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



<b>assume </b><a NAME="E1:3"/><i><font color="Green">E3</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="compts_1.html#NR2" target="_self">is_T2</a> 
 ;<br/>

<a NAME="E2:3"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>2</sub></font> st not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> holds <br/>ex <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Olive">b<sub>3</sub></font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">b<sub>4</sub></font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<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>4</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Olive">b<sub>4</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>

<b>assume </b><a NAME="E1:3_1"/><i><font color="Green">E4</font></i>: 
not <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>4</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="ref" href="topmetr.html#T2" target="_self">Th2</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>7</sub></font>, <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="E4:3_1"/><i><font color="Green">E5</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">c<sub>8</sub></font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>8</sub></font> )
 <b>by </b><i><a class="txt" href="topmetr.html#E1:3"><i><font color="Green">E3</font></i></a>, <a class="txt" href="topmetr.html#E1:3_1"><i><font color="Green">E4</font></i></a>, <a class="ref" href="compts_1.html#D4">COMPTS_1:def 4</a></i>;<br/>
<a NAME="E5:3_1"/><i><font color="Green">E6</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="txt" href="topmetr.html#E4:3_1"><i><font color="Green">E5</font></i></a>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>7</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>, <font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>8</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>2</sub></font> ;<br/>
<b>take </b>
<font color="Maroon">c<sub>9</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>10</sub></font>
;<br/>

<a NAME="E7:3_1"/>
( <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>1</sub></font> &amp; <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="topmetr.html#E4:3_1"><i><font color="Green">E5</font></i></a>, <a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>
<a NAME="E8:3_1"/><b>then </b>
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="ref" href="pre_topc.html#D9">PRE_TOPC:def 9</a></i>;<br/>
<b>hence </b><a NAME="E9:3_1"/>
( <font color="Maroon">c<sub>9</sub></font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">c<sub>10</sub></font> is <a href="pre_topc.html#V3">open</a> )
 <b>by </b><i><a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>

<b>thus </b><a NAME="E10:3_1"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> )
 <b>by </b><i><a class="txt" href="topmetr.html#E4:3_1"><i><font color="Green">E5</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>

<font color="Maroon">c<sub>9</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>10</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p3">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T16">XBOOLE_1:16</a></i>
<br/>.= 
<a href="xboole_0.html#K1">{}</a>  <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="topmetr.html#E5:3_1"><i><font color="Green">E6</font></i></a>, <a class="ref" href="xboole_1.html#T16">XBOOLE_1:16</a></i>
<br/>.= 
 <a href="xboole_0.html#K1">{}</a> 

;<br/>
<b>hence </b><a NAME="E12:3_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:3"/>
<font color="Maroon">c<sub>2</sub></font> <a href="compts_1.html#NR2" target="_self">is_T2</a> 
 <b>by </b><i><a class="ref" href="compts_1.html#D4">COMPTS_1:def 4</a></i>;<br/>


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