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<b>let </b><font color="Maroon">S1</font> be    <a href="orders_2.html#L1">RelStr</a> , <font color="Maroon">S2</font> be    <a href="orders_2.html#L1">RelStr</a> ;<br/><b>let </b><font color="Maroon">D</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S1</font>;<br/><b>let </b><font color="Maroon">f</font> be   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">S1</font>,<font color="Maroon">S2</font>;<br/>





<b>assume </b><a NAME="E1:18"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">f</font> is <a href="orders_3.html#V5">monotone</a>
 ;<br/>

<b>let </b><font color="Maroon">q</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:18"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">q</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#K4">uparrow</a> <font color="Maroon">D</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="E4:18"/><i><font color="Green">E34</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#K4">uparrow</a> <font color="Maroon">D</font> &amp; <font color="Maroon">q</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/>
<b>reconsider </b><font color="Maroon">s1</font> = <font color="Maroon">S1</font>, <font color="Maroon">s2</font> = <font color="Maroon">S2</font> as  non <a href="struct_0.html#V3">empty</a>  <a href="orders_2.html#L1">RelStr</a>  <b>by </b><i>, , <a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">f1</font> = <font color="Maroon">f</font> as   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">s1</font>,<font color="Maroon">s2</font> ;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">s1</font> <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">y</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">s1</font><b> such that </b><br/><a NAME="E9:18"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">y</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> )
 <b>by </b><i>, <a class="ref" href="waybel_0.html#D16">WAYBEL_0:def 16</a></i>;<br/>
<a NAME="E10:18"/>
<font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">s2</font>
 
;<br/>
<b>then reconsider </b><font color="Maroon">q1</font> = <font color="Maroon">q</font>, <font color="Maroon">fy</font> = <font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">s2</font> <b>by </b><i/>;<br/>
<a NAME="E12:18"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">fy</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">q1</font>
 
<b>by </b><i>, , , <a class="ref" href="orders_3.html#D5">ORDERS_3:def 5</a></i>;<br/>
<a NAME="E13:18"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">s2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
;<br/>
<a NAME="E14:18"/><b>then </b>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">s1</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E15:18"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</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">D</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<b>hence </b><a NAME="E16:18"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K4">uparrow</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">D</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="waybel_0.html#D16">WAYBEL_0:def 16</a></i>;<br/>


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