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

<b>set </b><font color="Maroon">A</font> = <font color="Maroon">F<sub>1</sub></font>();<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="waybel_0.html#NM8">LATTICE</a> st <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b</font></span><a href="tarski.html#K4">]</a></span> &amp; ( for <font color="Olive">f</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a<sub>2</sub></font> iff ( <font color="Olive">f</font> <a href="hidden.html#R2">in</a>  <a href="orders_3.html#K1">MonFuncs</a> <font color="Olive">a</font>,<font color="Olive">b</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">a</font>,<font color="Olive">b</font>,<font color="Olive">f</font>] ) ) ) );<br/>
<div><i><font color="Green">E41</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:9_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>1</sub></font>()</span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/><b>then consider </b><font color="Maroon">a</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">b</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:9_1_1"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/><b>reconsider </b><font color="Maroon">a</font> = <font color="Maroon">a</font>, <font color="Maroon">b</font> = <font color="Maroon">b</font> as   <a href="waybel_0.html#NM8">LATTICE</a> <b>by </b><i>, </i>;<br/><b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b><font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">a<sub>1</sub></font>];<br/><b>consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:9_1_1"/><i><font color="Green">E46</font></i>: 
for <font color="Olive">f</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y</font> iff ( <font color="Olive">f</font> <a href="hidden.html#R2">in</a>  <a href="orders_3.html#K1">MonFuncs</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">f</font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1">XBOOLE_0:sch 1</a>();<br/></i><b>take </b><font color="Maroon">y</font> = <font color="Maroon">y</font>;<br/><b>thus </b><a NAME="E7:9_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">y</font>]
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<b>consider </b><font color="Maroon">F</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E3:9_1"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>1</sub></font>()</span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>and </b><br/><a NAME="E4:9_1"/><i><font color="Green">E48</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>1</sub></font>()</span><a href="zfmisc_1.html#K2">:]</a></span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>]
 <b>from </b><i><a class="ref" href="zfrefle1.html#S1">ZFREFLE1:sch 1</a>();<br/></i>
<div><i><font color="Green">E49</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be   <a href="waybel_0.html#NM8">LATTICE</a>, <font color="Maroon">b</font> be   <a href="waybel_0.html#NM8">LATTICE</a>;<br/><b>assume </b><a NAME="E1:9_1_2"/>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>() )
 ;<br/><a NAME="E2:9_1_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>1</sub></font>()</span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><b>then consider </b><font color="Maroon">a'</font> being   <a href="waybel_0.html#NM8">LATTICE</a>, <font color="Maroon">b'</font> being   <a href="waybel_0.html#NM8">LATTICE</a><b> such that </b><br/><a NAME="E4:9_1_2"/><i><font color="Green">E50</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a'</font>,<font color="Maroon">b'</font></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E5:9_1_2"/><i><font color="Green">E51</font></i>: 
for <font color="Olive">f</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> iff ( <font color="Olive">f</font> <a href="hidden.html#R2">in</a>  <a href="orders_3.html#K1">MonFuncs</a> <font color="Maroon">a'</font>,<font color="Maroon">b'</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">a'</font>,<font color="Maroon">b'</font>,<font color="Olive">f</font>] ) )
 <b>by </b><i/>;<br/><a NAME="E6:9_1_2"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a'</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b'</font> )
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><b>let </b><font color="Maroon">f</font> be    <a href="hidden.html#M1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:9_1_2_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span>
 ;<br/><b>hence </b><a NAME="E2:9_1_2_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">f</font>]
 <b>by </b><i>, </i>;<br/><b>thus </b><a NAME="E3:9_1_2_1"/>
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a>  <a href="orders_3.html#K1">MonFuncs</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>
 <b>by </b><i>, , </i>;<br/><a NAME="E4:9_1_2_1"/><b>then </b>
 ex <font color="Olive">g</font> being  <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">a</font>,<font color="Maroon">b</font> st <br/>( <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Olive">g</font> &amp; <font color="Olive">g</font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">a</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">b</font> &amp; <font color="Olive">g</font> is <a href="orders_3.html#V5">monotone</a> )
 <b>by </b><i><a class="ref" href="orders_3.html#D6">ORDERS_3:def 6</a></i>;<br/><b>hence </b><a NAME="E5:9_1_2_1"/>
( <font color="Maroon">f</font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">a</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">b</font> &amp; <font color="Maroon">f</font> is  <a href="orders_3.html#V5">monotone</a> <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">a</font>,<font color="Maroon">b</font> )
 ;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E8:9_1_2"/>
<font color="Maroon">f</font> is  <a href="orders_3.html#V5">monotone</a> <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">a</font>,<font color="Maroon">b</font>
 ;<br/><b>then reconsider </b><font color="Maroon">g</font> = <font color="Maroon">f</font> as  <a href="orders_3.html#V5">monotone</a> <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">a</font>,<font color="Maroon">b</font> ;<br/><a NAME="E10:9_1_2"/>
( the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  implies (  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">a</font> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">g</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">b</font> ) )
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E11:9_1_2"/><b>then </b>
<font color="Maroon">g</font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">a</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/><a NAME="E12:9_1_2"/><b>then </b>
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a>  <a href="orders_3.html#K1">MonFuncs</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="orders_3.html#D6">ORDERS_3:def 6</a></i>;<br/><b>hence </b><a NAME="E13:9_1_2"/>
( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">f</font>] implies <font color="Maroon">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b>for <font color="Olive">a</font> being  <a href="waybel_0.html#NM8">LATTICE</a>  st <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> holds <br/><font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">a</font>;<br/>
<div><i><font color="Green">E54</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:9_1_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>()
 ;<br/><b>then reconsider </b><font color="Maroon">a</font> = <font color="Maroon">x</font> as   <a href="waybel_0.html#NM8">LATTICE</a> <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">b</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">a</font> as    <a href="hidden.html#M1">set</a>  ;<br/><b>take </b><font color="Maroon">b</font> = <font color="Maroon">b</font>;<br/><b>thus </b><a NAME="E4:9_1_3"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">b</font>]
 ;<br/></div><b>end;</b></div>
<b>consider </b><font color="Maroon">D</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E8:9_1"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">D</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>()
 <b>and </b><br/><a NAME="E9:9_1"/><i><font color="Green">E55</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">x</font>,<font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>]
 <b>from </b><i><a class="ref" href="zfrefle1.html#S1">ZFREFLE1:sch 1</a>();<br/></i>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">a<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span>;<br/>
<a NAME="E10:9_1"/><i><font color="Green">E70</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()<br/> for <font color="Olive">f</font>, <font color="Olive">g</font> being  <a href="funct_1.html#NM1">Function</a>  st <font color="Olive">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">a</font>,<font color="Olive">b</font>) &amp; <font color="Olive">g</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b</font>,<font color="Olive">c</font>) holds <br/><font color="Olive">g</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Olive">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">a</font>,<font color="Olive">c</font>)
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>(), <font color="Maroon">b</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>(), <font color="Maroon">c</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>();<br/><b>let </b><font color="Maroon">f</font> be   <a href="funct_1.html#NM1">Function</a>, <font color="Maroon">g</font> be   <a href="funct_1.html#NM1">Function</a>;<br/>



<b>assume </b><a NAME="E1:9_1_4"/><i><font color="Green">E71</font></i>: 
( <font color="Maroon">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">g</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">c</font></span><a href="tarski.html#K4">]</a></span> )
 ;<br/>

<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">a</font>, <font color="Maroon">y</font> = <font color="Maroon">b</font>, <font color="Maroon">z</font> = <font color="Maroon">c</font> as   <a href="waybel_0.html#NM8">LATTICE</a> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">f'</font> = <font color="Maroon">f</font> as  <a href="orders_3.html#V5">monotone</a> <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">x</font>,<font color="Maroon">y</font> <b>by </b><i>, </i>;<br/>
<b>reconsider </b><font color="Maroon">g'</font> = <font color="Maroon">g</font> as  <a href="orders_3.html#V5">monotone</a> <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">y</font>,<font color="Maroon">z</font> <b>by </b><i>, </i>;<br/>
<a NAME="E5:9_1_4"/>
( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">f'</font>] &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">y</font>,<font color="Maroon">z</font>,<font color="Maroon">g'</font>] )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E6:9_1_4"/><b>then </b>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">a</font>,<font color="Maroon">c</font>,<font color="Maroon">g'</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f'</font>]
 
<b>by </b><i/>;<br/>
<a NAME="E7:9_1_4"/><b>then </b>
( <font color="Maroon">g'</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f'</font> is  <a href="orders_3.html#V5">monotone</a> <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">x</font>,<font color="Maroon">z</font> implies <font color="Maroon">g'</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">z</font></span><a href="tarski.html#K4">]</a></span> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E8:9_1_4"/>
<font color="Maroon">g</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">a</font>,<font color="Maroon">c</font>)
 <b>by </b><i><a class="ref" href="yellow_2.html#T14">YELLOW_2:14</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:9_1"/><i><font color="Green">E74</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() holds  <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">a</font>,<font color="Olive">b</font>) <a href="tarski.html#R1">c=</a>  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">a</font>),<font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b</font>)
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>(), <font color="Maroon">b</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>();<br/><b>let </b><font color="Maroon">f</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>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">a</font>, <font color="Maroon">y</font> = <font color="Maroon">b</font> as   <a href="waybel_0.html#NM8">LATTICE</a> <b>by </b><i/>;<br/>
<b>assume </b><a NAME="E2:9_1_5"/>
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span>
 ;<br/>

<a NAME="E3:9_1_5"/><b>then </b>
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">x</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">y</font>
 
<b>by </b><i/>;<br/>
<a NAME="E4:9_1_5"/><b>then </b>
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> <span class="p1">(<span class="default"><font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">y</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E5:9_1_5"/>
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">a</font>),<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">b</font>)
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<div><i><font color="Green">E75</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_6"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>();<br/><b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">a</font> as   <a href="waybel_0.html#NM8">LATTICE</a> <b>by </b><i/>;<br/><a NAME="E2:9_1_6"/>
<font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">x</font>
 <b>by </b><i/>;<br/><a NAME="E3:9_1_6"/><b>then </b>
(  <a href="partfun1.html#K6">id</a> <span class="p1">(<span class="default"><font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="grcat_1.html#K7">id</a> <font color="Maroon">x</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">x</font>, <a href="grcat_1.html#K7">id</a> <font color="Maroon">x</font>] )
 <b>by </b><i>, <a class="ref" href="grcat_1.html#D11">GRCAT_1:def 11</a></i>;<br/><b>hence </b><a NAME="E4:9_1_6"/>
 <a href="partfun1.html#K6">id</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">a</font>) <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">a</font>,<font color="Maroon">a</font>)
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>consider </b><font color="Maroon">C</font> being  <a href="altcat_1.html#V6">strict</a> <a href="yellow18.html#V4">concrete</a> <a href="altcat_1.html#NM4">category</a><b> such that </b><br/><a NAME="E14:9_1"/><i><font color="Green">E76</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>()
 <b>and </b><br/><a NAME="E15:9_1"/>
for <font color="Olive">a</font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">C</font> holds   <a href="yellow18.html#K3">the_carrier_of</a> <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">a</font>)
 <b>and </b><br/><a NAME="E16:9_1"/><i><font color="Green">E77</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">C</font> holds  <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b</font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">a</font>,<font color="Olive">b</font>)
 <b>from </b><i><a class="ref" href="yellow18.html#S16">YELLOW18:sch 16</a>(, , );<br/></i>
<b>take </b>
<font color="Maroon">C</font>
;<br/>

<b>thus </b><a NAME="E17:9_1"/>
( <font color="Maroon">C</font> is <a href="altcat_1.html#V9">semi-functional</a> &amp; <font color="Maroon">C</font> is <a href="yellow18.html#V3">set-id-inheriting</a> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow21.html#D4" target="_self">YELLOW21:def 4</a><br/>

<b>thus </b><a NAME="E18:9_1"/>
for <font color="Olive">a</font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">C</font> holds  <font color="Olive">a</font> is   <a href="waybel_0.html#NM8">LATTICE</a>
 <b>by </b><i>, </i>;<br/>

<b>thus </b><a NAME="E19:9_1"/>
for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">C</font><br/> for <font color="Olive">A</font>, <font color="Olive">B</font> being  <a href="waybel_0.html#NM8">LATTICE</a>  st <font color="Olive">A</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font> &amp; <font color="Olive">B</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> holds <br/><span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b</font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="tarski.html#R1">c=</a>  <a href="orders_3.html#K1">MonFuncs</a> <font color="Olive">A</font>,<font color="Olive">B</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">C</font>, <font color="Maroon">b</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">C</font>;<br/><b>let </b><font color="Maroon">A</font> be   <a href="waybel_0.html#NM8">LATTICE</a>, <font color="Maroon">B</font> be   <a href="waybel_0.html#NM8">LATTICE</a>;<br/>



<b>assume </b><a NAME="E1:9_1_7"/><i><font color="Green">E78</font></i>: 
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> )
 ;<br/>

<b>let </b><font color="Maroon">f</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/>

<a NAME="E2:9_1_7"/>
<span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font></span><a href="tarski.html#K4">]</a></span>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E3:9_1_7"/>
(  not <font color="Maroon">f</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="altcat_1.html#K1">^&gt;</a></span> or <font color="Maroon">f</font> <a href="hidden.html#R2">in</a>  <a href="orders_3.html#K1">MonFuncs</a> <font color="Maroon">A</font>,<font color="Maroon">B</font> )
 <b>by </b><i>, , </i>;<br/>


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

<b>thus </b><a NAME="E20:9_1"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>()
 <b>by </b><i/>;<br/>

<b>let </b><font color="Maroon">a</font> be   <a href="waybel_0.html#NM8">LATTICE</a>, <font color="Maroon">b</font> be   <a href="waybel_0.html#NM8">LATTICE</a>;<br/><b>let </b><font color="Maroon">f</font> be  <a href="orders_3.html#V5">monotone</a> <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">a</font>,<font color="Maroon">b</font>;<br/>



<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:9_1_8"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">C</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>
 ;<br/><a NAME="E2:9_1_8"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">C</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D4">FUNCT_1:def 4</a></i>;<br/><a NAME="E3:9_1_8"/><b>then </b><i><font color="Green">E80</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>1</sub></font>()</span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><b>hence </b><a NAME="E4:9_1_8"/>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>() )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">a</font>, <font color="Maroon">y</font> = <font color="Maroon">b</font> as   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">C</font> <b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> = 
<span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="altcat_1.html#K1">^&gt;</a></span>

<br/>.= 
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i/>
;<br/><b>hence </b><a NAME="E7:9_1_8"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">f</font>]
 <b>by </b><i>, , </i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E22:9_1"/><i><font color="Green">E81</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>() )
 ;<br/>

<b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">a</font>, <font color="Maroon">y</font> = <font color="Maroon">b</font> as   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">C</font> <b>by </b><i/>;<br/>
the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">C</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> = 
<span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="altcat_1.html#K1">^&gt;</a></span>

<br/>.= 
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i/>
;<br/>
<b>hence </b><a NAME="E25:9_1"/>
(  not <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">f</font>] or <font color="Maroon">f</font> <a href="hidden.html#R2">in</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">C</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> )
 <b>by </b><i>, </i>;<br/>


</div>
