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<b>let </b><font color="Maroon">S</font> be  non <a href="struct_0.html#V3">empty</a> <a href="waybel19.html#NM1">TopPoset</a>, <font color="Maroon">T</font> be  non <a href="struct_0.html#V3">empty</a> <a href="waybel19.html#NM1">TopPoset</a>;<br/><b>let </b><font color="Maroon">X</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="waybel_0.html#V2">filtered</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">f</font> be  <a href="orders_3.html#V5">monotone</a> <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">S</font>,<font color="Maroon">T</font>;<br/><b>let </b><font color="Maroon">Y</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="waybel_0.html#V2">filtered</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>;<br/>







<b>assume </b><a NAME="E1:39"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font>
 ;<br/>

<b>set </b><font color="Maroon">N</font> = <font color="Maroon">f</font> <a href="waybel_9.html#K7">*</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="yellow_9.html#K3">opp+id</a> </span>)</span>;<br/>
<b>set </b><font color="Maroon">M</font> = <font color="Maroon">Y</font> <a href="yellow_9.html#K3">opp+id</a> ;<br/>
<a NAME="E2:39"/><i><font color="Green">E30</font></i>: 
(  <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="waybel_9.html#K7">*</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="yellow_9.html#K3">opp+id</a> </span>)</span></span>)</span>,the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="waybel_9.html#K7">*</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="yellow_9.html#K3">opp+id</a> </span>)</span></span>)</span> #) <a href="hidden.html#R1">=</a>  <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="yellow_9.html#K3">opp+id</a> </span>)</span>,the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="yellow_9.html#K3">opp+id</a> </span>)</span> #) &amp; the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="waybel_9.html#K7">*</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="yellow_9.html#K3">opp+id</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="yellow_9.html#K3">opp+id</a> </span>)</span> )
 
<b>by </b><i><a class="ref" href="waybel_9.html#D8">WAYBEL_9:def 8</a></i>;<br/>
<a NAME="E3:39"/><i><font color="Green">E31</font></i>: 
( the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">Y</font> <a href="yellow_9.html#K3">opp+id</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">Y</font> <a href="yellow_9.html#K3">opp+id</a>  is  <a href="yellow_0.html#V4">full</a>  <a href="yellow_0.html#M1">SubRelStr</a> of <font color="Maroon">T</font> <a href="yellow_7.html#NK1" target="_self">opp</a>  &amp; the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon">Y</font> <a href="yellow_9.html#K3">opp+id</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="partfun1.html#K6">id</a> <font color="Maroon">Y</font> )
 
<b>by </b><i><a class="ref" href="waybel19.html#T27">WAYBEL19:27</a>, <a class="ref" href="yellow_9.html#T7">YELLOW_9:7</a></i>;<br/>
<a NAME="E4:39"/><i><font color="Green">E38</font></i>: 
( the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="yellow_9.html#K3">opp+id</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> &amp; <font color="Maroon">X</font> <a href="yellow_9.html#K3">opp+id</a>  is  <a href="yellow_0.html#V4">full</a>  <a href="yellow_0.html#M1">SubRelStr</a> of <font color="Maroon">S</font> <a href="yellow_7.html#NK1" target="_self">opp</a>  &amp; the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="yellow_9.html#K3">opp+id</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="partfun1.html#K6">id</a> <font color="Maroon">X</font> )
 
<b>by </b><i><a class="ref" href="waybel19.html#T27">WAYBEL19:27</a>, <a class="ref" href="yellow_9.html#T7">YELLOW_9:7</a></i>;<br/>
<a NAME="E5:39"/><b>then </b><i><font color="Green">E39</font></i>: 
( the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="waybel_9.html#K7">*</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="yellow_9.html#K3">opp+id</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">X</font> &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i>, <a class="ref" href="relat_1.html#T94">RELAT_1:94</a></i>;<br/>
<a NAME="E6:39"/><b>then </b><i><font color="Green">E42</font></i>: 
(  <a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="waybel_9.html#K7">*</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="yellow_9.html#K3">opp+id</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font> &amp;  <a href="relset_1.html#K4">dom</a> the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="waybel_9.html#K7">*</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="yellow_9.html#K3">opp+id</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> )
 
<b>by </b><i>, , <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relat_1.html#T148">RELAT_1:148</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">g</font> = <font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">X</font> as   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="waybel_9.html#K7">*</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="yellow_9.html#K3">opp+id</a> </span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">Y</font> <a href="yellow_9.html#K3">opp+id</a> </span>)</span> <b>by </b><i>, , , , , <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/>
<b>take </b>
<font color="Maroon">g</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_6.html#D12">YELLOW_6:def 12</a><br/>

<b>thus </b><a NAME="E8:39"/>
the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="waybel_9.html#K7">*</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="yellow_9.html#K3">opp+id</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon">Y</font> <a href="yellow_9.html#K3">opp+id</a> </span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">g</font>
 <b>by </b><i>, , , <a class="txt" href="waybel21.html#E1:39"><i><font color="Green">E26</font></i></a>, <a class="ref" href="relat_1.html#T79">RELAT_1:79</a></i>;<br/>

<b>let </b><font color="Maroon">m</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">Y</font> <a href="yellow_9.html#K3">opp+id</a> </span>)</span>;<br/>

<b>consider </b><font color="Maroon">n</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E10:39"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> &amp; <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> )
 <b>by </b><i>, , <a class="ref" href="funct_2.html#T115">FUNCT_2:115</a></i>;<br/>
<b>reconsider </b><font color="Maroon">n</font> = <font color="Maroon">n</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="waybel_9.html#K7">*</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="yellow_9.html#K3">opp+id</a> </span>)</span></span>)</span> <b>by </b><i>, <a class="ref" href="waybel21.html#D1" target="_self">Def1</a>, <a class="ref" href="yellow_9.html#T7">YELLOW_9:7</a></i>;<br/>
<b>take </b>
<font color="Maroon">n</font>
;<br/>

<b>let </b><font color="Maroon">p</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="waybel_9.html#K7">*</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="yellow_9.html#K3">opp+id</a> </span>)</span></span>)</span>;<br/>

<a NAME="E12:39"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 
<b>by </b><i>, </i>;<br/>
<b>then reconsider </b><font color="Maroon">n'</font> = <font color="Maroon">n</font>, <font color="Maroon">p'</font> = <font color="Maroon">p</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> <b>by </b><i><a class="ref" href="waybel21.html#D1" target="_self">Def1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">fp</font> = <font color="Maroon">f</font> <a href="yellow_6.html#K3">.</a> <font color="Maroon">p'</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">Y</font> <a href="yellow_9.html#K3">opp+id</a> </span>)</span> <b>by </b><i>, , , , <a class="ref" href="funct_2.html#T43">FUNCT_2:43</a></i>;<br/>
<a NAME="E15:39"/>
<font color="Maroon">X</font> <a href="yellow_9.html#K3">opp+id</a>  is    <a href="yellow_0.html#M1">SubRelStr</a> of <font color="Maroon">S</font> <a href="yellow_7.html#NK1" target="_self">opp</a> 
 
<b>by </b><i><a class="ref" href="yellow_9.html#T7">YELLOW_9:7</a></i>;<br/>
<a NAME="E16:39"/><b>then </b><i><font color="Green">E44</font></i>: 
( <font color="Maroon">f</font> <a href="waybel_9.html#K7">*</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="yellow_9.html#K3">opp+id</a> </span>)</span> is    <a href="yellow_0.html#M1">SubRelStr</a> of <font color="Maroon">S</font> <a href="yellow_7.html#NK1" target="_self">opp</a>  &amp; <font color="Maroon">n'</font> <a href="lattice3.html#K8">~</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">n'</font> &amp; <font color="Maroon">p'</font> <a href="lattice3.html#K8">~</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">p'</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E17:39"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">Y</font> <a href="yellow_9.html#K3">opp+id</a>  is  <a href="yellow_0.html#V4">full</a>  <a href="yellow_0.html#M1">SubRelStr</a> of <font color="Maroon">T</font> <a href="yellow_7.html#NK1" target="_self">opp</a> 
 
<b>by </b><i><a class="ref" href="yellow_9.html#T7">YELLOW_9:7</a></i>;<br/>
<b>assume </b><a NAME="E18:39"/>
<font color="Maroon">n</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">p</font>
 ;<br/>

<a NAME="E19:39"/><b>then </b>
<font color="Maroon">n'</font> <a href="lattice3.html#K8">~</a>  <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">p'</font> <a href="lattice3.html#K8">~</a> 
 
<b>by </b><i>, <a class="ref" href="yellow_0.html#T60">YELLOW_0:60</a></i>;<br/>
<a NAME="E20:39"/><b>then </b>
<font color="Maroon">n'</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">p'</font>
 
<b>by </b><i><a class="ref" href="lattice3.html#T9">LATTICE3:9</a></i>;<br/>
<a NAME="E21:39"/><b>then </b>
<font color="Maroon">f</font> <a href="yellow_6.html#K3">.</a> <font color="Maroon">n'</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">f</font> <a href="yellow_6.html#K3">.</a> <font color="Maroon">p'</font>
 
<b>by </b><i><a class="ref" href="waybel_1.html#D2">WAYBEL_1:def 2</a></i>;<br/>
<a NAME="E22:39"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="yellow_6.html#K3">.</a> <font color="Maroon">n'</font></span>)</span> <a href="lattice3.html#K8">~</a>  <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="yellow_6.html#K3">.</a> <font color="Maroon">p'</font></span>)</span> <a href="lattice3.html#K8">~</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="yellow_6.html#K3">.</a> <font color="Maroon">n'</font></span>)</span> <a href="lattice3.html#K8">~</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="yellow_6.html#K3">.</a> <font color="Maroon">p'</font></span>)</span> <a href="lattice3.html#K8">~</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">fp</font> &amp; the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">Y</font> <a href="yellow_9.html#K3">opp+id</a> </span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i><a class="ref" href="waybel21.html#D1" target="_self">Def1</a>, <a class="ref" href="lattice3.html#T9">LATTICE3:9</a></i>;<br/>
<a NAME="E23:39"/><b>then </b>
<font color="Maroon">fp</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">m</font>
 
<b>by </b><i>, <a class="ref" href="yellow_0.html#T61">YELLOW_0:61</a></i>;<br/>
<b>hence </b><a NAME="E24:39"/>
<font color="Maroon">m</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">g</font> <a href="yellow_6.html#K3">.</a> <font color="Maroon">p</font>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/>


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