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<b>let </b><font color="Maroon">S</font> be  <a href="lattice3.html#V3">complete</a> <a href="waybel_3.html#V3">continuous</a> <a href="waybel_0.html#NM8">LATTICE</a>, <font color="Maroon">T</font> be  <a href="lattice3.html#V3">complete</a> <a href="waybel_3.html#V3">continuous</a> <a href="waybel_0.html#NM8">LATTICE</a>;<br/>

<b>consider </b><font color="Maroon">X</font> being  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> , <font color="Maroon">p</font> being  <a href="waybel_1.html#V6">projection</a> <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <font color="Maroon">X</font></span>)</span>,<span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <font color="Maroon">X</font></span>)</span><b> such that </b><br/><a NAME="E2:62"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">p</font> is <a href="waybel_0.html#V22">directed-sups-preserving</a> &amp; <font color="Maroon">T</font>, <a href="yellow_2.html#K1">Image</a> <font color="Maroon">p</font> <a href="waybel_1.html#R5">are_isomorphic</a>  )
 <b>by </b><i><a class="ref" href="waybel15.html#T20">WAYBEL15:20</a></i>;<br/>
<a NAME="E3:62"/>
( <font color="Maroon">p</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">p</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">p</font> &amp; <span class="p1">(<span class="default"><a href="grcat_1.html#K7">id</a> <font color="Maroon">S</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><a href="grcat_1.html#K7">id</a> <font color="Maroon">S</font></span>)</span> <a href="funct_2.html#R2">=</a>  <a href="grcat_1.html#K7">id</a> <font color="Maroon">S</font> )
 
<b>by </b><i><a class="ref" href="quantal1.html#D9">QUANTAL1:def 9</a></i>;<br/>
<a NAME="E4:62"/><b>then </b>
<span class="p1">(<span class="default"><a href="waybel27.html#K4" target="_self">UPS</a> <span class="p2">(<span class="default"><a href="grcat_1.html#K7">id</a> <font color="Maroon">S</font></span>)</span>,<font color="Maroon">p</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><a href="waybel27.html#K4" target="_self">UPS</a> <span class="p2">(<span class="default"><a href="grcat_1.html#K7">id</a> <font color="Maroon">S</font></span>)</span>,<font color="Maroon">p</font></span>)</span> <a href="funct_2.html#R2">=</a>  <a href="waybel27.html#K4" target="_self">UPS</a> <span class="p1">(<span class="default"><a href="grcat_1.html#K7">id</a> <font color="Maroon">S</font></span>)</span>,<font color="Maroon">p</font>
 
<b>by </b><i>, <a class="ref" href="waybel27.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E5:62"/><b>then </b>
 <a href="waybel27.html#K4" target="_self">UPS</a> <span class="p1">(<span class="default"><a href="grcat_1.html#K7">id</a> <font color="Maroon">S</font></span>)</span>,<font color="Maroon">p</font> is  <a href="quantal1.html#V11">idempotent</a> <a href="waybel_0.html#V22">directed-sups-preserving</a> <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="waybel27.html#K2" target="_self">UPS</a> <font color="Maroon">S</font>,<span class="p2">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <font color="Maroon">X</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="waybel27.html#K2" target="_self">UPS</a> <font color="Maroon">S</font>,<span class="p2">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <font color="Maroon">X</font></span>)</span></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="quantal1.html#D9">QUANTAL1:def 9</a></i>;<br/>
<a NAME="E6:62"/><b>then </b><i><font color="Green">E38</font></i>: 
 <a href="waybel27.html#K4" target="_self">UPS</a> <span class="p1">(<span class="default"><a href="grcat_1.html#K7">id</a> <font color="Maroon">S</font></span>)</span>,<font color="Maroon">p</font> is  <a href="waybel_0.html#V22">directed-sups-preserving</a> <a href="waybel_1.html#V6">projection</a> <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="waybel27.html#K2" target="_self">UPS</a> <font color="Maroon">S</font>,<span class="p2">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <font color="Maroon">X</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="waybel27.html#K2" target="_self">UPS</a> <font color="Maroon">S</font>,<span class="p2">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <font color="Maroon">X</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="waybel_1.html#D13">WAYBEL_1:def 13</a></i>;<br/>
<a NAME="E7:62"/>
 <a href="yellow_1.html#K3">BoolePoset</a> <font color="Maroon">X</font>,<span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> 1</span>)</span> <a href="yellow_1.html#K6">|^</a> <font color="Maroon">X</font> <a href="waybel_1.html#R5">are_isomorphic</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E8:62"/><b>then </b>
(  <a href="waybel27.html#K2" target="_self">UPS</a> <font color="Maroon">S</font>,<span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <font color="Maroon">X</font></span>)</span>, <a href="waybel27.html#K2" target="_self">UPS</a> <font color="Maroon">S</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> 1</span>)</span> <a href="yellow_1.html#K6">|^</a> <font color="Maroon">X</font></span>)</span> <a href="waybel_1.html#R5">are_isomorphic</a>  &amp;  <a href="waybel27.html#K2" target="_self">UPS</a> <font color="Maroon">S</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> 1</span>)</span> <a href="yellow_1.html#K6">|^</a> <font color="Maroon">X</font></span>)</span>,<span class="p1">(<span class="default"><a href="waybel27.html#K2" target="_self">UPS</a> <font color="Maroon">S</font>,<span class="p2">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> 1</span>)</span></span>)</span> <a href="yellow_1.html#K6">|^</a> <font color="Maroon">X</font> <a href="waybel_1.html#R5">are_isomorphic</a>  )
 
<b>by </b><i><a class="txt" href="waybel27.html#E6:62"><i><font color="Green">E38</font></i></a>, </i>;<br/>
<a NAME="E9:62"/><b>then </b>
 <a href="waybel27.html#K2" target="_self">UPS</a> <font color="Maroon">S</font>,<span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <font color="Maroon">X</font></span>)</span>,<span class="p1">(<span class="default"><a href="waybel27.html#K2" target="_self">UPS</a> <font color="Maroon">S</font>,<span class="p2">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> 1</span>)</span></span>)</span> <a href="yellow_1.html#K6">|^</a> <font color="Maroon">X</font> <a href="waybel_1.html#R5">are_isomorphic</a> 
 
<b>by </b><i><a class="ref" href="waybel_1.html#T8">WAYBEL_1:8</a></i>;<br/>
<a NAME="E10:62"/><b>then </b><i><font color="Green">E39</font></i>: 
<span class="p1">(<span class="default"><a href="waybel27.html#K2" target="_self">UPS</a> <font color="Maroon">S</font>,<span class="p2">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> 1</span>)</span></span>)</span> <a href="yellow_1.html#K6">|^</a> <font color="Maroon">X</font>, <a href="waybel27.html#K2" target="_self">UPS</a> <font color="Maroon">S</font>,<span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <font color="Maroon">X</font></span>)</span> <a href="waybel_1.html#R5">are_isomorphic</a> 
 
<b>by </b><i><a class="ref" href="waybel_1.html#T7">WAYBEL_1:7</a></i>;<br/>
<b>consider </b><font color="Maroon">L</font> being  <a href="waybel11.html#V4">Scott</a>  <a href="yellow_9.html#M1">TopAugmentation</a> of <font color="Maroon">S</font>;<br/>
<a NAME="E12:62"/>
(  <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">L</font> #) <a href="hidden.html#R1">=</a>  <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">S</font> #) &amp;  <a href="yellow_1.html#K2">InclPoset</a> <span class="p1">(<span class="default"><a href="waybel11.html#K5">sigma</a> <font color="Maroon">L</font></span>)</span> is <a href="waybel_3.html#V3">continuous</a> )
 
<b>by </b><i><a class="ref" href="waybel14.html#T36">WAYBEL14:36</a>, <a class="ref" href="yellow_9.html#D4">YELLOW_9:def 4</a></i>;<br/>
<a NAME="E13:62"/><b>then </b><i><font color="Green">E48</font></i>: 
 <a href="yellow_1.html#K2">InclPoset</a> <span class="p1">(<span class="default"><a href="waybel11.html#K5">sigma</a> <font color="Maroon">S</font></span>)</span> is <a href="waybel_3.html#V3">continuous</a>
 
<b>by </b><i><a class="ref" href="yellow_9.html#T52">YELLOW_9:52</a></i>;<br/>
<a NAME="E14:62"/>
 <a href="waybel27.html#K2" target="_self">UPS</a> <font color="Maroon">S</font>,<span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> 1</span>)</span>, <a href="yellow_1.html#K2">InclPoset</a> <span class="p1">(<span class="default"><a href="waybel11.html#K5">sigma</a> <font color="Maroon">S</font></span>)</span> <a href="waybel_1.html#R5">are_isomorphic</a> 
 
<b>by </b><i><a class="txt" href="waybel27.html#E2:62"><i><font color="Green">E36</font></i></a></i>;<br/>
<a NAME="E15:62"/><b>then </b>
 <a href="yellow_1.html#K2">InclPoset</a> <span class="p1">(<span class="default"><a href="waybel11.html#K5">sigma</a> <font color="Maroon">S</font></span>)</span>, <a href="waybel27.html#K2" target="_self">UPS</a> <font color="Maroon">S</font>,<span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> 1</span>)</span> <a href="waybel_1.html#R5">are_isomorphic</a> 
 
<b>by </b><i><a class="ref" href="waybel_1.html#T7">WAYBEL_1:7</a></i>;<br/>
<a NAME="E16:62"/><b>then </b>
(  <a href="waybel27.html#K2" target="_self">UPS</a> <font color="Maroon">S</font>,<span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> 1</span>)</span> is <a href="waybel_3.html#V3">continuous</a> &amp;  <a href="waybel27.html#K2" target="_self">UPS</a> <font color="Maroon">S</font>,<span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> 1</span>)</span> is <a href="lattice3.html#V3">complete</a> )
 
<b>by </b><i>, <a class="ref" href="waybel15.html#T11">WAYBEL15:11</a></i>;<br/>
<a NAME="E17:62"/><b>then </b>
for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font> holds  <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p2">(<span class="default"><a href="waybel27.html#K2" target="_self">UPS</a> <font color="Maroon">S</font>,<span class="p3">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> 1</span>)</span></span>)</span></span>)</span> <a href="yellow16.html#K2">.</a> <font color="Olive">x</font> is  <a href="lattice3.html#V3">complete</a> <a href="waybel_3.html#V3">continuous</a> <a href="waybel_0.html#NM8">LATTICE</a>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/>
<a NAME="E18:62"/><b>then </b>
<font color="Maroon">X</font> <a href="waybel26.html#NK1" target="_self">-POS_prod</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p2">(<span class="default"><a href="waybel27.html#K2" target="_self">UPS</a> <font color="Maroon">S</font>,<span class="p3">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> 1</span>)</span></span>)</span></span>)</span> is <a href="waybel_3.html#V3">continuous</a>
 
<b>by </b><i><a class="ref" href="waybel20.html#T34">WAYBEL20:34</a></i>;<br/>
<a NAME="E19:62"/><b>then </b>
<span class="p1">(<span class="default"><a href="waybel27.html#K2" target="_self">UPS</a> <font color="Maroon">S</font>,<span class="p2">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> 1</span>)</span></span>)</span> <a href="yellow_1.html#K6">|^</a> <font color="Maroon">X</font> is <a href="waybel_3.html#V3">continuous</a>
 
<b>by </b><i><a class="ref" href="yellow_1.html#D5">YELLOW_1:def 5</a></i>;<br/>
<a NAME="E20:62"/><b>then </b>
 <a href="waybel27.html#K2" target="_self">UPS</a> <font color="Maroon">S</font>,<span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <font color="Maroon">X</font></span>)</span> is  <a href="waybel_3.html#V3">continuous</a> <a href="waybel_0.html#NM8">LATTICE</a>
 
<b>by </b><i><a class="txt" href="waybel27.html#E13:62"><i><font color="Green">E48</font></i></a>, <a class="ref" href="waybel15.html#T11">WAYBEL15:11</a></i>;<br/>
<a NAME="E21:62"/><b>then </b><i><font color="Green">E51</font></i>: 
 <a href="yellow_2.html#K1">Image</a> <span class="p1">(<span class="default"><a href="waybel27.html#K4" target="_self">UPS</a> <span class="p2">(<span class="default"><a href="grcat_1.html#K7">id</a> <font color="Maroon">S</font></span>)</span>,<font color="Maroon">p</font></span>)</span> is <a href="waybel_3.html#V3">continuous</a>
 
<b>by </b><i>, <a class="ref" href="waybel15.html#T17">WAYBEL15:17</a></i>;<br/>
<a NAME="E22:62"/>
 <a href="yellow_2.html#K1">Image</a> <font color="Maroon">p</font> is  non <a href="struct_0.html#V3">empty</a> <a href="lattice3.html#V3">complete</a> <a href="orders_2.html#NM1">Poset</a>
 
<b>by </b><i>, <a class="ref" href="waybel20.html#T18">WAYBEL20:18</a></i>;<br/>
<a NAME="E23:62"/><b>then </b>
 <a href="waybel27.html#K2" target="_self">UPS</a> <font color="Maroon">S</font>,<font color="Maroon">T</font>, <a href="waybel27.html#K2" target="_self">UPS</a> <font color="Maroon">S</font>,<span class="p1">(<span class="default"><a href="yellow_2.html#K1">Image</a> <font color="Maroon">p</font></span>)</span> <a href="waybel_1.html#R5">are_isomorphic</a> 
 
<b>by </b><i>, <a class="txt" href="waybel27.html#E6:62"><i><font color="Green">E38</font></i></a></i>;<br/>
<a NAME="E24:62"/><b>then </b>
 <a href="waybel27.html#K2" target="_self">UPS</a> <font color="Maroon">S</font>,<font color="Maroon">T</font>, <a href="yellow_2.html#K1">Image</a> <span class="p1">(<span class="default"><a href="waybel27.html#K4" target="_self">UPS</a> <span class="p2">(<span class="default"><a href="grcat_1.html#K7">id</a> <font color="Maroon">S</font></span>)</span>,<font color="Maroon">p</font></span>)</span> <a href="waybel_1.html#R5">are_isomorphic</a> 
 
<b>by </b><i>, <a class="txt" href="waybel27.html#E10:62"><i><font color="Green">E39</font></i></a></i>;<br/>
<a NAME="E25:62"/><b>then </b>
 <a href="yellow_2.html#K1">Image</a> <span class="p1">(<span class="default"><a href="waybel27.html#K4" target="_self">UPS</a> <span class="p2">(<span class="default"><a href="grcat_1.html#K7">id</a> <font color="Maroon">S</font></span>)</span>,<font color="Maroon">p</font></span>)</span>, <a href="waybel27.html#K2" target="_self">UPS</a> <font color="Maroon">S</font>,<font color="Maroon">T</font> <a href="waybel_1.html#R5">are_isomorphic</a> 
 
<b>by </b><i><a class="ref" href="waybel_1.html#T7">WAYBEL_1:7</a></i>;<br/>
<b>hence </b><a NAME="E26:62"/>
 <a href="waybel27.html#K2" target="_self">UPS</a> <font color="Maroon">S</font>,<font color="Maroon">T</font> is <a href="waybel_3.html#V3">continuous</a>
 <b>by </b><i>, <a class="ref" href="waybel15.html#T11">WAYBEL15:11</a></i>;<br/>


</div>
