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<span class="kw">let </span><font color="Maroon" title="c1">n</font>, <font color="Maroon" title="c2">k</font>, <font color="Maroon" title="c3">l</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</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#K3" title="NAT_1:func.3">min*</a> <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c3">l</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="nat_1.html#K4" title="NAT_1:func.4">min</a> (<font color="Maroon" title="c1">n</font>,<span class="p1">(<span class="default"><a href="nat_1.html#K4" title="NAT_1:func.4">min</a> (<font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c3">l</font>)</span>)</span>)</span><br/>

<a NAME="E1:7"/><span class="lab"><font color="Green" title="E5">A1</font></span>: 
 <a href="seq_4.html#K4" title="SEQ_4:func.4">min</a> <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c3">l</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></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"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c3">l</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></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="E2:7"/>
<span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c3">l</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c1">n</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c3">l</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="enumset1.html#T2" onmouseover="rs('enumset1/T2')" onmouseout="rh()">ENUMSET1:2</a></span>;<br/>
<a NAME="E3:7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E6">A2</font></span>: 
 <a href="seq_4.html#K4" title="SEQ_4:func.4">min</a> <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c3">l</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c1">n</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>,<span class="p1">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">min</a> <span class="p2"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c3">l</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="stirl2_1.html#T2" target="_self" onmouseover="rs('stirl2_1/T2')" onmouseout="rh()">Th2</a></span>;<br/>
<a NAME="E4:7"/>
 <a href="seq_4.html#K4" title="SEQ_4:func.4">min</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c3">l</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="nat_1.html#K4" title="NAT_1:func.4">min</a> (<font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c3">l</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="stirl2_1.html#T6" target="_self" onmouseover="rs('stirl2_1/T6')" onmouseout="rh()">Th6</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:7"/>
 <a href="nat_1.html#K3" title="NAT_1:func.3">min*</a> <span class="p1"><a href="domain_1.html#K8" title="DOMAIN_1:func.8">{</a><span class="default"><font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c3">l</font></span><a href="domain_1.html#K8" title="DOMAIN_1:func.8">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="nat_1.html#K4" title="NAT_1:func.4">min</a> (<font color="Maroon" title="c1">n</font>,<span class="p1">(<span class="default"><a href="nat_1.html#K4" title="NAT_1:func.4">min</a> (<font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c3">l</font>)</span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:7"><span class="lab"><font color="Green" title="E6">A2</font></span></a>, <a class="txt" href="#E1:7"><span class="lab"><font color="Green" title="E5">A1</font></span></a>, <a class="ref" href="stirl2_1.html#T5" target="_self" onmouseover="rs('stirl2_1/T5')" onmouseout="rh()">Th5</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/>


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