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<b>let </b><font color="Maroon">P</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="orders_3.html#NM1">POSet_set</a>;<br/><b>let </b><font color="Maroon">o1</font> be   <a href="altcat_1.html#NM1">object</a> of <span class="p1">(<span class="default"><a href="orders_3.html#K4" target="_self">POSAltCat</a> <font color="Maroon">P</font></span>)</span>, <font color="Maroon">o2</font> be   <a href="altcat_1.html#NM1">object</a> of <span class="p1">(<span class="default"><a href="orders_3.html#K4" target="_self">POSAltCat</a> <font color="Maroon">P</font></span>)</span>;<br/><b>let </b><font color="Maroon">A</font> be    <a href="orders_3.html#M1" target="_self">Element</a> of <font color="Maroon">P</font>, <font color="Maroon">B</font> be    <a href="orders_3.html#M1" target="_self">Element</a> of <font color="Maroon">P</font>;<br/>





<b>assume </b><a NAME="E1:41"/><i><font color="Green">E20</font></i>: 
( <font color="Maroon">o1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> &amp; <font color="Maroon">o2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> )
 ;<br/>

<b>let </b><font color="Maroon">x</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="E2:41"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">o1</font>,<font color="Maroon">o2</font></span><a href="altcat_1.html#K1">^&gt;</a></span>
 ;<br/>

<a NAME="E3:41"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="altcat_1.html#U1">Arrows</a> of <span class="p1">(<span class="default"><a href="orders_3.html#K4" target="_self">POSAltCat</a> <font color="Maroon">P</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">o1</font>,<font color="Maroon">o2</font>
 
;<br/>
<a NAME="E4:41"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="orders_3.html#K1" target="_self">MonFuncs</a> <font color="Maroon">A</font>,<font color="Maroon">B</font>
 
<b>by </b><i><a class="txt" href="orders_3.html#E1:41"><i><font color="Green">E20</font></i></a>, <a class="txt" href="orders_3.html#E3"><i><font color="Green">Lemma19</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">f</font> being   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">A</font>,<font color="Maroon">B</font><b> such that </b><br/><a NAME="E6:41"/><i><font color="Green">E21</font></i>: 
( <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">f</font> <a href="hidden.html#R2">in</a>  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B</font> &amp; <font color="Maroon">f</font> is <a href="orders_3.html#V5" target="_self">monotone</a> )
 <b>by </b><i/>;<br/>
<b>thus </b><a NAME="E7:41"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B</font>
 <b>by </b><i><a class="txt" href="orders_3.html#E6:41"><i><font color="Green">E21</font></i></a></i>;<br/>


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