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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="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E1:31_2_1"/>
not <font color="Maroon">c<sub>2</sub></font> is <a href="tex_2.html#V2" target="_self">proper</a>
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>3</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="E3:31_2_1"/><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E4:31_2_1"/>
not <font color="Maroon">c<sub>3</sub></font> is <a href="tex_2.html#V1" target="_self">proper</a>
 <b>by </b><i><a class="ref" href="tex_2.html#D3" target="_self">Def3</a></i>;<br/>
<b>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>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="txt" href="tex_2.html#E3:31_2_1"><i><font color="Green">E11</font></i></a></i>;<br/>
<a NAME="E6:31_2_1"/>
<font color="Maroon">c<sub>4</sub></font> is <a href="realset1.html#V1">trivial</a>
 
;<br/>
<b>hence </b><a NAME="E7:31_2_1"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="realset2.html#V3">trivial</a>
 <b>by </b><i><a class="ref" href="realset2.html#D5">REALSET2:def 5</a></i>;<br/>


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