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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#V2">reflexive</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#V2">reflexive</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">P</font> be  <a href="waybel_0.html#V13">upper</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>;<br/>





<b>assume </b><a NAME="E1:5"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">f</font> is <a href="orders_3.html#V5">monotone</a>
 ;<br/>

<a NAME="E2:5"/>
for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">P</font> &amp; <font color="Olive">y</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Olive">x</font> holds <br/><font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">P</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_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">S</font>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font>;<br/>

<b>assume </b><a NAME="E1:5_1"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">P</font> &amp; <font color="Maroon">y</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">x</font> )
 ;<br/>

<a NAME="E2:5_1"/><b>then </b><i><font color="Green">E47</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
 
<b>by </b><i>, <a class="ref" href="waybel_1.html#D2">WAYBEL_1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">fy</font> = <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font>, <font color="Maroon">fx</font> = <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font> ;<br/>
<a NAME="E4:5_1"/>
<font color="Maroon">fx</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 
<b>by </b><i>, <a class="ref" href="funct_2.html#T46">FUNCT_2:46</a></i>;<br/>
<a NAME="E5:5_1"/><b>then </b>
<font color="Maroon">fy</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 
<b>by </b><i>, <a class="ref" href="waybel_0.html#D20">WAYBEL_0:def 20</a></i>;<br/>
<b>hence </b><a NAME="E6:5_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T46">FUNCT_2:46</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:5"/>
<font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">P</font> is <a href="waybel_0.html#V13">upper</a>
 <b>by </b><i><a class="ref" href="waybel_0.html#D20">WAYBEL_0:def 20</a></i>;<br/>


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