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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:3_1"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="realset1.html#V1">trivial</a>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>3</sub></font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E3:3_1"/><i><font color="Green">E3</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="tex_2.html#D1">TEX_2:def 1</a></i>;<br/><b>thus </b><a NAME="E4:3_1"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="txt" href="borsuk_4.html#E3:3_1"><i><font color="Green">E3</font></i></a></i>;<br/></div>
<b>end;</b></div>

<b>given </b><font color="Maroon">c<sub>3</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><a NAME="E2:3"/><i><font color="Green">E3</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="domain_1.html#K6">}</a></span>
 ;<br/>

<b>thus </b><a NAME="E3:3"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="realset1.html#V1">trivial</a>
 <b>by </b><i><a class="txt" href="borsuk_4.html#E2:3"><i><font color="Green">E3</font></i></a></i>;<br/>


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