<?xml version="1.0"?>
<div class="add">

<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>set </b><font color="Maroon">G</font> = 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>;<br/>
<b>set </b><font color="Maroon">C</font> = the <a href="altcat_1.html#U2">Comp</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>thus </b><a NAME="E1:40_1_1"/>
the <a href="altcat_1.html#U2">Comp</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="altcat_1.html#V3">associative</a>
  <i><font color="Red">:: according to </font></i><a class="ref" href="altcat_1.html#D17">ALTCAT_1:def 17</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be   <a href="struct_0.html#NM1">Element</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">j</font> be   <a href="struct_0.html#NM1">Element</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">k</font> be   <a href="struct_0.html#NM1">Element</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">l</font> be   <a href="struct_0.html#NM1">Element</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#D7">ALTCAT_1:def 7</a><br/>

<b>reconsider </b><font color="Maroon">i'</font> = <font color="Maroon">i</font>, <font color="Maroon">j'</font> = <font color="Maroon">j</font>, <font color="Maroon">k'</font> = <font color="Maroon">k</font>, <font color="Maroon">l'</font> = <font color="Maroon">l</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>let </b><font color="Maroon">f</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">g</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">h</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E2:40_1_1_1"/>
( <font color="Maroon">f</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">i</font>,<font color="Maroon">j</font> &amp; <font color="Maroon">g</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">j</font>,<font color="Maroon">k</font> &amp; <font color="Maroon">h</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">k</font>,<font color="Maroon">l</font> )
 ;<br/>

<a NAME="E3:40_1_1_1"/><b>then </b><i><font color="Green">E20</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">i'</font>,<font color="Maroon">j'</font> &amp; <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">j'</font>,<font color="Maroon">k'</font> &amp; <font color="Maroon">h</font> <a href="hidden.html#R2">in</a>  <a href="orders_3.html#K1" target="_self">MonFuncs</a> <font color="Maroon">k'</font>,<font color="Maroon">l'</font> )
 
<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="struct_0.html#NM6">Function</a> of <font color="Maroon">i'</font>,<font color="Maroon">j'</font><b> such that </b><br/><a NAME="E5:40_1_1_1"/><i><font color="Green">E21</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">i'</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">j'</font> &amp; <font color="Maroon">f'</font> is <a href="orders_3.html#V5" target="_self">monotone</a> )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">g'</font> being   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">j'</font>,<font color="Maroon">k'</font><b> such that </b><br/><a NAME="E7:40_1_1_1"/><i><font color="Green">E30</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">j'</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">k'</font> &amp; <font color="Maroon">g'</font> is <a href="orders_3.html#V5" target="_self">monotone</a> )
 <b>by </b><i><a class="txt" href="orders_3.html#E3:40_1_1_1"><i><font color="Green">E20</font></i></a>, </i>;<br/>
<b>consider </b><font color="Maroon">h'</font> being   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">k'</font>,<font color="Maroon">l'</font><b> such that </b><br/><a NAME="E9:40_1_1_1"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h'</font> &amp; <font color="Maroon">h'</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">k'</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">l'</font> &amp; <font color="Maroon">h'</font> is <a href="orders_3.html#V5" target="_self">monotone</a> )
 <b>by </b><i><a class="txt" href="orders_3.html#E3:40_1_1_1"><i><font color="Green">E20</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>, <font color="Maroon">h'</font> = <font color="Maroon">h</font> as   <a href="funct_1.html#NM1">Function</a> <b>by </b><i><a class="txt" href="orders_3.html#E5:40_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="E11:40_1_1_1"/><i><font color="Green">E40</font></i>: 
<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">i'</font>,<font color="Maroon">k'</font>
 
<b>by </b><i><a class="txt" href="orders_3.html#E3:40_1_1_1"><i><font color="Green">E20</font></i></a>, </i>;<br/>
<a NAME="E12:40_1_1_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">h'</font> <a href="funct_1.html#NK3" target="_self">*</a> <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">j'</font>,<font color="Maroon">l'</font>
 
<b>by </b><i><a class="txt" href="orders_3.html#E3:40_1_1_1"><i><font color="Green">E20</font></i></a>, </i>;<br/>
<a NAME="E13:40_1_1_1"/><i><font color="Green">E42</font></i>: 
the <a href="altcat_1.html#U2">Comp</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="altcat_1.html#K4">.</a> <font color="Maroon">i</font>,<font color="Maroon">k</font>,<font color="Maroon">l</font> <a href="hidden.html#R1">=</a>  <a href="altcat_1.html#K6">FuncComp</a> <span class="p1">(<span class="default"><a href="orders_3.html#K1" target="_self">MonFuncs</a> <font color="Maroon">i'</font>,<font color="Maroon">k'</font></span>)</span>,<span class="p1">(<span class="default"><a href="orders_3.html#K1" target="_self">MonFuncs</a> <font color="Maroon">k'</font>,<font color="Maroon">l'</font></span>)</span>
 
<b>by </b><i><a class="txt" href="orders_3.html#E3"><i><font color="Green">Lemma19</font></i></a></i>;<br/>
<a NAME="E14:40_1_1_1"/><i><font color="Green">E45</font></i>: 
the <a href="altcat_1.html#U2">Comp</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="altcat_1.html#K4">.</a> <font color="Maroon">i</font>,<font color="Maroon">j</font>,<font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="altcat_1.html#K6">FuncComp</a> <span class="p1">(<span class="default"><a href="orders_3.html#K1" target="_self">MonFuncs</a> <font color="Maroon">i'</font>,<font color="Maroon">j'</font></span>)</span>,<span class="p1">(<span class="default"><a href="orders_3.html#K1" target="_self">MonFuncs</a> <font color="Maroon">j'</font>,<font color="Maroon">k'</font></span>)</span>
 
<b>by </b><i><a class="txt" href="orders_3.html#E3"><i><font color="Green">Lemma19</font></i></a></i>;<br/>
<a NAME="E15:40_1_1_1"/><i><font color="Green">E46</font></i>: 
the <a href="altcat_1.html#U2">Comp</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="altcat_1.html#K4">.</a> <font color="Maroon">i</font>,<font color="Maroon">j</font>,<font color="Maroon">l</font> <a href="hidden.html#R1">=</a>  <a href="altcat_1.html#K6">FuncComp</a> <span class="p1">(<span class="default"><a href="orders_3.html#K1" target="_self">MonFuncs</a> <font color="Maroon">i'</font>,<font color="Maroon">j'</font></span>)</span>,<span class="p1">(<span class="default"><a href="orders_3.html#K1" target="_self">MonFuncs</a> <font color="Maroon">j'</font>,<font color="Maroon">l'</font></span>)</span>
 
<b>by </b><i><a class="txt" href="orders_3.html#E3"><i><font color="Green">Lemma19</font></i></a></i>;<br/>
<a NAME="E16:40_1_1_1"/><i><font color="Green">E47</font></i>: 
the <a href="altcat_1.html#U2">Comp</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="altcat_1.html#K4">.</a> <font color="Maroon">j</font>,<font color="Maroon">k</font>,<font color="Maroon">l</font> <a href="hidden.html#R1">=</a>  <a href="altcat_1.html#K6">FuncComp</a> <span class="p1">(<span class="default"><a href="orders_3.html#K1" target="_self">MonFuncs</a> <font color="Maroon">j'</font>,<font color="Maroon">k'</font></span>)</span>,<span class="p1">(<span class="default"><a href="orders_3.html#K1" target="_self">MonFuncs</a> <font color="Maroon">k'</font>,<font color="Maroon">l'</font></span>)</span>
 
<b>by </b><i><a class="txt" href="orders_3.html#E3"><i><font color="Green">Lemma19</font></i></a></i>;<br/>
<a NAME="E17:40_1_1_1"/><i><font color="Green">E48</font></i>: 
<span class="p1">(<span class="default">the <a href="altcat_1.html#U2">Comp</a> of <span class="p2">(<span class="default"><a href="orders_3.html#K4" target="_self">POSAltCat</a> <font color="Maroon">P</font></span>)</span> <a href="altcat_1.html#K4">.</a> <font color="Maroon">i</font>,<font color="Maroon">j</font>,<font color="Maroon">k</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">g</font>,<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g'</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">f'</font>
 
<b>by </b><i><a class="txt" href="orders_3.html#E3:40_1_1_1"><i><font color="Green">E20</font></i></a>, , <a class="ref" href="altcat_1.html#T13">ALTCAT_1:13</a></i>;<br/>
<a NAME="E18:40_1_1_1"/><i><font color="Green">E49</font></i>: 
<span class="p1">(<span class="default">the <a href="altcat_1.html#U2">Comp</a> of <span class="p2">(<span class="default"><a href="orders_3.html#K4" target="_self">POSAltCat</a> <font color="Maroon">P</font></span>)</span> <a href="altcat_1.html#K4">.</a> <font color="Maroon">i</font>,<font color="Maroon">k</font>,<font color="Maroon">l</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">h</font>,<span class="p1">(<span class="default"><font color="Maroon">g'</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">f'</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">h'</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">g'</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">f'</font></span>)</span>
 
<b>by </b><i><a class="txt" href="orders_3.html#E3:40_1_1_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="orders_3.html#D3" target="_self">Def3</a>, , <a class="ref" href="altcat_1.html#T13">ALTCAT_1:13</a></i>;<br/>
<a NAME="E19:40_1_1_1"/><i><font color="Green">E50</font></i>: 
<span class="p1">(<span class="default">the <a href="altcat_1.html#U2">Comp</a> of <span class="p2">(<span class="default"><a href="orders_3.html#K4" target="_self">POSAltCat</a> <font color="Maroon">P</font></span>)</span> <a href="altcat_1.html#K4">.</a> <font color="Maroon">j</font>,<font color="Maroon">k</font>,<font color="Maroon">l</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">h</font>,<font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h'</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">g'</font>
 
<b>by </b><i><a class="txt" href="orders_3.html#E3:40_1_1_1"><i><font color="Green">E20</font></i></a>, , <a class="ref" href="altcat_1.html#T13">ALTCAT_1:13</a></i>;<br/>
<a NAME="E20:40_1_1_1"/>
<span class="p1">(<span class="default">the <a href="altcat_1.html#U2">Comp</a> of <span class="p2">(<span class="default"><a href="orders_3.html#K4" target="_self">POSAltCat</a> <font color="Maroon">P</font></span>)</span> <a href="altcat_1.html#K4">.</a> <font color="Maroon">i</font>,<font color="Maroon">j</font>,<font color="Maroon">l</font></span>)</span> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">h'</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">g'</font></span>)</span>,<font color="Maroon">f'</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">h'</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">g'</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">f'</font>
 
<b>by </b><i><a class="txt" href="orders_3.html#E3:40_1_1_1"><i><font color="Green">E20</font></i></a>, , , <a class="ref" href="altcat_1.html#T13">ALTCAT_1:13</a></i>;<br/>
<b>hence </b><a NAME="E21:40_1_1_1"/>
<span class="p1">(<span class="default">the <a href="altcat_1.html#U2">Comp</a> of <span class="p2">(<span class="default"><a href="orders_3.html#K4" target="_self">POSAltCat</a> <font color="Maroon">P</font></span>)</span> <a href="altcat_1.html#K4">.</a> <font color="Maroon">i</font>,<font color="Maroon">k</font>,<font color="Maroon">l</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">h</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="altcat_1.html#U2">Comp</a> of <span class="p3">(<span class="default"><a href="orders_3.html#K4" target="_self">POSAltCat</a> <font color="Maroon">P</font></span>)</span> <a href="altcat_1.html#K4">.</a> <font color="Maroon">i</font>,<font color="Maroon">j</font>,<font color="Maroon">k</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">g</font>,<font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">the <a href="altcat_1.html#U2">Comp</a> of <span class="p2">(<span class="default"><a href="orders_3.html#K4" target="_self">POSAltCat</a> <font color="Maroon">P</font></span>)</span> <a href="altcat_1.html#K4">.</a> <font color="Maroon">i</font>,<font color="Maroon">j</font>,<font color="Maroon">l</font></span>)</span> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="altcat_1.html#U2">Comp</a> of <span class="p3">(<span class="default"><a href="orders_3.html#K4" target="_self">POSAltCat</a> <font color="Maroon">P</font></span>)</span> <a href="altcat_1.html#K4">.</a> <font color="Maroon">j</font>,<font color="Maroon">k</font>,<font color="Maroon">l</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">h</font>,<font color="Maroon">g</font></span>)</span>,<font color="Maroon">f</font>
 <b>by </b><i><a class="txt" href="orders_3.html#E7:40_1_1_1"><i><font color="Green">E30</font></i></a>, <a class="txt" href="orders_3.html#E9:40_1_1_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="orders_3.html#T4" target="_self">Th4</a>, <a class="ref" href="relat_1.html#T55">RELAT_1:55</a></i>;<br/>


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

<b>thus </b><a NAME="E2:40_1_1"/>
the <a href="altcat_1.html#U2">Comp</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="altcat_1.html#V5">with_left_units</a>
  <i><font color="Red">:: according to </font></i><a class="ref" href="altcat_1.html#D18">ALTCAT_1:def 18</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">j</font> be   <a href="struct_0.html#NM1">Element</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#D9">ALTCAT_1:def 9</a><br/>

<b>reconsider </b><font color="Maroon">j'</font> = <font color="Maroon">j</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>take </b><font color="Maroon">e</font> =  <a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">j'</font>;<br/>

<a NAME="E2:40_1_1_2"/>
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">j</font>,<font color="Maroon">j</font> <a href="hidden.html#R1">=</a>  <a href="orders_3.html#K1" target="_self">MonFuncs</a> <font color="Maroon">j'</font>,<font color="Maroon">j'</font>
 
<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="E3:40_1_1_2"/>
<font color="Maroon">e</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">j</font>,<font color="Maroon">j</font>
 <b>by </b><i><a class="ref" href="orders_3.html#D1" target="_self">Def1</a></i>;<br/>

<a NAME="E4:40_1_1_2"/><i><font color="Green">E107</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a>  <a href="orders_3.html#K1" target="_self">MonFuncs</a> <font color="Maroon">j'</font>,<font color="Maroon">j'</font>
 
<b>by </b><i><a class="ref" href="orders_3.html#D1" target="_self">Def1</a></i>;<br/>
<b>then consider </b><font color="Maroon">e'</font> being   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">j'</font>,<font color="Maroon">j'</font><b> such that </b><br/><a NAME="E6:40_1_1_2"/><i><font color="Green">E108</font></i>: 
( <font color="Maroon">e</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">e'</font> &amp; <font color="Maroon">e'</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">j'</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">j'</font> &amp; <font color="Maroon">e'</font> is <a href="orders_3.html#V5" target="_self">monotone</a> )
 <b>by </b><i/>;<br/>
<b>let </b><font color="Maroon">i</font> be   <a href="struct_0.html#NM1">Element</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">f</font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>reconsider </b><font color="Maroon">i'</font> = <font color="Maroon">i</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="E8:40_1_1_2"/>
<font color="Maroon">f</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">i</font>,<font color="Maroon">j</font>
 ;<br/>

<a NAME="E9:40_1_1_2"/><b>then </b><i><font color="Green">E109</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">i'</font>,<font color="Maroon">j'</font>
 
<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="struct_0.html#NM6">Function</a> of <font color="Maroon">i'</font>,<font color="Maroon">j'</font><b> such that </b><br/><a NAME="E11:40_1_1_2"/><i><font color="Green">E110</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">i'</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">j'</font> &amp; <font color="Maroon">f'</font> is <a href="orders_3.html#V5" target="_self">monotone</a> )
 <b>by </b><i/>;<br/>
<a NAME="E12:40_1_1_2"/><i><font color="Green">E111</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">f'</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">j'</font>
 
;<br/>
<a NAME="E13:40_1_1_2"/><i><font color="Green">E112</font></i>: 
the <a href="altcat_1.html#U2">Comp</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="altcat_1.html#K4">.</a> <font color="Maroon">i</font>,<font color="Maroon">j</font>,<font color="Maroon">j</font> <a href="hidden.html#R1">=</a>  <a href="altcat_1.html#K6">FuncComp</a> <span class="p1">(<span class="default"><a href="orders_3.html#K1" target="_self">MonFuncs</a> <font color="Maroon">i'</font>,<font color="Maroon">j'</font></span>)</span>,<span class="p1">(<span class="default"><a href="orders_3.html#K1" target="_self">MonFuncs</a> <font color="Maroon">j'</font>,<font color="Maroon">j'</font></span>)</span>
 
<b>by </b><i><a class="txt" href="orders_3.html#E3"><i><font color="Green">Lemma19</font></i></a></i>;<br/>
<a NAME="E14:40_1_1_2"/>
<font color="Maroon">e'</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font>
 
<b>by </b><i>, , <a class="ref" href="orders_3.html#D3" target="_self">Def3</a>, <a class="ref" href="relat_1.html#T79">RELAT_1:79</a></i>;<br/>
<b>hence </b><a NAME="E15:40_1_1_2"/>
<span class="p1">(<span class="default">the <a href="altcat_1.html#U2">Comp</a> of <span class="p2">(<span class="default"><a href="orders_3.html#K4" target="_self">POSAltCat</a> <font color="Maroon">P</font></span>)</span> <a href="altcat_1.html#K4">.</a> <font color="Maroon">i</font>,<font color="Maroon">j</font>,<font color="Maroon">j</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">e</font>,<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font>
 <b>by </b><i>, , <a class="ref" href="orders_3.html#D2" target="_self">Def2</a>, , , <a class="ref" href="altcat_1.html#T13">ALTCAT_1:13</a></i>;<br/>


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

<b>thus </b><a NAME="E3:40_1_1"/>
the <a href="altcat_1.html#U2">Comp</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="altcat_1.html#V4">with_right_units</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be   <a href="struct_0.html#NM1">Element</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#D8">ALTCAT_1:def 8</a><br/>

<b>reconsider </b><font color="Maroon">i'</font> = <font color="Maroon">i</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>take </b><font color="Maroon">e</font> =  <a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">i'</font>;<br/>

<a NAME="E2:40_1_1_3"/>
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">i</font>,<font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="orders_3.html#K1" target="_self">MonFuncs</a> <font color="Maroon">i'</font>,<font color="Maroon">i'</font>
 
<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="E3:40_1_1_3"/>
<font color="Maroon">e</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">i</font>,<font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="orders_3.html#D1" target="_self">Def1</a></i>;<br/>

<a NAME="E4:40_1_1_3"/><i><font color="Green">E115</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a>  <a href="orders_3.html#K1" target="_self">MonFuncs</a> <font color="Maroon">i'</font>,<font color="Maroon">i'</font>
 
<b>by </b><i><a class="ref" href="orders_3.html#D1" target="_self">Def1</a></i>;<br/>
<b>then consider </b><font color="Maroon">e'</font> being   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">i'</font>,<font color="Maroon">i'</font><b> such that </b><br/><a NAME="E6:40_1_1_3"/><i><font color="Green">E116</font></i>: 
( <font color="Maroon">e</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">e'</font> &amp; <font color="Maroon">e'</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">i'</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">i'</font> &amp; <font color="Maroon">e'</font> is <a href="orders_3.html#V5" target="_self">monotone</a> )
 <b>by </b><i/>;<br/>
<b>let </b><font color="Maroon">j</font> be   <a href="struct_0.html#NM1">Element</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">f</font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>reconsider </b><font color="Maroon">j'</font> = <font color="Maroon">j</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="E8:40_1_1_3"/>
<font color="Maroon">f</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">i</font>,<font color="Maroon">j</font>
 ;<br/>

<a NAME="E9:40_1_1_3"/><b>then </b><i><font color="Green">E134</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">i'</font>,<font color="Maroon">j'</font>
 
<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="struct_0.html#NM6">Function</a> of <font color="Maroon">i'</font>,<font color="Maroon">j'</font><b> such that </b><br/><a NAME="E11:40_1_1_3"/><i><font color="Green">E135</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">i'</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">j'</font> &amp; <font color="Maroon">f'</font> is <a href="orders_3.html#V5" target="_self">monotone</a> )
 <b>by </b><i/>;<br/>
<a NAME="E12:40_1_1_3"/><i><font color="Green">E136</font></i>: 
 <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">i'</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E13:40_1_1_3"/><i><font color="Green">E137</font></i>: 
the <a href="altcat_1.html#U2">Comp</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="altcat_1.html#K4">.</a> <font color="Maroon">i</font>,<font color="Maroon">i</font>,<font color="Maroon">j</font> <a href="hidden.html#R1">=</a>  <a href="altcat_1.html#K6">FuncComp</a> <span class="p1">(<span class="default"><a href="orders_3.html#K1" target="_self">MonFuncs</a> <font color="Maroon">i'</font>,<font color="Maroon">i'</font></span>)</span>,<span class="p1">(<span class="default"><a href="orders_3.html#K1" target="_self">MonFuncs</a> <font color="Maroon">i'</font>,<font color="Maroon">j'</font></span>)</span>
 
<b>by </b><i><a class="txt" href="orders_3.html#E3"><i><font color="Green">Lemma19</font></i></a></i>;<br/>
<a NAME="E14:40_1_1_3"/>
<font color="Maroon">f'</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">e'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font>
 
<b>by </b><i>, , <a class="ref" href="orders_3.html#D3" target="_self">Def3</a>, <a class="ref" href="relat_1.html#T78">RELAT_1:78</a></i>;<br/>
<b>hence </b><a NAME="E15:40_1_1_3"/>
<span class="p1">(<span class="default">the <a href="altcat_1.html#U2">Comp</a> of <span class="p2">(<span class="default"><a href="orders_3.html#K4" target="_self">POSAltCat</a> <font color="Maroon">P</font></span>)</span> <a href="altcat_1.html#K4">.</a> <font color="Maroon">i</font>,<font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">f</font>,<font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font>
 <b>by </b><i>, , <a class="ref" href="orders_3.html#D2" target="_self">Def2</a>, , , <a class="ref" href="altcat_1.html#T13">ALTCAT_1:13</a></i>;<br/>


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


</div>
