<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><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#NK1" title="NUMBERS:NK.1">NAT</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="nat_1.html#K4" title="NAT_1:func.4">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></span><br/>



<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="3_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="nat_1.html#K4" title="NAT_1:func.4">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></span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></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"><span class="kw">suppose </span></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>  )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="nat_1.html#K4" title="NAT_1:func.4">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></span><br/><div class="add"><a NAME="E2:3_1_1_1"/><span class="kw">then </span>
( (  <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>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &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>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  ) 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>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &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>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  ) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#D1" onmouseover="rs('nat_1/D1')" onmouseout="rh()">NAT_1:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E3:3_1_1_1"/>
 <a href="nat_1.html#K4" title="NAT_1:func.4">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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#D9" onmouseover="rs('xxreal_0/D9')" onmouseout="rh()">XXREAL_0:def 9</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_1_1_2"><span class="kw">suppose </span></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>  )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="nat_1.html#K4" title="NAT_1:func.4">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></span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><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#NK1" title="NUMBERS:NK.1">NAT</a> ;<br/><a NAME="E3:3_1_1_2"/><span class="lab"><font color="Green" title="E3">A1</font></span>: 
 <a href="seq_4.html#K4" title="SEQ_4:func.4">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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="stirl2_1.html#T1" target="_self" onmouseover="rs('stirl2_1/T1')" onmouseout="rh()">Th1</a></span>;<br/><a NAME="E4:3_1_1_2"/><span class="lab"><font color="Green" title="E4">A2</font></span>: 
 <a href="seq_4.html#K4" title="SEQ_4:func.4">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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="stirl2_1.html#T1" target="_self" onmouseover="rs('stirl2_1/T1')" onmouseout="rh()">Th1</a></span>;<br/><a NAME="E5:3_1_1_2"/>
 <a href="seq_4.html#K4" title="SEQ_4:func.4">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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="stirl2_1.html#T1" target="_self" onmouseover="rs('stirl2_1/T1')" onmouseout="rh()">Th1</a></span>;<br/><span class="kw">hence </span><a NAME="E6:3_1_1_2"/>
 <a href="nat_1.html#K4" title="NAT_1:func.4">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:3_1_1_2"><span class="lab"><font color="Green" title="E3">A1</font></span></a>, <a class="txt" href="#E4:3_1_1_2"><span class="lab"><font color="Green" title="E4">A2</font></span></a>, <a class="ref" href="stirl2_1.html#T2" target="_self" onmouseover="rs('stirl2_1/T2')" onmouseout="rh()">Th2</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E2:3"/>
 <a href="nat_1.html#K4" title="NAT_1:func.4">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>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
