" title="E37">A28</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">Ls</font> is <a href="ordinal1.html#V6" title="ORDINAL1:attr.6">c=-linear</a>
 
<b>by </b><i><a class="ref" href="ordinal1.html#D9" title="ORDINAL1:def.9">ORDINAL1:def 9</a></i>;<br/>
<a NAME="E10:35_3"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">Ls</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</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="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></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_3_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c13">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="c7">Ls</font> or <font color="Maroon" title="c13">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</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="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></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L</font> )</span><br/>

<b>assume </b><a NAME="E1:35_3_3"/>
<font color="Maroon" title="c13">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="c7">Ls</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c13">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</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="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></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L</font></span><br/>

<b>then consider </b><font color="Maroon" title="c14">o</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:35_3_3"/><i><font color="Green" title="E38">A29</font></i>: 
( <font color="Maroon" title="c14">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c7">Ls</font> &amp; <font color="Maroon" title="c13">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">Ls</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">o</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/>
<b>reconsider </b><font color="Maroon" title="c15">o</font> = <font color="Maroon" title="c14">o</font> as   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> <b>by </b><i><a class="txt" href="#E3:35_3_3"><i><font color="Green" title="E38">A29</font></i></a></i>;<br/>
<a NAME="E5:35_3_3"/>
<font color="Maroon" title="c7">Ls</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice8.html#K5" title="LATTICE8:func.5">ConsecutiveDelta2</a> <font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c15">o</font>
 
<b>by </b><i><a class="txt" href="#E3:35_3"><i><font color="Green" title="E33">A6</font></i></a>, <a class="txt" href="#E3:35_3_3"><i><font color="Green" title="E38">A29</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c16">h</font> = <font color="Maroon" title="c7">Ls</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">o</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="c15">o</font></span>)</span>,<font color="Maroon" title="c2">L</font> <b>by </b><i><a class="txt" href="#E1:35_3"><i><font color="Green" title="E32">A5</font></i></a>, <a class="txt" href="#E3:35_3"><i><font color="Green" title="E33">A6</font></i></a>, <a class="txt" href="#E3:35_3_3"><i><font color="Green" title="E38">A29</font></i></a></i>;<br/>
<a NAME="E7:35_3_3"/><i><font color="Green" title="E39">A30</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c16">h</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="c15">o</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="c15">o</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="E8:35_3_3"/><i><font color="Green" title="E40">A31</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c16">h</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L</font>
 
;<br/>
<a NAME="E9:35_3_3"/>
<font color="Maroon" title="c15">o</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c6">O1</font>
 
<b>by </b><i><a class="txt" href="#E3:35_3"><i><font color="Green" title="E33">A6</font></i></a>, <a class="txt" href="#E3:35_3_3"><i><font color="Green" title="E38">A29</font></i></a>, <a class="ref" href="ordinal1.html#D2" title="ORDINAL1:def.2">ORDINAL1:def 2</a></i>;<br/>
<a NAME="E10:35_3_3"/><b>then </b>
 <a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c15">o</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="c6">O1</font>
 
<b>by </b><i><a class="ref" href="lattice8.html#T22" target="_self" title="LATTICE8:th.22">Th22</a></i>;<br/>
<a NAME="E11:35_3_3"/><b>then </b>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c16">h</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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="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></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="txt" href="#E7:35_3_3"><i><font color="Green" title="E39">A30</font></i></a>, <a class="ref" href="zfmisc_1.html#T119" title="ZFMISC_1:th.119">ZFMISC_1:119</a></i>;<br/>
<b>hence </b><a NAME="E12:35_3_3"/>
<font color="Maroon" title="c13">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</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="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></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L</font>
 <b>by </b><i><a class="txt" href="#E3:35_3_3"><i><font color="Green" title="E38">A29</font></i></a>, <a class="txt" href="#E8:35_3_3"><i><font color="Green" title="E40">A31</font></i></a>, <a class="ref" href="partfun1.html#D5" title="PARTFUN1:def.5">PARTFUN1:def 5</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E11:35_3"/><b>then </b>
 <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> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</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="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></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L</font>
 
<b>by </b><i><a class="txt" href="#E9:35_3"><i><font color="Green" title="E37">A28</font></i></a>, <a class="ref" href="hahnban.html#T13" title="HAHNBAN:th.13">HAHNBAN:13</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c13">g</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E13:35_3"/><i><font color="Green" title="E38">A32</font></i>: 
(  <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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">g</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c13">g</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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="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></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c13">g</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L</font> )
 <b>by </b><i><a class="ref" href="partfun1.html#D5" title="PARTFUN1:def.5">PARTFUN1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c14">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="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> <b>by </b><i><a class="txt" href="#E13:35_3"><i><font color="Green" title="E38">A32</font></i></a></i>;<br/>
<a NAME="E15:35_3"/>
<font color="Maroon" title="c7">Ls</font> is <a href="funcop_1.html#V1" title="FUNCOP_1:attr.1">Function-yielding</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_3_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c15">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="funcop_1.html#D6" title="FUNCOP_1:def.6">FUNCOP_1:def 6</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c7">Ls</font> or <font color="Maroon" title="c7">Ls</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x</font> is    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  )</span><br/>

<b>assume </b><a NAME="E1:35_3_4"/><i><font color="Green" title="E39">A33</font></i>: 
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c7">Ls</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">Ls</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x</font> is    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> </span><br/>

<b>then reconsider </b><font color="Maroon" title="c16">o</font> = <font color="Maroon" title="c15">x</font> as   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> ;<br/>
<a NAME="E3:35_3_4"/>
<font color="Maroon" title="c7">Ls</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice8.html#K5" title="LATTICE8:func.5">ConsecutiveDelta2</a> <font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c16">o</font>
 
<b>by </b><i><a class="txt" href="#E3:35_3"><i><font color="Green" title="E33">A6</font></i></a>, <a class="txt" href="#E1:35_3_4"><i><font color="Green" title="E39">A33</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:35_3_4"/>
<font color="Maroon" title="c7">Ls</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x</font> is   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>
 <b>by </b><i><a class="txt" href="#E1:35_3"><i><font color="Green" title="E32">A5</font></i></a>, <a class="txt" href="#E3:35_3"><i><font color="Green" title="E33">A6</font></i></a>, <a class="txt" href="#E1:35_3_4"><i><font color="Green" title="E39">A33</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c15">LsF</font> = <font color="Maroon" title="c7">Ls</font> as  <a href="funcop_1.html#V1" title="FUNCOP_1:attr.1">Function-yielding</a> <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> ;<br/>
<a NAME="E17:35_3"/><i><font color="Green" title="E39">A34</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c14">f</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> <span class="p2">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <font color="Maroon" title="c15">LsF</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="lattice5.html#T1" title="LATTICE5:th.1">LATTICE5:1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c16">o1</font> = <font color="Maroon" title="c6">O1</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> <b>by </b><i><a class="txt" href="#E1:35_3"><i><font color="Green" title="E32">A5</font></i></a></i>;<br/>
<b>set </b><font color="Maroon" title="c17">YY</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p3">(<span class="default"><a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> <font color="Maroon" title="c1">A</font>,<font color="Olive" title="b1">O2</font></span>)</span>,<span class="p3">(<span class="default"><a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> <font color="Maroon" title="c1">A</font>,<font color="Olive" title="b1">O2</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> where <font color="Olive" title="b1">O2</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c16">o1</font> : verum </span>}</span> ;<br/>
<a NAME="E19:35_3"/><i><font color="Green" title="E40">A35</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <font color="Maroon" title="c7">Ls</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p3">(<span class="default"><a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> <font color="Maroon" title="c1">A</font>,<font color="Olive" title="b1">O2</font></span>)</span>,<span class="p3">(<span class="default"><a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> <font color="Maroon" title="c1">A</font>,<font color="Olive" title="b1">O2</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> where <font color="Olive" title="b1">O2</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c16">o1</font> : verum </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_3_5"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:35_3_5"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <font color="Maroon" title="c7">Ls</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p3">(<span class="default"><a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> <font color="Maroon" title="c1">A</font>,<font color="Olive" title="b1">O2</font></span>)</span>,<span class="p3">(<span class="default"><a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> <font color="Maroon" title="c1">A</font>,<font color="Olive" title="b1">O2</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> where <font color="Olive" title="b1">O2</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c16">o1</font> : verum </span>}</span> 
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">{<span class="default"> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p3">(<span class="default"><a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> <font color="Maroon" title="c1">A</font>,<font color="Olive" title="b1">O2</font></span>)</span>,<span class="p3">(<span class="default"><a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> <font color="Maroon" title="c1">A</font>,<font color="Olive" title="b1">O2</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> where <font color="Olive" title="b1">O2</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c16">o1</font> : verum </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <font color="Maroon" title="c7">Ls</font></span>)</span></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_3_5_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c18">Z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <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> <span class="p1">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <font color="Maroon" title="c7">Ls</font></span>)</span> or <font color="Maroon" title="c18">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p3">(<span class="default"><a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> <font color="Maroon" title="c1">A</font>,<font color="Olive" title="b1">O2</font></span>)</span>,<span class="p3">(<span class="default"><a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> <font color="Maroon" title="c1">A</font>,<font color="Olive" title="b1">O2</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> where <font color="Olive" title="b1">O2</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c16">o1</font> : verum </span>}</span>  )</span><br/>

<b>assume </b><a NAME="E1:35_3_5_1"/>
<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> <span class="p1">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <font color="Maroon" title="c7">Ls</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c18">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p3">(<span class="default"><a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> <font color="Maroon" title="c1">A</font>,<font color="Olive" title="b1">O2</font></span>)</span>,<span class="p3">(<span class="default"><a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> <font color="Maroon" title="c1">A</font>,<font color="Olive" title="b1">O2</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> where <font color="Olive" title="b1">O2</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c16">o1</font> : verum </span>}</span> </span><br/>

<b>then consider </b><font color="Maroon" title="c19">o</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:35_3_5_1"/><i><font color="Green" title="E41">A36</font></i>: 
( <font color="Maroon" title="c19">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <font color="Maroon" title="c7">Ls</font></span>)</span> &amp; <font color="Maroon" title="c18">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <font color="Maroon" title="c7">Ls</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">o</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="E4:35_3_5_1"/><i><font color="Green" title="E42">A37</font></i>: 
<font color="Maroon" title="c19">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c15">LsF</font>
 
<b>by </b><i><a class="txt" href="#E3:35_3_5_1"><i><font color="Green" title="E41">A36</font></i></a>, <a class="ref" href="funct_6.html#T89" title="FUNCT_6:th.89">FUNCT_6:89</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c20">o'</font> = <font color="Maroon" title="c19">o</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c16">o1</font> <b>by </b><i><a class="txt" href="#E3:35_3"><i><font color="Green" title="E33">A6</font></i></a></i>;<br/>
<a NAME="E6:35_3_5_1"/>
<font color="Maroon" title="c7">Ls</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">o'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice8.html#K5" title="LATTICE8:func.5">ConsecutiveDelta2</a> <font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c20">o'</font>
 
<b>by </b><i><a class="txt" href="#E3:35_3"><i><font color="Green" title="E33">A6</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c21">ls</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">o'</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">o'</font></span>)</span>,<font color="Maroon" title="c2">L</font> <b>by </b><i><a class="txt" href="#E1:35_3"><i><font color="Green" title="E32">A5</font></i></a></i>;<br/>
<font color="Maroon" title="c18">Z</font> = 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c21">ls</font>
<b>by </b><i><a class="txt" href="#E3:35_3_5_1"><i><font color="Green" title="E41">A36</font></i></a>, <a class="txt" href="#E4:35_3_5_1"><i><font color="Green" title="E42">A37</font></i></a>, <a class="ref" href="funct_6.html#T31" title="FUNCT_6:th.31">FUNCT_6:31</a></i>
<br/>.= 
<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">o'</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">o'</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/>
<b>hence </b><a NAME="E9:35_3_5_1"/>
<font color="Maroon" title="c18">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p3">(<span class="default"><a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> <font color="Maroon" title="c1">A</font>,<font color="Olive" title="b1">O2</font></span>)</span>,<span class="p3">(<span class="default"><a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> <font color="Maroon" title="c1">A</font>,<font color="Olive" title="b1">O2</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> where <font color="Olive" title="b1">O2</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c16">o1</font> : verum </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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

<b>let </b><font color="Maroon" title="c18">Z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c18">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p3">(<span class="default"><a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> <font color="Maroon" title="c1">A</font>,<font color="Olive" title="b1">O2</font></span>)</span>,<span class="p3">(<span class="default"><a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> <font color="Maroon" title="c1">A</font>,<font color="Olive" title="b1">O2</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> where <font color="Olive" title="b1">O2</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c16">o1</font> : verum </span>}</span>  or <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> <span class="p1">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <font color="Maroon" title="c7">Ls</font></span>)</span> )</span><br/>

<b>assume </b><a NAME="E2:35_3_5"/>
<font color="Maroon" title="c18">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p3">(<span class="default"><a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> <font color="Maroon" title="c1">A</font>,<font color="Olive" title="b1">O2</font></span>)</span>,<span class="p3">(<span class="default"><a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> <font color="Maroon" title="c1">A</font>,<font color="Olive" title="b1">O2</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> where <font color="Olive" title="b1">O2</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c16">o1</font> : verum </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <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> <span class="p1">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <font color="Maroon" title="c7">Ls</font></span>)</span></span><br/>

<b>then consider </b><font color="Maroon" title="c19">o</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c16">o1</font><b> such that </b><br/><a NAME="E4:35_3_5"/><i><font color="Green" title="E41">A38</font></i>: 
<font color="Maroon" title="c18">Z</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="c19">o</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="c19">o</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 ;<br/>
<a NAME="E5:35_3_5"/>
<font color="Maroon" title="c19">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c15">LsF</font>
 
<b>by </b><i><a class="txt" href="#E3:35_3"><i><font color="Green" title="E33">A6</font></i></a></i>;<br/>
<a NAME="E6:35_3_5"/><b>then </b><i><font color="Green" title="E42">A39</font></i>: 
<font color="Maroon" title="c19">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <font color="Maroon" title="c15">LsF</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_6.html#T89" title="FUNCT_6:th.89">FUNCT_6:89</a></i>;<br/>
<a NAME="E7:35_3_5"/>
<font color="Maroon" title="c7">Ls</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice8.html#K5" title="LATTICE8:func.5">ConsecutiveDelta2</a> <font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c19">o</font>
 
<b>by </b><i><a class="txt" href="#E3:35_3"><i><font color="Green" title="E33">A6</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c20">ls</font> = <font color="Maroon" title="c7">Ls</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">o</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="c19">o</font></span>)</span>,<font color="Maroon" title="c2">L</font> <b>by </b><i><a class="txt" href="#E1:35_3"><i><font color="Green" title="E32">A5</font></i></a></i>;<br/>
<font color="Maroon" title="c18">Z</font> = 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c20">ls</font>
<b>by </b><i><a class="txt" href="#E4:35_3_5"><i><font color="Green" title="E41">A38</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <font color="Maroon" title="c7">Ls</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">o</font>
<b>by </b><i><a class="txt" href="#E3:35_3"><i><font color="Green" title="E33">A6</font></i></a>, <a class="ref" href="funct_6.html#T31" title="FUNCT_6:th.31">FUNCT_6:31</a></i>
;<br/>
<b>hence </b><a NAME="E10:35_3_5"/>
<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> <span class="p1">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <font color="Maroon" title="c7">Ls</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E6:35_3_5"><i><font color="Green" title="E42">A39</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E20:35_3"/>
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">O1</font>
 
<b>by </b><i><a class="txt" href="#E1:35_3"><i><font color="Green" title="E32">A5</font></i></a>, <a class="ref" href="ordinal3.html#T10" title="ORDINAL3:th.10">ORDINAL3:10</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c18">RTs</font> =  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">Ts</font> as  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>  <b>by </b><i><a class="txt" href="#E6:35_3"><i><font color="Green" title="E35">A8</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/>
<a NAME="E22:35_3"/>
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">RTs</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">RTs</font> holds <br/><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font> <a href="xboole_0.html#R3" title="XBOOLE_0:pred.3">are_c=-comparable</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_3_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c19">x</font>, <font color="Maroon" title="c20">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c19">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">RTs</font> &amp; <font color="Maroon" title="c20">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">RTs</font> implies <font color="Maroon" title="c19">x</font>,<font color="Maroon" title="c20">y</font> <a href="xboole_0.html#R3" title="XBOOLE_0:pred.3">are_c=-comparable</a>  )</span><br/>

<b>assume </b><a NAME="E1:35_3_6"/><i><font color="Green" title="E41">A40</font></i>: 
( <font color="Maroon" title="c19">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">RTs</font> &amp; <font color="Maroon" title="c20">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">RTs</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c19">x</font>,<font color="Maroon" title="c20">y</font> <a href="xboole_0.html#R3" title="XBOOLE_0:pred.3">are_c=-comparable</a> </span><br/>

<b>then consider </b><font color="Maroon" title="c21">o1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:35_3_6"/><i><font color="Green" title="E42">A41</font></i>: 
( <font color="Maroon" title="c21">o1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c8">Ts</font> &amp; <font color="Maroon" title="c8">Ts</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">o1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">x</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/>
<b>consider </b><font color="Maroon" title="c22">o2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:35_3_6"/><i><font color="Green" title="E43">A42</font></i>: 
( <font color="Maroon" title="c22">o2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c8">Ts</font> &amp; <font color="Maroon" title="c8">Ts</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">o2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">y</font> )
 <b>by </b><i><a class="txt" href="#E1:35_3_6"><i><font color="Green" title="E41">A40</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="c23">o1'</font> = <font color="Maroon" title="c21">o1</font>, <font color="Maroon" title="c24">o2'</font> = <font color="Maroon" title="c22">o2</font> as   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> <b>by </b><i><a class="txt" href="#E3:35_3_6"><i><font color="Green" title="E42">A41</font></i></a>, <a class="txt" href="#E5:35_3_6"><i><font color="Green" title="E43">A42</font></i></a></i>;<br/>
<a NAME="E7:35_3_6"/><i><font color="Green" title="E44">A43</font></i>: 
<font color="Maroon" title="c8">Ts</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">o1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c23">o1'</font>
 
<b>by </b><i><a class="txt" href="#E6:35_3"><i><font color="Green" title="E35">A8</font></i></a>, <a class="txt" href="#E3:35_3_6"><i><font color="Green" title="E42">A41</font></i></a></i>;<br/>
<a NAME="E8:35_3_6"/><i><font color="Green" title="E45">A44</font></i>: 
<font color="Maroon" title="c8">Ts</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">o2'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c24">o2'</font>
 
<b>by </b><i><a class="txt" href="#E6:35_3"><i><font color="Green" title="E35">A8</font></i></a>, <a class="txt" href="#E5:35_3_6"><i><font color="Green" title="E43">A42</font></i></a></i>;<br/>
<a NAME="E9:35_3_6"/>
( <font color="Maroon" title="c23">o1'</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c24">o2'</font> or <font color="Maroon" title="c24">o2'</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c23">o1'</font> )
 
;<br/>
<a NAME="E10:35_3_6"/><b>then </b>
( <font color="Maroon" title="c19">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c20">y</font> or <font color="Maroon" title="c20">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c19">x</font> )
 
<b>by </b><i><a class="txt" href="#E3:35_3_6"><i><font color="Green" title="E42">A41</font></i></a>, <a class="txt" href="#E5:35_3_6"><i><font color="Green" title="E43">A42</font></i></a>, <a class="txt" href="#E7:35_3_6"><i><font color="Green" title="E44">A43</font></i></a>, <a class="txt" href="#E8:35_3_6"><i><font color="Green" title="E45">A44</font></i></a>, <a class="ref" href="lattice8.html#T22" target="_self" title="LATTICE8:th.22">Th22</a></i>;<br/>
<b>hence </b><a NAME="E11:35_3_6"/>
<font color="Maroon" title="c19">x</font>,<font color="Maroon" title="c20">y</font> <a href="xboole_0.html#R3" title="XBOOLE_0:pred.3">are_c=-comparable</a> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D9" title="XBOOLE_0:def.9">XBOOLE_0:def 9</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E23:35_3"/><b>then </b><i><font color="Green" title="E41">A45</font></i>: 
<font color="Maroon" title="c18">RTs</font> is <a href="ordinal1.html#V6" title="ORDINAL1:attr.6">c=-linear</a>
 
<b>by </b><i><a class="ref" href="ordinal1.html#D9" title="ORDINAL1:def.9">ORDINAL1:def 9</a></i>;<br/>
<a NAME="E24:35_3"/><i><font color="Green" title="E42">A46</font></i>: 
<span class="p1">{<span class="default"> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p3">(<span class="default"><a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> <font color="Maroon" title="c1">A</font>,<font color="Olive" title="b1">O2</font></span>)</span>,<span class="p3">(<span class="default"><a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> <font color="Maroon" title="c1">A</font>,<font color="Olive" title="b1">O2</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> where <font color="Olive" title="b1">O2</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c16">o1</font> : verum </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b1">a</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> where <font color="Olive" title="b1">a</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c18">RTs</font> : <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">RTs</font> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_3_7"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon" title="c19">XX</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b1">a</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> where <font color="Olive" title="b1">a</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c18">RTs</font> : <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">RTs</font> </span>}</span> ;<br/>
<b>thus </b><a NAME="E1:35_3_7"/>
<span class="p1">{<span class="default"> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p3">(<span class="default"><a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> <font color="Maroon" title="c1">A</font>,<font color="Olive" title="b1">O2</font></span>)</span>,<span class="p3">(<span class="default"><a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> <font color="Maroon" title="c1">A</font>,<font color="Olive" title="b1">O2</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> where <font color="Olive" title="b1">O2</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c16">o1</font> : verum </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b1">a</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> where <font color="Olive" title="b1">a</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c18">RTs</font> : <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">RTs</font> </span>}</span> 
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">{<span class="default"> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b1">a</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> where <font color="Olive" title="b1">a</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c18">RTs</font> : <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">RTs</font> </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p3">(<span class="default"><a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> <font color="Maroon" title="c1">A</font>,<font color="Olive" title="b1">O2</font></span>)</span>,<span class="p3">(<span class="default"><a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> <font color="Maroon" title="c1">A</font>,<font color="Olive" title="b1">O2</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> where <font color="Olive" title="b1">O2</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c16">o1</font> : verum </span>}</span> </span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_3_7_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c20">Z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c20">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p3">(<span class="default"><a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> <font color="Maroon" title="c1">A</font>,<font color="Olive" title="b1">O2</font></span>)</span>,<span class="p3">(<span class="default"><a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> <font color="Maroon" title="c1">A</font>,<font color="Olive" title="b1">O2</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> where <font color="Olive" title="b1">O2</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c16">o1</font> : verum </span>}</span>  or <font color="Maroon" title="c20">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b1">a</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> where <font color="Olive" title="b1">a</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c18">RTs</font> : <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">RTs</font> </span>}</span>  )</span><br/>

<b>assume </b><a NAME="E1:35_3_7_1"/>
<font color="Maroon" title="c20">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p3">(<span class="default"><a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> <font color="Maroon" title="c1">A</font>,<font color="Olive" title="b1">O2</font></span>)</span>,<span class="p3">(<span class="default"><a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> <font color="Maroon" title="c1">A</font>,<font color="Olive" title="b1">O2</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> where <font color="Olive" title="b1">O2</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c16">o1</font> : verum </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c20">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b1">a</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> where <font color="Olive" title="b1">a</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c18">RTs</font> : <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">RTs</font> </span>}</span> </span><br/>

<b>then consider </b><font color="Maroon" title="c21">o</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c16">o1</font><b> such that </b><br/><a NAME="E3:35_3_7_1"/><i><font color="Green" title="E43">A47</font></i>: 
<font color="Maroon" title="c20">Z</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">o</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">o</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 ;<br/>
<a NAME="E4:35_3_7_1"/>
<font color="Maroon" title="c8">Ts</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c21">o</font>
 
<b>by </b><i><a class="txt" href="#E6:35_3"><i><font color="Green" title="E35">A8</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c22">CoS</font> =  <a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c21">o</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c18">RTs</font> <b>by </b><i><a class="txt" href="#E6:35_3"><i><font color="Green" title="E35">A8</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="E6:35_3_7_1"/>
<font color="Maroon" title="c20">Z</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="c22">CoS</font>,<font color="Maroon" title="c22">CoS</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="txt" href="#E3:35_3_7_1"><i><font color="Green" title="E43">A47</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:35_3_7_1"/>
<font color="Maroon" title="c20">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b1">a</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> where <font color="Olive" title="b1">a</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c18">RTs</font> : <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">RTs</font> </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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

<b>let </b><font color="Maroon" title="c20">Z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c20">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b1">a</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> where <font color="Olive" title="b1">a</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c18">RTs</font> : <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">RTs</font> </span>}</span>  or <font color="Maroon" title="c20">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p3">(<span class="default"><a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> <font color="Maroon" title="c1">A</font>,<font color="Olive" title="b1">O2</font></span>)</span>,<span class="p3">(<span class="default"><a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> <font color="Maroon" title="c1">A</font>,<font color="Olive" title="b1">O2</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> where <font color="Olive" title="b1">O2</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c16">o1</font> : verum </span>}</span>  )</span><br/>

<b>assume </b><a NAME="E2:35_3_7"/>
<font color="Maroon" title="c20">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b1">a</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> where <font color="Olive" title="b1">a</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c18">RTs</font> : <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">RTs</font> </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c20">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p3">(<span class="default"><a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> <font color="Maroon" title="c1">A</font>,<font color="Olive" title="b1">O2</font></span>)</span>,<span class="p3">(<span class="default"><a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> <font color="Maroon" title="c1">A</font>,<font color="Olive" title="b1">O2</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> where <font color="Olive" title="b1">O2</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c16">o1</font> : verum </span>}</span> </span><br/>

<b>then consider </b><font color="Maroon" title="c21">a</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c18">RTs</font><b> such that </b><br/><a NAME="E4:35_3_7"/><i><font color="Green" title="E43">A48</font></i>: 
( <font color="Maroon" title="c20">Z</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="c21">a</font>,<font color="Maroon" title="c21">a</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp; <font color="Maroon" title="c21">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">RTs</font> )
 ;<br/>
<b>consider </b><font color="Maroon" title="c22">o</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E6:35_3_7"/><i><font color="Green" title="E44">A49</font></i>: 
( <font color="Maroon" title="c22">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c8">Ts</font> &amp; <font color="Maroon" title="c21">a</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="c22">o</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/>
<b>reconsider </b><font color="Maroon" title="c23">o'</font> = <font color="Maroon" title="c22">o</font> as   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> <b>by </b><i><a class="txt" href="#E6:35_3_7"><i><font color="Green" title="E44">A49</font></i></a></i>;<br/>
<a NAME="E8:35_3_7"/><i><font color="Green" title="E45">A50</font></i>: 
<font color="Maroon" title="c21">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c23">o'</font>
 
<b>by </b><i><a class="txt" href="#E6:35_3"><i><font color="Green" title="E35">A8</font></i></a>, <a class="txt" href="#E6:35_3_7"><i><font color="Green" title="E44">A49</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c24">O</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c16">o1</font><b> such that </b><br/><a NAME="E10:35_3_7"/><i><font color="Green" title="E46">A51</font></i>: 
<font color="Maroon" title="c24">O</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">o'</font>
 <b>by </b><i><a class="txt" href="#E6:35_3"><i><font color="Green" title="E35">A8</font></i></a>, <a class="txt" href="#E6:35_3_7"><i><font color="Green" title="E44">A49</font></i></a></i>;<br/>
<b>thus </b><a NAME="E11:35_3_7"/>
<font color="Maroon" title="c20">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p3">(<span class="default"><a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> <font color="Maroon" title="c1">A</font>,<font color="Olive" title="b1">O2</font></span>)</span>,<span class="p3">(<span class="default"><a href="lattice8.html#K3" title="LATTICE8:func.3">ConsecutiveSet2</a> <font color="Maroon" title="c1">A</font>,<font color="Olive" title="b1">O2</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> where <font color="Olive" title="b1">O2</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c16">o1</font> : verum </span>}</span> 
 <b>by </b><i><a class="txt" href="#E4:35_3_7"><i><font color="Green" title="E43">A48</font></i></a>, <a class="txt" href="#E8:35_3_7"><i><font color="Green" title="E45">A50</font></i></a>, <a class="txt" href="#E10:35_3_7"><i><font color="Green" title="E46">A51</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<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="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></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> = 
<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="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p3">(<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></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></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
<b>by </b><i><a class="txt" href="#E1:35_3"><i><font color="Green" title="E32">A5</font></i></a>, <a class="txt" href="#E6:35_3"><i><font color="Green" title="E35">A8</font></i></a>, <a class="ref" href="lattice8.html#T17" target="_self" title="LATTICE8:th.17">Th17</a></i>
<br/>.= 
<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="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c18">RTs</font></span>)</span>,<span class="p2">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c18">RTs</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
<b>by </b><i><a class="txt" href="#E1:35_3"><i><font color="Green" title="E32">A5</font></i></a>, <a class="txt" href="#E6:35_3"><i><font color="Green" title="E35">A8</font></i></a>, <a class="ref" href="lattice8.html#T17" target="_self" title="LATTICE8:th.17">Th17</a></i>
<br/>.= 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c14">f</font>
<b>by </b><i><a class="txt" href="#E17:35_3"><i><font color="Green" title="E39">A34</font></i></a>, <a class="txt" href="#E19:35_3"><i><font color="Green" title="E40">A35</font></i></a>, <a class="txt" href="#E23:35_3"><i><font color="Green" title="E41">A45</font></i></a>, <a class="txt" href="#E24:35_3"><i><font color="Green" title="E42">A46</font></i></a>, <a class="ref" href="lattice5.html#T3" title="LATTICE5:th.3">LATTICE5:3</a></i>
;<br/>
<b>hence </b><a NAME="E26:35_3"/>
 <a href="lattice8.html#K5" title="LATTICE8:func.5">ConsecutiveDelta2</a> <font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c6">O1</font> is   <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">O1</font></span>)</span>,<font color="Maroon" title="c2">L</font>
 <b>by </b><i><a class="txt" href="#E4:35_3"><i><font color="Green" title="E34">A7</font></i></a>, <a class="txt" href="#E13:35_3"><i><font color="Green" title="E38">A32</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11" title="RELSET_1:th.11">RELSET_1:11</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E4:35"/>
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="#E1:35"><i><font color="Green" title="E29">A1</font></i></a>, <a class="txt" href="#E2:35"><i><font color="Green" title="E30">A2</font></i></a>, <a class="txt" href="#E3:35"><i><font color="Green" title="E31">A4</font></i></a>);<br/></i>
<b>hence </b><a NAME="E5:35"/>
 <a href="lattice8.html#K5" title="LATTICE8:func.5">ConsecutiveDelta2</a> <font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c5">O</font> is   <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="c5">O</font></span>)</span>,<font color="Maroon" title="c2">L</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
