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

<b>set </b><font color="Maroon">c<sub>1</sub></font> =  <a href="waybel18.html#K9">Sierpinski_Space</a> ;<br/>
<b>set </b><font color="Maroon">c<sub>2</sub></font> =  <a href="yellow_1.html#K3">BoolePoset</a> 1;<br/>
<b>let </b><font color="Maroon">c<sub>3</sub></font> be  <a href="waybel18.html#V2">injective</a> <a href="t_0topsp.html#NM1">T_0-TopSpace</a>;<br/>

<b>consider </b><font color="Maroon">c<sub>4</sub></font> being  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E2:38"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="waybel18.html#R1">is_Retract_of</a>  <a href="waybel18.html#K3">product</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9">Sierpinski_Space</a> </span>)</span>
 <b>by </b><i><a class="ref" href="waybel18.html#T20">WAYBEL18:20</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="struct_0.html#NM7">Function</a> of <span class="p1">(<span class="default"><a href="waybel18.html#K3">product</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9">Sierpinski_Space</a> </span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="waybel18.html#K3">product</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9">Sierpinski_Space</a> </span>)</span></span>)</span><b> such that </b><br/><a NAME="E4:38"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>5</sub></font> is <a href="pre_topc.html#V5">continuous</a>
 <b>and </b><br/><a NAME="E5:38"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>and </b><br/><a NAME="E6:38"/><i><font color="Green">E24</font></i>: 
 <a href="waybel18.html#K7">Image</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="borsuk_3.html#R1">are_homeomorphic</a> 
 <b>by </b><i><a class="txt" href="waybel25.html#E2:38"><i><font color="Green">E21</font></i></a>, <a class="ref" href="waybel18.html#D8">WAYBEL18:def 8</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>6</sub></font> being  <a href="waybel11.html#V4">Scott</a>  <a href="yellow_9.html#M1">TopAugmentation</a> of  <a href="yellow_1.html#K5">product</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> 1</span>)</span></span>)</span>;<br/>
<a NAME="E8:38"/><i><font color="Green">E25</font></i>: 
the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="waybel18.html#K3">product</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9">Sierpinski_Space</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="waybel18.html#T16">WAYBEL18:16</a></i>;<br/>
<a NAME="E9:38"/><i><font color="Green">E26</font></i>: 
 <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>6</sub></font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>6</sub></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 <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> 1</span>)</span></span>)</span></span>)</span>,the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> 1</span>)</span></span>)</span></span>)</span> #)
 
<b>by </b><i><a class="ref" href="yellow_9.html#D4">YELLOW_9:def 4</a></i>;<br/>
<a NAME="E10:38"/><i><font color="Green">E27</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>6</sub></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="waybel18.html#K3">product</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9">Sierpinski_Space</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="waybel25.html#T3" target="_self">Th3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>5</sub></font> as   <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>6</sub></font> ;<br/>
<a NAME="E12:38"/>
(  <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>6</sub></font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>6</sub></font> #),the <a href="pre_topc.html#U1">topology</a> of <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>6</sub></font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>6</sub></font> #) #) <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>6</sub></font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>6</sub></font> #) implies  <a href="waybel25.html#K1" target="_self">Omega</a> <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>6</sub></font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>6</sub></font> #) <a href="hidden.html#R1">=</a>  <a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">c<sub>6</sub></font> )
 
<b>by </b><i><a class="ref" href="waybel25.html#T13" target="_self">Th13</a></i>;<br/>
<a NAME="E13:38"/><b>then </b><i><font color="Green">E28</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"><a href="waybel25.html#K1" target="_self">Omega</a> <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel18.html#K3">product</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9">Sierpinski_Space</a> </span>)</span></span>)</span>,the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="waybel18.html#K3">product</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9">Sierpinski_Space</a> </span>)</span></span>)</span> #)</span>)</span>,the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel18.html#K3">product</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9">Sierpinski_Space</a> </span>)</span></span>)</span>,the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="waybel18.html#K3">product</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9">Sierpinski_Space</a> </span>)</span></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"><a href="yellow_1.html#K5">product</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> 1</span>)</span></span>)</span></span>)</span>,the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> 1</span>)</span></span>)</span></span>)</span> #)
 
<b>by </b><i><a class="txt" href="waybel25.html#E8:38"><i><font color="Green">E25</font></i></a>, <a class="txt" href="waybel25.html#E10:38"><i><font color="Green">E27</font></i></a>, <a class="ref" href="waybel25.html#T16" target="_self">Th16</a></i>;<br/>
<a NAME="E14:38"/>
<font color="Maroon">c<sub>7</sub></font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i><a class="txt" href="waybel25.html#E4:38"><i><font color="Green">E22</font></i></a>, <a class="txt" href="waybel25.html#E8:38"><i><font color="Green">E25</font></i></a>, <a class="txt" href="waybel25.html#E10:38"><i><font color="Green">E27</font></i></a>, <a class="ref" href="yellow12.html#T36">YELLOW12:36</a></i>;<br/>
<a NAME="E15:38"/><b>then </b><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>7</sub></font> is <a href="waybel_0.html#V22">directed-sups-preserving</a>
 
<b>by </b><i><a class="ref" href="waybel17.html#T22">WAYBEL17:22</a></i>;<br/>
<a NAME="E16:38"/>
<font color="Maroon">c<sub>7</sub></font> is <a href="waybel_1.html#V6">projection</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:38_1"/>
( <font color="Maroon">c<sub>7</sub></font> is <a href="quantal1.html#V11">idempotent</a> &amp; <font color="Maroon">c<sub>7</sub></font> is <a href="orders_3.html#V5">monotone</a> )
 <b>by </b><i><a class="txt" href="waybel25.html#E5:38"><i><font color="Green">E23</font></i></a>, <a class="txt" href="waybel25.html#E15:38"><i><font color="Green">E29</font></i></a>, <a class="ref" href="quantal1.html#D9">QUANTAL1:def 9</a>, <a class="ref" href="waybel17.html#T3">WAYBEL17:3</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_1.html#D13">WAYBEL_1:def 13</a><br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</sub></font> as  <a href="waybel_1.html#V6">projection</a> <a href="struct_0.html#NM7">Function</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> 1</span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> 1</span>)</span></span>)</span></span>)</span> <b>by </b><i><a class="txt" href="waybel25.html#E9:38"><i><font color="Green">E26</font></i></a>, <a class="txt" href="waybel25.html#E25"><i><font color="Green">Lemma19</font></i></a></i>;<br/>
<a NAME="E18:38"/>
<font color="Maroon">c<sub>8</sub></font> is <a href="waybel_0.html#V22">directed-sups-preserving</a>
 
<b>by </b><i><a class="txt" href="waybel25.html#E9:38"><i><font color="Green">E26</font></i></a>, <a class="txt" href="waybel25.html#E15:38"><i><font color="Green">E29</font></i></a>, <a class="ref" href="waybel21.html#T6">WAYBEL21:6</a></i>;<br/>
<a NAME="E19:38"/><b>then </b><i><font color="Green">E30</font></i>: 
 <a href="yellow_2.html#K1">Image</a> <font color="Maroon">c<sub>8</sub></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="waybel15.html#T17">WAYBEL15:17</a>, <a class="ref" href="yellow_2.html#T37">YELLOW_2:37</a></i>;<br/>
<a NAME="E20:38"/><i><font color="Green">E31</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"><a href="waybel25.html#K1" target="_self">Omega</a> <span class="p2">(<span class="default"><a href="waybel18.html#K7">Image</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>,the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <span class="p2">(<span class="default"><a href="waybel18.html#K7">Image</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> #), <a href="waybel25.html#K1" target="_self">Omega</a> <span class="p1">(<span class="default"><a href="waybel18.html#K7">Image</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="waybel_1.html#R5">are_isomorphic</a> 
 
<b>by </b><i><a class="ref" href="waybel13.html#T26">WAYBEL13:26</a></i>;<br/>
<a NAME="E21:38"/>
 <a href="waybel25.html#K1" target="_self">Omega</a> <span class="p1">(<span class="default"><a href="waybel18.html#K7">Image</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>, <a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">c<sub>3</sub></font> <a href="waybel_1.html#R5">are_isomorphic</a> 
 
<b>by </b><i><a class="txt" href="waybel25.html#E6:38"><i><font color="Green">E24</font></i></a>, <a class="ref" href="waybel25.html#T19" target="_self">Th19</a></i>;<br/>
<a NAME="E22:38"/><b>then </b><i><font color="Green">E32</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"><a href="waybel25.html#K1" target="_self">Omega</a> <span class="p2">(<span class="default"><a href="waybel18.html#K7">Image</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>,the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <span class="p2">(<span class="default"><a href="waybel18.html#K7">Image</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> #), <a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">c<sub>3</sub></font> <a href="waybel_1.html#R5">are_isomorphic</a> 
 
<b>by </b><i><a class="txt" href="waybel25.html#E20:38"><i><font color="Green">E31</font></i></a>, <a class="ref" href="waybel_1.html#T8">WAYBEL_1:8</a></i>;<br/>
<i><font color="Green">E33</font></i>: the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_2.html#K1">Image</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> = 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>
<br/>.= 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel18.html#K7">Image</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="waybel18.html#T10">WAYBEL18:10</a></i>
<br/>.= 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <span class="p2">(<span class="default"><a href="waybel18.html#K7">Image</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="waybel25.html#E15"><i><font color="Green">Lemma10</font></i></a></i>
;<br/>
<a NAME="E24:38"/><i><font color="Green">E34</font></i>: 
the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_2.html#K1">Image</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> 1</span>)</span></span>)</span></span>)</span> <a href="toler_1.html#K1">|_2</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_2.html#K1">Image</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_0.html#D14">YELLOW_0:def 14</a></i>;<br/>
<a NAME="E25:38"/>
 <a href="waybel25.html#K1" target="_self">Omega</a> <span class="p1">(<span class="default"><a href="waybel18.html#K7">Image</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> is  <a href="yellow_0.html#V4">full</a>  <a href="yellow_0.html#M1">SubRelStr</a> of  <a href="waybel25.html#K1" target="_self">Omega</a> <span class="p1">(<span class="default"><a href="waybel18.html#K3">product</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9">Sierpinski_Space</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="waybel25.html#T17" target="_self">Th17</a></i>;<br/>
<a NAME="E26:38"/><b>then </b>
the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <span class="p2">(<span class="default"><a href="waybel18.html#K7">Image</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <span class="p2">(<span class="default"><a href="waybel18.html#K3">product</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9">Sierpinski_Space</a> </span>)</span></span>)</span></span>)</span> <a href="toler_1.html#K1">|_2</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <span class="p2">(<span class="default"><a href="waybel18.html#K7">Image</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_0.html#D14">YELLOW_0:def 14</a></i>;<br/>
<b>hence </b><a NAME="E27:38"/>
 <a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">c<sub>3</sub></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="txt" href="waybel25.html#E13:38"><i><font color="Green">E28</font></i></a>, <a class="txt" href="waybel25.html#E19:38"><i><font color="Green">E30</font></i></a>, <a class="txt" href="waybel25.html#E22:38"><i><font color="Green">E32</font></i></a>, <a class="txt" href="waybel25.html#E23:38"><i><font color="Green">E33</font></i></a>, <a class="txt" href="waybel25.html#E24:38"><i><font color="Green">E34</font></i></a>, <a class="ref" href="waybel15.html#T11">WAYBEL15:11</a>, <a class="ref" href="waybel20.html#T18">WAYBEL20:18</a>, <a class="ref" href="yellow14.html#T12">YELLOW14:12</a>, <a class="ref" href="yellow14.html#T13">YELLOW14:13</a></i>;<br/>


</div>
