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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/>



<b>set </b><font color="Maroon">c<sub>3</sub></font> =  <a href="waybel18.html#K9">Sierpinski_Space</a> ;<br/>
<b>set </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>2</sub></font> <a href="waybel26.html#NK2" target="_self">-TOP_prod</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9">Sierpinski_Space</a> </span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>5</sub></font> =  <a href="waybel26.html#K1" target="_self">oContMaps</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel26.html#NK2" target="_self">-TOP_prod</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9">Sierpinski_Space</a> </span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> =  <a href="waybel26.html#K1" target="_self">oContMaps</a> <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>7</sub></font> = <font color="Maroon">c<sub>2</sub></font> <a href="waybel26.html#NK1" target="_self">-POS_prod</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p2">(<span class="default"><a href="waybel26.html#K1" target="_self">oContMaps</a> <font color="Maroon">c<sub>1</sub></font>,<a href="waybel18.html#K9">Sierpinski_Space</a> </span>)</span></span>)</span>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="waybel26.html#K1" target="_self">oContMaps</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel26.html#NK2" target="_self">-TOP_prod</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9">Sierpinski_Space</a> </span>)</span></span>)</span></span>)</span>) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  =  <a href="funct_6.html#K10">commute</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<b>consider </b><font color="Maroon">c<sub>8</sub></font> being    <a href="pboole.html#M1">ManySortedSet</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel26.html#K1" target="_self">oContMaps</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel26.html#NK2" target="_self">-TOP_prod</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9">Sierpinski_Space</a> </span>)</span></span>)</span></span>)</span><b> such that </b><br/><a NAME="E2:46"/><i><font color="Green">E38</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="waybel26.html#K1" target="_self">oContMaps</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel26.html#NK2" target="_self">-TOP_prod</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9">Sierpinski_Space</a> </span>)</span></span>)</span></span>)</span> holds <font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>)
 <b>from </b><i><a class="ref" href="pboole.html#S5">PBOOLE:sch 5</a>();<br/></i>
<a NAME="E3:46"/><i><font color="Green">E39</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>8</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="waybel26.html#K1" target="_self">oContMaps</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel26.html#NK2" target="_self">-TOP_prod</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9">Sierpinski_Space</a> </span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E4:46"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>8</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel26.html#NK1" target="_self">-POS_prod</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p3">(<span class="default"><a href="waybel26.html#K1" target="_self">oContMaps</a> <font color="Maroon">c<sub>1</sub></font>,<a href="waybel18.html#K9">Sierpinski_Space</a> </span>)</span></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:46_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>10</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:46_1"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>10</sub></font> as  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel26.html#NK2" target="_self">-TOP_prod</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</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="txt" href="waybel26.html#E3:46"><i><font color="Green">E39</font></i></a>, <a class="txt" href="waybel26.html#E3:46_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="waybel26.html#T2" target="_self">Th2</a></i>;<br/>
<a NAME="E5:46_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="funct_6.html#K10">commute</a> <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="txt" href="waybel26.html#E2:46"><i><font color="Green">E38</font></i></a>, <a class="txt" href="waybel26.html#E3:46"><i><font color="Green">E39</font></i></a>, <a class="txt" href="waybel26.html#E3:46_1"><i><font color="Green">E40</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>9</sub></font> as   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel26.html#K1" target="_self">oContMaps</a> <font color="Maroon">c<sub>1</sub></font>,<a href="waybel18.html#K9">Sierpinski_Space</a> </span>)</span> <b>by </b><i><a class="ref" href="waybel26.html#T32" target="_self">Th32</a></i>;<br/>
<a NAME="E7:46_1"/>
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>12</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel26.html#K1" target="_self">oContMaps</a> <font color="Maroon">c<sub>1</sub></font>,<a href="waybel18.html#K9">Sierpinski_Space</a> </span>)</span> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E8:46_1"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">c<sub>2</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel26.html#K1" target="_self">oContMaps</a> <font color="Maroon">c<sub>1</sub></font>,<a href="waybel18.html#K9">Sierpinski_Space</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/>
<a NAME="E9:46_1"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel26.html#K1" target="_self">oContMaps</a> <font color="Maroon">c<sub>1</sub></font>,<a href="waybel18.html#K9">Sierpinski_Space</a> </span>)</span> <a href="yellow_1.html#K6">|^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_1.html#T28">YELLOW_1:28</a></i>;<br/>
<b>hence </b><a NAME="E10:46_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel26.html#NK1" target="_self">-POS_prod</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p3">(<span class="default"><a href="waybel26.html#K1" target="_self">oContMaps</a> <font color="Maroon">c<sub>1</sub></font>,<a href="waybel18.html#K9">Sierpinski_Space</a> </span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="yellow_1.html#D5">YELLOW_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>8</sub></font> as   <a href="struct_0.html#NM7">Function</a> of <span class="p1">(<span class="default"><a href="waybel26.html#K1" target="_self">oContMaps</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel26.html#NK2" target="_self">-TOP_prod</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9">Sierpinski_Space</a> </span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel26.html#NK1" target="_self">-POS_prod</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p3">(<span class="default"><a href="waybel26.html#K1" target="_self">oContMaps</a> <font color="Maroon">c<sub>1</sub></font>,<a href="waybel18.html#K9">Sierpinski_Space</a> </span>)</span></span>)</span></span>)</span> <b>by </b><i><a class="txt" href="waybel26.html#E3:46"><i><font color="Green">E39</font></i></a>, <a class="ref" href="funct_2.html#T4">FUNCT_2:4</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>9</sub></font>
;<br/>

<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel26.html#NK1" target="_self">-POS_prod</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p3">(<span class="default"><a href="waybel26.html#K1" target="_self">oContMaps</a> <font color="Maroon">c<sub>1</sub></font>,<a href="waybel18.html#K9">Sierpinski_Space</a> </span>)</span></span>)</span></span>)</span>) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  =  <a href="funct_6.html#K10">commute</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<b>consider </b><font color="Maroon">c<sub>10</sub></font> being    <a href="pboole.html#M1">ManySortedSet</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel26.html#NK1" target="_self">-POS_prod</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p3">(<span class="default"><a href="waybel26.html#K1" target="_self">oContMaps</a> <font color="Maroon">c<sub>1</sub></font>,<a href="waybel18.html#K9">Sierpinski_Space</a> </span>)</span></span>)</span></span>)</span><b> such that </b><br/><a NAME="E7:46"/><i><font color="Green">E40</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel26.html#NK1" target="_self">-POS_prod</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p3">(<span class="default"><a href="waybel26.html#K1" target="_self">oContMaps</a> <font color="Maroon">c<sub>1</sub></font>,<a href="waybel18.html#K9">Sierpinski_Space</a> </span>)</span></span>)</span></span>)</span> holds <font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b<sub>1</sub></font>)
 <b>from </b><i><a class="ref" href="pboole.html#S5">PBOOLE:sch 5</a>();<br/></i>
<a NAME="E8:46"/><i><font color="Green">E41</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel26.html#NK1" target="_self">-POS_prod</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p3">(<span class="default"><a href="waybel26.html#K1" target="_self">oContMaps</a> <font color="Maroon">c<sub>1</sub></font>,<a href="waybel18.html#K9">Sierpinski_Space</a> </span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E9:46"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>10</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel26.html#K1" target="_self">oContMaps</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel26.html#NK2" target="_self">-TOP_prod</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9">Sierpinski_Space</a> </span>)</span></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>11</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:46_2"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>12</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:46_2"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E4:46_2"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel26.html#K1" target="_self">oContMaps</a> <font color="Maroon">c<sub>1</sub></font>,<a href="waybel18.html#K9">Sierpinski_Space</a> </span>)</span> <a href="yellow_1.html#K6">|^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel26.html#E8:46"><i><font color="Green">E41</font></i></a>, <a class="txt" href="waybel26.html#E3:46_2"><i><font color="Green">E42</font></i></a>, <a class="ref" href="yellow_1.html#D5">YELLOW_1:def 5</a></i>;<br/>
<a NAME="E5:46_2"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">c<sub>2</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel26.html#K1" target="_self">oContMaps</a> <font color="Maroon">c<sub>1</sub></font>,<a href="waybel18.html#K9">Sierpinski_Space</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_1.html#T28">YELLOW_1:28</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>13</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E7:46_2"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>13</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel26.html#K1" target="_self">oContMaps</a> <font color="Maroon">c<sub>1</sub></font>,<a href="waybel18.html#K9">Sierpinski_Space</a> </span>)</span> )
 <b>by </b><i><a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/>
<a NAME="E8:46_2"/>
( <font color="Maroon">c<sub>13</sub></font> is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel26.html#K1" target="_self">oContMaps</a> <font color="Maroon">c<sub>1</sub></font>,<a href="waybel18.html#K9">Sierpinski_Space</a> </span>)</span> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a>  <a href="funct_6.html#K10">commute</a> <font color="Maroon">c<sub>13</sub></font> )
 
<b>by </b><i><a class="txt" href="waybel26.html#E7:46"><i><font color="Green">E40</font></i></a>, <a class="txt" href="waybel26.html#E8:46"><i><font color="Green">E41</font></i></a>, <a class="txt" href="waybel26.html#E3:46_2"><i><font color="Green">E42</font></i></a>, <a class="txt" href="waybel26.html#E7:46_2"><i><font color="Green">E43</font></i></a>, <a class="ref" href="funct_2.html#T4">FUNCT_2:4</a></i>;<br/>
<a NAME="E9:46_2"/><b>then </b>
<font color="Maroon">c<sub>11</sub></font> is  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel26.html#NK2" target="_self">-TOP_prod</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</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="waybel26.html#T34" target="_self">Th34</a></i>;<br/>
<a NAME="E10:46_2"/><b>then </b>
<font color="Maroon">c<sub>11</sub></font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="waybel26.html#K1" target="_self">oContMaps</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel26.html#NK2" target="_self">-TOP_prod</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9">Sierpinski_Space</a> </span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="waybel26.html#T2" target="_self">Th2</a></i>;<br/>
<b>hence </b><a NAME="E11:46_2"/>
<font color="Maroon">c<sub>11</sub></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="waybel26.html#K1" target="_self">oContMaps</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel26.html#NK2" target="_self">-TOP_prod</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9">Sierpinski_Space</a> </span>)</span></span>)</span></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>10</sub></font> as   <a href="struct_0.html#NM7">Function</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel26.html#NK1" target="_self">-POS_prod</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p3">(<span class="default"><a href="waybel26.html#K1" target="_self">oContMaps</a> <font color="Maroon">c<sub>1</sub></font>,<a href="waybel18.html#K9">Sierpinski_Space</a> </span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="waybel26.html#K1" target="_self">oContMaps</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel26.html#NK2" target="_self">-TOP_prod</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9">Sierpinski_Space</a> </span>)</span></span>)</span></span>)</span> <b>by </b><i><a class="txt" href="waybel26.html#E8:46"><i><font color="Green">E41</font></i></a>, <a class="ref" href="funct_2.html#T4">FUNCT_2:4</a></i>;<br/>
<a NAME="E11:46"/><i><font color="Green">E42</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel26.html#K1" target="_self">oContMaps</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel26.html#NK2" target="_self">-TOP_prod</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</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="tarski.html#R1">c=</a>  <a href="funct_2.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel26.html#NK2" target="_self">-TOP_prod</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</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="waybel26.html#T33" target="_self">Th33</a></i>;<br/>
<i><font color="Green">E43</font></i>: the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel26.html#NK2" target="_self">-TOP_prod</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9">Sierpinski_Space</a> </span>)</span></span>)</span> = 
 <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</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#D3">WAYBEL18:def 3</a></i>
<br/>.= 
 <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K2">--&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <a href="waybel18.html#K9">Sierpinski_Space</a> </span>)</span>
<b>by </b><i><a class="ref" href="waybel26.html#T31" target="_self">Th31</a></i>
<br/>.= 
 <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">c<sub>2</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <a href="waybel18.html#K9">Sierpinski_Space</a> 
<b>by </b><i><a class="ref" href="card_3.html#T20">CARD_3:20</a></i>
;<br/>
<i><font color="Green">E44</font></i>: the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel26.html#NK1" target="_self">-POS_prod</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p3">(<span class="default"><a href="waybel26.html#K1" target="_self">oContMaps</a> <font color="Maroon">c<sub>1</sub></font>,<a href="waybel18.html#K9">Sierpinski_Space</a> </span>)</span></span>)</span></span>)</span> = 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel26.html#K1" target="_self">oContMaps</a> <font color="Maroon">c<sub>1</sub></font>,<a href="waybel18.html#K9">Sierpinski_Space</a> </span>)</span> <a href="yellow_1.html#K6">|^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="yellow_1.html#D5">YELLOW_1:def 5</a></i>
<br/>.= 
 <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">c<sub>2</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel26.html#K1" target="_self">oContMaps</a> <font color="Maroon">c<sub>1</sub></font>,<a href="waybel18.html#K9">Sierpinski_Space</a> </span>)</span>
<b>by </b><i><a class="ref" href="yellow_1.html#T28">YELLOW_1:28</a></i>
;<br/>
<a NAME="E14:46"/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel26.html#K1" target="_self">oContMaps</a> <font color="Maroon">c<sub>1</sub></font>,<a href="waybel18.html#K9">Sierpinski_Space</a> </span>)</span> <a href="tarski.html#R1">c=</a>  <a href="funct_2.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <a href="waybel18.html#K9">Sierpinski_Space</a> 
 
<b>by </b><i><a class="ref" href="waybel26.html#T33" target="_self">Th33</a></i>;<br/>
<a NAME="E15:46"/><b>then </b><i><font color="Green">E45</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel26.html#NK1" target="_self">-POS_prod</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p3">(<span class="default"><a href="waybel26.html#K1" target="_self">oContMaps</a> <font color="Maroon">c<sub>1</sub></font>,<a href="waybel18.html#K9">Sierpinski_Space</a> </span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><a href="funct_2.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <a href="waybel18.html#K9">Sierpinski_Space</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="waybel26.html#E13:46"><i><font color="Green">E44</font></i></a>, <a class="ref" href="funct_5.html#T63">FUNCT_5:63</a></i>;<br/>
<a NAME="E16:46"/><i><font color="Green">E46</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">Omega</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel26.html#NK2" target="_self">-TOP_prod</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</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">Omega</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel26.html#NK2" target="_self">-TOP_prod</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</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> <font color="Maroon">c<sub>2</sub></font> <a href="waybel26.html#NK1" target="_self">-POS_prod</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p2">(<span class="default"><a href="waybel25.html#K1">Omega</a> <a href="waybel18.html#K9">Sierpinski_Space</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="waybel25.html#T14">WAYBEL25:14</a></i>;<br/>
<a NAME="E17:46"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>9</sub></font> is <a href="orders_3.html#V5">monotone</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>12</sub></font>, <font color="Maroon">c<sub>13</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="waybel26.html#K1" target="_self">oContMaps</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel26.html#NK2" target="_self">-TOP_prod</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9">Sierpinski_Space</a> </span>)</span></span>)</span></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_1.html#D2">WAYBEL_1:def 2</a><br/>

<b>assume </b><a NAME="E1:46_5"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>13</sub></font>
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>12</sub></font>, <font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>13</sub></font> as  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="waybel25.html#K1">Omega</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel26.html#NK2" target="_self">-TOP_prod</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9">Sierpinski_Space</a> </span>)</span></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="waybel26.html#T1" target="_self">Th1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>12</sub></font>, <font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>13</sub></font> as  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel26.html#NK2" target="_self">-TOP_prod</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</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="waybel26.html#T2" target="_self">Th2</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_5_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>18</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font>;<br/><a NAME="E1:46_5_1"/><i><font color="Green">E49</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p2">(<span class="default"><a href="waybel26.html#K1" target="_self">oContMaps</a> <font color="Maroon">c<sub>1</sub></font>,<a href="waybel18.html#K9">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="yellow16.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a>  <a href="waybel26.html#K1" target="_self">oContMaps</a> <font color="Maroon">c<sub>1</sub></font>,<a href="waybel18.html#K9">Sierpinski_Space</a> 
 <b>by </b><i><a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>19</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="waybel_3.html#K7">.</a> <font color="Maroon">c<sub>18</sub></font>, <font color="Maroon">c<sub>20</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="waybel_3.html#K7">.</a> <font color="Maroon">c<sub>18</sub></font> as  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="waybel25.html#K1">Omega</a> <a href="waybel18.html#K9">Sierpinski_Space</a> </span>)</span> <b>by </b><i><a class="ref" href="waybel26.html#T1" target="_self">Th1</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_5_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>21</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:46_5_1_1"/>
<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>22</sub></font> = <font color="Maroon">c<sub>21</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/><b>reconsider </b><font color="Maroon">c<sub>23</sub></font> = <font color="Maroon">c<sub>14</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>22</sub></font>, <font color="Maroon">c<sub>24</sub></font> = <font color="Maroon">c<sub>15</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>22</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel26.html#NK1" target="_self">-POS_prod</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p3">(<span class="default"><a href="waybel25.html#K1">Omega</a> <a href="waybel18.html#K9">Sierpinski_Space</a> </span>)</span></span>)</span></span>)</span> <b>by </b><i><a class="txt" href="waybel26.html#E16:46"><i><font color="Green">E46</font></i></a></i>;<br/><a NAME="E4:46_5_1_1"/>
( <font color="Maroon">c<sub>12</sub></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="waybel26.html#K1" target="_self">oContMaps</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel26.html#NK2" target="_self">-TOP_prod</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9">Sierpinski_Space</a> </span>)</span></span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>13</sub></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="waybel26.html#K1" target="_self">oContMaps</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel26.html#NK2" target="_self">-TOP_prod</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9">Sierpinski_Space</a> </span>)</span></span>)</span></span>)</span> )
 ;<br/><a NAME="E5:46_5_1_1"/><b>then </b>
( <font color="Maroon">c<sub>9</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a>  <a href="funct_6.html#K10">commute</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a>  <a href="funct_6.html#K10">commute</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <font color="Maroon">c<sub>2</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <a href="waybel18.html#K9">Sierpinski_Space</a> </span>)</span> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <font color="Maroon">c<sub>2</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <a href="waybel18.html#K9">Sierpinski_Space</a> </span>)</span> )
 <b>by </b><i><a class="txt" href="waybel26.html#E2:46"><i><font color="Green">E38</font></i></a>, <a class="txt" href="waybel26.html#E11:46"><i><font color="Green">E42</font></i></a>, <a class="txt" href="waybel26.html#E12:46"><i><font color="Green">E43</font></i></a></i>;<br/><a NAME="E6:46_5_1_1"/><b>then </b><i><font color="Green">E50</font></i>: 
( <font color="Maroon">c<sub>19</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="waybel18.html#K5">.</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <a href="waybel18.html#K5">.</a> <font color="Maroon">c<sub>18</sub></font> )
 <b>by </b><i><a class="ref" href="funct_6.html#T86">FUNCT_6:86</a></i>;<br/><a NAME="E7:46_5_1_1"/>
<font color="Maroon">c<sub>14</sub></font> <a href="yellow_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="txt" href="waybel26.html#E1:46_5"><i><font color="Green">E48</font></i></a>, <a class="ref" href="waybel26.html#T3" target="_self">Th3</a></i>;<br/><a NAME="E8:46_5_1_1"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1">Omega</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel26.html#NK2" target="_self">-TOP_prod</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9">Sierpinski_Space</a> </span>)</span></span>)</span></span>)</span> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">b<sub>2</sub></font> )
 <b>by </b><i><a class="ref" href="yellow_2.html#D1">YELLOW_2:def 1</a></i>;<br/><a NAME="E9:46_5_1_1"/><b>then </b>
<font color="Maroon">c<sub>23</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>24</sub></font>
 <b>by </b><i><a class="txt" href="waybel26.html#E16:46"><i><font color="Green">E46</font></i></a>, <a class="ref" href="yellow_0.html#T1">YELLOW_0:1</a></i>;<br/><a NAME="E10:46_5_1_1"/><b>then </b>
( <font color="Maroon">c<sub>23</sub></font> <a href="waybel_3.html#K7">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>24</sub></font> <a href="waybel_3.html#K7">.</a> <font color="Maroon">c<sub>18</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p2">(<span class="default"><a href="waybel25.html#K1">Omega</a> <a href="waybel18.html#K9">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="yellow16.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a>  <a href="waybel25.html#K1">Omega</a> <a href="waybel18.html#K9">Sierpinski_Space</a>  )
 <b>by </b><i><a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a>, <a class="ref" href="waybel_3.html#T28">WAYBEL_3:28</a></i>;<br/><b>hence </b><a NAME="E11:46_5_1_1"/>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1">Omega</a> <a href="waybel18.html#K9">Sierpinski_Space</a> </span>)</span> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">b<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="waybel26.html#E6:46_5_1_1"><i><font color="Green">E50</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E4:46_5_1"/><b>then </b>
<font color="Maroon">c<sub>19</sub></font> <a href="yellow_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>20</sub></font>
 <b>by </b><i><a class="ref" href="yellow_2.html#D1">YELLOW_2:def 1</a></i>;<br/><b>hence </b><a NAME="E5:46_5_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="waybel_3.html#K7">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="waybel_3.html#K7">.</a> <font color="Maroon">c<sub>18</sub></font>
 <b>by </b><i><a class="txt" href="waybel26.html#E1:46_5_1"><i><font color="Green">E49</font></i></a>, <a class="ref" href="waybel26.html#T3" target="_self">Th3</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:46_5"/>
<font color="Maroon">c<sub>9</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>9</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="ref" href="waybel_3.html#T28">WAYBEL_3:28</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E18:46"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>11</sub></font> is <a href="orders_3.html#V5">monotone</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>12</sub></font>, <font color="Maroon">c<sub>13</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel26.html#NK1" target="_self">-POS_prod</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p3">(<span class="default"><a href="waybel26.html#K1" target="_self">oContMaps</a> <font color="Maroon">c<sub>1</sub></font>,<a href="waybel18.html#K9">Sierpinski_Space</a> </span>)</span></span>)</span></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_1.html#D2">WAYBEL_1:def 2</a><br/>

<b>assume </b><a NAME="E1:46_6"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>13</sub></font>
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>11</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font>, <font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>11</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font> as  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="waybel25.html#K1">Omega</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel26.html#NK2" target="_self">-TOP_prod</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9">Sierpinski_Space</a> </span>)</span></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="waybel26.html#T1" target="_self">Th1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_6_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>16</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:46_6_1"/>
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>16</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/><b>reconsider </b><font color="Maroon">c<sub>18</sub></font> = <font color="Maroon">c<sub>14</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>17</sub></font>, <font color="Maroon">c<sub>19</sub></font> = <font color="Maroon">c<sub>15</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>17</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel26.html#NK1" target="_self">-POS_prod</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p3">(<span class="default"><a href="waybel25.html#K1">Omega</a> <a href="waybel18.html#K9">Sierpinski_Space</a> </span>)</span></span>)</span></span>)</span> <b>by </b><i><a class="txt" href="waybel26.html#E16:46"><i><font color="Green">E46</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_6_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>20</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font>;<br/><a NAME="E1:46_6_1_1"/><i><font color="Green">E50</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p2">(<span class="default"><a href="waybel26.html#K1" target="_self">oContMaps</a> <font color="Maroon">c<sub>1</sub></font>,<a href="waybel18.html#K9">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="yellow16.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a>  <a href="waybel26.html#K1" target="_self">oContMaps</a> <font color="Maroon">c<sub>1</sub></font>,<a href="waybel18.html#K9">Sierpinski_Space</a> 
 <b>by </b><i><a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>21</sub></font> = <font color="Maroon">c<sub>12</sub></font> <a href="waybel_3.html#K7">.</a> <font color="Maroon">c<sub>20</sub></font>, <font color="Maroon">c<sub>22</sub></font> = <font color="Maroon">c<sub>13</sub></font> <a href="waybel_3.html#K7">.</a> <font color="Maroon">c<sub>20</sub></font> as  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="waybel25.html#K1">Omega</a> <a href="waybel18.html#K9">Sierpinski_Space</a> </span>)</span> <b>by </b><i><a class="ref" href="waybel26.html#T1" target="_self">Th1</a></i>;<br/><a NAME="E3:46_6_1_1"/>
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel26.html#NK1" target="_self">-POS_prod</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p3">(<span class="default"><a href="waybel26.html#K1" target="_self">oContMaps</a> <font color="Maroon">c<sub>1</sub></font>,<a href="waybel18.html#K9">Sierpinski_Space</a> </span>)</span></span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel26.html#NK1" target="_self">-POS_prod</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p3">(<span class="default"><a href="waybel26.html#K1" target="_self">oContMaps</a> <font color="Maroon">c<sub>1</sub></font>,<a href="waybel18.html#K9">Sierpinski_Space</a> </span>)</span></span>)</span></span>)</span> )
 ;<br/><a NAME="E4:46_6_1_1"/><b>then </b>
( <font color="Maroon">c<sub>11</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a>  <a href="funct_6.html#K10">commute</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a>  <a href="funct_6.html#K10">commute</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><a href="funct_2.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <a href="waybel18.html#K9">Sierpinski_Space</a> </span>)</span> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><a href="funct_2.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <a href="waybel18.html#K9">Sierpinski_Space</a> </span>)</span> )
 <b>by </b><i><a class="txt" href="waybel26.html#E7:46"><i><font color="Green">E40</font></i></a>, <a class="txt" href="waybel26.html#E15:46"><i><font color="Green">E45</font></i></a></i>;<br/><a NAME="E5:46_6_1_1"/><b>then </b><i><font color="Green">E51</font></i>: 
( <font color="Maroon">c<sub>18</sub></font> <a href="waybel_3.html#K7">.</a> <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>17</sub></font> &amp; <font color="Maroon">c<sub>19</sub></font> <a href="waybel_3.html#K7">.</a> <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>17</sub></font> )
 <b>by </b><i><a class="ref" href="funct_6.html#T86">FUNCT_6:86</a></i>;<br/><a NAME="E6:46_6_1_1"/>
<font color="Maroon">c<sub>12</sub></font> <a href="waybel_3.html#K7">.</a> <font color="Maroon">c<sub>20</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>13</sub></font> <a href="waybel_3.html#K7">.</a> <font color="Maroon">c<sub>20</sub></font>
 <b>by </b><i><a class="txt" href="waybel26.html#E1:46_6"><i><font color="Green">E49</font></i></a>, <a class="ref" href="waybel_3.html#T28">WAYBEL_3:28</a></i>;<br/><a NAME="E7:46_6_1_1"/><b>then </b>
<font color="Maroon">c<sub>21</sub></font> <a href="yellow_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>22</sub></font>
 <b>by </b><i><a class="txt" href="waybel26.html#E1:46_6_1_1"><i><font color="Green">E50</font></i></a>, <a class="ref" href="waybel26.html#T3" target="_self">Th3</a></i>;<br/><a NAME="E8:46_6_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p2">(<span class="default"><a href="waybel25.html#K1">Omega</a> <a href="waybel18.html#K9">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="yellow16.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a>  <a href="waybel25.html#K1">Omega</a> <a href="waybel18.html#K9">Sierpinski_Space</a>  &amp; ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1">Omega</a> <a href="waybel18.html#K9">Sierpinski_Space</a> </span>)</span> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>17</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>17</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">b<sub>2</sub></font> ) )
 <b>by </b><i><a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a>, <a class="ref" href="yellow_2.html#D1">YELLOW_2:def 1</a></i>;<br/><b>hence </b><a NAME="E9:46_6_1_1"/>
<font color="Maroon">c<sub>18</sub></font> <a href="waybel_3.html#K7">.</a> <font color="Maroon">c<sub>20</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>19</sub></font> <a href="waybel_3.html#K7">.</a> <font color="Maroon">c<sub>20</sub></font>
 <b>by </b><i><a class="txt" href="waybel26.html#E5:46_6_1_1"><i><font color="Green">E51</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E5:46_6_1"/><b>then </b>
<font color="Maroon">c<sub>18</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>19</sub></font>
 <b>by </b><i><a class="ref" href="waybel_3.html#T28">WAYBEL_3:28</a></i>;<br/><a NAME="E6:46_6_1"/><b>then </b>
<font color="Maroon">c<sub>14</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>17</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>15</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="txt" href="waybel26.html#E16:46"><i><font color="Green">E46</font></i></a>, <a class="ref" href="yellow_0.html#T1">YELLOW_0:1</a></i>;<br/><b>hence </b><a NAME="E7:46_6_1"/>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1">Omega</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel26.html#NK2" target="_self">-TOP_prod</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9">Sierpinski_Space</a> </span>)</span></span>)</span></span>)</span> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">b<sub>2</sub></font> )
 ;<br/></div><b>end;</b></div>
<a NAME="E4:46_6"/><b>then </b>
<font color="Maroon">c<sub>14</sub></font> <a href="yellow_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>15</sub></font>
 
<b>by </b><i><a class="ref" href="yellow_2.html#D1">YELLOW_2:def 1</a></i>;<br/>
<b>hence </b><a NAME="E5:46_6"/>
<font color="Maroon">c<sub>11</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>11</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="ref" href="waybel26.html#T3" target="_self">Th3</a></i>;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_7"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>12</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel26.html#NK1" target="_self">-POS_prod</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p3">(<span class="default"><a href="waybel26.html#K1" target="_self">oContMaps</a> <font color="Maroon">c<sub>1</sub></font>,<a href="waybel18.html#K9">Sierpinski_Space</a> </span>)</span></span>)</span></span>)</span>;<br/><a NAME="E1:46_7"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">c<sub>2</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel26.html#K1" target="_self">oContMaps</a> <font color="Maroon">c<sub>1</sub></font>,<a href="waybel18.html#K9">Sierpinski_Space</a> </span>)</span>
 <b>by </b><i><a class="txt" href="waybel26.html#E13:46"><i><font color="Green">E44</font></i></a></i>;<br/><a NAME="E2:46_7"/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel26.html#K1" target="_self">oContMaps</a> <font color="Maroon">c<sub>1</sub></font>,<a href="waybel18.html#K9">Sierpinski_Space</a> </span>)</span> <a href="tarski.html#R1">c=</a>  <a href="funct_2.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <a href="waybel18.html#K9">Sierpinski_Space</a> 
 <b>by </b><i><a class="ref" href="waybel26.html#T33" target="_self">Th33</a></i>;<br/><a NAME="E3:46_7"/><b>then </b>
 <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">c<sub>2</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel26.html#K1" target="_self">oContMaps</a> <font color="Maroon">c<sub>1</sub></font>,<a href="waybel18.html#K9">Sierpinski_Space</a> </span>)</span> <a href="tarski.html#R1">c=</a>  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><a href="funct_2.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <a href="waybel18.html#K9">Sierpinski_Space</a> </span>)</span>
 <b>by </b><i><a class="ref" href="funct_5.html#T63">FUNCT_5:63</a></i>;<br/><a NAME="E4:46_7"/><b>then </b><i><font color="Green">E50</font></i>: 
 <a href="funct_6.html#K10">commute</a> <span class="p1">(<span class="default"><a href="funct_6.html#K10">commute</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="waybel26.html#E1:46_7"><i><font color="Green">E49</font></i></a>, <a class="ref" href="funct_6.html#T87">FUNCT_6:87</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font> = 
<font color="Maroon">c<sub>9</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
<br/>.= 
 <a href="funct_6.html#K10">commute</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="waybel26.html#E2:46"><i><font color="Green">E38</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>12</sub></font>
<b>by </b><i><a class="txt" href="waybel26.html#E7:46"><i><font color="Green">E40</font></i></a>, <a class="txt" href="waybel26.html#E4:46_7"><i><font color="Green">E50</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="grcat_1.html#K7">id</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel26.html#NK1" target="_self">-POS_prod</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p4">(<span class="default"><a href="waybel26.html#K1" target="_self">oContMaps</a> <font color="Maroon">c<sub>1</sub></font>,<a href="waybel18.html#K9">Sierpinski_Space</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font>
<b>by </b><i><a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>
;<br/></div><b>end;</b></div>
<a NAME="E20:46"/><b>then </b><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>11</sub></font> <a href="funct_2.html#R2">=</a>  <a href="grcat_1.html#K7">id</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel26.html#NK1" target="_self">-POS_prod</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p3">(<span class="default"><a href="waybel26.html#K1" target="_self">oContMaps</a> <font color="Maroon">c<sub>1</sub></font>,<a href="waybel18.html#K9">Sierpinski_Space</a> </span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_8"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>12</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="waybel26.html#K1" target="_self">oContMaps</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel26.html#NK2" target="_self">-TOP_prod</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9">Sierpinski_Space</a> </span>)</span></span>)</span></span>)</span>;<br/><a NAME="E1:46_8"/>
<font color="Maroon">c<sub>12</sub></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="waybel26.html#K1" target="_self">oContMaps</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel26.html#NK2" target="_self">-TOP_prod</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9">Sierpinski_Space</a> </span>)</span></span>)</span></span>)</span>
 ;<br/><a NAME="E2:46_8"/><b>then </b><i><font color="Green">E50</font></i>: 
 <a href="funct_6.html#K10">commute</a> <span class="p1">(<span class="default"><a href="funct_6.html#K10">commute</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="waybel26.html#E11:46"><i><font color="Green">E42</font></i></a>, <a class="txt" href="waybel26.html#E12:46"><i><font color="Green">E43</font></i></a>, <a class="ref" href="funct_6.html#T87">FUNCT_6:87</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font> = 
<font color="Maroon">c<sub>11</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
<br/>.= 
 <a href="funct_6.html#K10">commute</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="waybel26.html#E7:46"><i><font color="Green">E40</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>12</sub></font>
<b>by </b><i><a class="txt" href="waybel26.html#E2:46"><i><font color="Green">E38</font></i></a>, <a class="txt" href="waybel26.html#E2:46_8"><i><font color="Green">E50</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="grcat_1.html#K7">id</a> <span class="p2">(<span class="default"><a href="waybel26.html#K1" target="_self">oContMaps</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel26.html#NK2" target="_self">-TOP_prod</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9">Sierpinski_Space</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font>
<b>by </b><i><a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>
;<br/></div><b>end;</b></div>
<a NAME="E22:46"/><b>then </b>
<font color="Maroon">c<sub>11</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>9</sub></font> <a href="funct_2.html#R2">=</a>  <a href="grcat_1.html#K7">id</a> <span class="p1">(<span class="default"><a href="waybel26.html#K1" target="_self">oContMaps</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel26.html#NK2" target="_self">-TOP_prod</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9">Sierpinski_Space</a> </span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>;<br/>
<b>hence </b><a NAME="E23:46"/>
<font color="Maroon">c<sub>9</sub></font> is <a href="waybel_0.html#V23">isomorphic</a>
 <b>by </b><i><a class="txt" href="waybel26.html#E17:46"><i><font color="Green">E47</font></i></a>, <a class="txt" href="waybel26.html#E18:46"><i><font color="Green">E48</font></i></a>, <a class="txt" href="waybel26.html#E20:46"><i><font color="Green">E49</font></i></a>, <a class="ref" href="yellow16.html#T17">YELLOW16:17</a></i>;<br/>

<b>let </b><font color="Maroon">c<sub>12</sub></font> be  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel26.html#NK2" target="_self">-TOP_prod</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9">Sierpinski_Space</a> </span>)</span></span>)</span>;<br/>

<a NAME="E24:46"/>
<font color="Maroon">c<sub>12</sub></font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="waybel26.html#K1" target="_self">oContMaps</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel26.html#NK2" target="_self">-TOP_prod</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9">Sierpinski_Space</a> </span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="waybel26.html#T2" target="_self">Th2</a></i>;<br/>
<b>hence </b><a NAME="E25:46"/>
<font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a>  <a href="funct_6.html#K10">commute</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="waybel26.html#E2:46"><i><font color="Green">E38</font></i></a></i>;<br/>


</div>
