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<b>consider </b><font color="Maroon">c<sub>2</sub></font> being    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span>;<br/>
<a NAME="E2:28_1_1"/>
not the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> is <a href="realset1.html#V1">trivial</a>
 
<b>by </b><i><a class="ref" href="realset2.html#D5">REALSET2:def 5</a></i>;<br/>
<a NAME="E3:28_1_1"/><b>then </b><i><font color="Green">E23</font></i>: 
not the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span> is <a href="xboole_0.html#V1">empty</a>
 
<b>by </b><i><a class="ref" href="realset1.html#T4">REALSET1:4</a></i>;<br/>
<a NAME="E4:28_1_1"/><b>then </b><i><font color="Green">E24</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> &amp; not <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>2</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="uproots.html#E3:28_1_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>take </b><font color="Maroon">c<sub>4</sub></font> = <span class="p1"><a href="algseq_1.html#K5">&lt;%</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="algseq_1.html#K5">%&gt;</a></span>;<br/>

<a NAME="E6:28_1_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="normsp_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <span class="p1">(<span class="default"><a href="polynom3.html#K12">0_.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="normsp_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="ref" href="polynom3.html#T28">POLYNOM3:28</a>, <a class="ref" href="polynom5.html#T33">POLYNOM5:33</a></i>;<br/>
<a NAME="E7:28_1_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="polynom3.html#K12">0_.</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="uproots.html#E4:28_1_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E8:28_1_1"/>
<font color="Maroon">c<sub>4</sub></font> is <a href="uproots.html#V1" target="_self">non-zero</a>
 <b>by </b><i><a class="ref" href="uproots.html#D5" target="_self">Def5</a></i>;<br/>


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