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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="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>











<b>assume </b><b>that </b><br/><a NAME="E1:11"/><i><font color="Green">E8</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E2:11"/><i><font color="Green">E9</font></i>: 
<font color="Maroon">c<sub>4</sub></font> is    <a href="connsp_2.html#M1" target="_self">a_neighborhood</a> of <font color="Maroon">c<sub>3</sub></font>
 <b>and </b><br/><a NAME="E3:11"/><i><font color="Green">E10</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> )
 ;<br/>

<a NAME="E4:11"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="connsp_2.html#E2:11"><i><font color="Green">E9</font></i></a>, <a class="ref" href="connsp_2.html#D1" target="_self">Def1</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>2</sub></font></span>)</span><b> such that </b><br/><a NAME="E6:11"/><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>7</sub></font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E7:11"/><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>and </b><br/><a NAME="E8:11"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="ref" href="tops_1.html#T54">TOPS_1:54</a></i>;<br/>
<a NAME="E9:11"/>
<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 <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="connsp_2.html#E6:11"><i><font color="Green">E11</font></i></a>, <a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<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="E11:11"/><i><font color="Green">E14</font></i>: 
<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>and </b><br/><a NAME="E12:11"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <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> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="pre_topc.html#D9">PRE_TOPC:def 9</a></i>;<br/>
<a NAME="E13:11"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="connsp_2.html#E12:11"><i><font color="Green">E15</font></i></a>, <a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>8</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E15:11"/>
<font color="Maroon">c<sub>9</sub></font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="txt" href="connsp_2.html#E11:11"><i><font color="Green">E14</font></i></a>, <a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>
<a NAME="E16:11"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>2</sub></font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="txt" href="connsp_2.html#E1:11"><i><font color="Green">E8</font></i></a>, <a class="ref" href="tops_1.html#T38">TOPS_1:38</a></i>;<br/>
<a NAME="E17:11"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="connsp_2.html#E3:11"><i><font color="Green">E10</font></i></a>, <a class="txt" href="connsp_2.html#E7:11"><i><font color="Green">E12</font></i></a>, <a class="txt" href="connsp_2.html#E8:11"><i><font color="Green">E13</font></i></a>, <a class="txt" href="connsp_2.html#E13:11"><i><font color="Green">E16</font></i></a>, <a class="ref" href="tops_1.html#T54">TOPS_1:54</a></i>;<br/>
<b>hence </b><a NAME="E18:11"/>
<font color="Maroon">c<sub>5</sub></font> is    <a href="connsp_2.html#M1" target="_self">a_neighborhood</a> of <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="connsp_2.html#D1" target="_self">Def1</a></i>;<br/>


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