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<b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="tsp_1.html#M2">SubSpace</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>reconsider </b><font color="Maroon">c<sub>3</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="tsp_2.html#E19"><i><font color="Green">Lemma15</font></i></a></i>;<br/>
<b>assume </b><a NAME="E2:21_5_1"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="tex_2.html#V2">proper</a>
 ;<br/>

<b>assume </b><a NAME="E3:21_5_1"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="tsp_2.html#V2" target="_self">maximal_T_0</a>
 ;<br/>

<b>assume </b><a NAME="E4:21_5_1"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="tsep_1.html#V1">open</a>
 ;<br/>

<a NAME="E5:21_5_1"/><b>then </b><i><font color="Green">E19</font></i>: 
<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="tsep_1.html#T16">TSEP_1:16</a></i>;<br/>
<a NAME="E6:21_5_1"/>
<font color="Maroon">c<sub>3</sub></font> is <a href="tsp_2.html#V1" target="_self">maximal_T_0</a>
 
<b>by </b><i><a class="txt" href="tsp_2.html#E3:21_5_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="tsp_2.html#T11" target="_self">Th11</a></i>;<br/>
<a NAME="E7:21_5_1"/><b>then </b>
not <font color="Maroon">c<sub>3</sub></font> is <a href="tex_2.html#V1">proper</a>
 
<b>by </b><i><a class="txt" href="tsp_2.html#E5:21_5_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="tsp_2.html#T3" target="_self">Th3</a></i>;<br/>
<b>hence </b><a NAME="E8:21_5_1"/>
contradiction
 <b>by </b><i><a class="txt" href="tsp_2.html#E2:21_5_1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="tex_2.html#T13">TEX_2:13</a></i>;<br/>


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