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<div class="add">

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">s</font> being  <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">L</font>,<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> st <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">s</font> &amp; <font color="Olive">s</font> is <a href="orders_3.html#V5">monotone</a> &amp; ( for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> holds  <font color="Olive">s</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="tarski.html#R1">c=</a>  <a href="waybel_0.html#K5">downarrow</a> <font color="Olive">x</font> ) );<br/>
<b>consider </b><font color="Maroon">S</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E2:44_1_1"/><i><font color="Green">E31</font></i>: 
for <font color="Olive">a</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> iff ( <font color="Olive">a</font> <a href="hidden.html#R2">in</a>  <a href="partfun1.html#K4">PFuncs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>,the <a href="struct_0.html#U1">carrier</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">S<sub>1</sub></font>[<font color="Olive">a</font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1">XBOOLE_0:sch 1</a>();<br/></i>
<a NAME="E3:44_1_1"/><i><font color="Green">E32</font></i>: 
for <font color="Olive">a</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> iff  ex <font color="Olive">s</font> being  <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">L</font>,<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> st <br/>( <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">s</font> &amp; <font color="Olive">s</font> is <a href="orders_3.html#V5">monotone</a> &amp; ( for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> holds  <font color="Olive">s</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="tarski.html#R1">c=</a>  <a href="waybel_0.html#K5">downarrow</a> <font color="Olive">x</font> ) ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>thus </b><a NAME="E1:44_1_1_1"/>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> implies  ex <font color="Olive">s</font> being  <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">L</font>,<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> st <br/>( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">s</font> &amp; <font color="Olive">s</font> is <a href="orders_3.html#V5">monotone</a> &amp; ( for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> holds  <font color="Olive">s</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="tarski.html#R1">c=</a>  <a href="waybel_0.html#K5">downarrow</a> <font color="Olive">x</font> ) ) )
 <b>by </b><i/>;<br/>

<b>given </b><font color="Maroon">s</font> being   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">L</font>,<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><a NAME="E2:44_1_1_1"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> &amp; <font color="Maroon">s</font> is <a href="orders_3.html#V5">monotone</a> &amp; ( for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> holds  <font color="Maroon">s</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="tarski.html#R1">c=</a>  <a href="waybel_0.html#K5">downarrow</a> <font color="Olive">x</font> ) )
 ;<br/>

<a NAME="E3:44_1_1_1"/>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="partfun1.html#K4">PFuncs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>,the <a href="struct_0.html#U1">carrier</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="partfun1.html#T119">PARTFUN1:119</a></i>;<br/>
<b>hence </b><a NAME="E4:44_1_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 <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> ex <font color="Olive">f</font>, <font color="Olive">g</font> being  <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">L</font>,<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> st <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">f</font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">g</font> &amp; <font color="Olive">f</font> <a href="yellow_2.html#R1">&lt;=</a> <font color="Olive">g</font> );<br/>
<b>consider </b><font color="Maroon">R</font> being    <a href="relset_1.html#M2">Relation</a> of <font color="Maroon">S</font>,<font color="Maroon">S</font><b> such that </b><br/><a NAME="E5:44_1_1"/><i><font color="Green">E34</font></i>: 
for <font color="Olive">c</font>, <font color="Olive">d</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">c</font>,<font color="Olive">d</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> iff ( <font color="Olive">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> &amp; <font color="Olive">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">c</font>,<font color="Olive">d</font>] ) )
 <b>from </b><i><a class="ref" href="relset_1.html#S1">RELSET_1:sch 1</a>();<br/></i>
<a NAME="E6:44_1_1"/><i><font color="Green">E37</font></i>: 
for <font color="Olive">c</font>, <font color="Olive">d</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">c</font>,<font color="Olive">d</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> iff  ex <font color="Olive">f</font>, <font color="Olive">g</font> being  <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">L</font>,<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> st <br/>( <font color="Olive">c</font> <a href="hidden.html#R1">=</a> <font color="Olive">f</font> &amp; <font color="Olive">d</font> <a href="hidden.html#R1">=</a> <font color="Olive">g</font> &amp; <font color="Olive">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> &amp; <font color="Olive">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> &amp; <font color="Olive">f</font> <a href="yellow_2.html#R1">&lt;=</a> <font color="Olive">g</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">d</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:44_1_1_2_1"/><i><font color="Green">E40</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">d</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 ;<br/><a NAME="E2:44_1_1_2_1"/><b>then </b><i><font color="Green">E50</font></i>: 
( <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> &amp; <font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> )
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">f</font> being   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">L</font>,<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>, <font color="Maroon">g</font> being   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">L</font>,<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="E4:44_1_1_2_1"/><i><font color="Green">E74</font></i>: 
( <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> &amp; <font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> &amp; <font color="Maroon">f</font> <a href="yellow_2.html#R1">&lt;=</a> <font color="Maroon">g</font> )
 <b>by </b><i>, </i>;<br/><b>thus </b><a NAME="E5:44_1_1_2_1"/>
 ex <font color="Olive">f</font>, <font color="Olive">g</font> being  <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">L</font>,<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> st <br/>( <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Olive">f</font> &amp; <font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <font color="Olive">g</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> &amp; <font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> &amp; <font color="Olive">f</font> <a href="yellow_2.html#R1">&lt;=</a> <font color="Olive">g</font> )
 <b>by </b><i>, <a class="txt" href="waybel_4.html#E2:44_1_1"><i><font color="Green">E31</font></i></a></i>;<br/></div>
<b>end;</b></div>

<b>given </b><font color="Maroon">f</font> being   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">L</font>,<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>, <font color="Maroon">g</font> being   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">L</font>,<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><a NAME="E2:44_1_1_2"/><i><font color="Green">E75</font></i>: 
( <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> &amp; <font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> &amp; <font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> &amp; <font color="Maroon">f</font> <a href="yellow_2.html#R1">&lt;=</a> <font color="Maroon">g</font> )
 ;<br/>

<b>thus </b><a NAME="E3:44_1_1_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">d</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>take </b>
 <a href="orders_2.html#G1">RelStr</a>(# <font color="Maroon">S</font>,<font color="Maroon">R</font> #)
;<br/>

<b>thus </b><a NAME="E7:44_1_1"/>
for <font color="Olive">a</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( ( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <a href="orders_2.html#G1">RelStr</a>(# <font color="Maroon">S</font>,<font color="Maroon">R</font> #) implies  ex <font color="Olive">s</font> being  <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">L</font>,<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> st <br/>( <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">s</font> &amp; <font color="Olive">s</font> is <a href="orders_3.html#V5">monotone</a> &amp; ( for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> holds  <font color="Olive">s</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="tarski.html#R1">c=</a>  <a href="waybel_0.html#K5">downarrow</a> <font color="Olive">x</font> ) ) ) &amp; (  ex <font color="Olive">s</font> being  <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">L</font>,<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> st <br/>( <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">s</font> &amp; <font color="Olive">s</font> is <a href="orders_3.html#V5">monotone</a> &amp; ( for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> holds  <font color="Olive">s</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="tarski.html#R1">c=</a>  <a href="waybel_0.html#K5">downarrow</a> <font color="Olive">x</font> ) ) implies <font color="Olive">a</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <a href="orders_2.html#G1">RelStr</a>(# <font color="Maroon">S</font>,<font color="Maroon">R</font> #) ) &amp; ( for <font color="Olive">c</font>, <font color="Olive">d</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">c</font>,<font color="Olive">d</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <a href="orders_2.html#G1">RelStr</a>(# <font color="Maroon">S</font>,<font color="Maroon">R</font> #) iff  ex <font color="Olive">f</font>, <font color="Olive">g</font> being  <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">L</font>,<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> st <br/>( <font color="Olive">c</font> <a href="hidden.html#R1">=</a> <font color="Olive">f</font> &amp; <font color="Olive">d</font> <a href="hidden.html#R1">=</a> <font color="Olive">g</font> &amp; <font color="Olive">c</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <a href="orders_2.html#G1">RelStr</a>(# <font color="Maroon">S</font>,<font color="Maroon">R</font> #) &amp; <font color="Olive">d</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <a href="orders_2.html#G1">RelStr</a>(# <font color="Maroon">S</font>,<font color="Maroon">R</font> #) &amp; <font color="Olive">f</font> <a href="yellow_2.html#R1">&lt;=</a> <font color="Olive">g</font> ) ) ) )
 <b>by </b><i>, </i>;<br/>


</div>
