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<b>let </b><font color="Maroon" title="c1">Ke</font>, <font color="Maroon" title="c2">Ne</font> be   <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/>



<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:3_1_1"/>
( <font color="Maroon" title="c1">Ke</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> or <font color="Maroon" title="c2">Ne</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> or ( not <font color="Maroon" title="c1">Ke</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; not <font color="Maroon" title="c2">Ne</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_1_1_1"><b>suppose </b></a><a NAME="E1:3_1_1_1"/>
( <font color="Maroon" title="c1">Ke</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> or <font color="Maroon" title="c2">Ne</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )
 ;<br/><div class="add"><a NAME="E2:3_1_1_1"/><b>then </b>
( (  <a href="nat_1.html#K3" title="NAT_1:func.3">min*</a> <font color="Maroon" title="c1">Ke</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 &amp;  <a href="nat_1.html#K3" title="NAT_1:func.3">min*</a> <font color="Maroon" title="c2">Ne</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 ) or (  <a href="nat_1.html#K3" title="NAT_1:func.3">min*</a> <font color="Maroon" title="c2">Ne</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 &amp;  <a href="nat_1.html#K3" title="NAT_1:func.3">min*</a> <font color="Maroon" title="c1">Ke</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 ) )
 <b>by </b><i><a class="ref" href="nat_1.html#D1" title="NAT_1:def.1">NAT_1:def 1</a></i>;<br/><b>hence </b><a NAME="E3:3_1_1_1"/>
 <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <span class="p1">(<span class="default"><a href="nat_1.html#K3" title="NAT_1:func.3">min*</a> <font color="Maroon" title="c1">Ke</font></span>)</span>,<span class="p1">(<span class="default"><a href="nat_1.html#K3" title="NAT_1:func.3">min*</a> <font color="Maroon" title="c2">Ne</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="nat_1.html#K3" title="NAT_1:func.3">min*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">Ke</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c2">Ne</font></span>)</span>
 <b>by </b><i><a class="ref" href="xxreal_0.html#D8" title="XXREAL_0:def.8">XXREAL_0:def 8</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_1_1_2"><b>suppose </b></a><a NAME="E1:3_1_1_2"/>
( not <font color="Maroon" title="c1">Ke</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; not <font color="Maroon" title="c2">Ne</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c3">K</font> = <font color="Maroon" title="c1">Ke</font>, <font color="Maroon" title="c4">N</font> = <font color="Maroon" title="c2">Ne</font> as  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/><a NAME="E3:3_1_1_2"/>
(  <a href="seq_4.html#K6" title="SEQ_4:func.6">min</a> <font color="Maroon" title="c3">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="nat_1.html#K3" title="NAT_1:func.3">min*</a> <font color="Maroon" title="c1">Ke</font> &amp;  <a href="seq_4.html#K6" title="SEQ_4:func.6">min</a> <font color="Maroon" title="c4">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="nat_1.html#K3" title="NAT_1:func.3">min*</a> <font color="Maroon" title="c2">Ne</font> &amp;  <a href="seq_4.html#K6" title="SEQ_4:func.6">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">K</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c4">N</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="nat_1.html#K3" title="NAT_1:func.3">min*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">Ke</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c2">Ne</font></span>)</span> )
 <b>by </b><i><a class="ref" href="stirl2_1.html#T1" target="_self" title="STIRL2_1:th.1">Th1</a></i>;<br/><b>hence </b><a NAME="E4:3_1_1_2"/>
 <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <span class="p1">(<span class="default"><a href="nat_1.html#K3" title="NAT_1:func.3">min*</a> <font color="Maroon" title="c1">Ke</font></span>)</span>,<span class="p1">(<span class="default"><a href="nat_1.html#K3" title="NAT_1:func.3">min*</a> <font color="Maroon" title="c2">Ne</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="nat_1.html#K3" title="NAT_1:func.3">min*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">Ke</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c2">Ne</font></span>)</span>
 <b>by </b><i><a class="ref" href="stirl2_1.html#T2" target="_self" title="STIRL2_1:th.2">Th2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:3"/>
 <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <span class="p1">(<span class="default"><a href="nat_1.html#K3" title="NAT_1:func.3">min*</a> <font color="Maroon" title="c1">Ke</font></span>)</span>,<span class="p1">(<span class="default"><a href="nat_1.html#K3" title="NAT_1:func.3">min*</a> <font color="Maroon" title="c2">Ne</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="nat_1.html#K3" title="NAT_1:func.3">min*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">Ke</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c2">Ne</font></span>)</span>
 ;<br/>


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