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<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">Ne</font> be   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K5">NAT</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:15"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">Ne</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">n</font>
 <b>and </b><br/><a NAME="E2:15"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:15_1_1"/>
(  not <font color="Maroon">Ne</font> is <a href="xboole_0.html#V1">empty</a> or <font color="Maroon">Ne</font> is <a href="xboole_0.html#V1">empty</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_1_1"><b>suppose </b></a><a NAME="E1:15_1_1_1"/>
not <font color="Maroon">Ne</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:15_1_1_1"/>
 <a href="henmodel.html#K1">min*</a> <font color="Maroon">Ne</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_1_2"><b>suppose </b></a><a NAME="E1:15_1_1_2"/>
<font color="Maroon">Ne</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><a NAME="E2:15_1_1_2"/><b>then </b>
(  <a href="henmodel.html#K1">min*</a> <font color="Maroon">Ne</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> 1 is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  )
 <b>by </b><i>, <a class="ref" href="henmodel.html#D1">HENMODEL:def 1</a>, <a class="ref" href="nat_1.html#T60">NAT_1:60</a></i>;<br/><b>hence </b><a NAME="E3:15_1_1_2"/>
 <a href="henmodel.html#K1">min*</a> <font color="Maroon">Ne</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> 1
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:15"/>
 <a href="henmodel.html#K1">min*</a> <font color="Maroon">Ne</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> 1
 ;<br/>


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