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

<b>let </b><font color="Maroon">S</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="struct_0.html#L1">1-sorted</a> ;<br/><b>let </b><font color="Maroon">T</font> be  <a href="lattice3.html#V3">complete</a> <a href="waybel_0.html#NM8">LATTICE</a>;<br/><b>let </b><font color="Maroon">f</font> be   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">S</font>,<font color="Maroon">T</font>, <font color="Maroon">g</font> be   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">S</font>,<font color="Maroon">T</font>, <font color="Maroon">h</font> be   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">S</font>,<font color="Maroon">T</font>;<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/>







<b>assume </b><a NAME="E1:42"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">h</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K2">"/\"</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">f</font>,<font color="Maroon">g</font></span><a href="domain_1.html#K7">}</a></span>,<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>
 ;<br/>

<a NAME="E2:42"/>
 <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="E5:42"/><i><font color="Green">E43</font></i>: 
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>reconsider </b><font color="Maroon">f'</font> = <font color="Maroon">f</font>, <font color="Maroon">g'</font> = <font color="Maroon">g</font> as   <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>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">f'</font> = <font color="Maroon">f'</font>, <font color="Maroon">g'</font> = <font color="Maroon">g'</font> as   <a href="struct_0.html#NM1">Element</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/>
<b>reconsider </b><font color="Maroon">DU</font> = <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">f'</font>,<font color="Maroon">g'</font></span><a href="domain_1.html#K7">}</a></span> 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> ;<br/>
<font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> = 
<span class="p1">(<span class="default"><a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">DU</font></span>)</span> <a href="waybel_3.html#K7">.</a> <font color="Maroon">i</font>
<b>by </b><i>, <a class="ref" href="yellow_1.html#D5">YELLOW_1:def 5</a></i>
<br/>.= 
 <a href="yellow_0.html#K2">"/\"</a> <span class="p1">(<span class="default"><a href="card_3.html#K5">pi</a> <span class="p2"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">f</font>,<font color="Maroon">g</font></span><a href="domain_1.html#K7">}</a></span>,<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>, </i>
<br/>.= 
 <a href="yellow_0.html#K2">"/\"</a> <span class="p1">(<span class="default"><a href="card_3.html#K5">pi</a> <span class="p2"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">f</font>,<font color="Maroon">g</font></span><a href="domain_1.html#K7">}</a></span>,<font color="Maroon">i</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/>.= 
 <a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span></span><a href="domain_1.html#K7">}</a></span>
<b>by </b><i><a class="ref" href="card_3.html#T26">CARD_3:26</a></i>
;<br/>
<b>hence </b><a NAME="E10:42"/>
<font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span></span><a href="domain_1.html#K7">}</a></span>
 ;<br/>


</div>
