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

<span class="kw">consider </span><font color="Maroon" title="c3">A</font>, <font color="Maroon" title="c4">B</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">sequence</a> of <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a><span class="kw"> such that </span><br/><a NAME="E2:3_1_1"/><span class="lab"><font color="Green" title="E4">A1</font></span>: 
( <font color="Maroon" title="c3">A</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><font color="Maroon" title="c1">a</font></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span> &amp; <font color="Maroon" title="c4">B</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><font color="Maroon" title="c2">b</font></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span> &amp; (  for <font color="Olive" title="b1">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds <br/> ( <font color="Maroon" title="c3">A</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">B</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Maroon" title="c4">B</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">B</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b1">i</font></span>)</span> ) ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="ntalgo_1.html#E2"><span class="lab"><font color="Green" title="E2">Lm1</font></span></a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">m</font> = <font color="Maroon" title="c3">A</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p1">(<span class="default"><a href="nat_1.html#K3" title="NAT_1:func.3">min*</a> <span class="p2"> { <span class="default"> <font color="Olive" title="b1">i</font> where <font color="Olive" title="b1">i</font> is   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : <font color="Maroon" title="c4">B</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  </span> } </span> </span>)</span>;<br/>
<a NAME="E3:3_1_1"/>
<font color="Maroon" title="c3">A</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p1">(<span class="default"><a href="nat_1.html#K3" title="NAT_1:func.3">min*</a> <span class="p2"> { <span class="default"> <font color="Olive" title="b1">i</font> where <font color="Olive" title="b1">i</font> is   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : <font color="Maroon" title="c4">B</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  </span> } </span> </span>)</span> is    <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> 
 
;<br/>
<span class="kw">hence </span><a NAME="E4:3_1_1"/>
 ex <font color="Olive">b<sub>1</sub></font> being    <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>  ex <font color="Olive" title="b2">A</font>, <font color="Olive" title="b3">B</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">sequence</a> of <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> st <br/>( <font color="Olive" title="b2">A</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><font color="Maroon" title="c1">a</font></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span> &amp; <font color="Olive" title="b3">B</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><font color="Maroon" title="c2">b</font></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span> &amp; (  for <font color="Olive" title="b4">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds <br/> ( <font color="Olive" title="b2">A</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">B</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b4">i</font> &amp; <font color="Olive" title="b3">B</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">A</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b4">i</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">B</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b4">i</font></span>)</span> ) ) &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">A</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p1">(<span class="default"><a href="nat_1.html#K3" title="NAT_1:func.3">min*</a> <span class="p2"> { <span class="default"> <font color="Olive" title="b4">i</font> where <font color="Olive" title="b4">i</font> is   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : <font color="Olive" title="b3">B</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  </span> } </span> </span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:3_1_1"><span class="lab"><font color="Green" title="E4">A1</font></span></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>
