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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   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font>, <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>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="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E2:3"/>
( <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>2</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> is    <a href="connsp_2.html#M1" target="_self">a_neighborhood</a> of <font color="Maroon">c<sub>2</sub></font> implies <font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>6</sub></font> is    <a href="connsp_2.html#M1" target="_self">a_neighborhood</a> of <font color="Maroon">c<sub>2</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:3_1"/>
( <font color="Maroon">c<sub>2</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> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>6</sub></font> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="connsp_2.html#D1" target="_self">CONNSP_2:def 1</a><br/>

<a NAME="E2:3_1"/><b>then </b><i><font color="Green">E3</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E3:3_1"/>
<span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="tops_1.html#T49">TOPS_1:49</a></i>;<br/>
<b>hence </b><a NAME="E4:3_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>6</sub></font> is    <a href="connsp_2.html#M1" target="_self">a_neighborhood</a> of <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="connsp_2.html#E2:3_1"><i><font color="Green">E3</font></i></a>, <a class="ref" href="connsp_2.html#D1" target="_self">Def1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:3"/>
( <font color="Maroon">c<sub>3</sub></font> is    <a href="connsp_2.html#M1" target="_self">a_neighborhood</a> of <font color="Maroon">c<sub>2</sub></font> &amp; <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>2</sub></font> implies <font color="Maroon">c<sub>3</sub></font> <a href="subset_1.html#K4">\/</a> <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>2</sub></font> )
 ;<br/>


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