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<div class="add">

<b>let </b><font color="Maroon">S</font> be  non <a href="struct_0.html#V3">empty</a> <a href="orders_2.html#NM1">Poset</a>, <font color="Maroon">T</font> be  non <a href="struct_0.html#V3">empty</a> <a href="orders_2.html#NM1">Poset</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>;<br/>



<b>assume </b><a NAME="E1:134"/><i><font color="Green">E28</font></i>: 
for <font color="Olive">X</font> being  <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">S</font> holds  <font color="Maroon">f</font> <a href="waybel_0.html#R4" target="_self">preserves_sup_of</a> <font color="Olive">X</font>
 ;<br/>

<b>let </b><font color="Maroon">X</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D37" target="_self">WAYBEL_0:def 37</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E2:134"/><i><font color="Green">E30</font></i>: 
( not <font color="Maroon">X</font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Maroon">X</font> is <a href="waybel_0.html#V1" target="_self">directed</a> )
 <b>and </b><br/><a NAME="E3:134"/><i><font color="Green">E31</font></i>: 
 <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">X</font>,<font color="Maroon">S</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D31" target="_self">WAYBEL_0:def 31</a><br/>

<b>reconsider </b><font color="Maroon">Y</font> = <font color="Maroon">X</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="waybel_0.html#V1" target="_self">directed</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font> <b>by </b><i/>;<br/>
<a NAME="E5:134"/>
 <a href="waybel_0.html#K3" target="_self">downarrow</a> <font color="Maroon">Y</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">S</font>
 
;<br/>
<a NAME="E6:134"/><b>then </b>
( <font color="Maroon">f</font> <a href="waybel_0.html#R4" target="_self">preserves_sup_of</a>  <a href="waybel_0.html#K3" target="_self">downarrow</a> <font color="Maroon">X</font> &amp;  <a href="yellow_0.html#R1">ex_sup_of</a>  <a href="waybel_0.html#K3" target="_self">downarrow</a> <font color="Maroon">X</font>,<font color="Maroon">S</font> )
 
<b>by </b><i><a class="ref" href="waybel_0.html#D6" target="_self">Def6</a>, , </i>;<br/>
<a NAME="E7:134"/><b>then </b><i><font color="Green">E32</font></i>: 
(  <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K3" target="_self">downarrow</a> <font color="Maroon">X</font></span>)</span>,<font color="Maroon">T</font> &amp;  <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> <span class="p2">(<span class="default"><a href="waybel_0.html#K3" target="_self">downarrow</a> <font color="Maroon">X</font></span>)</span></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> <span class="p2">(<span class="default"><a href="waybel_0.html#K3" target="_self">downarrow</a> <font color="Maroon">X</font></span>)</span></span>)</span> &amp;  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K3" target="_self">downarrow</a> <font color="Maroon">X</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">X</font> )
 
<b>by </b><i>, , <a class="txt" href="waybel_0.html#E9:134"><i><font color="Green">E34</font></i></a></i>;<br/>
<a NAME="E8:134"/>
<font color="Maroon">X</font> <a href="tarski.html#R1">c=</a>  <a href="waybel_0.html#K3" target="_self">downarrow</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="waybel_0.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E9:134"/><b>then </b><i><font color="Green">E34</font></i>: 
<font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K3" target="_self">downarrow</a> <font color="Maroon">X</font></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T156">RELAT_1:156</a></i>;<br/>
<a NAME="E10:134"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K3" target="_self">downarrow</a> <font color="Maroon">X</font></span>)</span> <a href="lattice3.html#R2">is_&lt;=_than</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="waybel_0.html#D11" target="_self">Def11</a>, <a class="ref" href="yellow_0.html#T30">YELLOW_0:30</a></i>;<br/>
<a NAME="E11:134"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font> <a href="lattice3.html#R2">is_&lt;=_than</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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="134_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 <font color="Maroon">T</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D9">LATTICE3:def 9</a><br/>

<b>assume </b><a NAME="E1:134_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font>
 ;<br/>

<b>hence </b><a NAME="E2:134_1"/>
<font color="Maroon">x</font> <a href="orders_2.html#R1">&lt;=</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="lattice3.html#D9">LATTICE3:def 9</a></i>;<br/>


</div><b>end;</b></div>
<div><i><font color="Green">E40</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="134_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font>;<br/><b>assume </b><a NAME="E1:134_2"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Maroon">b</font>
 ;<br/><a NAME="E2:134_2"/>
<font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K3" target="_self">downarrow</a> <font color="Maroon">X</font></span>)</span> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Maroon">b</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="134_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D9">LATTICE3:def 9</a><br/>

<b>assume </b><a NAME="E1:134_2_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K3" target="_self">downarrow</a> <font color="Maroon">X</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:134_2_1"/><i><font color="Green">E76</font></i>: 
( <font color="Maroon">x</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">x</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K3" target="_self">downarrow</a> <font color="Maroon">X</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E4:134_2_1"/>
 <a href="waybel_0.html#K3" target="_self">downarrow</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">z</font> where <font color="Olive">z</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> :  ex <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> st <br/>( <font color="Olive">z</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">y</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> ) </span>}</span> 
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">z</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E6:134_2_1"/><i><font color="Green">E135</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> &amp;  ex <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> st <br/>( <font color="Maroon">z</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">y</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> ) )
 <b>by </b><i><a class="ref" href="waybel_0.html#T11" target="_self">Th11</a></i>;<br/>
<b>consider </b><font color="Maroon">y</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E8:134_2_1"/><i><font color="Green">E136</font></i>: 
( <font color="Maroon">z</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 <b>by </b><i/>;<br/>
<a NAME="E9:134_2_1"/>
( <font color="Maroon">f</font> is <a href="orders_3.html#V5">monotone</a> &amp; <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font> )
 
<b>by </b><i><a class="ref" href="waybel_0.html#D6" target="_self">Def6</a>, <a class="ref" href="waybel_0.html#T12" target="_self">Th12</a>, , <a class="ref" href="funct_2.html#T43">FUNCT_2:43</a></i>;<br/>
<a NAME="E10:134_2_1"/><b>then </b>
( <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> &amp; <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">b</font> )
 
<b>by </b><i><a class="ref" href="waybel_0.html#D14" target="_self">Def14</a>, <a class="ref" href="waybel_0.html#T12" target="_self">Th12</a>, <a class="ref" href="lattice3.html#D9">LATTICE3:def 9</a>, <a class="ref" href="orders_3.html#D5">ORDERS_3:def 5</a></i>;<br/>
<b>hence </b><a NAME="E11:134_2_1"/>
<font color="Maroon">a</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="waybel_0.html#T11" target="_self">Th11</a>, , <a class="ref" href="orders_2.html#T26">ORDERS_2:26</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:134_2"/>
<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> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="waybel_0.html#D11" target="_self">Def11</a>, <a class="ref" href="yellow_0.html#T30">YELLOW_0:30</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E13:134"/>
 <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">T</font>
 <b>by </b><i>, <a class="ref" href="yellow_0.html#T15">YELLOW_0:15</a></i>;<br/>

<b>hence </b><a NAME="E14:134"/>
 <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="waybel_0.html#D13" target="_self">Def13</a>, <a class="ref" href="yellow_0.html#D9">YELLOW_0:def 9</a></i>;<br/>


</div>
