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<b>let </b><font color="Maroon">f</font> be   <a href="struct_0.html#NM6">Function</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">S</font></span>)</span></span>)</span>,<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> be   <a href="struct_0.html#NM6">Function</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">S</font></span>)</span></span>)</span>,<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>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:83_1_2"/><i><font color="Green">E50</font></i>: 
for <font color="Olive">I</font> being  <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">S</font>  ex <font color="Olive">J</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> st <br/>( <font color="Olive">I</font> <a href="hidden.html#R1">=</a> <font color="Olive">J</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">I</font> <a href="hidden.html#R1">=</a>  <a href="waybel_0.html#K3">downarrow</a> <font color="Olive">J</font> )
 <b>and </b><br/><a NAME="E2:83_1_2"/><i><font color="Green">E52</font></i>: 
for <font color="Olive">I</font> being  <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">S</font>  ex <font color="Olive">J</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> st <br/>( <font color="Olive">I</font> <a href="hidden.html#R1">=</a> <font color="Olive">J</font> &amp; <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">I</font> <a href="hidden.html#R1">=</a>  <a href="waybel_0.html#K3">downarrow</a> <font color="Olive">J</font> )
 ;<br/>

<b>set </b><font color="Maroon">P</font> =  <a href="yellow_1.html#K2">InclPoset</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K7">Ids</a> <font color="Maroon">S</font></span>)</span>;<br/>
<a NAME="E3:83_1_2"/><i><font color="Green">E54</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> 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">S</font></span>)</span></span>)</span> &amp;  <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 <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">S</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<i><font color="Green">E56</font></i>: 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">S</font></span>)</span></span>)</span> = 
the <a href="struct_0.html#U1">carrier</a> of <a href="orders_2.html#G1">RelStr</a>(# <span class="p1">(<span class="default"><a href="waybel_0.html#K7">Ids</a> <font color="Maroon">S</font></span>)</span>,<span class="p1">(<span class="default"><a href="yellow_1.html#K1">RelIncl</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K7">Ids</a> <font color="Maroon">S</font></span>)</span></span>)</span> #)
<b>by </b><i><a class="ref" href="yellow_1.html#D1">YELLOW_1:def 1</a></i>
<br/>.= 
 <a href="waybel_0.html#K7">Ids</a> <font color="Maroon">S</font>

;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_2_2"><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="E1:83_1_2_2"/>
<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_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K7">Ids</a> <font color="Maroon">S</font></span>)</span></span>)</span>
 ;<br/><a NAME="E2:83_1_2_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">X</font> where <font color="Olive">X</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">S</font> : verum </span>}</span> 
 <b>by </b><i><a class="ref" href="waybel23.html#T5" target="_self">Th5</a>, <a class="ref" href="waybel_0.html#D23">WAYBEL_0:def 23</a></i>;<br/><a NAME="E3:83_1_2_2"/><b>then </b>
 ex <font color="Olive">I</font> being  <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">S</font> st <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">I</font>
 ;<br/><b>then reconsider </b><font color="Maroon">I</font> = <font color="Maroon">x</font> as   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">S</font> ;<br/><b>consider </b><font color="Maroon">J1</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E6:83_1_2_2"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">I</font> <a href="hidden.html#R1">=</a> <font color="Maroon">J1</font>
 <b>and </b><br/><a NAME="E7:83_1_2_2"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">I</font> <a href="hidden.html#R1">=</a>  <a href="waybel_0.html#K3">downarrow</a> <font color="Maroon">J1</font>
 <b>by </b><i><a class="ref" href="waybel23.html#T3" target="_self">Th3</a></i>;<br/><b>consider </b><font color="Maroon">J2</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E9:83_1_2_2"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">I</font> <a href="hidden.html#R1">=</a> <font color="Maroon">J2</font>
 <b>and </b><br/><a NAME="E10:83_1_2_2"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">I</font> <a href="hidden.html#R1">=</a>  <a href="waybel_0.html#K3">downarrow</a> <font color="Maroon">J2</font>
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E11:83_1_2_2"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, , <a class="txt" href="waybel23.html#E1:83_1_2"><i><font color="Green">E50</font></i></a>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:83_1_2"/>
<font color="Maroon">f</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">g</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


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