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<b>let </b><font color="Maroon">c<sub>2</sub></font> be  non <a href="struct_0.html#V3">empty</a>  <a href="tsp_1.html#M2" target="_self">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  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> <b>by </b><i><a class="ref" href="tsep_1.html#T1">TSEP_1:1</a></i>;<br/>
<a NAME="E2:47_2_1"/>
<font color="Maroon">c<sub>1</sub></font> is    <a href="tsp_1.html#M2" target="_self">SubSpace</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="tsep_1.html#T2">TSEP_1:2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> as  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> <b>by </b><i><a class="ref" href="tsep_1.html#T1">TSEP_1:1</a></i>;<br/>
<b>assume </b><a NAME="E4:47_2_1"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="t_0topsp.html#V2">T_0</a>
 ;<br/>

<b>assume </b><a NAME="E5:47_2_1"/>
not <font color="Maroon">c<sub>2</sub></font> is <a href="tex_2.html#V2">proper</a>
 ;<br/>

<a NAME="E6:47_2_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="ref" href="tex_2.html#T13">TEX_2:13</a></i>;<br/>
<a NAME="E7:47_2_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="tex_2.html#T5">TEX_2:5</a></i>;<br/>
<a NAME="E8:47_2_1"/><b>then </b>
not <font color="Maroon">c<sub>3</sub></font> is <a href="tsp_1.html#V1" target="_self">T_0</a>
 
<b>by </b><i><a class="ref" href="tsp_1.html#T6" target="_self">Th6</a></i>;<br/>
<b>hence </b><a NAME="E9:47_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="tsp_1.html#E4:47_2_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="tsp_1.html#T15" target="_self">Th15</a></i>;<br/>


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