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

<b>let </b><font color="Maroon" title="c1">M</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">X</font>, <font color="Maroon" title="c3">Y</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="orders_2.html#NM1" title="ORDERS_2:NM.1">Poset</a>;<br/><b>let </b><font color="Maroon" title="c4">f</font> be  <a href="waybel_0.html#V22" title="WAYBEL_0:attr.22">directed-sups-preserving</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c2">X</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">Y</font> <a href="yellow_1.html#K6" title="YELLOW_1:func.6">|^</a> <font color="Maroon" title="c1">M</font></span>)</span>;<br/><b>let </b><font color="Maroon" title="c5">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">M</font>;<br/>







<a NAME="E1:57"/><i><font color="Green" title="E42">A1</font></i>: 
<font color="Maroon" title="c4">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">X</font>,<span class="p1">(<span class="default"><a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c1">M</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">Y</font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel27.html#E43"><i><font color="Green" title="E40">Lm1</font></i></a></i>;<br/>
<a NAME="E2:57"/><b>then </b>
<font color="Maroon" title="c4">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">X</font>, <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c1">M</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">Y</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T121" title="FUNCT_2:th.121">FUNCT_2:121</a></i>;<br/>
<a NAME="E3:57"/><b>then </b><i><font color="Green" title="E43">A2</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c1">M</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">Y</font>
 
<b>by </b><i><a class="ref" href="relset_1.html#T12" title="RELSET_1:th.12">RELSET_1:12</a></i>;<br/>
<a NAME="E4:57"/><i><font color="Green" title="E44">A3</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">X</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">Y</font>
 
<b>by </b><i><a class="txt" href="waybel27.html#E43"><i><font color="Green" title="E40">Lm1</font></i></a></i>;<br/>
<a NAME="E5:57"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">M</font>
 
<b>by </b><i><a class="txt" href="waybel27.html#E43"><i><font color="Green" title="E40">Lm1</font></i></a></i>;<br/>
<a NAME="E6:57"/><b>then </b>
<span class="p1">(<span class="default"><a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <font color="Maroon" title="c4">f</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c6">g</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E8:57"/><i><font color="Green" title="E45">A4</font></i>: 
( <span class="p1">(<span class="default"><a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">g</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">X</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">g</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">Y</font> )
 <b>by </b><i><a class="txt" href="waybel27.html#E4:57"><i><font color="Green" title="E44">A3</font></i></a>, <a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">g</font> = <font color="Maroon" title="c6">g</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c3">Y</font> <b>by </b><i><a class="txt" href="waybel27.html#E8:57"><i><font color="Green" title="E45">A4</font></i></a>, <a class="ref" href="funct_2.html#T4" title="FUNCT_2:th.4">FUNCT_2:4</a></i>;<br/>
<a NAME="E10:57"/><i><font color="Green" title="E46">A5</font></i>: 
<font color="Maroon" title="c3">Y</font> <a href="yellow_1.html#K6" title="YELLOW_1:func.6">|^</a> <font color="Maroon" title="c1">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <font color="Maroon" title="c3">Y</font></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_1.html#D5" title="YELLOW_1:def.5">YELLOW_1:def 5</a></i>;<br/>
<a NAME="E11:57"/><i><font color="Green" title="E47">A6</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <font color="Maroon" title="c3">Y</font></span>)</span> <a href="yellow16.html#K2" title="YELLOW16:func.2">.</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">Y</font>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/>
<a NAME="E12:57"/>
<font color="Maroon" title="c7">g</font> is <a href="waybel_0.html#V22" title="WAYBEL_0:attr.22">directed-sups-preserving</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">X</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D37" title="WAYBEL_0:def.37">WAYBEL_0:def 37</a><br/>

<b>assume </b><a NAME="E1:57_1"/>
( not <font color="Maroon" title="c8">A</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; <font color="Maroon" title="c8">A</font> is <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a> )
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c9">B</font> = <font color="Maroon" title="c8">A</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">X</font> ;<br/>
<a NAME="E3:57_1"/><i><font color="Green" title="E48">A7</font></i>: 
<font color="Maroon" title="c4">f</font> <a href="waybel_0.html#R4" title="WAYBEL_0:pred.4">preserves_sup_of</a> <font color="Maroon" title="c9">B</font>
 
<b>by </b><i><a class="ref" href="waybel_0.html#D37" title="WAYBEL_0:def.37">WAYBEL_0:def 37</a></i>;<br/>
<b>assume </b><a NAME="E4:57_1"/>
 <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Maroon" title="c8">A</font>,<font color="Maroon" title="c2">X</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D31" title="WAYBEL_0:def.31">WAYBEL_0:def 31</a><br/>

<a NAME="E5:57_1"/><b>then </b>
(  <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c9">B</font>,<font color="Maroon" title="c3">Y</font> <a href="yellow_1.html#K6" title="YELLOW_1:func.6">|^</a> <font color="Maroon" title="c1">M</font> &amp;  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c9">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c9">B</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="waybel27.html#E3:57_1"><i><font color="Green" title="E48">A7</font></i></a>, <a class="ref" href="waybel_0.html#D31" title="WAYBEL_0:def.31">WAYBEL_0:def 31</a></i>;<br/>
<a NAME="E6:57_1"/><b>then </b><i><font color="Green" title="E49">A8</font></i>: 
(  <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a>  <a href="yellow16.html#K1" title="YELLOW16:func.1">pi</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c8">A</font></span>)</span>,<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c3">Y</font> &amp;  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><a href="yellow16.html#K1" title="YELLOW16:func.1">pi</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c8">A</font></span>)</span>,<font color="Maroon" title="c5">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c8">A</font></span>)</span></span>)</span> <a href="waybel27.html#K1" title="WAYBEL27:func.1">.</a> <font color="Maroon" title="c5">i</font> )
 
<b>by </b><i><a class="txt" href="waybel27.html#E10:57"><i><font color="Green" title="E46">A5</font></i></a>, <a class="txt" href="waybel27.html#E11:57"><i><font color="Green" title="E47">A6</font></i></a>, <a class="ref" href="yellow16.html#T33" title="YELLOW16:th.33">YELLOW16:33</a>, <a class="ref" href="yellow16.html#T35" title="YELLOW16:th.35">YELLOW16:35</a></i>;<br/>
<a NAME="E7:57_1"/><i><font color="Green" title="E50">A9</font></i>: 
 <a href="yellow16.html#K1" title="YELLOW16:func.1">pi</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c8">A</font></span>)</span>,<font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">g</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c8">A</font>
 
<b>by </b><i><a class="txt" href="waybel27.html#E3:57"><i><font color="Green" title="E43">A2</font></i></a>, <a class="txt" href="waybel27.html#E8:57"><i><font color="Green" title="E45">A4</font></i></a>, <a class="ref" href="waybel27.html#T8" target="_self" title="WAYBEL27:th.8">Th8</a></i>;<br/>
<b>thus </b><a NAME="E8:57_1"/>
 <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Maroon" title="c7">g</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c8">A</font>,<font color="Maroon" title="c3">Y</font>
 <b>by </b><i><a class="txt" href="waybel27.html#E3:57"><i><font color="Green" title="E43">A2</font></i></a>, <a class="txt" href="waybel27.html#E8:57"><i><font color="Green" title="E45">A4</font></i></a>, <a class="txt" href="waybel27.html#E6:57_1"><i><font color="Green" title="E49">A8</font></i></a>, <a class="ref" href="waybel27.html#T8" target="_self" title="WAYBEL27:th.8">Th8</a></i>;<br/>

<b>thus </b><a NAME="E9:57_1"/>
 <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c8">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c8">A</font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel27.html#E1:57"><i><font color="Green" title="E42">A1</font></i></a>, <a class="txt" href="waybel27.html#E8:57"><i><font color="Green" title="E45">A4</font></i></a>, <a class="txt" href="waybel27.html#E6:57_1"><i><font color="Green" title="E49">A8</font></i></a>, <a class="txt" href="waybel27.html#E7:57_1"><i><font color="Green" title="E50">A9</font></i></a>, <a class="ref" href="funct_6.html#T86" title="FUNCT_6:th.86">FUNCT_6:86</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E13:57"/>
<span class="p1">(<span class="default"><a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> is  <a href="waybel_0.html#V22" title="WAYBEL_0:attr.22">directed-sups-preserving</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c3">Y</font>
 <b>by </b><i><a class="txt" href="waybel27.html#E8:57"><i><font color="Green" title="E45">A4</font></i></a></i>;<br/>


</div>
