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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="E1:21"/>
0 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">S</font>
 ;<br/>

<a NAME="E2:21"/><b>then </b><i><font color="Green" title="E4">A1</font></i>: 
 <a href="seq_4.html#K6" title="SEQ_4:func.6">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="seq_4.html#D6" title="SEQ_4:def.6">SEQ_4:def 6</a></i>;<br/>
<b>assume </b><a NAME="E3:21"/>
 <a href="seq_4.html#K6" title="SEQ_4:func.6">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="E4:21"/>
contradiction
 <b>by </b><i><a class="txt" href="abian.html#E2:21"><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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