<?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="orders_2.html#L1">RelStr</a> ;<br/>

<b>set </b><font color="Maroon">c<sub>2</sub></font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : verum </span>}</span> ;<br/>
<a NAME="E1:5"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : verum </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="zfmisc_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</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="E1:5_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : verum </span>}</span> 
 ;<br/>

<a NAME="E2:5_1"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="subset_1.html#K3">`</a> 
 
;<br/>
<b>hence </b><a NAME="E3:5_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>3</sub></font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : verum </span>}</span>  as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<b>set </b><font color="Maroon">c<sub>4</sub></font> =  <a href="waybel_9.html#G1">TopRelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="cantor_1.html#K1">UniCl</a> <span class="p2">(<span class="default"><a href="cantor_1.html#K2">FinMeetCl</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> #);<br/>
<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> =  <a href="pre_topc.html#G1">TopStruct</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="cantor_1.html#K1">UniCl</a> <span class="p2">(<span class="default"><a href="cantor_1.html#K2">FinMeetCl</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> #) as  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a> <b>by </b><i><a class="ref" href="cantor_1.html#T17">CANTOR_1:17</a>, <a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</a></i>;<br/>
<a NAME="E4:5"/><i><font color="Green">E4</font></i>: 
 <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <a href="waybel_9.html#G1">TopRelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="cantor_1.html#K1">UniCl</a> <span class="p2">(<span class="default"><a href="cantor_1.html#K2">FinMeetCl</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> #),the <a href="pre_topc.html#U1">topology</a> of <a href="waybel_9.html#G1">TopRelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="cantor_1.html#K1">UniCl</a> <span class="p2">(<span class="default"><a href="cantor_1.html#K2">FinMeetCl</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> #) #) <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 
;<br/>
<a NAME="E5:5"/>
 <a href="waybel_9.html#G1">TopRelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="cantor_1.html#K1">UniCl</a> <span class="p2">(<span class="default"><a href="cantor_1.html#K2">FinMeetCl</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> #) is <a href="pre_topc.html#V2">TopSpace-like</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:5_2"/>
the <a href="struct_0.html#U1">carrier</a> of <a href="waybel_9.html#G1">TopRelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="cantor_1.html#K1">UniCl</a> <span class="p2">(<span class="default"><a href="cantor_1.html#K2">FinMeetCl</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> #) <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <a href="waybel_9.html#G1">TopRelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="cantor_1.html#K1">UniCl</a> <span class="p2">(<span class="default"><a href="cantor_1.html#K2">FinMeetCl</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> #)
 <b>by </b><i><a class="txt" href="waybel19.html#E4:5"><i><font color="Green">E4</font></i></a>, <a class="ref" href="pre_topc.html#D1">PRE_TOPC:def 1</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="pre_topc.html#D1">PRE_TOPC:def 1</a><br/>

<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="struct_0.html#NM3">Subset-Family</a> of <a href="waybel_9.html#G1">TopRelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="cantor_1.html#K1">UniCl</a> <span class="p2">(<span class="default"><a href="cantor_1.html#K2">FinMeetCl</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> #);<br/><b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font> as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>5</sub></font> ;<br/><b>assume </b><a NAME="E2:5_2_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> the <a href="pre_topc.html#U1">topology</a> of <a href="waybel_9.html#G1">TopRelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="cantor_1.html#K1">UniCl</a> <span class="p2">(<span class="default"><a href="cantor_1.html#K2">FinMeetCl</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> #)
 ;<br/><a NAME="E3:5_2_1"/><b>then </b>
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="pre_topc.html#D1">PRE_TOPC:def 1</a></i>;<br/><b>hence </b><a NAME="E4:5_2_1"/>
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <a href="waybel_9.html#G1">TopRelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="cantor_1.html#K1">UniCl</a> <span class="p2">(<span class="default"><a href="cantor_1.html#K2">FinMeetCl</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> #)
 ;<br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <a href="waybel_9.html#G1">TopRelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="cantor_1.html#K1">UniCl</a> <span class="p2">(<span class="default"><a href="cantor_1.html#K2">FinMeetCl</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> #);<br/>

<b>assume </b><a NAME="E3:5_2"/>
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <a href="waybel_9.html#G1">TopRelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="cantor_1.html#K1">UniCl</a> <span class="p2">(<span class="default"><a href="cantor_1.html#K2">FinMeetCl</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> #) &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <a href="waybel_9.html#G1">TopRelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="cantor_1.html#K1">UniCl</a> <span class="p2">(<span class="default"><a href="cantor_1.html#K2">FinMeetCl</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> #) )
 ;<br/>

<a NAME="E4:5_2"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D1">PRE_TOPC:def 1</a></i>;<br/>
<b>hence </b><a NAME="E5:5_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <a href="waybel_9.html#G1">TopRelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="cantor_1.html#K1">UniCl</a> <span class="p2">(<span class="default"><a href="cantor_1.html#K2">FinMeetCl</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> #)
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> =  <a href="waybel_9.html#G1">TopRelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="cantor_1.html#K1">UniCl</a> <span class="p2">(<span class="default"><a href="cantor_1.html#K2">FinMeetCl</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> #) as  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#V2">TopSpace-like</a> <a href="waybel_9.html#V1">strict</a>  <a href="waybel_9.html#L1">TopRelStr</a>  ;<br/>
<b>take </b>
<font color="Maroon">c<sub>6</sub></font>
;<br/>

<a NAME="E7:5"/>
 <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 <font color="Maroon">c<sub>1</sub></font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> #)
 
;<br/>
<b>hence </b><a NAME="E8:5"/>
<font color="Maroon">c<sub>6</sub></font> is  <a href="pre_topc.html#V2">correct</a> <a href="waybel_9.html#V1">strict</a>  <a href="yellow_9.html#M1">TopAugmentation</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="yellow_9.html#D4">YELLOW_9:def 4</a></i>;<br/>

<a NAME="E9:5"/><i><font color="Green">E5</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is    <a href="cantor_1.html#M2">prebasis</a> of <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="cantor_1.html#T20">CANTOR_1:20</a></i>;<br/>
<a NAME="E10:5"/><b>then </b><i><font color="Green">E6</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="cantor_1.html#D5">CANTOR_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>7</sub></font> being    <a href="cantor_1.html#M1">Basis</a> of <font color="Maroon">c<sub>5</sub></font><b> such that </b><br/><a NAME="E12:5"/><i><font color="Green">E7</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">c=</a>  <a href="cantor_1.html#K2">FinMeetCl</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="waybel19.html#E9:5"><i><font color="Green">E5</font></i></a>, <a class="ref" href="cantor_1.html#D5">CANTOR_1:def 5</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>8</sub></font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>6</sub></font> : verum </span>}</span> ;<br/>
<a NAME="E13:5"/>
 <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 <font color="Maroon">c<sub>1</sub></font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> #)
 
;<br/>
<a NAME="E14:5"/><b>then </b><i><font color="Green">E8</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>6</sub></font> : verum </span>}</span> 
 
<b>by </b><i><a class="txt" href="waybel19.html#E2"><i><font color="Green">Lemma2</font></i></a></i>;<br/>
<a NAME="E15:5"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">c=</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>6</sub></font> &amp; the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a>  <a href="cantor_1.html#K1">UniCl</a> <font color="Maroon">c<sub>7</sub></font> )
 
<b>by </b><i><a class="ref" href="cantor_1.html#D2">CANTOR_1:def 2</a></i>;<br/>
<a NAME="E16:5"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> is    <a href="cantor_1.html#M1">Basis</a> of <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="cantor_1.html#D2">CANTOR_1:def 2</a></i>;<br/>
<b>hence </b><a NAME="E17:5"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>6</sub></font> : verum </span>}</span>  is    <a href="cantor_1.html#M2">prebasis</a> of <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="waybel19.html#E10:5"><i><font color="Green">E6</font></i></a>, <a class="txt" href="waybel19.html#E12:5"><i><font color="Green">E7</font></i></a>, <a class="txt" href="waybel19.html#E14:5"><i><font color="Green">E8</font></i></a>, <a class="ref" href="cantor_1.html#D5">CANTOR_1:def 5</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel19.html#D1" target="_self">WAYBEL19:def 1</a><br/>


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