<?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="E1:31"/><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="waybel_0.html#NM6">Semilattice</a>
 ;<br/>

<b>set </b><font color="Maroon">IT</font> =  <a href="yellow_1.html#K5">product</a> <font color="Maroon">J</font>;<br/>
<a NAME="E2:31"/>
for <font color="Olive">x</font>, <font color="Olive">y</font> being  <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>  ex <font color="Olive">z</font> being  <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> st <br/>( <font color="Olive">x</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Olive">z</font> &amp; <font color="Olive">y</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Olive">z</font> &amp; ( for <font color="Olive">z'</font> being  <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>  st <font color="Olive">x</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Olive">z'</font> &amp; <font color="Olive">y</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Olive">z'</font> holds <br/><font color="Olive">z</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Olive">z'</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</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>, <font color="Maroon">y</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>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="waybel_3.html#K6">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> = <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="waybel_3.html#K7">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="lattice3.html#K11">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="waybel_3.html#K7">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>;<br/>
<b>consider </b><font color="Maroon">z</font> being    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">I</font><b> such that </b><br/><a NAME="E2:31_1"/><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">z</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">i</font>)
 <b>from </b><i><a class="ref" href="pboole.html#S5">PBOOLE:sch 5</a>();<br/></i>
<a NAME="E3:31_1"/><i><font color="Green">E48</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E4:31_1"/>
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">z</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="waybel_3.html#K6">.</a> <font color="Olive">i</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font>;<br/>

<a NAME="E1:31_1_1"/>
<font color="Maroon">z</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x</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">y</font> <a href="waybel_3.html#K7">.</a> <font color="Maroon">i</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E2:31_1_1"/>
<font color="Maroon">z</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">J</font> <a href="waybel_3.html#K6">.</a> <font color="Maroon">i</font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">z</font> = <font color="Maroon">z</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>, <a class="ref" href="waybel_3.html#T27">WAYBEL_3:27</a></i>;<br/>
<b>take </b>
<font color="Maroon">z</font>
;<br/>

<a NAME="E6:31_1"/>
for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> holds <br/> ( <font color="Maroon">x</font> <a href="waybel_3.html#K7">.</a> <font color="Olive">i</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">z</font> <a href="waybel_3.html#K7">.</a> <font color="Olive">i</font> &amp; <font color="Maroon">y</font> <a href="waybel_3.html#K7">.</a> <font color="Olive">i</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">z</font> <a href="waybel_3.html#K7">.</a> <font color="Olive">i</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font>;<br/>

<a NAME="E1:31_1_2"/>
( <font color="Maroon">J</font> <a href="waybel_3.html#K6">.</a> <font color="Maroon">i</font> is  <a href="orders_2.html#V4">antisymmetric</a> <a href="lattice3.html#V2">with_infima</a>  <a href="orders_2.html#L1">RelStr</a>  &amp; <font color="Maroon">z</font> <a href="waybel_3.html#K7">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x</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">y</font> <a href="waybel_3.html#K7">.</a> <font color="Maroon">i</font></span>)</span> )
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E2:31_1_2"/>
( <font color="Maroon">x</font> <a href="waybel_3.html#K7">.</a> <font color="Maroon">i</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">z</font> <a href="waybel_3.html#K7">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">y</font> <a href="waybel_3.html#K7">.</a> <font color="Maroon">i</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">z</font> <a href="waybel_3.html#K7">.</a> <font color="Maroon">i</font> )
 <b>by </b><i><a class="ref" href="yellow_0.html#T23">YELLOW_0:23</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:31_1"/>
( <font color="Maroon">x</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">z</font> &amp; <font color="Maroon">y</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">z</font> )
 <b>by </b><i><a class="ref" href="waybel_3.html#T28">WAYBEL_3:28</a></i>;<br/>

<b>let </b><font color="Maroon">z'</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>assume </b><a NAME="E8:31_1"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">x</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">z'</font> &amp; <font color="Maroon">y</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">z'</font> )
 ;<br/>

<a NAME="E9:31_1"/>
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">z</font> <a href="waybel_3.html#K7">.</a> <font color="Olive">i</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">z'</font> <a href="waybel_3.html#K7">.</a> <font color="Olive">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font>;<br/>

<a NAME="E1:31_1_3"/>
( <font color="Maroon">J</font> <a href="waybel_3.html#K6">.</a> <font color="Maroon">i</font> is  <a href="orders_2.html#V4">antisymmetric</a> <a href="lattice3.html#V2">with_infima</a>  <a href="orders_2.html#L1">RelStr</a>  &amp; <font color="Maroon">x</font> <a href="waybel_3.html#K7">.</a> <font color="Maroon">i</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">z'</font> <a href="waybel_3.html#K7">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">z'</font> <a href="waybel_3.html#K7">.</a> <font color="Maroon">i</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">y</font> <a href="waybel_3.html#K7">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">z</font> <a href="waybel_3.html#K7">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x</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">y</font> <a href="waybel_3.html#K7">.</a> <font color="Maroon">i</font></span>)</span> )
 
<b>by </b><i>, , , <a class="ref" href="waybel_3.html#T28">WAYBEL_3:28</a></i>;<br/>
<b>hence </b><a NAME="E2:31_1_3"/>
<font color="Maroon">z</font> <a href="waybel_3.html#K7">.</a> <font color="Maroon">i</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">z'</font> <a href="waybel_3.html#K7">.</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T23">YELLOW_0:23</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E10:31_1"/>
<font color="Maroon">z</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">z'</font>
 <b>by </b><i><a class="ref" href="waybel_3.html#T28">WAYBEL_3:28</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:31"/>
 <a href="yellow_1.html#K5">product</a> <font color="Maroon">J</font> is <a href="lattice3.html#V2">with_infima</a>
 <b>by </b><i><a class="ref" href="lattice3.html#D11">LATTICE3:def 11</a></i>;<br/>


</div>
