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

<b>let </b><font color="Maroon">I</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">J</font> be  <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 <font color="Maroon">I</font>;<br/>



<b>assume </b><a NAME="E1:34"/><i><font color="Green">E41</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> holds  <font color="Maroon">J</font> <a href="waybel_3.html#K6">.</a> <font color="Olive">i</font> is  <a href="lattice3.html#V3">complete</a> <a href="waybel_1.html#V9">Heyting</a> <a href="waybel_0.html#NM8">LATTICE</a>
 ;<br/>

<b>set </b><font color="Maroon">H</font> =  <a href="yellow_1.html#K5">product</a> <font color="Maroon">J</font>;<br/>
<a NAME="E2:34"/><i><font color="Green">E42</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> holds  <font color="Maroon">J</font> <a href="waybel_3.html#K6">.</a> <font color="Olive">i</font> is   <a href="waybel_0.html#NM6">Semilattice</a>
 
<b>by </b><i/>;<br/>
<a NAME="E3:34"/><i><font color="Green">E48</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> holds  <font color="Maroon">J</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/>;<br/>
<b>then reconsider </b><font color="Maroon">H</font> =  <a href="yellow_1.html#K5">product</a> <font color="Maroon">J</font> as  <a href="lattice3.html#V3">complete</a> <a href="waybel_0.html#NM8">LATTICE</a> <b>by </b><i><a class="ref" href="waybel_3.html#T31">WAYBEL_3:31</a></i>;<br/>
<a NAME="E5:34"/>
<font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H</font>
 
;<br/>
<b>hence </b><a NAME="E6:34"/>
(  <a href="yellow_1.html#K5">product</a> <font color="Maroon">J</font> is <a href="lattice3.html#V3">complete</a> &amp;  <a href="yellow_1.html#K5">product</a> <font color="Maroon">J</font> is   <a href="waybel_0.html#NM8">LATTICE</a> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_1.html#D19">WAYBEL_1:def 19</a><br/>

<b>let </b><font color="Maroon">f</font> be   <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">J</font></span>)</span>;<br/>

<b>reconsider </b><font color="Maroon">g</font> = <font color="Maroon">f</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">H</font> ;<br/>
<b>reconsider </b><font color="Maroon">F</font> = <font color="Maroon">f</font> <a href="waybel_1.html#K4">"/\"</a>  as   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">H</font>,<font color="Maroon">H</font> ;<br/>
<a NAME="E9:34"/>
<font color="Maroon">F</font> is <a href="waybel_0.html#V18">sups-preserving</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">X</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">H</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D33">WAYBEL_0:def 33</a><br/>

<b>reconsider </b><font color="Maroon">Y</font> = <font color="Maroon">F</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font>, <font color="Maroon">X'</font> = <font color="Maroon">X</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">J</font></span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon">sX</font> =  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">X</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">J</font></span>)</span> ;<br/>
<b>assume </b><a NAME="E3:34_1"/>
 <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">X</font>,<font color="Maroon">H</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D31">WAYBEL_0:def 31</a><br/>

<b>thus </b><a NAME="E4:34_1"/>
 <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">F</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font>,<font color="Maroon">H</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T17">YELLOW_0:17</a></i>;<br/>

<b>reconsider </b><font color="Maroon">f1</font> =  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font></span>)</span>, <font color="Maroon">f2</font> = <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">X</font></span>)</span> 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">J</font></span>)</span> ;<br/>
<a NAME="E6:34_1"/><i><font color="Green">E52</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font> )
 
<b>by </b><i><a class="ref" href="waybel_3.html#T27">WAYBEL_3:27</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_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="E1:34_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f1</font>
 ;<br/><b>then reconsider </b><font color="Maroon">i</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> <b>by </b><i><a class="ref" href="waybel_3.html#T27">WAYBEL_3:27</a></i>;<br/><a NAME="E3:34_1_1"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">J</font> <a href="waybel_3.html#K6">.</a> <font color="Maroon">i</font> is  <a href="lattice3.html#V3">complete</a> <a href="waybel_1.html#V9">Heyting</a> <a href="waybel_0.html#NM8">LATTICE</a>
 <b>by </b><i/>;<br/><a NAME="E4:34_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="waybel_3.html#K7">.</a> <font color="Maroon">i</font></span>)</span> <a href="waybel_1.html#K4">"/\"</a>  is <a href="waybel_1.html#V5">lower_adjoint</a>
 <b>by </b><i><a class="ref" href="waybel_1.html#D19">WAYBEL_1:def 19</a></i>;<br/><a NAME="E5:34_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="waybel_3.html#K7">.</a> <font color="Maroon">i</font></span>)</span> <a href="waybel_1.html#K4">"/\"</a>  is <a href="waybel_0.html#V18">sups-preserving</a>
 <b>by </b><i>, <a class="ref" href="waybel_1.html#T14">WAYBEL_1:14</a></i>;<br/><a NAME="E6:34_1_1"/><b>then </b><i><font color="Green">E56</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="waybel_3.html#K7">.</a> <font color="Maroon">i</font></span>)</span> <a href="waybel_1.html#K4">"/\"</a>  <a href="waybel_0.html#R4">preserves_sup_of</a>  <a href="waybel_3.html#K5">pi</a> <font color="Maroon">X'</font>,<font color="Maroon">i</font> &amp;  <a href="yellow_0.html#R1">ex_sup_of</a>  <a href="waybel_3.html#K5">pi</a> <font color="Maroon">X'</font>,<font color="Maroon">i</font>,<font color="Maroon">J</font> <a href="waybel_3.html#K6">.</a> <font color="Maroon">i</font> )
 <b>by </b><i>, <a class="ref" href="waybel_0.html#D33">WAYBEL_0:def 33</a>, <a class="ref" href="yellow_0.html#T17">YELLOW_0:17</a></i>;<br/><a NAME="E7:34_1_1"/>
<font color="Maroon">f2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">X</font></span>)</span>
 <b>by </b><i><a class="ref" href="waybel_1.html#D18">WAYBEL_1:def 18</a></i>;<br/><b>then </b><i><font color="Green">E67</font></i>: <font color="Maroon">f2</font> <a href="waybel_3.html#K7">.</a> <font color="Maroon">i</font> = 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="waybel_3.html#K7">.</a> <font color="Maroon">i</font></span>)</span> <a href="lattice3.html#K11">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">sX</font> <a href="waybel_3.html#K7">.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="waybel_3.html#K7">.</a> <font color="Maroon">i</font></span>)</span> <a href="lattice3.html#K11">"/\"</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p2">(<span class="default"><a href="waybel_3.html#K5">pi</a> <font color="Maroon">X'</font>,<font color="Maroon">i</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="waybel_3.html#T32">WAYBEL_3:32</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="waybel_3.html#K7">.</a> <font color="Maroon">i</font></span>)</span> <a href="waybel_1.html#K4">"/\"</a> </span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p2">(<span class="default"><a href="waybel_3.html#K5">pi</a> <font color="Maroon">X'</font>,<font color="Maroon">i</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="waybel_1.html#D18">WAYBEL_1:def 18</a></i>
<br/>.= 
 <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="waybel_3.html#K7">.</a> <font color="Maroon">i</font></span>)</span> <a href="waybel_1.html#K4">"/\"</a> </span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><a href="waybel_3.html#K5">pi</a> <font color="Maroon">X'</font>,<font color="Maroon">i</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="waybel_0.html#D31">WAYBEL_0:def 31</a></i>
;<br/><a NAME="E9:34_1_1"/>
 <a href="waybel_3.html#K5">pi</a> <font color="Maroon">Y</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="Maroon">f</font> <a href="waybel_3.html#K7">.</a> <font color="Maroon">i</font></span>)</span> <a href="waybel_1.html#K4">"/\"</a> </span>)</span> <a href="relset_1.html#K10">.:</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>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_1_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:34_1_1_2"/>
 <a href="waybel_3.html#K5">pi</a> <font color="Maroon">Y</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="Maroon">f</font> <a href="waybel_3.html#K7">.</a> <font color="Maroon">i</font></span>)</span> <a href="waybel_1.html#K4">"/\"</a> </span>)</span> <a href="relset_1.html#K10">.:</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>
  <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="34_1_1_2_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="E1:34_1_1_2_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="waybel_3.html#K5">pi</a> <font color="Maroon">Y</font>,<font color="Maroon">i</font>
 ;<br/>

<b>then consider </b><font color="Maroon">f3</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E3:34_1_1_2_1"/><i><font color="Green">E69</font></i>: 
( <font color="Maroon">f3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f3</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>reconsider </b><font color="Maroon">f3</font> = <font color="Maroon">f3</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">J</font></span>)</span> <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">h</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:34_1_1_2_1"/><i><font color="Green">E70</font></i>: 
( <font color="Maroon">h</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font> &amp; <font color="Maroon">h</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">f3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">h</font> )
 <b>by </b><i>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<b>reconsider </b><font color="Maroon">h</font> = <font color="Maroon">h</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">J</font></span>)</span> <b>by </b><i/>;<br/>
<i><font color="Green">E71</font></i>: <font color="Maroon">f3</font> <a href="waybel_3.html#K7">.</a> <font color="Maroon">i</font> = 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="lattice3.html#K11">"/\"</a> <font color="Maroon">h</font></span>)</span> <a href="waybel_3.html#K7">.</a> <font color="Maroon">i</font>
<b>by </b><i>, <a class="ref" href="waybel_1.html#D18">WAYBEL_1:def 18</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="waybel_3.html#K7">.</a> <font color="Maroon">i</font></span>)</span> <a href="lattice3.html#K11">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="waybel_3.html#K7">.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="waybel_3.html#K7">.</a> <font color="Maroon">i</font></span>)</span> <a href="waybel_1.html#K4">"/\"</a> </span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="waybel_3.html#K7">.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i><a class="ref" href="waybel_1.html#D18">WAYBEL_1:def 18</a></i>
;<br/>
<a NAME="E9:34_1_1_2_1"/>
<font color="Maroon">h</font> <a href="waybel_3.html#K7">.</a> <font color="Maroon">i</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/>
<b>hence </b><a NAME="E10:34_1_1_2_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="Maroon">f</font> <a href="waybel_3.html#K7">.</a> <font color="Maroon">i</font></span>)</span> <a href="waybel_1.html#K4">"/\"</a> </span>)</span> <a href="relset_1.html#K10">.:</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>
 <b>by </b><i>, , <a class="ref" href="funct_2.html#T43">FUNCT_2:43</a></i>;<br/>


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

<b>thus </b><a NAME="E2:34_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="waybel_3.html#K7">.</a> <font color="Maroon">i</font></span>)</span> <a href="waybel_1.html#K4">"/\"</a> </span>)</span> <a href="relset_1.html#K10">.:</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> <a href="tarski.html#R1">c=</a>  <a href="waybel_3.html#K5">pi</a> <font color="Maroon">Y</font>,<font color="Maroon">i</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">b</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="E1:34_1_1_2_2"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="waybel_3.html#K7">.</a> <font color="Maroon">i</font></span>)</span> <a href="waybel_1.html#K4">"/\"</a> </span>)</span> <a href="relset_1.html#K10">.:</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>
 ;<br/>

<b>then consider </b><font color="Maroon">f4</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:34_1_1_2_2"/><i><font color="Green">E72</font></i>: 
( <font color="Maroon">f4</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="waybel_3.html#K7">.</a> <font color="Maroon">i</font></span>)</span> <a href="waybel_1.html#K4">"/\"</a> </span>)</span> &amp; <font color="Maroon">f4</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> &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="waybel_3.html#K7">.</a> <font color="Maroon">i</font></span>)</span> <a href="waybel_1.html#K4">"/\"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">f4</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<b>consider </b><font color="Maroon">f5</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E5:34_1_1_2_2"/><i><font color="Green">E73</font></i>: 
( <font color="Maroon">f5</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X'</font> &amp; <font color="Maroon">f4</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f5</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>reconsider </b><font color="Maroon">f5</font> = <font color="Maroon">f5</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">J</font></span>)</span> <b>by </b><i/>;<br/>
<i><font color="Green">E75</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="waybel_3.html#K7">.</a> <font color="Maroon">i</font></span>)</span> <a href="waybel_1.html#K4">"/\"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">f4</font> = 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="waybel_3.html#K7">.</a> <font color="Maroon">i</font></span>)</span> <a href="lattice3.html#K11">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">f5</font> <a href="waybel_3.html#K7">.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i>, <a class="ref" href="waybel_1.html#D18">WAYBEL_1:def 18</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="lattice3.html#K11">"/\"</a> <font color="Maroon">f5</font></span>)</span> <a href="waybel_3.html#K7">.</a> <font color="Maroon">i</font>
<b>by </b><i>, </i>
;<br/>
<a NAME="E8:34_1_1_2_2"/>
<font color="Maroon">f</font> <a href="lattice3.html#K11">"/\"</a> <font color="Maroon">f5</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="waybel_1.html#K4">"/\"</a> </span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">f5</font>
 
<b>by </b><i><a class="ref" href="waybel_1.html#D18">WAYBEL_1:def 18</a></i>;<br/>
<a NAME="E9:34_1_1_2_2"/><b>then </b>
<font color="Maroon">f</font> <a href="lattice3.html#K11">"/\"</a> <font color="Maroon">f5</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="waybel_1.html#K4">"/\"</a> </span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font>
 
<b>by </b><i>, <a class="ref" href="funct_2.html#T43">FUNCT_2:43</a></i>;<br/>
<b>hence </b><a NAME="E10:34_1_1_2_2"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="waybel_3.html#K5">pi</a> <font color="Maroon">Y</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><b>hence </b><a NAME="E10:34_1_1"/>
<font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, , <a class="ref" href="waybel_3.html#T32">WAYBEL_3:32</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:34_1"/>
 <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">X</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E10:34"/>
<font color="Maroon">f</font> <a href="waybel_1.html#K4">"/\"</a>  is <a href="waybel_1.html#V5">lower_adjoint</a>
 <b>by </b><i><a class="ref" href="waybel_1.html#T18">WAYBEL_1:18</a></i>;<br/>


</div>
