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

<b>let </b><font color="Maroon" title="c1">N</font> be  <a href="struct_0.html#V8" title="STRUCT_0:attr.8">finite</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"> <font color="Maroon" title="c1">N</font>, <a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K7" title="WAYBEL_0:func.7">Ids</a> <font color="Maroon" title="c1">N</font></span>)</span> <a href="waybel_1.html#R5" title="WAYBEL_1:pred.5">are_isomorphic</a> </span><br/>

<b>set </b><font color="Maroon" title="c2">I</font> =  <a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K7" title="WAYBEL_0:func.7">Ids</a> <font color="Maroon" title="c1">N</font></span>)</span>;<br/>
<b>take </b><font color="Maroon" title="c3">i</font> =  <a href="yellow_2.html#K3" title="YELLOW_2:func.3">IdsMap</a> <font color="Maroon" title="c1">N</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_1.html#D8" title="WAYBEL_1:def.8">WAYBEL_1:def 8</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">i</font> is <a href="waybel_0.html#V23" title="WAYBEL_0:attr.23">V212</a></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1"><b>now </b></a><div class="add"><b>thus </b><a NAME="E1:21_1"/>
( not <font color="Maroon" title="c1">N</font> is <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> &amp; not  <a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K7" title="WAYBEL_0:func.7">Ids</a> <font color="Maroon" title="c1">N</font></span>)</span> is <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> implies ( <font color="Maroon" title="c3">i</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp; <font color="Maroon" title="c3">i</font> is <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a> &amp;  ex <font color="Olive" title="b1">s</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K7" title="WAYBEL_0:func.7">Ids</a> <font color="Maroon" title="c1">N</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">N</font> st <br/>( <font color="Olive" title="b1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">i</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a>  &amp; <font color="Olive" title="b1">s</font> is <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a> ) ) )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( <font color="Maroon" title="c1">N</font> is <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> or  <a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K7" title="WAYBEL_0:func.7">Ids</a> <font color="Maroon" title="c1">N</font></span>)</span> is <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> ) implies ( <font color="Maroon" title="c1">N</font> is <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> &amp;  <a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K7" title="WAYBEL_0:func.7">Ids</a> <font color="Maroon" title="c1">N</font></span>)</span> is <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> ) )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:21_1_1"/>
( not <font color="Maroon" title="c1">N</font> is <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> &amp; not  <a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K7" title="WAYBEL_0:func.7">Ids</a> <font color="Maroon" title="c1">N</font></span>)</span> is <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">i</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp; <font color="Maroon" title="c3">i</font> is <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a> &amp;  ex <font color="Olive" title="b1">s</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K7" title="WAYBEL_0:func.7">Ids</a> <font color="Maroon" title="c1">N</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">N</font> st <br/>( <font color="Olive" title="b1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">i</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a>  &amp; <font color="Olive" title="b1">s</font> is <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a> ) )</span><br/>

<b>thus </b><a NAME="E2:21_1_1"/>
( <font color="Maroon" title="c3">i</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp; <font color="Maroon" title="c3">i</font> is <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">s</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K7" title="WAYBEL_0:func.7">Ids</a> <font color="Maroon" title="c1">N</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">N</font> st <br/>( <font color="Olive" title="b1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">i</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a>  &amp; <font color="Olive" title="b1">s</font> is <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a> )</span><br/>

<b>take </b><font color="Maroon" title="c4">s</font> =  <a href="yellow_2.html#K2" title="YELLOW_2:func.2">SupMap</a> <font color="Maroon" title="c1">N</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">i</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a>  &amp; <font color="Maroon" title="c4">s</font> is <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a> )</span><br/>

<a NAME="E3:21_1_1"/><i><font color="Green" title="E9">A1</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K7" title="WAYBEL_0:func.7">Ids</a> <font color="Maroon" title="c1">N</font></span>)</span></span>)</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="E4:21_1_1"/>
<span class="p1"><a href="waybel_1.html#K1" title="WAYBEL_1:func.1">[</a><span class="default"><font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">s</font></span><a href="waybel_1.html#K1" title="WAYBEL_1:func.1">]</a></span> is <a href="waybel_1.html#V3" title="WAYBEL_1:attr.3">Galois</a>
 
<b>by </b><i><a class="ref" href="waybel_1.html#T60" title="WAYBEL_1:th.60">WAYBEL_1:60</a></i>;<br/>
<a NAME="E5:21_1_1"/><b>then </b>
<font color="Maroon" title="c3">i</font> is <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>
 
<b>by </b><i><a class="ref" href="waybel_1.html#T26" title="WAYBEL_1:th.26">WAYBEL_1:26</a></i>;<br/>
<a NAME="E6:21_1_1"/><b>then </b><i><font color="Green" title="E10">A2</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K7" title="WAYBEL_0:func.7">Ids</a> <font color="Maroon" title="c1">N</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></i>;<br/>
<a NAME="E7:21_1_1"/>
for <font color="Olive" title="b1">y</font>, <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">i</font> &amp; <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y</font> iff ( <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">i</font> &amp; <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">i</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">x</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">y</font>, <font color="Maroon" title="c6">x</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="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">i</font> &amp; <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">y</font> iff ( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">i</font> &amp; <font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">i</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> ) )</span><br/>

<a NAME="E1:21_1_1_1"/><i><font color="Green" title="E11">A3</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">N</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<div><b>hereby </b> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">i</font> &amp; <font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">i</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> implies ( <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">i</font> &amp; <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">y</font> ) )</span>
<div class="add"><b>assume </b><b>that </b><br/><a NAME="E1:21_1_1_1_1"/><i><font color="Green" title="E12">A4</font></i>: 
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">i</font>
 <b>and </b><br/><a NAME="E2:21_1_1_1_1"/><i><font color="Green" title="E13">A5</font></i>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">i</font> &amp; <font color="Maroon" title="c3">i</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">y</font> )</span><br/><b>reconsider </b><font color="Maroon" title="c7">Y</font> = <font color="Maroon" title="c5">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K7" title="WAYBEL_0:func.7">Ids</a> <font color="Maroon" title="c1">N</font></span>)</span></span>)</span> <b>by </b><i><a class="txt" href="#E1:21_1_1_1_1"><i><font color="Green" title="E12">A4</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c8">Y1</font> = <font color="Maroon" title="c7">Y</font> as   <a href="waybel_0.html#NM3" title="WAYBEL_0:NM.3">Ideal</a> of <font color="Maroon" title="c1">N</font> <b>by </b><i><a class="ref" href="yellow_2.html#T43" title="YELLOW_2:th.43">YELLOW_2:43</a></i>;<br/><a NAME="E5:21_1_1_1_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">s</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">Y</font>
 <b>by </b><i><a class="txt" href="#E2:21_1_1_1_1"><i><font color="Green" title="E13">A5</font></i></a></i>;<br/><b>hence </b><a NAME="E6:21_1_1_1_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">i</font>
 <b>by </b><i><a class="txt" href="#E1:21_1_1_1"><i><font color="Green" title="E11">A3</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">i</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">y</font></span><br/><b>thus </b><font color="Maroon" title="c3">i</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> = 
<font color="Maroon" title="c3">i</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c8">Y1</font></span>)</span>
<b>by </b><i><a class="txt" href="#E2:21_1_1_1_1"><i><font color="Green" title="E13">A5</font></i></a>, <a class="ref" href="yellow_2.html#D3" title="YELLOW_2:def.3">YELLOW_2:def 3</a></i>
<br/>.= 
 <a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c8">Y1</font></span>)</span>
<b>by </b><i><a class="ref" href="yellow_2.html#D4" title="YELLOW_2:def.4">YELLOW_2:def 4</a></i>
<br/>.= 
<font color="Maroon" title="c5">y</font>
<b>by </b><i><a class="ref" href="waybel14.html#T5" title="WAYBEL14:th.5">WAYBEL14:5</a>, <a class="ref" href="waybel_3.html#T16" title="WAYBEL_3:th.16">WAYBEL_3:16</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>assume </b><b>that </b><br/><a NAME="E3:21_1_1_1"/><i><font color="Green" title="E12">A6</font></i>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">i</font>
 <b>and </b><br/><a NAME="E4:21_1_1_1"/><i><font color="Green" title="E13">A7</font></i>: 
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">i</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">i</font> &amp; <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">y</font> )</span><br/>

<b>reconsider </b><font color="Maroon" title="c7">X</font> = <font color="Maroon" title="c6">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">N</font> <b>by </b><i><a class="txt" href="#E3:21_1_1_1"><i><font color="Green" title="E12">A6</font></i></a></i>;<br/>
<a NAME="E6:21_1_1_1"/><i><font color="Green" title="E14">A8</font></i>: 
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">i</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">X</font>
 
<b>by </b><i><a class="txt" href="#E4:21_1_1_1"><i><font color="Green" title="E13">A7</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:21_1_1_1"/>
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">i</font>
 <b>by </b><i><a class="txt" href="#E6:21_1_1"><i><font color="Green" title="E10">A2</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">y</font></span><br/>

<b>reconsider </b><font color="Maroon" title="c8">Y</font> = <font color="Maroon" title="c5">y</font> as   <a href="waybel_0.html#NM3" title="WAYBEL_0:NM.3">Ideal</a> of <font color="Maroon" title="c1">N</font> <b>by </b><i><a class="txt" href="#E6:21_1_1_1"><i><font color="Green" title="E14">A8</font></i></a>, <a class="ref" href="yellow_2.html#T43" title="YELLOW_2:th.43">YELLOW_2:43</a></i>;<br/>
<b>thus </b><font color="Maroon" title="c4">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">y</font> = 
 <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c8">Y</font>
<b>by </b><i><a class="ref" href="yellow_2.html#D3" title="YELLOW_2:def.3">YELLOW_2:def 3</a></i>
<br/>.= 
 <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c7">X</font></span>)</span>
<b>by </b><i><a class="txt" href="#E4:21_1_1_1"><i><font color="Green" title="E13">A7</font></i></a>, <a class="ref" href="yellow_2.html#D4" title="YELLOW_2:def.4">YELLOW_2:def 4</a></i>
<br/>.= 
<font color="Maroon" title="c6">x</font>
<b>by </b><i><a class="ref" href="waybel_0.html#T34" title="WAYBEL_0:th.34">WAYBEL_0:34</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>hence </b><a NAME="E8:21_1_1"/>
<font color="Maroon" title="c4">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">i</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> 
 <b>by </b><i><a class="txt" href="#E3:21_1_1"><i><font color="Green" title="E9">A1</font></i></a>, <a class="txt" href="#E6:21_1_1"><i><font color="Green" title="E10">A2</font></i></a>, <a class="ref" href="funct_1.html#T54" title="FUNCT_1:th.54">FUNCT_1:54</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">s</font> is <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a></span><br/>

<b>thus </b><a NAME="E9:21_1_1"/>
<font color="Maroon" title="c4">s</font> is <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a>
 ; <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>thus </b><a NAME="E2:21_1"/>
( ( <font color="Maroon" title="c1">N</font> is <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> or  <a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K7" title="WAYBEL_0:func.7">Ids</a> <font color="Maroon" title="c1">N</font></span>)</span> is <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> ) implies ( <font color="Maroon" title="c1">N</font> is <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> &amp;  <a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K7" title="WAYBEL_0:func.7">Ids</a> <font color="Maroon" title="c1">N</font></span>)</span> is <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> ) )
 ; <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>hence </b><a NAME="E2:21"/>
<font color="Maroon" title="c3">i</font> is <a href="waybel_0.html#V23" title="WAYBEL_0:attr.23">V212</a>
 <b>by </b><i><a class="ref" href="waybel_0.html#D38" title="WAYBEL_0:def.38">WAYBEL_0:def 38</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>
