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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="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:20"/>
0 <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<a NAME="E2:20"/><b>then </b><i><font color="Green">E4</font></i>: 
 <a href="cqc_sim1.html#K10">min</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 0
 
<b>by </b><i><a class="ref" href="cqc_sim1.html#D8">CQC_SIM1:def 8</a></i>;<br/>
<b>assume </b><a NAME="E3:20"/>
 <a href="cqc_sim1.html#K10">min</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/>

<b>hence </b><a NAME="E4:20"/>
contradiction
 <b>by </b><i><a class="txt" href="abian.html#E2:20"><i><font color="Green">E4</font></i></a>, <a class="ref" href="nat_1.html#T19">NAT_1:19</a></i>;<br/>


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