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

<b>let </b><font color="Maroon">L</font> be  <a href="lattice3.html#V4">complete</a> <a href="lattices.html#NM1">Lattice</a>;<br/><b>let </b><font color="Maroon">f</font> be  <a href="quantal1.html#V14">monotone</a> <a href="quantal1.html#NM1">UnOp</a> of <font color="Maroon">L</font>;<br/><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/>





<b>assume </b><a NAME="E1:43"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">a</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font>
 ;<br/>

<b>set </b><font color="Maroon">cL</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>;<br/>
<b>set </b><font color="Maroon">ccL</font> =  <a href="card_1.html#K1">Card</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>;<br/>
<b>set </b><font color="Maroon">nccL</font> =  <a href="card_1.html#K2">nextcard</a> <span class="p1">(<span class="default"><a href="card_1.html#K1">Card</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font></span>)</span>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="ordinal1.html#NM2">Ordinal</a>) <b>-&gt; </b>  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> = <font color="Maroon">f</font>,<font color="Maroon">a<sub>1</sub></font> <a href="knaster.html#K5" target="_self">+.</a> <font color="Maroon">a</font>;<br/>
<b>consider </b><font color="Maroon">Ls</font> being   <a href="ordinal1.html#NM3">T-Sequence</a><b> such that </b><br/><a NAME="E3:43"/><i><font color="Green">E21</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">Ls</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K2">nextcard</a> <span class="p1">(<span class="default"><a href="card_1.html#K1">Card</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font></span>)</span> &amp; ( for <font color="Olive">O2</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Olive">O2</font> <a href="hidden.html#R2">in</a>  <a href="card_1.html#K2">nextcard</a> <span class="p1">(<span class="default"><a href="card_1.html#K1">Card</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font></span>)</span> holds <br/><font color="Maroon">Ls</font> <a href="funct_1.html#K1">.</a> <font color="Olive">O2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">O2</font>) ) )
 <b>from </b><i><a class="ref" href="ordinal2.html#S2">ORDINAL2:sch 2</a>();<br/></i>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:43_1"/><i><font color="Green">E22</font></i>: 
for <font color="Olive">O</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Olive">O</font> <a href="hidden.html#R2">in</a>  <a href="card_1.html#K2">nextcard</a> <span class="p1">(<span class="default"><a href="card_1.html#K1">Card</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font></span>)</span> holds <br/> not <font color="Maroon">f</font>,<font color="Olive">O</font> <a href="knaster.html#K5" target="_self">+.</a> <font color="Maroon">a</font> <a href="abian.html#R2">is_a_fixpoint_of</a> <font color="Maroon">f</font>
 ;<br/><a NAME="E2:43_1"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">Ls</font> is <a href="funct_1.html#V2">one-to-one</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x1</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">x2</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a><br/>

<b>assume </b><a NAME="E1:43_1_1"/><i><font color="Green">E24</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">Ls</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">Ls</font> &amp; <font color="Maroon">Ls</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Ls</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font> )
 ;<br/>

<b>then reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">x1</font>, <font color="Maroon">x2</font> = <font color="Maroon">x2</font> as   <a href="ordinal1.html#NM2">Ordinal</a> <b>by </b><i><a class="ref" href="ordinal1.html#T23">ORDINAL1:23</a></i>;<br/>
<a NAME="E3:43_1_1"/><i><font color="Green">E25</font></i>: 
( <font color="Maroon">Ls</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font>,<font color="Maroon">x1</font> <a href="knaster.html#K5" target="_self">+.</a> <font color="Maroon">a</font> &amp; <font color="Maroon">Ls</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font>,<font color="Maroon">x2</font> <a href="knaster.html#K5" target="_self">+.</a> <font color="Maroon">a</font> )
 
<b>by </b><i>, </i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:43_1_1_1"/>
( <font color="Maroon">x1</font> <a href="xboole_0.html#R2">c&lt;</a> <font color="Maroon">x2</font> or <font color="Maroon">x2</font> <a href="xboole_0.html#R2">c&lt;</a> <font color="Maroon">x1</font> or <font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_1_1_1_1"><b>suppose </b></a><a NAME="E1:43_1_1_1_1"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">x1</font> <a href="xboole_0.html#R2">c&lt;</a> <font color="Maroon">x2</font>
 ;<br/><div class="add"><a NAME="E2:43_1_1_1_1"/>
not <font color="Maroon">f</font>,<font color="Maroon">x2</font> <a href="knaster.html#K5" target="_self">+.</a> <font color="Maroon">a</font> <a href="abian.html#R2">is_a_fixpoint_of</a> <font color="Maroon">f</font>
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E3:43_1_1_1_1"/>
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font>
 <b>by </b><i>, , , , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_1_1_1_2"><b>suppose </b></a><a NAME="E1:43_1_1_1_2"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">x2</font> <a href="xboole_0.html#R2">c&lt;</a> <font color="Maroon">x1</font>
 ;<br/><div class="add"><a NAME="E2:43_1_1_1_2"/>
not <font color="Maroon">f</font>,<font color="Maroon">x1</font> <a href="knaster.html#K5" target="_self">+.</a> <font color="Maroon">a</font> <a href="abian.html#R2">is_a_fixpoint_of</a> <font color="Maroon">f</font>
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E3:43_1_1_1_2"/>
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font>
 <b>by </b><i>, , , , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_1_1_1_3"><b>suppose </b></a><a NAME="E1:43_1_1_1_3"/>
<font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:43_1_1_1_3"/>
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E3:43_1"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">Ls</font> <a href="tarski.html#R1">c=</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="43_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</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="E1:43_1_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">Ls</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:43_1_2"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">Ls</font> &amp; <font color="Maroon">Ls</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="ordinal1.html#NM2">Ordinal</a> <b>by </b><i>, <a class="ref" href="ordinal1.html#T23">ORDINAL1:23</a></i>;<br/>
<a NAME="E5:43_1_2"/>
<font color="Maroon">Ls</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font>,<font color="Maroon">x</font> <a href="knaster.html#K5" target="_self">+.</a> <font color="Maroon">a</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E6:43_1_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><a NAME="E4:43_1"/><b>then </b>
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="card_1.html#K2">nextcard</a> <span class="p2">(<span class="default"><a href="card_1.html#K1">Card</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font></span>)</span></span>)</span> <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_1.html#K1">Card</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 <b>by </b><i>, , <a class="ref" href="card_1.html#T26">CARD_1:26</a></i>;<br/><a NAME="E5:43_1"/><b>then </b>
(  <a href="card_1.html#K2">nextcard</a> <span class="p1">(<span class="default"><a href="card_1.html#K1">Card</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font></span>)</span> <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_1.html#K1">Card</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> &amp;  <a href="card_1.html#K1">Card</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> <a href="card_1.html#NR2" target="_self">&lt;`</a>  <a href="card_1.html#K2">nextcard</a> <span class="p1">(<span class="default"><a href="card_1.html#K1">Card</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font></span>)</span> )
 <b>by </b><i><a class="ref" href="card_1.html#T32">CARD_1:32</a>, <a class="ref" href="card_1.html#D5">CARD_1:def 5</a></i>;<br/><b>hence </b><a NAME="E6:43_1"/>
contradiction
 <b>by </b><i><a class="ref" href="card_1.html#T14">CARD_1:14</a></i>;<br/></div><b>end;</b></div>
<b>then consider </b><font color="Maroon">O</font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E6:43"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">O</font> <a href="hidden.html#R2">in</a>  <a href="card_1.html#K2">nextcard</a> <span class="p1">(<span class="default"><a href="card_1.html#K1">Card</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font></span>)</span> &amp; <font color="Maroon">f</font>,<font color="Maroon">O</font> <a href="knaster.html#K5" target="_self">+.</a> <font color="Maroon">a</font> <a href="abian.html#R2">is_a_fixpoint_of</a> <font color="Maroon">f</font> )
 ;<br/>
<b>take </b>
<font color="Maroon">O</font>
;<br/>

<a NAME="E7:43"/>
 <a href="card_1.html#K1">Card</a> <font color="Maroon">O</font> <a href="card_1.html#NR2" target="_self">&lt;`</a>  <a href="card_1.html#K2">nextcard</a> <span class="p1">(<span class="default"><a href="card_1.html#K1">Card</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="card_1.html#T25">CARD_1:25</a></i>;<br/>
<b>hence </b><a NAME="E8:43"/>
 <a href="card_1.html#K1">Card</a> <font color="Maroon">O</font> <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_1.html#K1">Card</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 <b>by </b><i><a class="ref" href="card_4.html#T23">CARD_4:23</a></i>;<br/>

<b>thus </b><a NAME="E9:43"/>
<font color="Maroon">f</font>,<font color="Maroon">O</font> <a href="knaster.html#K5" target="_self">+.</a> <font color="Maroon">a</font> <a href="abian.html#R2">is_a_fixpoint_of</a> <font color="Maroon">f</font>
 <b>by </b><i/>;<br/>


</div>
