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



<b>assume </b><a NAME="E1:4"/><i><font color="Green">E38</font></i>: 
not  <a href="henmodel.html#K1">min*</a> <font color="Maroon">Ne</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Ne</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Ke</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:4_1_1"/>
( <font color="Maroon">Ne</font> is <a href="xboole_0.html#V1">empty</a> or  not <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="4_1_1_1"><b>suppose </b></a><a NAME="E1:4_1_1_1"/>
<font color="Maroon">Ne</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:4_1_1_1"/>
 <a href="henmodel.html#K1">min*</a> <font color="Maroon">Ne</font> <a href="hidden.html#R1">=</a>  <a href="henmodel.html#K1">min*</a> <span class="p1">(<span class="default"><font color="Maroon">Ne</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">Ke</font></span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_1_1_2"><b>suppose </b></a><a NAME="E1:4_1_1_2"/>
not <font color="Maroon">Ne</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><a NAME="E2:4_1_1_2"/><b>then </b><i><font color="Green">E39</font></i>: 
 <a href="henmodel.html#K1">min*</a> <font color="Maroon">Ne</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Ne</font>
 <b>by </b><i><a class="ref" href="henmodel.html#D1">HENMODEL:def 1</a></i>;<br/><a NAME="E3:4_1_1_2"/><b>then </b>
not  <a href="henmodel.html#K1">min*</a> <font color="Maroon">Ne</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Ke</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E4:4_1_1_2"/><b>then </b><i><font color="Green">E40</font></i>: 
 <a href="henmodel.html#K1">min*</a> <font color="Maroon">Ne</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Ne</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">Ke</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E5:4_1_1_2"/><b>then </b>
(  <a href="henmodel.html#K1">min*</a> <span class="p1">(<span class="default"><font color="Maroon">Ne</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">Ke</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">Ne</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">Ke</font> &amp; <font color="Maroon">Ne</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">Ke</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Ne</font> )
 <b>by </b><i><a class="ref" href="xboole_1.html#T36">XBOOLE_1:36</a>, <a class="ref" href="henmodel.html#D1">HENMODEL:def 1</a></i>;<br/><a NAME="E6:4_1_1_2"/><b>then </b>
(  <a href="henmodel.html#K1">min*</a> <span class="p1">(<span class="default"><font color="Maroon">Ne</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">Ke</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="henmodel.html#K1">min*</a> <font color="Maroon">Ne</font> &amp;  <a href="henmodel.html#K1">min*</a> <font color="Maroon">Ne</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="henmodel.html#K1">min*</a> <span class="p1">(<span class="default"><font color="Maroon">Ne</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">Ke</font></span>)</span> )
 <b>by </b><i>, <a class="ref" href="henmodel.html#D1">HENMODEL:def 1</a></i>;<br/><b>hence </b><a NAME="E7:4_1_1_2"/>
 <a href="henmodel.html#K1">min*</a> <font color="Maroon">Ne</font> <a href="hidden.html#R1">=</a>  <a href="henmodel.html#K1">min*</a> <span class="p1">(<span class="default"><font color="Maroon">Ne</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">Ke</font></span>)</span>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:4"/>
 <a href="henmodel.html#K1">min*</a> <font color="Maroon">Ne</font> <a href="hidden.html#R1">=</a>  <a href="henmodel.html#K1">min*</a> <span class="p1">(<span class="default"><font color="Maroon">Ne</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">Ke</font></span>)</span>
 ;<br/>


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