<?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="t_0topsp.html#V2">T_0</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/>

<b>assume </b><a NAME="E1:12"/><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="group_1.html#V6">finite</a>
 ;<br/>

<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  =  <a href="setfam_1.html#K1">meet</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M2">Element</a> of the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> </span>}</span> ;<br/>
<a NAME="E2:12"/><i><font color="Green">E12</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</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> holds <br/><font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</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="yellow_1.html#K3">BoolePoset</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<a NAME="E1:12_1"/><i><font color="Green">E13</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D1">PRE_TOPC:def 1</a></i>;<br/>
<b>assume </b><a NAME="E2:12_1"/>
<font color="Maroon">c<sub>2</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/>

<a NAME="E3:12_1"/><b>then </b>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M2">Element</a> of the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="waybel31.html#E1:12_1"><i><font color="Green">E13</font></i></a></i>;<br/>
<a NAME="E4:12_1"/><b>then </b>
 <a href="setfam_1.html#K1">meet</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M2">Element</a> of the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> </span>}</span>  <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="setfam_1.html#T4">SETFAM_1:4</a></i>;<br/>
<a NAME="E5:12_1"/><b>then </b>
 <a href="setfam_1.html#K1">meet</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M2">Element</a> of the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> </span>}</span>  <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/>
<b>hence </b><a NAME="E6:12_1"/>
 <a href="setfam_1.html#K1">meet</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M2">Element</a> of the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> </span>}</span>  <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="yellow_1.html#K3">BoolePoset</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="waybel_7.html#T4">WAYBEL_7:4</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>2</sub></font> being   <a href="funct_2.html#NM1">Function</a> of 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"><a href="yellow_1.html#K3">BoolePoset</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span><b> such that </b><br/><a NAME="E4:12"/><i><font color="Green">E13</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</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> holds <br/><font color="Maroon">c<sub>2</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="funct_2.html#S2">FUNCT_2:sch 2</a>(<a class="txt" href="waybel31.html#E2:12"><i><font color="Green">E12</font></i></a>);<br/></i>
<a NAME="E5:12"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</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="yellow_1.html#K3">BoolePoset</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
;<br/>
<a NAME="E6:12"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font> <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>
 
<b>by </b><i><a class="ref" href="waybel_7.html#T4">WAYBEL_7:4</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>3</sub></font> =  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font> as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E8:12"/><i><font color="Green">E14</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>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</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="E1:12_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>5</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:12_2"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E4:12_2"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E5:12_2"/>
<font color="Maroon">c<sub>5</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>
 
<b>by </b><i><a class="txt" href="waybel31.html#E3:12_2"><i><font color="Green">E15</font></i></a></i>;<br/>
<a NAME="E6:12_2"/><b>then </b><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K1">meet</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M2">Element</a> of the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="waybel31.html#E4:12"><i><font color="Green">E13</font></i></a>, <a class="txt" href="waybel31.html#E4:12_2"><i><font color="Green">E16</font></i></a></i>;<br/>
<a NAME="E7:12_2"/>
<span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M2">Element</a> of the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> </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="12_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</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="E1:12_2_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M2">Element</a> of the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>7</sub></font> being    <a href="subset_1.html#M2">Element</a> of the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:12_2_1"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font>
 <b>and </b><br/><a NAME="E4:12_2_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/>
<b>thus </b><a NAME="E5:12_2_1"/>
<font color="Maroon">c<sub>6</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>
 <b>by </b><i><a class="txt" href="waybel31.html#E3:12_2_1"><i><font color="Green">E18</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M2">Element</a> of the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> </span>}</span>  as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font> ;<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>1</sub></font> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>assume </b><a NAME="E1:12_2_2"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>9</sub></font> being    <a href="subset_1.html#M2">Element</a> of the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:12_2_2"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font>
 <b>and </b><br/><a NAME="E4:12_2_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/><b>thus </b><a NAME="E5:12_2_2"/>
<font color="Maroon">c<sub>8</sub></font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="txt" href="waybel31.html#E3:12_2_2"><i><font color="Green">E18</font></i></a>, <a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/></div><b>end;</b></div>
<a NAME="E11:12_2"/><b>then </b><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>7</sub></font> is <a href="tops_2.html#V1">open</a>
 
<b>by </b><i><a class="ref" href="tops_2.html#D1">TOPS_2:def 1</a></i>;<br/>
<a NAME="E12:12_2"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i><a class="txt" href="waybel31.html#E1:12"><i><font color="Green">E11</font></i></a>, <a class="ref" href="group_1.html#D14">GROUP_1:def 14</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>8</sub></font> = the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font> as  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="ref" href="yellow15.html#T3">YELLOW15:3</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <font color="Maroon">a<sub>1</sub></font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] means <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E14:12_2"/>
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>8</sub></font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span>  is <a href="finset_1.html#V1">finite</a>
 
<b>from </b><i><a class="ref" href="pre_circ.html#S1">PRE_CIRC:sch 1</a>();<br/></i>
<a NAME="E15:12_2"/><b>then </b>
 <a href="setfam_1.html#K6">meet</a> <font color="Maroon">c<sub>7</sub></font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="txt" href="waybel31.html#E11:12_2"><i><font color="Green">E18</font></i></a>, <a class="ref" href="tops_2.html#T27">TOPS_2:27</a></i>;<br/>
<b>hence </b><a NAME="E16:12_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="waybel31.html#E6:12_2"><i><font color="Green">E17</font></i></a>, <a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:12"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V3">open</a> holds <br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> holds <br/>ex <font color="Olive">b<sub>3</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E1:12_3"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>4</sub></font> is <a href="pre_topc.html#V3">open</a>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E2:12_3"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<a NAME="E3:12_3"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="waybel31.html#E1:12_3"><i><font color="Green">E15</font></i></a>, <a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>
<a NAME="E4:12_3"/><b>then </b><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M2">Element</a> of the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="waybel31.html#E2:12_3"><i><font color="Green">E16</font></i></a></i>;<br/>
<a NAME="E5:12_3"/>
 <a href="setfam_1.html#K1">meet</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M2">Element</a> of the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> </span>}</span>  <a href="tarski.html#R1">c=</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="12_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</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="E1:12_3_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K1">meet</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M2">Element</a> of the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> </span>}</span> 
 ;<br/>

<a NAME="E2:12_3_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="waybel31.html#E4:12_3"><i><font color="Green">E17</font></i></a>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:12_3_1"/>
<font color="Maroon">c<sub>6</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/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> =  <a href="setfam_1.html#K1">meet</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M2">Element</a> of the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> </span>}</span>  as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<b>take </b>
<font color="Maroon">c<sub>6</sub></font>
;<br/>

<a NAME="E7:12_3"/>
<font color="Maroon">c<sub>5</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/>
<a NAME="E8:12_3"/><b>then </b><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E9:12_3"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="waybel31.html#E4:12"><i><font color="Green">E13</font></i></a></i>;<br/>
<b>hence </b><a NAME="E10:12_3"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="waybel31.html#E8:12_3"><i><font color="Green">E18</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_3_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>7</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:12_3_2"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M2">Element</a> of the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">c<sub>8</sub></font> being    <a href="subset_1.html#M2">Element</a> of the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:12_3_2"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font>
 <b>and </b><br/><a NAME="E4:12_3_2"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/><b>thus </b><a NAME="E5:12_3_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="waybel31.html#E3:12_3_2"><i><font color="Green">E19</font></i></a>, <a class="txt" href="waybel31.html#E4:12_3_2"><i><font color="Green">E20</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E12:12_3"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="waybel31.html#E4:12_3"><i><font color="Green">E17</font></i></a>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>

<b>thus </b><a NAME="E13:12_3"/>
<font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_3_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</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="E1:12_3_3"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/>

<b>hence </b><a NAME="E2:12_3_3"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="waybel31.html#E4:12_3"><i><font color="Green">E17</font></i></a>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>


</div><b>end;</b></div>


</div><b>end;</b></div>
<a NAME="E10:12"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font> is    <a href="cantor_1.html#M1">Basis</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="waybel31.html#E8:12"><i><font color="Green">E14</font></i></a>, <a class="ref" href="yellow_9.html#T32">YELLOW_9:32</a></i>;<br/>
<a NAME="E11:12"/><b>then </b><i><font color="Green">E15</font></i>: 
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="card_1.html#K1">Card</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is    <a href="cantor_1.html#M1">Basis</a> of <font color="Maroon">c<sub>1</sub></font> : verum </span>}</span> 
 
;<br/>
<a NAME="E12:12"/><b>then </b><i><font color="Green">E16</font></i>: 
 <a href="setfam_1.html#K1">meet</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="card_1.html#K1">Card</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is    <a href="cantor_1.html#M1">Basis</a> of <font color="Maroon">c<sub>1</sub></font> : verum </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="setfam_1.html#T4">SETFAM_1:4</a></i>;<br/>
<a NAME="E13:12"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="card_1.html#K1">Card</a> <font color="Olive">b<sub>2</sub></font></span>)</span> where B is    <a href="cantor_1.html#M1">Basis</a> of <font color="Maroon">c<sub>1</sub></font> : verum </span>}</span>  holds <br/> <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:12_4"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="card_1.html#K1">Card</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is    <a href="cantor_1.html#M1">Basis</a> of <font color="Maroon">c<sub>1</sub></font> : verum </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>5</sub></font> being    <a href="cantor_1.html#M1">Basis</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:12_4"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>
<a NAME="E4:12_4"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</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="E1:12_4_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>7</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:12_4_1"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E4:12_4_1"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</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>8</sub></font> = <font color="Maroon">c<sub>7</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="waybel31.html#E3:12_4_1"><i><font color="Green">E18</font></i></a></i>;<br/>
<a NAME="E6:12_4_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K1">meet</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M2">Element</a> of the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="waybel31.html#E4:12"><i><font color="Green">E13</font></i></a>, <a class="txt" href="waybel31.html#E4:12_4_1"><i><font color="Green">E19</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:12_4_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="waybel31.html#E1:12"><i><font color="Green">E11</font></i></a>, <a class="ref" href="yellow15.html#T32">YELLOW15:32</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:12_4"/>
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="waybel31.html#E3:12_4"><i><font color="Green">E17</font></i></a>, <a class="ref" href="card_1.html#T27">CARD_1:27</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E14:12"/><b>then </b>
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K1">meet</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="card_1.html#K1">Card</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is    <a href="cantor_1.html#M1">Basis</a> of <font color="Maroon">c<sub>1</sub></font> : verum </span>}</span> 
 
<b>by </b><i><a class="txt" href="waybel31.html#E11:12"><i><font color="Green">E15</font></i></a>, <a class="ref" href="setfam_1.html#T6">SETFAM_1:6</a></i>;<br/>
<b>then </b><i><font color="Green">E17</font></i>:  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> = 
 <a href="setfam_1.html#K1">meet</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="card_1.html#K1">Card</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is    <a href="cantor_1.html#M1">Basis</a> of <font color="Maroon">c<sub>1</sub></font> : verum </span>}</span> 
<b>by </b><i><a class="txt" href="waybel31.html#E12:12"><i><font color="Green">E16</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>
<br/>.= 
 <a href="waybel23.html#K2">weight</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="ref" href="waybel23.html#D5">WAYBEL23:def 5</a></i>
;<br/>
<a NAME="E16:12"/><i><font color="Green">E18</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_6"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><b>that </b><br/><a NAME="E1:12_6"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E2:12_6"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E3:12_6"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>5</sub></font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="waybel31.html#E1:12_6"><i><font color="Green">E19</font></i></a>, <a class="txt" href="waybel31.html#E2:12_6"><i><font color="Green">E20</font></i></a></i>;<br/><a NAME="E5:12_6"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K1">meet</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M2">Element</a> of the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> </span>}</span> 
 <b>by </b><i><a class="txt" href="waybel31.html#E4:12"><i><font color="Green">E13</font></i></a></i>;<br/><a NAME="E6:12_6"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K1">meet</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M2">Element</a> of the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> </span>}</span> 
 <b>by </b><i><a class="txt" href="waybel31.html#E4:12"><i><font color="Green">E13</font></i></a></i>;<br/><b>assume </b><a NAME="E7:12_6"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E9:12_6"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>8</sub></font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E10:12_6"/><i><font color="Green">E25</font></i>: 
( ( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> &amp; not <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> ) or ( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> &amp; not <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> ) )
 <b>by </b><i><a class="ref" href="t_0topsp.html#D7">T_0TOPSP:def 7</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_6_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_6_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:12_6_1_1"/>
( ( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> &amp; not <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> ) or ( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> &amp; not <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> ) )
 </span><b>by </b><i><a class="txt" href="waybel31.html#E10:12_6"><i><font color="Green">E25</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_6_1_1_1"><b>suppose </b></a><a NAME="E1:12_6_1_1_1"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> &amp; not <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> )
 ;<br/><div class="add"><a NAME="E2:12_6_1_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="waybel31.html#E9:12_6"><i><font color="Green">E24</font></i></a>, <a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/><a NAME="E3:12_6_1_1_1"/><b>then </b><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M2">Element</a> of the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> </span>}</span> 
 <b>by </b><i><a class="txt" href="waybel31.html#E1:12_6_1_1_1"><i><font color="Green">E26</font></i></a></i>;<br/><a NAME="E4:12_6_1_1_1"/><i><font color="Green">E28</font></i>: 
 <a href="setfam_1.html#K1">meet</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M2">Element</a> of the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> </span>}</span>  <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>8</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_6_1_1_1_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="E1:12_6_1_1_1_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K1">meet</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M2">Element</a> of the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> </span>}</span> 
 ;<br/>

<b>hence </b><a NAME="E2:12_6_1_1_1_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="waybel31.html#E3:12_6_1_1_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>


</div><b>end;</b></div><a NAME="E5:12_6_1_1_1"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="pre_topc.html#D1">PRE_TOPC:def 1</a></i>;<br/><a NAME="E6:12_6_1_1_1"/><b>then </b><i><font color="Green">E29</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M2">Element</a> of the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> </span>}</span> 
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_6_1_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>9</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:12_6_1_1_1_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M2">Element</a> of the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">c<sub>10</sub></font> being    <a href="subset_1.html#M2">Element</a> of the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:12_6_1_1_1_2"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font>
 <b>and </b><br/><a NAME="E4:12_6_1_1_1_2"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/><b>thus </b><a NAME="E5:12_6_1_1_1_2"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="waybel31.html#E3:12_6_1_1_1_2"><i><font color="Green">E30</font></i></a>, <a class="txt" href="waybel31.html#E4:12_6_1_1_1_2"><i><font color="Green">E31</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E8:12_6_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K1">meet</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M2">Element</a> of the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> </span>}</span> 
 <b>by </b><i><a class="txt" href="waybel31.html#E3:12_6"><i><font color="Green">E21</font></i></a>, <a class="txt" href="waybel31.html#E5:12_6"><i><font color="Green">E22</font></i></a>, <a class="txt" href="waybel31.html#E6:12_6"><i><font color="Green">E23</font></i></a>, <a class="txt" href="waybel31.html#E6:12_6_1_1_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/><b>hence </b><a NAME="E9:12_6_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="waybel31.html#E1:12_6_1_1_1"><i><font color="Green">E26</font></i></a>, <a class="txt" href="waybel31.html#E4:12_6_1_1_1"><i><font color="Green">E28</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_6_1_1_2"><b>suppose </b></a><a NAME="E1:12_6_1_1_2"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> &amp; not <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> )
 ;<br/><div class="add"><a NAME="E2:12_6_1_1_2"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="waybel31.html#E9:12_6"><i><font color="Green">E24</font></i></a>, <a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/><a NAME="E3:12_6_1_1_2"/><b>then </b><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M2">Element</a> of the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> </span>}</span> 
 <b>by </b><i><a class="txt" href="waybel31.html#E1:12_6_1_1_2"><i><font color="Green">E26</font></i></a></i>;<br/><a NAME="E4:12_6_1_1_2"/><i><font color="Green">E28</font></i>: 
 <a href="setfam_1.html#K1">meet</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M2">Element</a> of the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> </span>}</span>  <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>8</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_6_1_1_2_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="E1:12_6_1_1_2_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K1">meet</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M2">Element</a> of the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> </span>}</span> 
 ;<br/>

<b>hence </b><a NAME="E2:12_6_1_1_2_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="waybel31.html#E3:12_6_1_1_2"><i><font color="Green">E27</font></i></a>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>


</div><b>end;</b></div><a NAME="E5:12_6_1_1_2"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="pre_topc.html#D1">PRE_TOPC:def 1</a></i>;<br/><a NAME="E6:12_6_1_1_2"/><b>then </b><i><font color="Green">E29</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M2">Element</a> of the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> </span>}</span> 
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_6_1_1_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>9</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:12_6_1_1_2_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M2">Element</a> of the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">c<sub>10</sub></font> being    <a href="subset_1.html#M2">Element</a> of the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:12_6_1_1_2_2"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font>
 <b>and </b><br/><a NAME="E4:12_6_1_1_2_2"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/><b>thus </b><a NAME="E5:12_6_1_1_2_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="waybel31.html#E3:12_6_1_1_2_2"><i><font color="Green">E30</font></i></a>, <a class="txt" href="waybel31.html#E4:12_6_1_1_2_2"><i><font color="Green">E31</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E8:12_6_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K1">meet</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M2">Element</a> of the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> </span>}</span> 
 <b>by </b><i><a class="txt" href="waybel31.html#E3:12_6"><i><font color="Green">E21</font></i></a>, <a class="txt" href="waybel31.html#E5:12_6"><i><font color="Green">E22</font></i></a>, <a class="txt" href="waybel31.html#E6:12_6"><i><font color="Green">E23</font></i></a>, <a class="txt" href="waybel31.html#E6:12_6_1_1_2"><i><font color="Green">E29</font></i></a>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/><b>hence </b><a NAME="E9:12_6_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="waybel31.html#E1:12_6_1_1_2"><i><font color="Green">E26</font></i></a>, <a class="txt" href="waybel31.html#E4:12_6_1_1_2"><i><font color="Green">E28</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E12:12_6"/>
contradiction
 ;<br/></div><b>end;</b></div>
<a NAME="E18:12"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i><a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/>
<a NAME="E19:12"/><b>then </b>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>, <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R2">are_equipotent</a> 
 
<b>by </b><i><a class="txt" href="waybel31.html#E16:12"><i><font color="Green">E18</font></i></a>, <a class="ref" href="wellord2.html#D4">WELLORD2:def 4</a></i>;<br/>
<b>hence </b><a NAME="E20:12"/>
 <a href="waybel23.html#K2">weight</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="waybel31.html#E15:12"><i><font color="Green">E17</font></i></a>, <a class="ref" href="card_1.html#T21">CARD_1:21</a></i>;<br/>


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