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

<b>let </b><font color="Maroon" title="c1">S</font>, <font color="Maroon" title="c2">T</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="orders_2.html#V2" title="ORDERS_2:attr.2">reflexive</a> <a href="orders_2.html#V4" title="ORDERS_2:attr.4">antisymmetric</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> ;<br/><b>let </b><font color="Maroon" title="c3">f</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font>;<br/>



<b>assume </b><a NAME="E1:24"/><i><font color="Green" title="E20">A1</font></i>: 
<font color="Maroon" title="c3">f</font> is <a href="waybel_0.html#V21" title="WAYBEL_0:attr.21">filtered-infs-preserving</a>
 ;<br/>

<b>let </b><font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c5">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_1.html#D2" title="WAYBEL_1:def.2">WAYBEL_1:def 2</a><br/>

<b>assume </b><a NAME="E2:24"/><i><font color="Green" title="E21">A2</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">y</font>
 ;<br/>

<a NAME="E3:24"/><i><font color="Green" title="E22">A3</font></i>: 
( <font color="Maroon" title="c4">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">y</font> &amp; <font color="Maroon" title="c4">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c4">x</font> )
 
<b>by </b><i><a class="txt" href="waybel20.html#E2:24"><i><font color="Green" title="E21">A2</font></i></a></i>;<br/>
<a NAME="E4:24"/><b>then </b><i><font color="Green" title="E23">A4</font></i>: 
<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="lattice3.html#NR3" title="LATTICE3:NR.3">is_&gt;=_than</a> <font color="Maroon" title="c4">x</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T8" title="YELLOW_0:th.8">YELLOW_0:8</a></i>;<br/>
<a NAME="E5:24"/>
for <font color="Olive" title="b1">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font>  st <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="lattice3.html#NR3" title="LATTICE3:NR.3">is_&gt;=_than</a> <font color="Olive" title="b1">b</font> holds <br/><font color="Maroon" title="c4">x</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Olive" title="b1">b</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T8" title="YELLOW_0:th.8">YELLOW_0:8</a></i>;<br/>
<a NAME="E6:24"/><b>then </b><i><font color="Green" title="E24">A5</font></i>: 
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> &amp;  <a href="yellow_0.html#R2" title="YELLOW_0:pred.2">ex_inf_of</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>,<font color="Maroon" title="c1">S</font> )
 
<b>by </b><i><a class="txt" href="waybel20.html#E4:24"><i><font color="Green" title="E23">A4</font></i></a>, <a class="ref" href="yellow_0.html#T31" title="YELLOW_0:th.31">YELLOW_0:31</a></i>;<br/>
<a NAME="E7:24"/>
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> holds <br/> ex <font color="Olive" title="b3">z</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font> st <br/>( <font color="Olive" title="b3">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> &amp; <font color="Olive" title="b1">a</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Olive" title="b3">z</font> &amp; <font color="Olive" title="b2">b</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Olive" title="b3">z</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">a</font>, <font color="Maroon" title="c7">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font>;<br/>

<b>assume </b><a NAME="E1:24_1"/><i><font color="Green" title="E25">A6</font></i>: 
( <font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> &amp; <font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> )
 ;<br/>

<b>take </b>
<font color="Maroon" title="c4">x</font>
;<br/>

<b>thus </b><a NAME="E2:24_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/>

<b>thus </b><a NAME="E3:24_1"/>
( <font color="Maroon" title="c6">a</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c4">x</font> &amp; <font color="Maroon" title="c7">b</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c4">x</font> )
 <b>by </b><i><a class="txt" href="waybel20.html#E3:24"><i><font color="Green" title="E22">A3</font></i></a>, <a class="txt" href="waybel20.html#E1:24_1"><i><font color="Green" title="E25">A6</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:24"/><b>then </b>
( <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> is <a href="waybel_0.html#V2" title="WAYBEL_0:attr.2">filtered</a> &amp; not <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )
 
<b>by </b><i><a class="ref" href="waybel_0.html#D2" title="WAYBEL_0:def.2">WAYBEL_0:def 2</a></i>;<br/>
<a NAME="E9:24"/><b>then </b><i><font color="Green" title="E25">A7</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="waybel_0.html#R3" title="WAYBEL_0:pred.3">preserves_inf_of</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 
<b>by </b><i><a class="txt" href="waybel20.html#E1:24"><i><font color="Green" title="E20">A1</font></i></a>, <a class="ref" href="waybel_0.html#D36" title="WAYBEL_0:def.36">WAYBEL_0:def 36</a></i>;<br/>
<a NAME="E10:24"/><b>then </b><i><font color="Green" title="E26">A8</font></i>: 
 <a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p2"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x</font>
 
<b>by </b><i><a class="txt" href="waybel20.html#E6:24"><i><font color="Green" title="E24">A5</font></i></a>, <a class="ref" href="waybel_0.html#D30" title="WAYBEL_0:def.30">WAYBEL_0:def 30</a></i>;<br/>
<a NAME="E11:24"/><i><font color="Green" title="E27">A9</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E12:24"/><b>then </b><i><font color="Green" title="E28">A10</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">y</font></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 
<b>by </b><i><a class="txt" href="waybel20.html#E10:24"><i><font color="Green" title="E26">A8</font></i></a>, <a class="ref" href="funct_1.html#T118" title="FUNCT_1:th.118">FUNCT_1:118</a></i>;<br/>
<a NAME="E13:24"/>
<font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">y</font></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 
<b>by </b><i><a class="txt" href="waybel20.html#E11:24"><i><font color="Green" title="E27">A9</font></i></a>, <a class="ref" href="funct_1.html#T118" title="FUNCT_1:th.118">FUNCT_1:118</a></i>;<br/>
<a NAME="E14:24"/><b>then </b>
 <a href="yellow_0.html#R2" title="YELLOW_0:pred.2">ex_inf_of</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">y</font></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>,<font color="Maroon" title="c2">T</font>
 
<b>by </b><i><a class="txt" href="waybel20.html#E6:24"><i><font color="Green" title="E24">A5</font></i></a>, <a class="txt" href="waybel20.html#E9:24"><i><font color="Green" title="E25">A7</font></i></a>, <a class="ref" href="waybel_0.html#D30" title="WAYBEL_0:def.30">WAYBEL_0:def 30</a></i>;<br/>
<a NAME="E15:24"/><b>then </b>
<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">y</font></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="lattice3.html#NR3" title="LATTICE3:NR.3">is_&gt;=_than</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x</font>
 
<b>by </b><i><a class="txt" href="waybel20.html#E12:24"><i><font color="Green" title="E28">A10</font></i></a>, <a class="ref" href="yellow_0.html#D10" title="YELLOW_0:def.10">YELLOW_0:def 10</a></i>;<br/>
<b>hence </b><a NAME="E16:24"/>
<font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">y</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T8" title="YELLOW_0:th.8">YELLOW_0:8</a></i>;<br/>


</div>
