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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> be    <a href="yellow_8.html#M1">Basis</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>





<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1"><b>now </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="E1:55_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>4</sub></font>
 ;<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="E3:55_1"/><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>3</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="ref" href="yellow_8.html#D2">YELLOW_8:def 2</a></i>;<br/><b>take </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font>;<br/><a NAME="E4:55_1"/><i><font color="Green">E21</font></i>: 
 <a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="tops_1.html#T44">TOPS_1:44</a></i>;<br/><a NAME="E5:55_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="yellow13.html#E3:55_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="yellow_8.html#T21">YELLOW_8:21</a></i>;<br/><a NAME="E6:55_1"/><b>then </b>
 <a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="tops_1.html#T55">TOPS_1:55</a></i>;<br/><b>hence </b><a NAME="E7:55_1"/>
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</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> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="yellow13.html#E3:55_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="yellow13.html#E4:55_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a>, <a class="ref" href="yellow_8.html#T21">YELLOW_8:21</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:55"/>
<font color="Maroon">c<sub>3</sub></font> is    <a href="yellow13.html#M1" target="_self">basis</a> of <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="yellow13.html#D1" target="_self">Def1</a></i>;<br/>


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