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<b>set </b><font color="Maroon">A</font> =  <a href="orders_3.html#K4" target="_self">POSAltCat</a> <font color="Maroon">P</font>;<br/>
<b>thus </b><a NAME="E1:39_1_1"/>
 <a href="orders_3.html#K4" target="_self">POSAltCat</a> <font color="Maroon">P</font> is <a href="altcat_1.html#V2">transitive</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_1"><b>proof </b></a><div class="add">

<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>, <font color="Maroon">o3</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>; <i><font color="Red">:: according to </font></i><a class="ref" href="altcat_1.html#D4">ALTCAT_1:def 4</a><br/>

<b>reconsider </b><font color="Maroon">o1'</font> = <font color="Maroon">o1</font>, <font color="Maroon">o2'</font> = <font color="Maroon">o2</font>, <font color="Maroon">o3'</font> = <font color="Maroon">o3</font> as    <a href="orders_3.html#M1" target="_self">Element</a> of <font color="Maroon">P</font> <b>by </b><i><a class="txt" href="orders_3.html#E3"><i><font color="Green">Lemma19</font></i></a></i>;<br/>
<b>assume </b><a NAME="E2:39_1_1_1"/>
( <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> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">o2</font>,<font color="Maroon">o3</font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/>

<a NAME="E3:39_1_1_1"/><b>then </b>
( 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> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; 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">o2</font>,<font color="Maroon">o3</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 
;<br/>
<a NAME="E4:39_1_1_1"/><b>then </b><i><font color="Green">E20</font></i>: 
(  <a href="orders_3.html#K1" target="_self">MonFuncs</a> <font color="Maroon">o1'</font>,<font color="Maroon">o2'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="orders_3.html#K1" target="_self">MonFuncs</a> <font color="Maroon">o2'</font>,<font color="Maroon">o3'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i><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="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:39_1_1_1"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a>  <a href="orders_3.html#K1" target="_self">MonFuncs</a> <font color="Maroon">o1'</font>,<font color="Maroon">o2'</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon">g</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E8:39_1_1_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">g</font> <a href="hidden.html#R2">in</a>  <a href="orders_3.html#K1" target="_self">MonFuncs</a> <font color="Maroon">o2'</font>,<font color="Maroon">o3'</font>
 <b>by </b><i><a class="txt" href="orders_3.html#E4:39_1_1_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon">g'</font> being   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">o2'</font>,<font color="Maroon">o3'</font><b> such that </b><br/><a NAME="E10:39_1_1_1"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g'</font> &amp; <font color="Maroon">g'</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">o2'</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">o3'</font> &amp; <font color="Maroon">g'</font> is <a href="orders_3.html#V5" target="_self">monotone</a> )
 <b>by </b><i><a class="ref" href="orders_3.html#D2" target="_self">Def2</a>, </i>;<br/>
<b>consider </b><font color="Maroon">f'</font> being   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">o1'</font>,<font color="Maroon">o2'</font><b> such that </b><br/><a NAME="E12:39_1_1_1"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f'</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">o1'</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">o2'</font> &amp; <font color="Maroon">f'</font> is <a href="orders_3.html#V5" target="_self">monotone</a> )
 <b>by </b><i><a class="txt" href="orders_3.html#E6:39_1_1_1"><i><font color="Green">E21</font></i></a>, </i>;<br/>
<b>reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font>, <font color="Maroon">g</font> = <font color="Maroon">g</font> as   <a href="funct_1.html#NM1">Function</a> <b>by </b><i>, <a class="ref" href="orders_3.html#D3" target="_self">Def3</a></i>;<br/>
<a NAME="E14:39_1_1_1"/>
<font color="Maroon">g</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">f</font> <a href="hidden.html#R2">in</a>  <a href="orders_3.html#K1" target="_self">MonFuncs</a> <font color="Maroon">o1'</font>,<font color="Maroon">o3'</font>
 
<b>by </b><i><a class="txt" href="orders_3.html#E6:39_1_1_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="orders_3.html#D2" target="_self">Def2</a>, </i>;<br/>
<a NAME="E15:39_1_1_1"/><b>then </b>
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">o3</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="txt" href="orders_3.html#E3"><i><font color="Green">Lemma19</font></i></a></i>;<br/>
<b>hence </b><a NAME="E16:39_1_1_1"/>
not <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">o1</font>,<font color="Maroon">o3</font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>


</div><b>end;</b></div>

<a NAME="E2:39_1_1"/>
not the <a href="struct_0.html#U1">carrier</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> is <a href="xboole_0.html#V1">empty</a>
 
<b>by </b><i><a class="txt" href="orders_3.html#E3"><i><font color="Green">Lemma19</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:39_1_1"/>
not  <a href="orders_3.html#K4" target="_self">POSAltCat</a> <font color="Maroon">P</font> is <a href="struct_0.html#V3">empty</a>
 <b>by </b><i><a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</a></i>;<br/>


</div>
