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<b>let </b><font color="Maroon" title="c1">S</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:22"/>
0 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">S</font>
 ;<br/>

<a NAME="E2:22"/><b>then </b><i><font color="Green" title="E4">A1</font></i>: 
 <a href="cqc_sim1.html#K10" title="CQC_SIM1:func.10">min</a> <font color="Maroon" title="c1">S</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 0
 
<b>by </b><i><a class="ref" href="cqc_sim1.html#D8" title="CQC_SIM1:def.8">CQC_SIM1:def 8</a></i>;<br/>
<b>assume </b><a NAME="E3:22"/>
 <a href="cqc_sim1.html#K10" title="CQC_SIM1:func.10">min</a> <font color="Maroon" title="c1">S</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 ;<br/>

<b>hence </b><a NAME="E4:22"/>
contradiction
 <b>by </b><i><a class="txt" href="abian.html#E2:22"><i><font color="Green" title="E4">A1</font></i></a>, <a class="ref" href="nat_1.html#T3" title="NAT_1:th.3">NAT_1:3</a></i>;<br/>


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