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

<b>let </b><font color="Maroon">L</font> be  non <a href="struct_0.html#V3">empty</a> <a href="lattice3.html#V3">complete</a> <a href="orders_2.html#NM1">Poset</a>;<br/>

<b>set </b><font color="Maroon">i</font> = <span class="p1">(<span class="default"><a href="yellow_2.html#K3">IdsMap</a> <font color="Maroon">L</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><a href="yellow_2.html#K2">SupMap</a> <font color="Maroon">L</font></span>)</span>;<br/>
<a NAME="E1:99"/><i><font color="Green">E32</font></i>: 
<span class="p1">(<span class="default"><a href="yellow_2.html#K3">IdsMap</a> <font color="Maroon">L</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><a href="yellow_2.html#K2">SupMap</a> <font color="Maroon">L</font></span>)</span> is <a href="orders_3.html#V5">monotone</a>
 
<b>by </b><i><a class="ref" href="yellow_2.html#T14">YELLOW_2:14</a></i>;<br/>
<a NAME="E2:99"/>
<span class="p1"><a href="waybel_1.html#K1" target="_self">[</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_2.html#K3">IdsMap</a> <font color="Maroon">L</font></span>)</span>,<span class="p2">(<span class="default"><a href="yellow_2.html#K2">SupMap</a> <font color="Maroon">L</font></span>)</span></span><a href="waybel_1.html#K1" target="_self">]</a></span> is <a href="waybel_1.html#V3" target="_self">Galois</a>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:99"/>
<span class="p1">(<span class="default"><a href="yellow_2.html#K3">IdsMap</a> <font color="Maroon">L</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><a href="yellow_2.html#K2">SupMap</a> <font color="Maroon">L</font></span>)</span> <a href="waybel_1.html#NR11" target="_self">is_a_closure_operator</a> 
 <b>by </b><i><a class="txt" href="waybel_1.html#E7:99"><i><font color="Green">E60</font></i></a>, <a class="txt" href="waybel_1.html#E6:99"><i><font color="Green">E49</font></i></a></i>;<br/>

<b>take </b><font color="Maroon">f</font> = <span class="p1">(<span class="default"><a href="yellow_2.html#K2">SupMap</a> <font color="Maroon">L</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><a href="waybel_1.html#K3" target="_self">inclusion</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="yellow_2.html#K3">IdsMap</a> <font color="Maroon">L</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p3">(<span class="default"><a href="yellow_2.html#K2">SupMap</a> <font color="Maroon">L</font></span>)</span></span>)</span></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_1.html#D8" target="_self">WAYBEL_1:def 8</a><br/>

<div><i><font color="Green">E35</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="99_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">I</font> be   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font>;<br/><a NAME="E1:99_1"/>
<font color="Maroon">I</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K7">Ids</a> <font color="Maroon">L</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="yellow_2.html#T43">YELLOW_2:43</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow_2.html#K3">IdsMap</a> <font color="Maroon">L</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="yellow_2.html#K2">SupMap</a> <font color="Maroon">L</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">I</font> = 
<span class="p1">(<span class="default"><a href="yellow_2.html#K3">IdsMap</a> <font color="Maroon">L</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow_2.html#K2">SupMap</a> <font color="Maroon">L</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">I</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="yellow_2.html#K3">IdsMap</a> <font color="Maroon">L</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">I</font></span>)</span>
<b>by </b><i><a class="ref" href="yellow_2.html#D3">YELLOW_2:def 3</a></i>
<br/>.= 
 <a href="waybel_0.html#K5">downarrow</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">I</font></span>)</span>
<b>by </b><i><a class="ref" href="yellow_2.html#D4">YELLOW_2:def 4</a></i>
;<br/><br/></div><b>end;</b></div>
<div><i><font color="Green">E45</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="99_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_2.html#K1">Image</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="yellow_2.html#K3">IdsMap</a> <font color="Maroon">L</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p3">(<span class="default"><a href="yellow_2.html#K2">SupMap</a> <font color="Maroon">L</font></span>)</span></span>)</span></span>)</span>;<br/><b>let </b><font color="Maroon">I</font> be   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font>;<br/><b>assume </b><a NAME="E1:99_2"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font>
 ;<br/><b>hence </b><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">I</font> = 
<span class="p1">(<span class="default"><a href="yellow_2.html#K2">SupMap</a> <font color="Maroon">L</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_1.html#K3" target="_self">inclusion</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="yellow_2.html#K3">IdsMap</a> <font color="Maroon">L</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p4">(<span class="default"><a href="yellow_2.html#K2">SupMap</a> <font color="Maroon">L</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">I</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="yellow_2.html#K2">SupMap</a> <font color="Maroon">L</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">I</font>
<b>by </b><i>, <a class="ref" href="waybel_1.html#D7" target="_self">Def7</a></i>
<br/>.= 
 <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">I</font>
<b>by </b><i><a class="ref" href="yellow_2.html#D3">YELLOW_2:def 3</a></i>
;<br/><br/></div><b>end;</b></div>
<a NAME="E6:99"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">f</font> is <a href="funct_1.html#V2">one-to-one</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="99_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_2.html#K1">Image</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="yellow_2.html#K3">IdsMap</a> <font color="Maroon">L</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p3">(<span class="default"><a href="yellow_2.html#K2">SupMap</a> <font color="Maroon">L</font></span>)</span></span>)</span></span>)</span>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_2.html#K1">Image</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="yellow_2.html#K3">IdsMap</a> <font color="Maroon">L</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p3">(<span class="default"><a href="yellow_2.html#K2">SupMap</a> <font color="Maroon">L</font></span>)</span></span>)</span></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_1.html#D1" target="_self">WAYBEL_1:def 1</a><br/>

<b>assume </b><a NAME="E1:99_3"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font>
 ;<br/>

<b>consider </b><font color="Maroon">Ix</font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K7">Ids</a> <font color="Maroon">L</font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E3:99_3"/><i><font color="Green">E55</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow_2.html#K3">IdsMap</a> <font color="Maroon">L</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="yellow_2.html#K2">SupMap</a> <font color="Maroon">L</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">Ix</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="yellow_2.html#T12">YELLOW_2:12</a></i>;<br/>
<b>consider </b><font color="Maroon">Iy</font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K7">Ids</a> <font color="Maroon">L</font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E5:99_3"/><i><font color="Green">E57</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow_2.html#K3">IdsMap</a> <font color="Maroon">L</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="yellow_2.html#K2">SupMap</a> <font color="Maroon">L</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">Iy</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="yellow_2.html#T12">YELLOW_2:12</a></i>;<br/>
<b>reconsider </b><font color="Maroon">Ix</font> = <font color="Maroon">Ix</font>, <font color="Maroon">Iy</font> = <font color="Maroon">Iy</font> as   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> <b>by </b><i><a class="ref" href="yellow_2.html#T43">YELLOW_2:43</a></i>;<br/>
<a NAME="E7:99_3"/><i><font color="Green">E58</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow_2.html#K3">IdsMap</a> <font color="Maroon">L</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="yellow_2.html#K2">SupMap</a> <font color="Maroon">L</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">Ix</font> <a href="hidden.html#R1">=</a>  <a href="waybel_0.html#K5">downarrow</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">Ix</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow_2.html#K3">IdsMap</a> <font color="Maroon">L</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="yellow_2.html#K2">SupMap</a> <font color="Maroon">L</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">Iy</font> <a href="hidden.html#R1">=</a>  <a href="waybel_0.html#K5">downarrow</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">Iy</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="waybel_1.html#E8:99"><i><font color="Green">E61</font></i></a></i>;<br/>
<a NAME="E8:99_3"/>
( <font color="Maroon">x</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K7">Ids</a> <font color="Maroon">L</font></span>)</span></span>)</span> &amp; <font color="Maroon">y</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K7">Ids</a> <font color="Maroon">L</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="yellow_0.html#T59">YELLOW_0:59</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font>, <font color="Maroon">y</font> = <font color="Maroon">y</font> as   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> <b>by </b><i><a class="ref" href="yellow_2.html#T43">YELLOW_2:43</a></i>;<br/>
<a NAME="E10:99_3"/><i><font color="Green">E59</font></i>: 
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">x</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">y</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E11:99_3"/>
(  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <span class="p2">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">Ix</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">Ix</font> &amp;  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <span class="p2">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">Iy</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">Iy</font> )
 
<b>by </b><i><a class="ref" href="waybel_0.html#T34">WAYBEL_0:34</a></i>;<br/>
<b>hence </b><a NAME="E12:99_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 <b>by </b><i>, , , , </i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:99"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">f</font> is <a href="orders_3.html#V5">monotone</a>
 
<b>by </b><i><a class="ref" href="yellow_2.html#T14">YELLOW_2:14</a></i>;<br/>
<a NAME="E8:99"/><i><font color="Green">E61</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="99_4"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:99_4"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><br/>

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

<b>assume </b><a NAME="E2:99_4"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> ;<br/>
<i><font color="Green">E62</font></i>: <span class="p1">(<span class="default"><a href="yellow_2.html#K2">SupMap</a> <font color="Maroon">L</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">x</font></span>)</span> = 
 <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="ref" href="yellow_2.html#D3">YELLOW_2:def 3</a></i>
<br/>.= 
<font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="waybel_0.html#T34">WAYBEL_0:34</a></i>
;<br/>
<a NAME="E5:99_4"/><i><font color="Green">E63</font></i>: 
 <a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">x</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K7">Ids</a> <font color="Maroon">L</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_2.html#T43">YELLOW_2:43</a></i>;<br/>
<b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow_2.html#K3">IdsMap</a> <font color="Maroon">L</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="yellow_2.html#K2">SupMap</a> <font color="Maroon">L</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">x</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="yellow_2.html#K3">IdsMap</a> <font color="Maroon">L</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow_2.html#K2">SupMap</a> <font color="Maroon">L</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
<br/>.= 
 <a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">x</font>
<b>by </b><i>, <a class="ref" href="yellow_2.html#D4">YELLOW_2:def 4</a></i>
;<br/>
<a NAME="E7:99_4"/><b>then </b>
 <a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow_2.html#K3">IdsMap</a> <font color="Maroon">L</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="yellow_2.html#K2">SupMap</a> <font color="Maroon">L</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="funct_2.html#T6">FUNCT_2:6</a></i>;<br/>
<a NAME="E8:99_4"/><b>then </b><i><font color="Green">E64</font></i>: 
 <a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_2.html#K1">Image</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="yellow_2.html#K3">IdsMap</a> <font color="Maroon">L</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p3">(<span class="default"><a href="yellow_2.html#K2">SupMap</a> <font color="Maroon">L</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/>
<b>then </b><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">x</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="yellow_2.html#K2">SupMap</a> <font color="Maroon">L</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_1.html#K3" target="_self">inclusion</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="yellow_2.html#K3">IdsMap</a> <font color="Maroon">L</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p4">(<span class="default"><a href="yellow_2.html#K2">SupMap</a> <font color="Maroon">L</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="yellow_2.html#K2">SupMap</a> <font color="Maroon">L</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i>, <a class="ref" href="waybel_1.html#D7" target="_self">Def7</a></i>
;<br/>
<b>hence </b><a NAME="E10:99_4"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font>
 <b>by </b><i>, , <a class="ref" href="funct_2.html#T6">FUNCT_2:6</a></i>;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="99_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_2.html#K1">Image</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="yellow_2.html#K3">IdsMap</a> <font color="Maroon">L</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p3">(<span class="default"><a href="yellow_2.html#K2">SupMap</a> <font color="Maroon">L</font></span>)</span></span>)</span></span>)</span>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_2.html#K1">Image</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="yellow_2.html#K3">IdsMap</a> <font color="Maroon">L</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p3">(<span class="default"><a href="yellow_2.html#K2">SupMap</a> <font color="Maroon">L</font></span>)</span></span>)</span></span>)</span>;<br/><b>thus </b><a NAME="E1:99_5"/>
( <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">y</font> implies <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> )
 <b>by </b><i>, </i>;<br/><b>assume </b><a NAME="E2:99_5"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font>
 ;<br/><b>consider </b><font color="Maroon">Ix</font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K7">Ids</a> <font color="Maroon">L</font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E4:99_5"/><i><font color="Green">E66</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow_2.html#K3">IdsMap</a> <font color="Maroon">L</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="yellow_2.html#K2">SupMap</a> <font color="Maroon">L</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">Ix</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="yellow_2.html#T12">YELLOW_2:12</a></i>;<br/><b>consider </b><font color="Maroon">Iy</font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K7">Ids</a> <font color="Maroon">L</font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E6:99_5"/><i><font color="Green">E67</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow_2.html#K3">IdsMap</a> <font color="Maroon">L</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="yellow_2.html#K2">SupMap</a> <font color="Maroon">L</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">Iy</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="yellow_2.html#T12">YELLOW_2:12</a></i>;<br/><b>reconsider </b><font color="Maroon">Ix</font> = <font color="Maroon">Ix</font>, <font color="Maroon">Iy</font> = <font color="Maroon">Iy</font> as   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> <b>by </b><i><a class="ref" href="yellow_2.html#T43">YELLOW_2:43</a></i>;<br/><a NAME="E8:99_5"/><i><font color="Green">E68</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow_2.html#K3">IdsMap</a> <font color="Maroon">L</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="yellow_2.html#K2">SupMap</a> <font color="Maroon">L</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">Ix</font> <a href="hidden.html#R1">=</a>  <a href="waybel_0.html#K5">downarrow</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">Ix</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow_2.html#K3">IdsMap</a> <font color="Maroon">L</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="yellow_2.html#K2">SupMap</a> <font color="Maroon">L</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">Iy</font> <a href="hidden.html#R1">=</a>  <a href="waybel_0.html#K5">downarrow</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">Iy</font></span>)</span> )
 <b>by </b><i><a class="txt" href="waybel_1.html#E8:99"><i><font color="Green">E61</font></i></a></i>;<br/><a NAME="E9:99_5"/>
( <font color="Maroon">x</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K7">Ids</a> <font color="Maroon">L</font></span>)</span></span>)</span> &amp; <font color="Maroon">y</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K7">Ids</a> <font color="Maroon">L</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="yellow_0.html#T59">YELLOW_0:59</a></i>;<br/><b>then reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font>, <font color="Maroon">y'</font> = <font color="Maroon">y</font> as   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> <b>by </b><i><a class="ref" href="yellow_2.html#T43">YELLOW_2:43</a></i>;<br/><b>reconsider </b><font color="Maroon">i1</font> =  <a href="waybel_0.html#K5">downarrow</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">Ix</font></span>)</span>, <font color="Maroon">i2</font> =  <a href="waybel_0.html#K5">downarrow</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">Iy</font></span>)</span> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K7">Ids</a> <font color="Maroon">L</font></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="yellow_2.html#T43">YELLOW_2:43</a></i>;<br/><a NAME="E12:99_5"/><i><font color="Green">E69</font></i>: 
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x'</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">x'</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y'</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">y'</font> )
 <b>by </b><i/>;<br/><a NAME="E13:99_5"/>
(  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <span class="p2">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">Ix</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">Ix</font> &amp;  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <span class="p2">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">Iy</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">Iy</font> )
 <b>by </b><i><a class="ref" href="waybel_0.html#T34">WAYBEL_0:34</a></i>;<br/><a NAME="E14:99_5"/><b>then </b>
 <a href="waybel_0.html#K5">downarrow</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">Ix</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="waybel_0.html#K5">downarrow</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">Iy</font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel_1.html#E4:99_5"><i><font color="Green">E66</font></i></a>, <a class="txt" href="waybel_1.html#E6:99_5"><i><font color="Green">E67</font></i></a>, <a class="txt" href="waybel_1.html#E8:99_5"><i><font color="Green">E68</font></i></a>, <a class="txt" href="waybel_1.html#E12:99_5"><i><font color="Green">E69</font></i></a>, , <a class="ref" href="waybel_0.html#T21">WAYBEL_0:21</a></i>;<br/><a NAME="E15:99_5"/><b>then </b>
<font color="Maroon">i1</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">i2</font>
 <b>by </b><i><a class="ref" href="yellow_1.html#T3">YELLOW_1:3</a></i>;<br/><b>hence </b><a NAME="E16:99_5"/>
<font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="txt" href="waybel_1.html#E6:99_5"><i><font color="Green">E67</font></i></a>, <a class="txt" href="waybel_1.html#E8:99_5"><i><font color="Green">E68</font></i></a>, <a class="txt" href="waybel_1.html#E12:99_5"><i><font color="Green">E69</font></i></a>, <a class="ref" href="yellow_0.html#T61">YELLOW_0:61</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E10:99"/>
<font color="Maroon">f</font> is <a href="waybel_0.html#V23">isomorphic</a>
 <b>by </b><i>, , <a class="ref" href="waybel_0.html#T66">WAYBEL_0:66</a></i>;<br/>


</div>
