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<span class="kw">let </span><font color="Maroon" title="c1">K</font>, <font color="Maroon" title="c2">N</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#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="seq_4.html#K4" title="SEQ_4:func.4">min</a> <font color="Maroon" title="c1">K</font></span>)</span>,<span class="p1">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">min</a> <font color="Maroon" title="c2">N</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K4" title="SEQ_4:func.4">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">K</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c2">N</font></span>)</span></span><br/>



<span class="kw">set </span><font color="Maroon" title="c3">m</font> =  <a href="nat_1.html#K4" title="NAT_1:func.4">min</a> (<span class="p1">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">min</a> <font color="Maroon" title="c2">N</font></span>)</span>,<span class="p1">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">min</a> <font color="Maroon" title="c1">K</font></span>)</span>);<br/>
<a NAME="E1:2"/><span class="lab"><font color="Green" title="E3">A1</font></span>: 
 for <font color="Olive" title="b1">k</font> being   <a href="xxreal_0.html#NM1" title="XXREAL_0:NM.1">ExtReal</a>  st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">N</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c1">K</font> holds <br/> <a href="nat_1.html#K4" title="NAT_1:func.4">min</a> (<span class="p1">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">min</a> <font color="Maroon" title="c2">N</font></span>)</span>,<span class="p1">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">min</a> <font color="Maroon" title="c1">K</font></span>)</span>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">k</font> be   <a href="xxreal_0.html#NM1" title="XXREAL_0:NM.1">ExtReal</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"> ( <font color="Maroon" title="c4">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">N</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c1">K</font> implies  <a href="nat_1.html#K4" title="NAT_1:func.4">min</a> (<span class="p1">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">min</a> <font color="Maroon" title="c2">N</font></span>)</span>,<span class="p1">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">min</a> <font color="Maroon" title="c1">K</font></span>)</span>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">k</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:2_1"/>
<font color="Maroon" title="c4">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">N</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c1">K</font>
 ; <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="seq_4.html#K4" title="SEQ_4:func.4">min</a> <font color="Maroon" title="c2">N</font></span>)</span>,<span class="p1">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">min</a> <font color="Maroon" title="c1">K</font></span>)</span>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">k</font></span><br/>

<a NAME="E2:2_1"/><span class="kw">then </span>
( <font color="Maroon" title="c4">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">N</font> or <font color="Maroon" title="c4">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">K</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/>
<a NAME="E3:2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E4">A2</font></span>: 
(  <a href="seq_4.html#K4" title="SEQ_4:func.4">min</a> <font color="Maroon" title="c2">N</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">k</font> or  <a href="seq_4.html#K4" title="SEQ_4:func.4">min</a> <font color="Maroon" title="c1">K</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">k</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_2.html#D7" onmouseover="rs('xxreal_2/D7')" onmouseout="rh()">XXREAL_2:def 7</a></span>;<br/>
<a NAME="E4:2_1"/><span class="lab"><font color="Green" title="E5">A3</font></span>: 
 <a href="nat_1.html#K4" title="NAT_1:func.4">min</a> (<span class="p1">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">min</a> <font color="Maroon" title="c2">N</font></span>)</span>,<span class="p1">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">min</a> <font color="Maroon" title="c1">K</font></span>)</span>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="seq_4.html#K4" title="SEQ_4:func.4">min</a> <font color="Maroon" title="c1">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T17" onmouseover="rs('xxreal_0/T17')" onmouseout="rh()">XXREAL_0:17</a></span>;<br/>
<a NAME="E5:2_1"/>
 <a href="nat_1.html#K4" title="NAT_1:func.4">min</a> (<span class="p1">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">min</a> <font color="Maroon" title="c2">N</font></span>)</span>,<span class="p1">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">min</a> <font color="Maroon" title="c1">K</font></span>)</span>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="seq_4.html#K4" title="SEQ_4:func.4">min</a> <font color="Maroon" title="c2">N</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T17" onmouseover="rs('xxreal_0/T17')" onmouseout="rh()">XXREAL_0:17</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:2_1"/>
 <a href="nat_1.html#K4" title="NAT_1:func.4">min</a> (<span class="p1">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">min</a> <font color="Maroon" title="c2">N</font></span>)</span>,<span class="p1">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">min</a> <font color="Maroon" title="c1">K</font></span>)</span>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:2_1"><span class="lab"><font color="Green" title="E4">A2</font></span></a>, <a class="txt" href="#E4:2_1"><span class="lab"><font color="Green" title="E5">A3</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</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>
<a NAME="E2:2"/>
(  <a href="nat_1.html#K4" title="NAT_1:func.4">min</a> (<span class="p1">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">min</a> <font color="Maroon" title="c2">N</font></span>)</span>,<span class="p1">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">min</a> <font color="Maroon" title="c1">K</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K4" title="SEQ_4:func.4">min</a> <font color="Maroon" title="c2">N</font> or  <a href="nat_1.html#K4" title="NAT_1:func.4">min</a> (<span class="p1">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">min</a> <font color="Maroon" title="c2">N</font></span>)</span>,<span class="p1">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">min</a> <font color="Maroon" title="c1">K</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K4" title="SEQ_4:func.4">min</a> <font color="Maroon" title="c1">K</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T15" onmouseover="rs('xxreal_0/T15')" onmouseout="rh()">XXREAL_0:15</a></span>;<br/>
<a NAME="E3:2"/><span class="kw">then </span>
(  <a href="nat_1.html#K4" title="NAT_1:func.4">min</a> (<span class="p1">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">min</a> <font color="Maroon" title="c2">N</font></span>)</span>,<span class="p1">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">min</a> <font color="Maroon" title="c1">K</font></span>)</span>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">N</font> or  <a href="nat_1.html#K4" title="NAT_1:func.4">min</a> (<span class="p1">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">min</a> <font color="Maroon" title="c2">N</font></span>)</span>,<span class="p1">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">min</a> <font color="Maroon" title="c1">K</font></span>)</span>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">K</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_2.html#D7" onmouseover="rs('xxreal_2/D7')" onmouseout="rh()">XXREAL_2:def 7</a></span>;<br/>
<a NAME="E4:2"/><span class="kw">then </span>
 <a href="nat_1.html#K4" title="NAT_1:func.4">min</a> (<span class="p1">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">min</a> <font color="Maroon" title="c2">N</font></span>)</span>,<span class="p1">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">min</a> <font color="Maroon" title="c1">K</font></span>)</span>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">N</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c1">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:2"/>
 <a href="nat_1.html#K4" title="NAT_1:func.4">min</a> (<span class="p1">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">min</a> <font color="Maroon" title="c1">K</font></span>)</span>,<span class="p1">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">min</a> <font color="Maroon" title="c2">N</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K4" title="SEQ_4:func.4">min</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">K</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c2">N</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2"><span class="lab"><font color="Green" title="E3">A1</font></span></a>, <a class="ref" href="xxreal_2.html#D7" onmouseover="rs('xxreal_2/D7')" onmouseout="rh()">XXREAL_2:def 7</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>
