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<b>let </b><font color="Maroon">S</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="orders_2.html#L1">RelStr</a> , <font color="Maroon">T</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="orders_2.html#L1">RelStr</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>let </b><font color="Maroon">X</font> be  <a href="waybel_0.html#V1">directed</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font>;<br/>





<b>assume </b><a NAME="E1:20"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">f</font> is <a href="orders_3.html#V5">monotone</a>
 ;<br/>

<b>set </b><font color="Maroon">Y</font> = <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font>;<br/>
<a NAME="E2:20"/>
for <font color="Olive">y1</font>, <font color="Olive">y2</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font>  st <font color="Olive">y1</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> &amp; <font color="Olive">y2</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> holds <br/> ex <font color="Olive">z</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font> st <br/>( <font color="Olive">z</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> &amp; <font color="Olive">y1</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">z</font> &amp; <font color="Olive">y2</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">z</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y1</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font>, <font color="Maroon">y2</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font>;<br/>

<b>assume </b><a NAME="E1:20_1"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">y1</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> &amp; <font color="Maroon">y2</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>then consider </b><font color="Maroon">x1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:20_1"/>
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E4:20_1"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>and </b><br/><a NAME="E5:20_1"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</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">x2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:20_1"/>
<font color="Maroon">x2</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E8:20_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>and </b><br/><a NAME="E9:20_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">y2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</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">x1</font> = <font color="Maroon">x1</font>, <font color="Maroon">x2</font> = <font color="Maroon">x2</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> <b>by </b><i>, </i>;<br/>
<b>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="E12:20_1"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">x1</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">z</font> &amp; <font color="Maroon">x2</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">z</font> )
 <b>by </b><i>, , <a class="ref" href="waybel_0.html#D1">WAYBEL_0:def 1</a></i>;<br/>
<b>take </b>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font>
;<br/>

<b>thus </b><a NAME="E13:20_1"/>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</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="funct_2.html#T43">FUNCT_2:43</a></i>;<br/>

<b>thus </b><a NAME="E14:20_1"/>
( <font color="Maroon">y1</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font> &amp; <font color="Maroon">y2</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font> )
 <b>by </b><i>, , , , <a class="ref" href="orders_3.html#D5">ORDERS_3:def 5</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:20"/>
<font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font> is <a href="waybel_0.html#V1">directed</a>
 <b>by </b><i><a class="ref" href="waybel_0.html#D1">WAYBEL_0:def 1</a></i>;<br/>


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