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

<b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="pre_topc.html#L1">TopStruct</a> ;<br/>

<a NAME="E1:3_1_1"/><i><font color="Green">E3</font></i>: 
(  <a href="pre_topc.html#K2">[#]</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>2</sub></font> &amp;  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</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> )
 
;<br/>
<b>thus </b><a NAME="E2:3_1_1"/>
( <font color="Maroon">c<sub>2</sub></font> is    <a href="tsp_1.html#M1" target="_self">SubSpace</a> of <font color="Maroon">c<sub>1</sub></font> implies ( the <a href="struct_0.html#U1">carrier</a> of <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 <font color="Maroon">c<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>2</sub></font> holds <br/>( <font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V4">closed</a> iff ex <font color="Olive">b<sub>2</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>2</sub></font> is <a href="pre_topc.html#V4">closed</a> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> <a href="subset_1.html#K8">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> ) ) ) ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:3_1_1_1"/><i><font color="Green">E4</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is    <a href="tsp_1.html#M1" target="_self">SubSpace</a> of <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<a NAME="E2:3_1_1_1"/><b>then </b><i><font color="Green">E5</font></i>: 
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D9">PRE_TOPC:def 9</a></i>;<br/>
<b>thus </b><a NAME="E3:3_1_1_1"/>
the <a href="struct_0.html#U1">carrier</a> of <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 <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="tsp_1.html#E1:3_1_1"><i><font color="Green">E3</font></i></a>, <a class="txt" href="tsp_1.html#E1:3_1_1_1"><i><font color="Green">E4</font></i></a>, <a class="ref" href="pre_topc.html#D9">PRE_TOPC:def 9</a></i>;<br/>

<a NAME="E4:3_1_1_1"/><i><font color="Green">E6</font></i>: 
<span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="txt" href="tsp_1.html#E2:3_1_1_1"><i><font color="Green">E5</font></i></a>, <a class="ref" href="xboole_1.html#T37">XBOOLE_1:37</a></i>;<br/>
<b>thus </b><a NAME="E5:3_1_1_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>2</sub></font> holds <br/>( <font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V4">closed</a> iff ex <font color="Olive">b<sub>2</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>2</sub></font> is <a href="pre_topc.html#V4">closed</a> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> <a href="subset_1.html#K8">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>

<b>thus </b><a NAME="E1:3_1_1_1_1"/>
( <font color="Maroon">c<sub>3</sub></font> is <a href="pre_topc.html#V4">closed</a> implies ex <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 <br/>( <font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V4">closed</a> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="subset_1.html#K8">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:3_1_1_1_1_1"/>
<font color="Maroon">c<sub>3</sub></font> is <a href="pre_topc.html#V4">closed</a>
 ;<br/>

<a NAME="E2:3_1_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>3</sub></font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D6">PRE_TOPC:def 6</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>4</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="E4:3_1_1_1_1_1"/><i><font color="Green">E7</font></i>: 
<font color="Maroon">c<sub>4</sub></font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E5:3_1_1_1_1_1"/><i><font color="Green">E8</font></i>: 
<span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K8">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="tsp_1.html#E1:3_1_1_1"><i><font color="Green">E4</font></i></a>, <a class="ref" href="tsp_1.html#D1" target="_self">Def1</a></i>;<br/>
<b>take </b><font color="Maroon">c<sub>5</sub></font> = <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>4</sub></font>;<br/>

<a NAME="E6:3_1_1_1_1_1"/><i><font color="Green">E9</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T22">PRE_TOPC:22</a></i>;<br/>
<b>hence </b><a NAME="E7:3_1_1_1_1_1"/>
<font color="Maroon">c<sub>5</sub></font> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i><a class="txt" href="tsp_1.html#E4:3_1_1_1_1_1"><i><font color="Green">E7</font></i></a>, <a class="ref" href="pre_topc.html#D6">PRE_TOPC:def 6</a></i>;<br/>

<font color="Maroon">c<sub>3</sub></font> = 
<span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K8">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="tsp_1.html#E5:3_1_1_1_1_1"><i><font color="Green">E8</font></i></a>, <a class="ref" href="pre_topc.html#T22">PRE_TOPC:22</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T54">XBOOLE_1:54</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <a href="xboole_0.html#K1">{}</a> 
<b>by </b><i><a class="ref" href="xboole_1.html#T37">XBOOLE_1:37</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="tsp_1.html#E6:3_1_1_1_1_1"><i><font color="Green">E9</font></i></a>, <a class="ref" href="xboole_1.html#T52">XBOOLE_1:52</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="tsp_1.html#E4:3_1_1_1"><i><font color="Green">E6</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E9:3_1_1_1_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K8">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>
 ;<br/>


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

<b>given </b><font color="Maroon">c<sub>4</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><a NAME="E2:3_1_1_1_1"/><i><font color="Green">E7</font></i>: 
<font color="Maroon">c<sub>4</sub></font> is <a href="pre_topc.html#V4">closed</a>
 <b>and </b><br/><a NAME="E3:3_1_1_1_1"/><i><font color="Green">E8</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K8">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>set </b><font color="Maroon">c<sub>5</sub></font> = <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>3</sub></font>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_1_1_1_2"><b>now </b></a><div class="add"><b>take </b><font color="Maroon">c<sub>6</sub></font> = <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>4</sub></font>;<br/><b>thus </b><a NAME="E1:3_1_1_1_1_2"/>
<font color="Maroon">c<sub>6</sub></font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="txt" href="tsp_1.html#E2:3_1_1_1_1"><i><font color="Green">E7</font></i></a>, <a class="ref" href="pre_topc.html#D6">PRE_TOPC:def 6</a></i>;<br/><a NAME="E2:3_1_1_1_1_2"/><i><font color="Green">E9</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="pre_topc.html#T22">PRE_TOPC:22</a></i>;<br/><span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>3</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="tsp_1.html#E3:3_1_1_1_1"><i><font color="Green">E8</font></i></a>, <a class="ref" href="xboole_1.html#T54">XBOOLE_1:54</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <a href="xboole_0.html#K1">{}</a> 
<b>by </b><i><a class="ref" href="xboole_1.html#T37">XBOOLE_1:37</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="tsp_1.html#E2:3_1_1_1_1_2"><i><font color="Green">E9</font></i></a>, <a class="ref" href="xboole_1.html#T52">XBOOLE_1:52</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="txt" href="tsp_1.html#E4:3_1_1_1"><i><font color="Green">E6</font></i></a></i>
;<br/><b>hence </b><a NAME="E4:3_1_1_1_1_2"/>
<span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="subset_1.html#K8">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>
 ;<br/></div><b>end;</b></div>
<a NAME="E5:3_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>3</sub></font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="txt" href="tsp_1.html#E1:3_1_1_1"><i><font color="Green">E4</font></i></a>, <a class="ref" href="tsp_1.html#D1" target="_self">Def1</a></i>;<br/>
<b>hence </b><a NAME="E6:3_1_1_1_1"/>
<font color="Maroon">c<sub>3</sub></font> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i><a class="ref" href="pre_topc.html#D6">PRE_TOPC:def 6</a></i>;<br/>


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


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

<b>assume </b><a NAME="E3:3_1_1"/><i><font color="Green">E4</font></i>: 
( the <a href="struct_0.html#U1">carrier</a> of <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 <font color="Maroon">c<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>2</sub></font> holds <br/>( <font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V4">closed</a> iff ex <font color="Olive">b<sub>2</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>2</sub></font> is <a href="pre_topc.html#V4">closed</a> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> <a href="subset_1.html#K8">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> ) ) ) )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_1_2"><b>now </b></a><div class="add"><b>thus </b><a NAME="E1:3_1_1_2"/>
the <a href="struct_0.html#U1">carrier</a> of <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 <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="tsp_1.html#E3:3_1_1"><i><font color="Green">E4</font></i></a></i>;<br/><a NAME="E2:3_1_1_2"/><i><font color="Green">E5</font></i>: 
<span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="tsp_1.html#E3:3_1_1"><i><font color="Green">E4</font></i></a>, <a class="ref" href="xboole_1.html#T37">XBOOLE_1:37</a></i>;<br/><b>thus </b><a NAME="E3:3_1_1_2"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>2</sub></font> holds <br/>( <font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V3">open</a> iff ex <font color="Olive">b<sub>2</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>2</sub></font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> <a href="subset_1.html#K8">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>

<b>thus </b><a NAME="E1:3_1_1_2_1"/>
( <font color="Maroon">c<sub>3</sub></font> is <a href="pre_topc.html#V3">open</a> implies ex <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 <br/>( <font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="subset_1.html#K8">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_1_2_1_1"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">c<sub>4</sub></font> = <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>3</sub></font>;<br/>
<b>assume </b><a NAME="E1:3_1_1_2_1_1"/><i><font color="Green">E6</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="pre_topc.html#V3">open</a>
 ;<br/>

<a NAME="E2:3_1_1_2_1_1"/><i><font color="Green">E7</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T22">PRE_TOPC:22</a></i>;<br/>
<a NAME="E3:3_1_1_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>3</sub></font> is <a href="pre_topc.html#V4">closed</a>
 
<b>by </b><i><a class="txt" href="tsp_1.html#E1:3_1_1_2_1_1"><i><font color="Green">E6</font></i></a>, <a class="ref" href="pre_topc.html#D6">PRE_TOPC:def 6</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>5</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="E5:3_1_1_2_1_1"/><i><font color="Green">E8</font></i>: 
<font color="Maroon">c<sub>5</sub></font> is <a href="pre_topc.html#V4">closed</a>
 <b>and </b><br/><a NAME="E6:3_1_1_2_1_1"/><i><font color="Green">E9</font></i>: 
<span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K8">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="tsp_1.html#E3:3_1_1"><i><font color="Green">E4</font></i></a></i>;<br/>
<b>take </b><font color="Maroon">c<sub>6</sub></font> = <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>5</sub></font>;<br/>

<a NAME="E7:3_1_1_2_1_1"/><i><font color="Green">E10</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T22">PRE_TOPC:22</a></i>;<br/>
<b>thus </b><a NAME="E8:3_1_1_2_1_1"/>
<font color="Maroon">c<sub>6</sub></font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="txt" href="tsp_1.html#E5:3_1_1_2_1_1"><i><font color="Green">E8</font></i></a>, <a class="ref" href="pre_topc.html#D6">PRE_TOPC:def 6</a></i>;<br/>

<font color="Maroon">c<sub>3</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="tsp_1.html#E2:3_1_1_2_1_1"><i><font color="Green">E7</font></i></a>, <a class="txt" href="tsp_1.html#E6:3_1_1_2_1_1"><i><font color="Green">E9</font></i></a>, <a class="ref" href="xboole_1.html#T54">XBOOLE_1:54</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <a href="xboole_0.html#K1">{}</a> 
<b>by </b><i><a class="ref" href="xboole_1.html#T37">XBOOLE_1:37</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="tsp_1.html#E7:3_1_1_2_1_1"><i><font color="Green">E10</font></i></a>, <a class="ref" href="xboole_1.html#T52">XBOOLE_1:52</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="txt" href="tsp_1.html#E2:3_1_1_2"><i><font color="Green">E5</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E10:3_1_1_2_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="subset_1.html#K8">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>
 ;<br/>


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

<b>given </b><font color="Maroon">c<sub>4</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><a NAME="E2:3_1_1_2_1"/><i><font color="Green">E6</font></i>: 
<font color="Maroon">c<sub>4</sub></font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E3:3_1_1_2_1"/><i><font color="Green">E7</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K8">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>set </b><font color="Maroon">c<sub>5</sub></font> = <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>3</sub></font>;<br/>
<a NAME="E4:3_1_1_2_1"/><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"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T22">PRE_TOPC:22</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_1_2_1_2"><b>now </b></a><div class="add"><b>take </b><font color="Maroon">c<sub>6</sub></font> = <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>4</sub></font>;<br/><a NAME="E1:3_1_1_2_1_2"/><i><font color="Green">E9</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="pre_topc.html#T22">PRE_TOPC:22</a></i>;<br/><b>hence </b><a NAME="E2:3_1_1_2_1_2"/>
<font color="Maroon">c<sub>6</sub></font> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i><a class="txt" href="tsp_1.html#E2:3_1_1_2_1"><i><font color="Green">E6</font></i></a>, <a class="ref" href="pre_topc.html#D6">PRE_TOPC:def 6</a></i>;<br/><span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>3</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="tsp_1.html#E3:3_1_1_2_1"><i><font color="Green">E7</font></i></a>, <a class="ref" href="xboole_1.html#T54">XBOOLE_1:54</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <a href="xboole_0.html#K1">{}</a> 
<b>by </b><i><a class="ref" href="xboole_1.html#T37">XBOOLE_1:37</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="tsp_1.html#E1:3_1_1_2_1_2"><i><font color="Green">E9</font></i></a>, <a class="ref" href="xboole_1.html#T52">XBOOLE_1:52</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="txt" href="tsp_1.html#E2:3_1_1_2"><i><font color="Green">E5</font></i></a></i>
;<br/><b>hence </b><a NAME="E4:3_1_1_2_1_2"/>
<span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="subset_1.html#K8">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>
 ;<br/></div><b>end;</b></div>
<a NAME="E6:3_1_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>3</sub></font> is <a href="pre_topc.html#V4">closed</a>
 
<b>by </b><i><a class="txt" href="tsp_1.html#E3:3_1_1"><i><font color="Green">E4</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:3_1_1_2_1"/>
<font color="Maroon">c<sub>3</sub></font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="txt" href="tsp_1.html#E4:3_1_1_2_1"><i><font color="Green">E8</font></i></a>, <a class="ref" href="pre_topc.html#D6">PRE_TOPC:def 6</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E5:3_1_1"/>
<font color="Maroon">c<sub>2</sub></font> is    <a href="tsp_1.html#M1" target="_self">SubSpace</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="tsp_1.html#D1" target="_self">Def1</a></i>;<br/>


</div>
