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<b>let </b><font color="Maroon">L</font> be  non <a href="struct_0.html#V3">empty</a> <a href="orders_2.html#NM1">Poset</a>;<br/><b>let </b><font color="Maroon">p</font> be   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">L</font>,<font color="Maroon">L</font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:88"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">p</font> is <a href="quantal1.html#V11">idempotent</a>
 <b>and </b><br/><a NAME="E2:88"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">p</font> is <a href="orders_3.html#V5">monotone</a>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_1.html#D13" target="_self">WAYBEL_1:def 13</a><br/>

<b>let </b><font color="Maroon">Lc</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font>;<br/>

<b>assume </b><a NAME="E3:88"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">Lc</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">c</font> where <font color="Olive">c</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : <font color="Olive">c</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">p</font> <a href="funct_2.html#K3">.</a> <font color="Olive">c</font> </span>}</span> 
 ;<br/>

<b>let </b><font color="Maroon">pc</font> be   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">Lc</font></span>)</span>,<span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">Lc</font></span>)</span>;<br/>

<b>assume </b><a NAME="E4:88"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">pc</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">Lc</font>
 ;<br/>

<a NAME="E5:88"/><i><font color="Green">E49</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">pc</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">Lc</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="quantal1.html#D9">QUANTAL1:def 9</a>,<a class="ref" href="waybel_1.html#D13" target="_self">WAYBEL_1:def 13</a>,<a class="ref" href="waybel_1.html#D14" target="_self">WAYBEL_1:def 14</a>
<div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="88_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">Lc</font></span>)</span>;<br/><a NAME="E1:88_1_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">pc</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/><a NAME="E2:88_1_1"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">x</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T59">YELLOW_0:59</a></i>;<br/><font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span> = 
<font color="Maroon">pc</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">pc</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i>, , , <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">pc</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">pc</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
<b>by </b><i>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
;<br/><b>hence </b><a NAME="E4:88_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">pc</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">pc</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">pc</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, , , <a class="ref" href="yellow_2.html#T20">YELLOW_2:20</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E2:88_1"/>
<font color="Maroon">pc</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">pc</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">pc</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>;<br/><b>thus </b><a NAME="E3:88_1"/>
<font color="Maroon">pc</font> is <a href="orders_3.html#V5">monotone</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="88_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x1</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">Lc</font></span>)</span>, <font color="Maroon">x2</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">Lc</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_1.html#D2" target="_self">WAYBEL_1:def 2</a><br/>

<b>reconsider </b><font color="Maroon">x1'</font> = <font color="Maroon">x1</font>, <font color="Maroon">x2'</font> = <font color="Maroon">x2</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> <b>by </b><i><a class="ref" href="yellow_0.html#T59">YELLOW_0:59</a></i>;<br/>
<a NAME="E2:88_1_2"/><i><font color="Green">E57</font></i>: 
( <font color="Maroon">pc</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x1'</font> &amp; <font color="Maroon">pc</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x2'</font> )
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/>
<b>assume </b><a NAME="E3:88_1_2"/>
<font color="Maroon">x1</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">x2</font>
 ;<br/>

<a NAME="E4:88_1_2"/><b>then </b>
<font color="Maroon">x1'</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">x2'</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T60">YELLOW_0:60</a></i>;<br/>
<a NAME="E5:88_1_2"/><b>then </b>
<font color="Maroon">p</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x1'</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">p</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x2'</font>
 
<b>by </b><i><a class="txt" href="waybel_1.html#E2:88_1_2"><i><font color="Green">E57</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E6:88_1_2"/>
<font color="Maroon">pc</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x1</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">pc</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x2</font>
 <b>by </b><i>, <a class="ref" href="yellow_0.html#T61">YELLOW_0:61</a></i>;<br/>


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

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="88_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">Lc</font></span>)</span>;<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">L</font> <b>by </b><i><a class="ref" href="yellow_0.html#T59">YELLOW_0:59</a></i>;<br/><a NAME="E2:88_2"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">pc</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x'</font>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/><a NAME="E3:88_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">Lc</font></span>)</span>
 ;<br/><a NAME="E4:88_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Lc</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/><a NAME="E5:88_2"/><b>then </b>
 ex <font color="Olive">c</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">c</font> &amp; <font color="Olive">c</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">p</font> <a href="funct_2.html#K3">.</a> <font color="Olive">c</font> )
 <b>by </b><i><a class="txt" href="waybel_1.html#E2:88_2"><i><font color="Green">E58</font></i></a></i>;<br/><a NAME="E6:88_2"/><b>then </b>
<font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">pc</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, <a class="ref" href="yellow_0.html#T61">YELLOW_0:61</a></i>;<br/><b>hence </b><a NAME="E7:88_2"/>
<span class="p1">(<span class="default"><a href="grcat_1.html#K7">id</a> <span class="p2">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">Lc</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">pc</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="tmap_1.html#T91">TMAP_1:91</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:88"/>
 <a href="grcat_1.html#K7">id</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K5">subrelstr</a> <font color="Maroon">Lc</font></span>)</span> <a href="yellow_2.html#R1">&lt;=</a> <font color="Maroon">pc</font>
 <b>by </b><i><a class="ref" href="yellow_2.html#T10">YELLOW_2:10</a></i>;<br/>


</div>
