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<b>let </b><font color="Maroon">S</font> be  <a href="lattice3.html#V3">complete</a> <a href="waybel_9.html#NM1">TopLattice</a>, <font color="Maroon">T</font> be  <a href="lattice3.html#V3">complete</a> <a href="waybel_9.html#NM1">TopLattice</a>;<br/><b>let </b><font color="Maroon">F</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="waybel24.html#K3" target="_self">ContMaps</a> <font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span>;<br/><b>let </b><font color="Maroon">i</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font>;<br/>





<a NAME="E1:44"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="pboole.html#K9">--&gt;</a> <font color="Maroon">T</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">SYT</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="pboole.html#K9">--&gt;</a> <font color="Maroon">T</font> as  <a href="yellow_1.html#V1">RelStr-yielding</a> <a href="waybel_3.html#V4">non-Empty</a>  <a href="pboole.html#M1">ManySortedSet</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">SYT</font> = <font color="Maroon">SYT</font> as  <a href="yellow_1.html#V1">RelStr-yielding</a> <a href="waybel_3.html#V4">non-Empty</a> <a href="waybel_3.html#V5">reflexive-yielding</a>  <a href="pboole.html#M1">ManySortedSet</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> ;<br/>
<a NAME="E4:44"/>
 <a href="waybel24.html#K3" target="_self">ContMaps</a> <font color="Maroon">S</font>,<font color="Maroon">T</font> is  <a href="yellow_0.html#V4">full</a>  <a href="yellow_0.html#M1">SubRelStr</a> of <font color="Maroon">T</font> <a href="yellow_1.html#K6">|^</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
 
<b>by </b><i/>;<br/>
<a NAME="E5:44"/><b>then </b>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel24.html#K3" target="_self">ContMaps</a> <font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="yellow_1.html#K6">|^</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_0.html#D13">YELLOW_0:def 13</a></i>;<br/>
<a NAME="E6:44"/><b>then </b><i><font color="Green">E27</font></i>: 
<font color="Maroon">F</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="yellow_1.html#K6">|^</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E7:44"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">T</font> <a href="yellow_1.html#K6">|^</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="hidden.html#R1">=</a>  <a href="yellow_1.html#K5">product</a> <font color="Maroon">SYT</font>
 
<b>by </b><i><a class="ref" href="yellow_1.html#D5">YELLOW_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">X</font> = <font color="Maroon">F</font> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <font color="Maroon">SYT</font></span>)</span> <b>by </b><i>, <a class="ref" href="yellow_1.html#D5">YELLOW_1:def 5</a></i>;<br/>
<a NAME="E9:44"/><i><font color="Green">E44</font></i>: 
 <a href="waybel_3.html#K5">pi</a> <font color="Maroon">X</font>,<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> where <font color="Olive">f</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="yellow_1.html#K6">|^</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span>)</span> : <font color="Olive">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:44_1"/>
 <a href="waybel_3.html#K5">pi</a> <font color="Maroon">X</font>,<font color="Maroon">i</font> <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> where <font color="Olive">f</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="yellow_1.html#K6">|^</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span>)</span> : <font color="Olive">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> </span>}</span> 
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_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> ; <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="E1:44_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="waybel_3.html#K5">pi</a> <font color="Maroon">X</font>,<font color="Maroon">i</font>
 ;<br/>

<b>then consider </b><font color="Maroon">g</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E3:44_1_1"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">g</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> )
 <b>by </b><i><a class="ref" href="card_3.html#D6">CARD_3:def 6</a></i>;<br/>
<b>thus </b><a NAME="E4:44_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> where <font color="Olive">f</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="yellow_1.html#K6">|^</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span>)</span> : <font color="Olive">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> </span>}</span> 
 <b>by </b><i>, </i>;<br/>


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

<b>thus </b><a NAME="E2:44_1"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> where <font color="Olive">f</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="yellow_1.html#K6">|^</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span>)</span> : <font color="Olive">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="waybel_3.html#K5">pi</a> <font color="Maroon">X</font>,<font color="Maroon">i</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</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="E1:44_1_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> where <font color="Olive">f</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="yellow_1.html#K6">|^</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span>)</span> : <font color="Olive">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">g</font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="yellow_1.html#K6">|^</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span>)</span><b> such that </b><br/><a NAME="E3:44_1_2"/><i><font color="Green">E58</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">g</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> )
 ;<br/>
<b>thus </b><a NAME="E4:44_1_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="waybel_3.html#K5">pi</a> <font color="Maroon">X</font>,<font color="Maroon">i</font>
 <b>by </b><i>, <a class="ref" href="card_3.html#D6">CARD_3:def 6</a></i>;<br/>


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


</div><b>end;</b></div>
<a NAME="E10:44"/>
for <font color="Olive">i</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> holds  <font color="Maroon">SYT</font> <a href="waybel_3.html#K6">.</a> <font color="Olive">i</font> is  <a href="lattice3.html#V3">complete</a> <a href="waybel_0.html#NM8">LATTICE</a>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/>
<b>then </b><span class="p1">(<span class="default"><a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">F</font>,<span class="p2">(<span class="default"><font color="Maroon">T</font> <a href="yellow_1.html#K6">|^</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> = 
 <a href="yellow_0.html#K1">"\/"</a> <span class="p1">(<span class="default"><a href="waybel_3.html#K5">pi</a> <font color="Maroon">X</font>,<font color="Maroon">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">SYT</font> <a href="waybel_3.html#K6">.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i>, <a class="ref" href="waybel_3.html#T32">WAYBEL_3:32</a></i>
<br/>.= 
 <a href="yellow_0.html#K1">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> where <font color="Olive">f</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="yellow_1.html#K6">|^</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span>)</span> : <font color="Olive">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> </span>}</span> ,<font color="Maroon">T</font>
<b>by </b><i>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>
;<br/>
<b>hence </b><a NAME="E12:44"/>
<span class="p1">(<span class="default"><a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">F</font>,<span class="p2">(<span class="default"><font color="Maroon">T</font> <a href="yellow_1.html#K6">|^</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K1">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> where <font color="Olive">f</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="yellow_1.html#K6">|^</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font></span>)</span> : <font color="Olive">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> </span>}</span> ,<font color="Maroon">T</font>
 ;<br/>


</div>
