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

<b>let </b><font color="Maroon">o1</font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/><b>let </b><font color="Maroon">o2</font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/><b>let </b><font color="Maroon">b1</font> be    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o1</font>;<br/><b>let </b><font color="Maroon">b2</font> be    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o2</font>;<br/><b>let </b><font color="Maroon">G</font> be    <a href="finseq_1.html#M2">FinSequence</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K14">Bags</a> <span class="p2">(<span class="default"><font color="Maroon">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font></span>)</span></span>)</span> <a href="finseq_2.html#K3">*</a> ;<br/>









<b>assume </b><b>that </b><br/><a NAME="E1:18"/><i><font color="Green">E35</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">b1</font></span>)</span>
 <b>and </b><br/><a NAME="E2:18"/><i><font color="Green">E36</font></i>: 
for <font color="Olive">i</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">b1</font></span>)</span> holds <br/> ex <font color="Olive">a1'</font> being   <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o1</font> ex <font color="Olive">Fk</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="polynom1.html#K14">Bags</a> <span class="p1">(<span class="default"><font color="Maroon">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font></span>)</span> st <br/>( <font color="Olive">Fk</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="matrlin.html#K1">/.</a> <font color="Olive">i</font> &amp; <span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">b1</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Olive">a1'</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Olive">Fk</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">b2</font></span>)</span> &amp; ( for <font color="Olive">m</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Olive">Fk</font> holds <br/> ex <font color="Olive">a1''</font> being   <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o2</font> st <br/>( <span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">b2</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">m</font> <a href="hidden.html#R1">=</a> <font color="Olive">a1''</font> &amp; <font color="Olive">Fk</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">m</font> <a href="hidden.html#R1">=</a> <font color="Olive">a1'</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Olive">a1''</font> ) ) )
 ;<br/>

<a NAME="E3:18"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">b1</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/>
<a NAME="E4:18"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E5:18"/><b>then </b><i><font color="Green">E37</font></i>: 
not <font color="Maroon">G</font> is <a href="xboole_0.html#V1">empty</a>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/>
<a NAME="E6:18"/><b>then </b><i><font color="Green">E38</font></i>: 
<font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> 1</span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <span class="p1">(<span class="default"><a href="polynom3.html#K1">Del</a> <font color="Maroon">G</font>,1</span>)</span>
 
<b>by </b><i><a class="ref" href="polyalg1.html#T4">POLYALG1:4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">D</font> =  <a href="polynom3.html#K1">Del</a> <font color="Maroon">G</font>,1 as    <a href="finseq_1.html#M2">FinSequence</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K14">Bags</a> <span class="p2">(<span class="default"><font color="Maroon">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font></span>)</span></span>)</span> <a href="finseq_2.html#K3">*</a>  ;<br/>
<a NAME="E8:18"/><i><font color="Green">E39</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G</font>
 
<b>by </b><i>, <a class="ref" href="finseq_5.html#T6">FINSEQ_5:6</a></i>;<br/>
<b>then consider </b><font color="Maroon">a1'</font> being    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o1</font>, <font color="Maroon">Fk</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="polynom1.html#K14">Bags</a> <span class="p1">(<span class="default"><font color="Maroon">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font></span>)</span><b> such that </b><br/><a NAME="E10:18"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">Fk</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="matrlin.html#K1">/.</a> 1 &amp; <span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">b1</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">a1'</font> )
 <b>and </b><br/><a NAME="E11:18"/><i><font color="Green">E41</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">Fk</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">b2</font></span>)</span>
 <b>and </b><br/><a NAME="E12:18"/>
for <font color="Olive">m</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">Fk</font> holds <br/> ex <font color="Olive">a1''</font> being   <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o2</font> st <br/>( <span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">b2</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">m</font> <a href="hidden.html#R1">=</a> <font color="Olive">a1''</font> &amp; <font color="Maroon">Fk</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a1'</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Olive">a1''</font> )
 <b>by </b><i>, </i>;<br/>
<b>reconsider </b><font color="Maroon">G1</font> = <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> 1 as    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K14">Bags</a> <span class="p2">(<span class="default"><font color="Maroon">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font></span>)</span></span>)</span> <a href="finseq_2.html#K3">*</a>  <b>by </b><i>, , <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<i><font color="Green">E42</font></i>:  <a href="dtconstr.html#K15">FlattenSeq</a> <font color="Maroon">G</font> = 
<span class="p1">(<span class="default"><a href="dtconstr.html#K15">FlattenSeq</a> <span class="p2"><a href="matrlin.html#K3">&lt;*</a><span class="default"><font color="Maroon">G1</font></span><a href="matrlin.html#K3">*&gt;</a></span></span>)</span> <a href="wsierp_1.html#K4">^</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15">FlattenSeq</a> <font color="Maroon">D</font></span>)</span>
<b>by </b><i>, <a class="ref" href="dtconstr.html#T21">DTCONSTR:21</a></i>
<br/>.= 
<font color="Maroon">G1</font> <a href="wsierp_1.html#K4">^</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15">FlattenSeq</a> <font color="Maroon">D</font></span>)</span>
<b>by </b><i><a class="ref" href="dtconstr.html#T13">DTCONSTR:13</a></i>
;<br/>
<a NAME="E15:18"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">Fk</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> 1
 
<b>by </b><i>, , <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<a NAME="E16:18"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">b2</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/>
<a NAME="E17:18"/><b>then </b>
<font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i>, , <a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/>
<a NAME="E18:18"/><b>then </b>
 <a href="dtconstr.html#K15">FlattenSeq</a> <font color="Maroon">G</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T48">FINSEQ_1:48</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">S</font> =  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15">FlattenSeq</a> <font color="Maroon">G</font></span>)</span> as  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K14">Bags</a> <span class="p2">(<span class="default"><font color="Maroon">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a>, <a class="ref" href="relat_1.html#T64">RELAT_1:64</a></i>;<br/>
<b>set </b><font color="Maroon">F</font> =  <a href="dtconstr.html#K15">FlattenSeq</a> <font color="Maroon">G</font>;<br/>
<a NAME="E20:18"/>
 <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <span class="p2">(<span class="default"><font color="Maroon">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="polynom1.html#K14">Bags</a> <span class="p1">(<span class="default"><font color="Maroon">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="orders_1.html#T100">ORDERS_1:100</a></i>;<br/>
<a NAME="E21:18"/><b>then </b>
 <a href="polynom1.html#K17">BagOrder</a> <span class="p1">(<span class="default"><font color="Maroon">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font></span>)</span> <a href="orders_1.html#R3">linearly_orders</a>  <a href="polynom1.html#K14">Bags</a> <span class="p1">(<span class="default"><font color="Maroon">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="orders_1.html#T133">ORDERS_1:133</a></i>;<br/>
<a NAME="E22:18"/><b>then </b><i><font color="Green">E44</font></i>: 
 <a href="polynom1.html#K17">BagOrder</a> <span class="p1">(<span class="default"><font color="Maroon">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font></span>)</span> <a href="orders_1.html#R3">linearly_orders</a> <font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="orders_1.html#T134">ORDERS_1:134</a></i>;<br/>
<a NAME="E23:18"/>
for <font color="Olive">n</font>, <font color="Olive">m</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15">FlattenSeq</a> <font color="Maroon">G</font></span>)</span> &amp; <font color="Olive">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15">FlattenSeq</a> <font color="Maroon">G</font></span>)</span> &amp; <font color="Olive">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">m</font> holds <br/>( <span class="p1">(<span class="default"><a href="dtconstr.html#K15">FlattenSeq</a> <font color="Maroon">G</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15">FlattenSeq</a> <font color="Maroon">G</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">m</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="dtconstr.html#K15">FlattenSeq</a> <font color="Maroon">G</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">n</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="dtconstr.html#K15">FlattenSeq</a> <font color="Maroon">G</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">m</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K17">BagOrder</a> <span class="p1">(<span class="default"><font color="Maroon">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be   <a href="nat_1.html#NM1">Nat</a>, <font color="Maroon">m</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:18_2"/><i><font color="Green">E58</font></i>: 
( <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15">FlattenSeq</a> <font color="Maroon">G</font></span>)</span> &amp; <font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15">FlattenSeq</a> <font color="Maroon">G</font></span>)</span> )
 <b>and </b><br/><a NAME="E2:18_2"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_2_1"/><i><font color="Green">E61</font></i>: 
<span class="p1">(<span class="default"><a href="dtconstr.html#K15">FlattenSeq</a> <font color="Maroon">G</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15">FlattenSeq</a> <font color="Maroon">G</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">m</font>
 ;<br/><a NAME="E2:18_2_1"/><i><font color="Green">E64</font></i>: 
<span class="p1">(<span class="default"><a href="dtconstr.html#K15">FlattenSeq</a> <font color="Maroon">G</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15">FlattenSeq</a> <font color="Maroon">G</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E3:18_2_1"/><i><font color="Green">E65</font></i>: 
<span class="p1">(<span class="default"><a href="dtconstr.html#K15">FlattenSeq</a> <font color="Maroon">G</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15">FlattenSeq</a> <font color="Maroon">G</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font>
 <b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><b>consider </b><font color="Maroon">i1</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j1</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E5:18_2_1"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">i1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G</font>
 <b>and </b><br/><a NAME="E6:18_2_1"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">j1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i1</font></span>)</span>
 <b>and </b><br/><a NAME="E7:18_2_1"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K7">Sum</a> <span class="p2">(<span class="default"><a href="polynom1.html#K5">Card</a> <span class="p3">(<span class="default"><font color="Maroon">G</font> <a href="finseq_1.html#K17">|</a> <span class="p4">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">j1</font>
 <b>and </b><br/><a NAME="E8:18_2_1"/><i><font color="Green">E70</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15">FlattenSeq</a> <font color="Maroon">G</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="polynom1.html#T32">POLYNOM1:32</a></i>;<br/><b>consider </b><font color="Maroon">i2</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j2</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E10:18_2_1"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">i2</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G</font>
 <b>and </b><br/><a NAME="E11:18_2_1"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">j2</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i2</font></span>)</span>
 <b>and </b><br/><a NAME="E12:18_2_1"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K7">Sum</a> <span class="p2">(<span class="default"><a href="polynom1.html#K5">Card</a> <span class="p3">(<span class="default"><font color="Maroon">G</font> <a href="finseq_1.html#K17">|</a> <span class="p4">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">j2</font>
 <b>and </b><br/><a NAME="E13:18_2_1"/><i><font color="Green">E74</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15">FlattenSeq</a> <font color="Maroon">G</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font>
 <b>by </b><i>, <a class="ref" href="polynom1.html#T32">POLYNOM1:32</a></i>;<br/><b>consider </b><font color="Maroon">a1'</font> being    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o1</font>, <font color="Maroon">Fk</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="polynom1.html#K14">Bags</a> <span class="p1">(<span class="default"><font color="Maroon">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font></span>)</span><b> such that </b><br/><a NAME="E15:18_2_1"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">Fk</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="matrlin.html#K1">/.</a> <font color="Maroon">i1</font>
 <b>and </b><br/><a NAME="E16:18_2_1"/><i><font color="Green">E76</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">b1</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a1'</font>
 <b>and </b><br/><a NAME="E17:18_2_1"/><i><font color="Green">E78</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">Fk</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">b2</font></span>)</span>
 <b>and </b><br/><a NAME="E18:18_2_1"/><i><font color="Green">E81</font></i>: 
for <font color="Olive">r</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">r</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">Fk</font> holds <br/> ex <font color="Olive">a1''</font> being   <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o2</font> st <br/>( <span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">b2</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">r</font> <a href="hidden.html#R1">=</a> <font color="Olive">a1''</font> &amp; <font color="Maroon">Fk</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a1'</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Olive">a1''</font> )
 <b>by </b><i>, , </i>;<br/><a NAME="E19:18_2_1"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">j1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">Fk</font>
 <b>by </b><i>, , <a class="txt" href="polynom6.html#E5:18"><i><font color="Green">E37</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><b>then consider </b><font color="Maroon">a1''</font> being    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o2</font><b> such that </b><br/><a NAME="E21:18_2_1"/><i><font color="Green">E83</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">b2</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a1''</font>
 <b>and </b><br/><a NAME="E22:18_2_1"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">Fk</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a1'</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">a1''</font>
 <b>by </b><i><a class="txt" href="polynom6.html#E10:18"><i><font color="Green">E40</font></i></a></i>;<br/><b>consider </b><font color="Maroon">a2'</font> being    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o1</font>, <font color="Maroon">Fk'</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="polynom1.html#K14">Bags</a> <span class="p1">(<span class="default"><font color="Maroon">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font></span>)</span><b> such that </b><br/><a NAME="E24:18_2_1"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">Fk'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="matrlin.html#K1">/.</a> <font color="Maroon">i2</font>
 <b>and </b><br/><a NAME="E25:18_2_1"/><i><font color="Green">E87</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">b1</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a2'</font>
 <b>and </b><br/><a NAME="E26:18_2_1"/><i><font color="Green">E88</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">Fk'</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">b2</font></span>)</span>
 <b>and </b><br/><a NAME="E27:18_2_1"/><i><font color="Green">E89</font></i>: 
for <font color="Olive">r</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">r</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">Fk'</font> holds <br/> ex <font color="Olive">a2''</font> being   <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o2</font> st <br/>( <span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">b2</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">r</font> <a href="hidden.html#R1">=</a> <font color="Olive">a2''</font> &amp; <font color="Maroon">Fk'</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a2'</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Olive">a2''</font> )
 <b>by </b><i>, , </i>;<br/><a NAME="E28:18_2_1"/><i><font color="Green">E90</font></i>: 
<font color="Maroon">j2</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">Fk'</font>
 <b>by </b><i>, , <a class="txt" href="polynom6.html#E22:18"><i><font color="Green">E44</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><b>then consider </b><font color="Maroon">a2''</font> being    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o2</font><b> such that </b><br/><a NAME="E30:18_2_1"/><i><font color="Green">E133</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">b2</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a2''</font>
 <b>and </b><br/><a NAME="E31:18_2_1"/><i><font color="Green">E134</font></i>: 
<font color="Maroon">Fk'</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a2'</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">a2''</font>
 <b>by </b><i/>;<br/><a NAME="E32:18_2_1"/>
<font color="Maroon">Fk</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i1</font>
 <b>by </b><i>, <a class="txt" href="polynom6.html#E5:18"><i><font color="Green">E37</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E33:18_2_1"/><b>then </b><i><font color="Green">E136</font></i>: 
<span class="p1">(<span class="default"><a href="dtconstr.html#K15">FlattenSeq</a> <font color="Maroon">G</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Fk</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j1</font>
 <b>by </b><i>, , <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E34:18_2_1"/>
<font color="Maroon">Fk'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i2</font>
 <b>by </b><i>, <a class="txt" href="polynom6.html#E22:18"><i><font color="Green">E44</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E35:18_2_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="dtconstr.html#K15">FlattenSeq</a> <font color="Maroon">G</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Fk'</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j2</font>
 <b>by </b><i>, <a class="txt" href="polynom6.html#E2:18"><i><font color="Green">E36</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E36:18_2_1"/><b>then </b><i><font color="Green">E137</font></i>: 
( <font color="Maroon">a1'</font> <a href="pboole.html#R6">=</a> <font color="Maroon">a2'</font> &amp; <font color="Maroon">a1''</font> <a href="pboole.html#R6">=</a> <font color="Maroon">a2''</font> )
 <b>by </b><i>, , , <a class="txt" href="polynom6.html#E15:18"><i><font color="Green">E43</font></i></a>, <a class="ref" href="polynom6.html#D1" target="_self">Def1</a>, , </i>;<br/><a NAME="E37:18_2_1"/><i><font color="Green">E138</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">b1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a1'</font>
 <b>by </b><i>, , <a class="txt" href="polynom6.html#E6:18"><i><font color="Green">E38</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E38:18_2_1"/>
<span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">b1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a2'</font>
 <b>by </b><i>, , <a class="ref" href="polynom6.html#T2" target="_self">Th2</a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E39:18_2_1"/><b>then </b><i><font color="Green">E141</font></i>: 
<font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font>
 <b>by </b><i>, , , , , <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/><a NAME="E40:18_2_1"/>
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">Fk</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">Fk</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E41:18_2_1"/><b>then </b><i><font color="Green">E142</font></i>: 
<font color="Maroon">j1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">b2</font></span>)</span>
 <b>by </b><i><a class="txt" href="polynom6.html#E8:18"><i><font color="Green">E39</font></i></a>, <a class="txt" href="polynom6.html#E11:18"><i><font color="Green">E41</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E42:18_2_1"/><b>then </b><i><font color="Green">E143</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">b2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a1''</font>
 <b>by </b><i><a class="txt" href="polynom6.html#E14:18"><i><font color="Green">E42</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E43:18_2_1"/>
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">Fk'</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">Fk'</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E44:18_2_1"/><b>then </b><i><font color="Green">E144</font></i>: 
<font color="Maroon">j2</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">b2</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E45:18_2_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">b2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a2''</font>
 <b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><b>hence </b><a NAME="E46:18_2_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="polynom6.html#T1" target="_self">Th1</a>, <a class="txt" href="polynom6.html#E1:18"><i><font color="Green">E35</font></i></a>, , , , , , <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:18_2"/>
<span class="p1">(<span class="default"><a href="dtconstr.html#K15">FlattenSeq</a> <font color="Maroon">G</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15">FlattenSeq</a> <font color="Maroon">G</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">m</font>
 ;<br/>

<a NAME="E5:18_2"/><i><font color="Green">E145</font></i>: 
<span class="p1">(<span class="default"><a href="dtconstr.html#K15">FlattenSeq</a> <font color="Maroon">G</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15">FlattenSeq</a> <font color="Maroon">G</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
 
<b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<a NAME="E6:18_2"/><i><font color="Green">E149</font></i>: 
<span class="p1">(<span class="default"><a href="dtconstr.html#K15">FlattenSeq</a> <font color="Maroon">G</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15">FlattenSeq</a> <font color="Maroon">G</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font>
 
<b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">Fn</font> = <span class="p1">(<span class="default"><a href="dtconstr.html#K15">FlattenSeq</a> <font color="Maroon">G</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font>, <font color="Maroon">Fm</font> = <span class="p1">(<span class="default"><a href="dtconstr.html#K15">FlattenSeq</a> <font color="Maroon">G</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">m</font> as   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font> <b>by </b><i><a class="ref" href="polynom1.html#D14">POLYNOM1:def 14</a></i>;<br/>
<b>reconsider </b><font color="Maroon">Fn'</font> = <font color="Maroon">Fn</font>, <font color="Maroon">Fm'</font> = <font color="Maroon">Fm</font> as    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <span class="p1">(<span class="default"><font color="Maroon">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font></span>)</span> ;<br/>
<b>consider </b><font color="Maroon">a1</font> being    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o1</font>, <font color="Maroon">a2</font> being    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o2</font><b> such that </b><br/><a NAME="E10:18_2"/><i><font color="Green">E150</font></i>: 
<font color="Maroon">Fn'</font> <a href="pboole.html#R6">=</a> <font color="Maroon">a1</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">a2</font>
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">c1</font> being    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o1</font>, <font color="Maroon">c2</font> being    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o2</font><b> such that </b><br/><a NAME="E12:18_2"/><i><font color="Green">E151</font></i>: 
<font color="Maroon">Fm'</font> <a href="pboole.html#R6">=</a> <font color="Maroon">c1</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">c2</font>
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">i1</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j1</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E14:18_2"/><i><font color="Green">E179</font></i>: 
<font color="Maroon">i1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G</font>
 <b>and </b><br/><a NAME="E15:18_2"/><i><font color="Green">E180</font></i>: 
<font color="Maroon">j1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i1</font></span>)</span>
 <b>and </b><br/><a NAME="E16:18_2"/><i><font color="Green">E181</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K7">Sum</a> <span class="p2">(<span class="default"><a href="polynom1.html#K5">Card</a> <span class="p3">(<span class="default"><font color="Maroon">G</font> <a href="finseq_1.html#K17">|</a> <span class="p4">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">j1</font>
 <b>and </b><br/><a NAME="E17:18_2"/><i><font color="Green">E182</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15">FlattenSeq</a> <font color="Maroon">G</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="polynom1.html#T32">POLYNOM1:32</a></i>;<br/>
<b>consider </b><font color="Maroon">i2</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j2</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E19:18_2"/><i><font color="Green">E183</font></i>: 
<font color="Maroon">i2</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G</font>
 <b>and </b><br/><a NAME="E20:18_2"/><i><font color="Green">E184</font></i>: 
<font color="Maroon">j2</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i2</font></span>)</span>
 <b>and </b><br/><a NAME="E21:18_2"/><i><font color="Green">E185</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K7">Sum</a> <span class="p2">(<span class="default"><a href="polynom1.html#K5">Card</a> <span class="p3">(<span class="default"><font color="Maroon">G</font> <a href="finseq_1.html#K17">|</a> <span class="p4">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">j2</font>
 <b>and </b><br/><a NAME="E22:18_2"/><i><font color="Green">E186</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15">FlattenSeq</a> <font color="Maroon">G</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font>
 <b>by </b><i>, <a class="ref" href="polynom1.html#T32">POLYNOM1:32</a></i>;<br/>
<b>consider </b><font color="Maroon">a1'</font> being    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o1</font>, <font color="Maroon">Fk</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="polynom1.html#K14">Bags</a> <span class="p1">(<span class="default"><font color="Maroon">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font></span>)</span><b> such that </b><br/><a NAME="E24:18_2"/><i><font color="Green">E187</font></i>: 
<font color="Maroon">Fk</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="matrlin.html#K1">/.</a> <font color="Maroon">i1</font>
 <b>and </b><br/><a NAME="E25:18_2"/><i><font color="Green">E188</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">b1</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a1'</font>
 <b>and </b><br/><a NAME="E26:18_2"/><i><font color="Green">E189</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">Fk</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">b2</font></span>)</span>
 <b>and </b><br/><a NAME="E27:18_2"/><i><font color="Green">E190</font></i>: 
for <font color="Olive">r</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">r</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">Fk</font> holds <br/> ex <font color="Olive">a1''</font> being   <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o2</font> st <br/>( <span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">b2</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">r</font> <a href="hidden.html#R1">=</a> <font color="Olive">a1''</font> &amp; <font color="Maroon">Fk</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a1'</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Olive">a1''</font> )
 <b>by </b><i>, , </i>;<br/>
<a NAME="E28:18_2"/><i><font color="Green">E191</font></i>: 
<font color="Maroon">j1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">Fk</font>
 
<b>by </b><i>, <a class="ref" href="polynom6.html#T1" target="_self">Th1</a>, <a class="txt" href="polynom6.html#E6:18"><i><font color="Green">E38</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<b>then consider </b><font color="Maroon">a1''</font> being    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o2</font><b> such that </b><br/><a NAME="E30:18_2"/><i><font color="Green">E192</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">b2</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a1''</font>
 <b>and </b><br/><a NAME="E31:18_2"/><i><font color="Green">E193</font></i>: 
<font color="Maroon">Fk</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a1'</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">a1''</font>
 <b>by </b><i><a class="txt" href="polynom6.html#E11:18"><i><font color="Green">E41</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">a2'</font> being    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o1</font>, <font color="Maroon">Fk'</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="polynom1.html#K14">Bags</a> <span class="p1">(<span class="default"><font color="Maroon">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font></span>)</span><b> such that </b><br/><a NAME="E33:18_2"/><i><font color="Green">E194</font></i>: 
<font color="Maroon">Fk'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="matrlin.html#K1">/.</a> <font color="Maroon">i2</font>
 <b>and </b><br/><a NAME="E34:18_2"/><i><font color="Green">E195</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">b1</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a2'</font>
 <b>and </b><br/><a NAME="E35:18_2"/><i><font color="Green">E196</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">Fk'</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">b2</font></span>)</span>
 <b>and </b><br/><a NAME="E36:18_2"/><i><font color="Green">E197</font></i>: 
for <font color="Olive">r</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">r</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">Fk'</font> holds <br/> ex <font color="Olive">a2''</font> being   <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o2</font> st <br/>( <span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">b2</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">r</font> <a href="hidden.html#R1">=</a> <font color="Olive">a2''</font> &amp; <font color="Maroon">Fk'</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a2'</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Olive">a2''</font> )
 <b>by </b><i>, , </i>;<br/>
<a NAME="E37:18_2"/><i><font color="Green">E198</font></i>: 
<font color="Maroon">j2</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">Fk'</font>
 
<b>by </b><i>, <a class="txt" href="polynom6.html#E1:18"><i><font color="Green">E35</font></i></a>, <a class="ref" href="polynom6.html#T2" target="_self">Th2</a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<b>then consider </b><font color="Maroon">a2''</font> being    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o2</font><b> such that </b><br/><a NAME="E39:18_2"/><i><font color="Green">E199</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">b2</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a2''</font>
 <b>and </b><br/><a NAME="E40:18_2"/><i><font color="Green">E200</font></i>: 
<font color="Maroon">Fk'</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a2'</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">a2''</font>
 <b>by </b><i/>;<br/>
<a NAME="E41:18_2"/>
<font color="Maroon">Fk</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i1</font>
 
<b>by </b><i>, <a class="txt" href="polynom6.html#E6:18"><i><font color="Green">E38</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<a NAME="E42:18_2"/><b>then </b><i><font color="Green">E201</font></i>: 
<font color="Maroon">Fn'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Fk</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j1</font>
 
<b>by </b><i>, <a class="ref" href="polynom6.html#T1" target="_self">Th1</a>, , <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<a NAME="E43:18_2"/>
<font color="Maroon">Fk'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i2</font>
 
<b>by </b><i>, <a class="ref" href="polynom6.html#T2" target="_self">Th2</a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<a NAME="E44:18_2"/><b>then </b>
<font color="Maroon">Fm'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Fk'</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j2</font>
 
<b>by </b><i>, <a class="txt" href="polynom6.html#E1:18"><i><font color="Green">E35</font></i></a>, <a class="txt" href="polynom6.html#E5:18"><i><font color="Green">E37</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<a NAME="E45:18_2"/><b>then </b><i><font color="Green">E202</font></i>: 
( <font color="Maroon">a1'</font> <a href="pboole.html#R6">=</a> <font color="Maroon">a1</font> &amp; <font color="Maroon">a1''</font> <a href="pboole.html#R6">=</a> <font color="Maroon">a2</font> &amp; <font color="Maroon">a2'</font> <a href="pboole.html#R6">=</a> <font color="Maroon">c1</font> &amp; <font color="Maroon">a2''</font> <a href="pboole.html#R6">=</a> <font color="Maroon">c2</font> )
 
<b>by </b><i>, , <a class="txt" href="polynom6.html#E22:18"><i><font color="Green">E44</font></i></a>, , , </i>;<br/>
<a NAME="E46:18_2"/><b>then </b><i><font color="Green">E203</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">b1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a1</font>
 
<b>by </b><i>, , <a class="txt" href="polynom6.html#E8:18"><i><font color="Green">E39</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<a NAME="E47:18_2"/><i><font color="Green">E204</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">b1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c1</font>
 
<b>by </b><i>, , , , <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<a NAME="E48:18_2"/>
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">Fk</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">Fk</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E49:18_2"/><b>then </b><i><font color="Green">E205</font></i>: 
<font color="Maroon">j1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">b2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="polynom6.html#E10:18"><i><font color="Green">E40</font></i></a>, <a class="txt" href="polynom6.html#E14:18"><i><font color="Green">E42</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E50:18_2"/><b>then </b><i><font color="Green">E206</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">b2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a2</font>
 
<b>by </b><i><a class="txt" href="polynom6.html#E15:18"><i><font color="Green">E43</font></i></a>, , <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<a NAME="E51:18_2"/>
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">Fk'</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">Fk'</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E52:18_2"/><b>then </b><i><font color="Green">E207</font></i>: 
<font color="Maroon">j2</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">b2</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E53:18_2"/><b>then </b><i><font color="Green">E208</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">b2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c2</font>
 
<b>by </b><i><a class="ref" href="polynom6.html#D1" target="_self">Def1</a>, , <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_2"><b>now </b></a><div class="add"><b>consider </b><font color="Maroon">S</font> being  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K14">Bags</a> <font color="Maroon">o1</font></span>)</span><b> such that </b><br/><a NAME="E2:18_2_2"/><i><font color="Green">E209</font></i>: 
 <a href="polynom1.html#K20">divisors</a> <font color="Maroon">b1</font> <a href="hidden.html#R1">=</a>  <a href="triang_1.html#K2">SgmX</a> <span class="p1">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">o1</font></span>)</span>,<font color="Maroon">S</font>
 <b>and </b><br/><a NAME="E3:18_2_2"/>
for <font color="Olive">p</font> being  <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">o1</font> holds <br/> ( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> iff <font color="Olive">p</font> <a href="polynom1.html#R3">divides</a> <font color="Maroon">b1</font> )
 <b>by </b><i><a class="ref" href="polynom1.html#D18">POLYNOM1:def 18</a></i>;<br/><a NAME="E4:18_2_2"/>
 <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">o1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o1</font>
 <b>by </b><i><a class="ref" href="orders_1.html#T100">ORDERS_1:100</a></i>;<br/><a NAME="E5:18_2_2"/><b>then </b>
 <a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">o1</font> <a href="orders_1.html#R3">linearly_orders</a>  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o1</font>
 <b>by </b><i><a class="ref" href="orders_1.html#T133">ORDERS_1:133</a></i>;<br/><a NAME="E6:18_2_2"/><b>then </b><i><font color="Green">E210</font></i>: 
 <a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">o1</font> <a href="orders_1.html#R3">linearly_orders</a> <font color="Maroon">S</font>
 <b>by </b><i><a class="ref" href="orders_1.html#T134">ORDERS_1:134</a></i>;<br/><b>consider </b><font color="Maroon">T</font> being  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K14">Bags</a> <font color="Maroon">o2</font></span>)</span><b> such that </b><br/><a NAME="E8:18_2_2"/><i><font color="Green">E212</font></i>: 
 <a href="polynom1.html#K20">divisors</a> <font color="Maroon">b2</font> <a href="hidden.html#R1">=</a>  <a href="triang_1.html#K2">SgmX</a> <span class="p1">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">o2</font></span>)</span>,<font color="Maroon">T</font>
 <b>and </b><br/><a NAME="E9:18_2_2"/>
for <font color="Olive">p</font> being  <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">o2</font> holds <br/> ( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T</font> iff <font color="Olive">p</font> <a href="polynom1.html#R3">divides</a> <font color="Maroon">b2</font> )
 <b>by </b><i><a class="ref" href="polynom1.html#D18">POLYNOM1:def 18</a></i>;<br/><a NAME="E10:18_2_2"/>
 <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">o2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o2</font>
 <b>by </b><i><a class="ref" href="orders_1.html#T100">ORDERS_1:100</a></i>;<br/><a NAME="E11:18_2_2"/><b>then </b>
 <a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">o2</font> <a href="orders_1.html#R3">linearly_orders</a>  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o2</font>
 <b>by </b><i><a class="ref" href="orders_1.html#T133">ORDERS_1:133</a></i>;<br/><a NAME="E12:18_2_2"/><b>then </b><i><font color="Green">E213</font></i>: 
 <a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">o2</font> <a href="orders_1.html#R3">linearly_orders</a> <font color="Maroon">T</font>
 <b>by </b><i><a class="ref" href="orders_1.html#T134">ORDERS_1:134</a></i>;<br/><div><i><font color="Green">E214</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_2_1"><b>now </b></a><div class="add"><b>assume </b><b>that </b><br/><a NAME="E1:18_2_2_1"/><i><font color="Green">E215</font></i>: 
not <font color="Maroon">i1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i2</font>
 <b>and </b><br/><a NAME="E2:18_2_2_1"/><i><font color="Green">E216</font></i>: 
(  not <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> or  not <font color="Maroon">j1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j2</font> )
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_2_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:18_2_2_1_1"/>
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> or <font color="Maroon">i1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">i2</font> )
 </span><b>by </b><i>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_2_2_1_1_1"><b>suppose </b></a><a NAME="E1:18_2_2_1_1_1"/>
<font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:18_2_2_1_1_1"/>
contradiction
 <b>by </b><i>, , <a class="txt" href="polynom6.html#E2:18"><i><font color="Green">E36</font></i></a>, , <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_2_2_1_1_2"><b>suppose </b></a><a NAME="E1:18_2_2_1_1_2"/><i><font color="Green">E217</font></i>: 
<font color="Maroon">i1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">i2</font>
 ;<br/><div class="add"><a NAME="E2:18_2_2_1_1_2"/><i><font color="Green">E218</font></i>: 
<font color="Maroon">j2</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i2</font></span>)</span>
 <b>by </b><i><a class="txt" href="polynom6.html#E1:18"><i><font color="Green">E35</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E3:18_2_2_1_1_2"/><i><font color="Green">E219</font></i>: 
 <a href="polynom1.html#K5">Card</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">i2</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K5">Card</a> <font color="Maroon">G</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">i2</font>
 <b>by </b><i><a class="ref" href="polynom3.html#T16">POLYNOM3:16</a></i>;<br/><a NAME="E4:18_2_2_1_1_2"/><i><font color="Green">E220</font></i>: 
 <a href="polynom1.html#K5">Card</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K5">Card</a> <font color="Maroon">G</font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="polynom3.html#T16">POLYNOM3:16</a></i>;<br/><a NAME="E5:18_2_2_1_1_2"/>
<font color="Maroon">i2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1
 <b>by </b><i>, <a class="ref" href="jordan3.html#T12">JORDAN3:12</a></i>;<br/><a NAME="E6:18_2_2_1_1_2"/><b>then </b><i><font color="Green">E221</font></i>: 
 <a href="wsierp_1.html#K7">Sum</a> <span class="p1">(<span class="default"><a href="polynom1.html#K5">Card</a> <span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">i2</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="wsierp_1.html#K7">Sum</a> <span class="p1">(<span class="default"><a href="polynom1.html#K5">Card</a> <span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="finseq_1.html#K17">|</a> <span class="p3">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="polynom3.html#T18">POLYNOM3:18</a></i>;<br/><a NAME="E7:18_2_2_1_1_2"/>
<font color="Maroon">i2</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E8:18_2_2_1_1_2"/><b>then </b><i><font color="Green">E222</font></i>: 
<font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i><a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/><a NAME="E9:18_2_2_1_1_2"/>
<font color="Maroon">i2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/><a NAME="E10:18_2_2_1_1_2"/><b>then </b><i><font color="Green">E223</font></i>: 
<font color="Maroon">i2</font> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i2</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><a NAME="E11:18_2_2_1_1_2"/>
<font color="Maroon">i2</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E12:18_2_2_1_1_2"/><b>then </b><i><font color="Green">E224</font></i>: 
<font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font>
 <b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E13:18_2_2_1_1_2"/>
<font color="Maroon">i2</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E14:18_2_2_1_1_2"/><b>then </b><i><font color="Green">E225</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font>
 <b>by </b><i><a class="ref" href="binarith.html#T53">BINARITH:53</a></i>;<br/><b>reconsider </b><font color="Maroon">G1</font> = <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i2</font> as    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K14">Bags</a> <span class="p2">(<span class="default"><font color="Maroon">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font></span>)</span></span>)</span> <a href="finseq_2.html#K3">*</a>  <b>by </b><i>, <a class="ref" href="polynom6.html#T2" target="_self">Th2</a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><b>reconsider </b><font color="Maroon">GG1</font> = <span class="p1"><a href="matrlin.html#K3">&lt;*</a><span class="default"><font color="Maroon">G1</font></span><a href="matrlin.html#K3">*&gt;</a></span> as    <a href="finseq_1.html#M2">FinSequence</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K14">Bags</a> <span class="p2">(<span class="default"><font color="Maroon">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font></span>)</span></span>)</span> <a href="finseq_2.html#K3">*</a>  ;<br/><a NAME="E17:18_2_2_1_1_2"/><i><font color="Green">E227</font></i>: 
 <a href="card_1.html#K4">card</a> <font color="Maroon">G1</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a> 
 <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/><a NAME="E18:18_2_2_1_1_2"/>
<font color="Maroon">G</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">GG1</font>
 <b>by </b><i>, , <a class="ref" href="polynom3.html#T19">POLYNOM3:19</a></i>;<br/><a NAME="E19:18_2_2_1_1_2"/><b>then </b>
 <a href="polynom1.html#K5">Card</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">i2</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K5">Card</a> <span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="finseq_1.html#K17">|</a> <span class="p3">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><a href="polynom1.html#K5">Card</a> <font color="Maroon">GG1</font></span>)</span>
 <b>by </b><i><a class="ref" href="polynom1.html#T13">POLYNOM1:13</a></i>;<br/><a NAME="E20:18_2_2_1_1_2"/><b>then </b>
 <a href="polynom1.html#K5">Card</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">i2</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K5">Card</a> <span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="finseq_1.html#K17">|</a> <span class="p3">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">G1</font></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="polynom1.html#T12">POLYNOM1:12</a></i>;<br/><a NAME="E21:18_2_2_1_1_2"/><b>then </b>
 <a href="wsierp_1.html#K7">Sum</a> <span class="p1">(<span class="default"><a href="polynom1.html#K5">Card</a> <span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">i2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K7">Sum</a> <span class="p2">(<span class="default"><a href="polynom1.html#K5">Card</a> <span class="p3">(<span class="default"><font color="Maroon">G</font> <a href="finseq_1.html#K17">|</a> <span class="p4">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i2</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="rvsum_1.html#T104">RVSUM_1:104</a></i>;<br/><a NAME="E22:18_2_2_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="wsierp_1.html#K7">Sum</a> <span class="p2">(<span class="default"><a href="polynom1.html#K5">Card</a> <span class="p3">(<span class="default"><font color="Maroon">G</font> <a href="finseq_1.html#K17">|</a> <span class="p4">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">j2</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="wsierp_1.html#K7">Sum</a> <span class="p1">(<span class="default"><a href="polynom1.html#K5">Card</a> <span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="finseq_1.html#K17">|</a> <font color="Maroon">i2</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E23:18_2_2_1_1_2"/><b>then </b><i><font color="Green">E228</font></i>: 
<span class="p1">(<span class="default"><a href="wsierp_1.html#K7">Sum</a> <span class="p2">(<span class="default"><a href="polynom1.html#K5">Card</a> <span class="p3">(<span class="default"><font color="Maroon">G</font> <a href="finseq_1.html#K17">|</a> <span class="p4">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">j2</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="wsierp_1.html#K7">Sum</a> <span class="p1">(<span class="default"><a href="polynom1.html#K5">Card</a> <span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="finseq_1.html#K17">|</a> <span class="p3">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E24:18_2_2_1_1_2"/>
 <a href="wsierp_1.html#K7">Sum</a> <span class="p1">(<span class="default"><a href="polynom1.html#K5">Card</a> <span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="finseq_1.html#K17">|</a> <span class="p3">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K7">Sum</a> <span class="p2">(<span class="default"><a href="polynom1.html#K5">Card</a> <span class="p3">(<span class="default"><font color="Maroon">G</font> <a href="finseq_1.html#K17">|</a> <span class="p4">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">j1</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><b>hence </b><a NAME="E25:18_2_2_1_1_2"/>
contradiction
 <b>by </b><i>, , <a class="txt" href="polynom6.html#E2:18"><i><font color="Green">E36</font></i></a>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_2_2_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:18_2_2_2"/>
( <font color="Maroon">i1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i2</font> or ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j2</font> ) )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_2_2_2_1"><b>case </b></a><a NAME="E1:18_2_2_2_1"/><i><font color="Green">E229</font></i>: 
<font color="Maroon">i1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i2</font>
 ;<br/><div class="add"><a NAME="E2:18_2_2_2_1"/><b>then </b><i><font color="Green">E230</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">b1</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">b1</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i2</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">o1</font>
 <b>by </b><i>, , , , , <a class="ref" href="triang_1.html#D2">TRIANG_1:def 2</a></i>;<br/><a NAME="E3:18_2_2_2_1"/><i><font color="Green">E231</font></i>: 
<font color="Maroon">a1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c1</font>
 <b>by </b><i>, , , , , , <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/><a NAME="E4:18_2_2_2_1"/>
<font color="Maroon">a1</font> <a href="polynom1.html#R2">&lt;='</a> <font color="Maroon">c1</font>
 <b>by </b><i><a class="txt" href="polynom6.html#E8:18"><i><font color="Green">E39</font></i></a>, , , , <a class="ref" href="polynom1.html#D16">POLYNOM1:def 16</a></i>;<br/><b>hence </b><a NAME="E5:18_2_2_2_1"/>
<font color="Maroon">a1</font> <a href="polynom1.html#R1">&lt;</a> <font color="Maroon">c1</font>
 <b>by </b><i>, <a class="ref" href="polynom1.html#D12">POLYNOM1:def 12</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_2_2_2_2"><b>case </b></a><b>that </b><a NAME="E1:18_2_2_2_2"/><i><font color="Green">E232</font></i>: 
<font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font>
 <b>and </b><br/><a NAME="E2:18_2_2_2_2"/><i><font color="Green">E233</font></i>: 
<font color="Maroon">j1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j2</font>
 ;<br/><div class="add"><a NAME="E3:18_2_2_2_2"/><i><font color="Green">E234</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">b2</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j1</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">b2</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j2</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">o2</font>
 <b>by </b><i>, <a class="txt" href="polynom6.html#E1:18_2"><i><font color="Green">E58</font></i></a>, , , , <a class="ref" href="triang_1.html#D2">TRIANG_1:def 2</a></i>;<br/><b>thus </b><a NAME="E4:18_2_2_2_2"/>
<font color="Maroon">a1</font> <a href="pboole.html#R6">=</a> <font color="Maroon">c1</font>
 <b>by </b><i><a class="txt" href="polynom6.html#E8:18"><i><font color="Green">E39</font></i></a>, , , </i>;<br/><a NAME="E5:18_2_2_2_2"/><i><font color="Green">E235</font></i>: 
<font color="Maroon">a2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c2</font>
 <b>by </b><i>, , <a class="txt" href="polynom6.html#E1:18_2"><i><font color="Green">E58</font></i></a>, <a class="txt" href="polynom6.html#E2:18_2"><i><font color="Green">E59</font></i></a>, , <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/><a NAME="E6:18_2_2_2_2"/>
<font color="Maroon">a2</font> <a href="polynom1.html#R2">&lt;='</a> <font color="Maroon">c2</font>
 <b>by </b><i><a class="txt" href="polynom6.html#E15:18"><i><font color="Green">E43</font></i></a>, <a class="ref" href="polynom6.html#D1" target="_self">Def1</a>, , , <a class="ref" href="polynom1.html#D16">POLYNOM1:def 16</a></i>;<br/><b>hence </b><a NAME="E7:18_2_2_2_2"/>
<font color="Maroon">a2</font> <a href="polynom1.html#R1">&lt;</a> <font color="Maroon">c2</font>
 <b>by </b><i>, <a class="ref" href="polynom1.html#D12">POLYNOM1:def 12</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E55:18_2"/><b>then </b>
( <font color="Maroon">Fn</font> <a href="polynom1.html#R1">&lt;</a> <font color="Maroon">Fm</font> or <font color="Maroon">Fn</font> <a href="pboole.html#R6">=</a> <font color="Maroon">Fm</font> )
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E56:18_2"/><b>then </b>
<font color="Maroon">Fn</font> <a href="polynom1.html#R2">&lt;='</a> <font color="Maroon">Fm</font>
 
<b>by </b><i><a class="ref" href="polynom1.html#D12">POLYNOM1:def 12</a></i>;<br/>
<b>hence </b><a NAME="E57:18_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="dtconstr.html#K15">FlattenSeq</a> <font color="Maroon">G</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="dtconstr.html#K15">FlattenSeq</a> <font color="Maroon">G</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">m</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K17">BagOrder</a> <span class="p1">(<span class="default"><font color="Maroon">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font></span>)</span>
 <b>by </b><i><a class="ref" href="polynom1.html#D16">POLYNOM1:def 16</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E24:18"/><b>then </b><i><font color="Green">E236</font></i>: 
 <a href="dtconstr.html#K15">FlattenSeq</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="triang_1.html#K2">SgmX</a> <span class="p1">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <span class="p2">(<span class="default"><font color="Maroon">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font></span>)</span></span>)</span>,<font color="Maroon">S</font>
 
<b>by </b><i>, <a class="ref" href="triang_1.html#D2">TRIANG_1:def 2</a></i>;<br/>
<a NAME="E25:18"/>
for <font color="Olive">p</font> being  <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font> holds <br/> ( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> iff <font color="Olive">p</font> <a href="polynom1.html#R3">divides</a> <font color="Maroon">b1</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">b2</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_3"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">T</font> being  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K14">Bags</a> <font color="Maroon">o1</font></span>)</span><b> such that </b><br/><a NAME="E2:18_3"/><i><font color="Green">E237</font></i>: 
 <a href="polynom1.html#K20">divisors</a> <font color="Maroon">b1</font> <a href="hidden.html#R1">=</a>  <a href="triang_1.html#K2">SgmX</a> <span class="p1">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">o1</font></span>)</span>,<font color="Maroon">T</font>
 <b>and </b><br/><a NAME="E3:18_3"/><i><font color="Green">E238</font></i>: 
for <font color="Olive">q</font> being  <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">o1</font> holds <br/> ( <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T</font> iff <font color="Olive">q</font> <a href="polynom1.html#R3">divides</a> <font color="Maroon">b1</font> )
 <b>by </b><i><a class="ref" href="polynom1.html#D18">POLYNOM1:def 18</a></i>;<br/>
<b>consider </b><font color="Maroon">W</font> being  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K14">Bags</a> <font color="Maroon">o2</font></span>)</span><b> such that </b><br/><a NAME="E5:18_3"/><i><font color="Green">E240</font></i>: 
 <a href="polynom1.html#K20">divisors</a> <font color="Maroon">b2</font> <a href="hidden.html#R1">=</a>  <a href="triang_1.html#K2">SgmX</a> <span class="p1">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">o2</font></span>)</span>,<font color="Maroon">W</font>
 <b>and </b><br/><a NAME="E6:18_3"/><i><font color="Green">E241</font></i>: 
for <font color="Olive">q</font> being  <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">o2</font> holds <br/> ( <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> iff <font color="Olive">q</font> <a href="polynom1.html#R3">divides</a> <font color="Maroon">b2</font> )
 <b>by </b><i><a class="ref" href="polynom1.html#D18">POLYNOM1:def 18</a></i>;<br/>
<a NAME="E7:18_3"/>
 <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">o1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o1</font>
 
<b>by </b><i><a class="ref" href="orders_1.html#T100">ORDERS_1:100</a></i>;<br/>
<a NAME="E8:18_3"/><b>then </b>
 <a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">o1</font> <a href="orders_1.html#R3">linearly_orders</a>  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o1</font>
 
<b>by </b><i><a class="ref" href="orders_1.html#T133">ORDERS_1:133</a></i>;<br/>
<a NAME="E9:18_3"/><b>then </b>
 <a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">o1</font> <a href="orders_1.html#R3">linearly_orders</a> <font color="Maroon">T</font>
 
<b>by </b><i><a class="ref" href="orders_1.html#T134">ORDERS_1:134</a></i>;<br/>
<a NAME="E10:18_3"/><b>then </b><i><font color="Green">E242</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">o1</font></span>)</span>,<font color="Maroon">T</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">T</font>
 
<b>by </b><i><a class="ref" href="triang_1.html#D2">TRIANG_1:def 2</a></i>;<br/>
<a NAME="E11:18_3"/>
 <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">o2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o2</font>
 
<b>by </b><i><a class="ref" href="orders_1.html#T100">ORDERS_1:100</a></i>;<br/>
<a NAME="E12:18_3"/><b>then </b>
 <a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">o2</font> <a href="orders_1.html#R3">linearly_orders</a>  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o2</font>
 
<b>by </b><i><a class="ref" href="orders_1.html#T133">ORDERS_1:133</a></i>;<br/>
<a NAME="E13:18_3"/><b>then </b>
 <a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">o2</font> <a href="orders_1.html#R3">linearly_orders</a> <font color="Maroon">W</font>
 
<b>by </b><i><a class="ref" href="orders_1.html#T134">ORDERS_1:134</a></i>;<br/>
<a NAME="E14:18_3"/><b>then </b><i><font color="Green">E243</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">o2</font></span>)</span>,<font color="Maroon">W</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font>
 
<b>by </b><i><a class="ref" href="triang_1.html#D2">TRIANG_1:def 2</a></i>;<br/>
<b>let </b><font color="Maroon">p</font> be   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font>;<br/>

<b>thus </b><a NAME="E15:18_3"/>
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> implies <font color="Maroon">p</font> <a href="polynom1.html#R3">divides</a> <font color="Maroon">b1</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">b2</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_3_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:18_3_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:18_3_1"/><i><font color="Green">E244</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15">FlattenSeq</a> <font color="Maroon">G</font></span>)</span>
 <b>and </b><br/><a NAME="E4:18_3_1"/><i><font color="Green">E245</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15">FlattenSeq</a> <font color="Maroon">G</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E6:18_3_1"/><i><font color="Green">E246</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G</font>
 <b>and </b><br/><a NAME="E7:18_3_1"/><i><font color="Green">E247</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
 <b>and </b><br/><a NAME="E8:18_3_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K7">Sum</a> <span class="p2">(<span class="default"><a href="polynom1.html#K5">Card</a> <span class="p3">(<span class="default"><font color="Maroon">G</font> <a href="finseq_1.html#K17">|</a> <span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font>
 <b>and </b><br/><a NAME="E9:18_3_1"/><i><font color="Green">E248</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15">FlattenSeq</a> <font color="Maroon">G</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="polynom6.html#T1" target="_self">Th1</a>, <a class="ref" href="polynom1.html#T32">POLYNOM1:32</a></i>;<br/>
<b>consider </b><font color="Maroon">a1'</font> being    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o1</font>, <font color="Maroon">Fk</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="polynom1.html#K14">Bags</a> <span class="p1">(<span class="default"><font color="Maroon">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font></span>)</span><b> such that </b><br/><a NAME="E11:18_3_1"/><i><font color="Green">E249</font></i>: 
<font color="Maroon">Fk</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="matrlin.html#K1">/.</a> <font color="Maroon">i</font>
 <b>and </b><br/><a NAME="E12:18_3_1"/><i><font color="Green">E250</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">b1</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a1'</font>
 <b>and </b><br/><a NAME="E13:18_3_1"/><i><font color="Green">E251</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">Fk</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">b2</font></span>)</span>
 <b>and </b><br/><a NAME="E14:18_3_1"/><i><font color="Green">E252</font></i>: 
for <font color="Olive">m</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">Fk</font> holds <br/> ex <font color="Olive">a1''</font> being   <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o2</font> st <br/>( <span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">b2</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">m</font> <a href="hidden.html#R1">=</a> <font color="Olive">a1''</font> &amp; <font color="Maroon">Fk</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a1'</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Olive">a1''</font> )
 <b>by </b><i>, , </i>;<br/>
<a NAME="E15:18_3_1"/><i><font color="Green">E253</font></i>: 
<font color="Maroon">Fk</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 
<b>by </b><i>, <a class="txt" href="polynom6.html#E2:18"><i><font color="Green">E36</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<b>then consider </b><font color="Maroon">a1''</font> being    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o2</font><b> such that </b><br/><a NAME="E17:18_3_1"/><i><font color="Green">E254</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">b2</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a1''</font>
 <b>and </b><br/><a NAME="E18:18_3_1"/><i><font color="Green">E255</font></i>: 
<font color="Maroon">Fk</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a1'</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">a1''</font>
 <b>by </b><i>, <a class="txt" href="polynom6.html#E8:18"><i><font color="Green">E39</font></i></a></i>;<br/>
<a NAME="E19:18_3_1"/><i><font color="Green">E256</font></i>: 
<font color="Maroon">p</font> <a href="pboole.html#R6">=</a> <font color="Maroon">a1'</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">a1''</font>
 
<b>by </b><i>, , <a class="txt" href="polynom6.html#E1:18"><i><font color="Green">E35</font></i></a>, <a class="txt" href="polynom6.html#E10:18"><i><font color="Green">E40</font></i></a>, <a class="txt" href="polynom6.html#E14:18"><i><font color="Green">E42</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<a NAME="E20:18_3_1"/>
<span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">b1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a1'</font>
 
<b>by </b><i>, , <a class="txt" href="polynom6.html#E5:18"><i><font color="Green">E37</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<a NAME="E21:18_3_1"/><b>then </b><i><font color="Green">E257</font></i>: 
<font color="Maroon">a1'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T</font>
 
<b>by </b><i>, , , , <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/>
<b>reconsider </b><font color="Maroon">a11'</font> = <font color="Maroon">a1'</font> as   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">o1</font> ;<br/>
<a NAME="E23:18_3_1"/><i><font color="Green">E259</font></i>: 
<font color="Maroon">a11'</font> <a href="polynom1.html#R3">divides</a> <font color="Maroon">b1</font>
 
<b>by </b><i>, <a class="txt" href="polynom6.html#E22:18"><i><font color="Green">E44</font></i></a></i>;<br/>
<a NAME="E24:18_3_1"/>
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">Fk</font></span>)</span>
 
<b>by </b><i>, <a class="txt" href="polynom6.html#E10:18"><i><font color="Green">E40</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E25:18_3_1"/><b>then </b><i><font color="Green">E260</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">b2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="polynom6.html#E6:18"><i><font color="Green">E38</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E26:18_3_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">b2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a1''</font>
 
<b>by </b><i><a class="txt" href="polynom6.html#E11:18"><i><font color="Green">E41</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<a NAME="E27:18_3_1"/><b>then </b><i><font color="Green">E261</font></i>: 
<font color="Maroon">a1''</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font>
 
<b>by </b><i>, , , <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/>
<b>reconsider </b><font color="Maroon">a11''</font> = <font color="Maroon">a1''</font> as   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">o2</font> ;<br/>
<a NAME="E29:18_3_1"/>
<font color="Maroon">a11''</font> <a href="polynom1.html#R3">divides</a> <font color="Maroon">b2</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E30:18_3_1"/>
<font color="Maroon">p</font> <a href="polynom1.html#R3">divides</a> <font color="Maroon">b1</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">b2</font>
 <b>by </b><i><a class="txt" href="polynom6.html#E15:18"><i><font color="Green">E43</font></i></a>, <a class="ref" href="polynom6.html#T2" target="_self">Th2</a>, </i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E16:18_3"/>
( <font color="Maroon">p</font> <a href="polynom1.html#R3">divides</a> <font color="Maroon">b1</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">b2</font> implies <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_3_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:18_3_2"/>
<font color="Maroon">p</font> <a href="polynom1.html#R3">divides</a> <font color="Maroon">b1</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">b2</font>
 ;<br/>

<b>then consider </b><font color="Maroon">p1</font> being    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o1</font>, <font color="Maroon">p2</font> being    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o2</font><b> such that </b><br/><a NAME="E3:18_3_2"/><i><font color="Green">E265</font></i>: 
<font color="Maroon">p1</font> <a href="polynom1.html#R3">divides</a> <font color="Maroon">b1</font>
 <b>and </b><br/><a NAME="E4:18_3_2"/><i><font color="Green">E266</font></i>: 
<font color="Maroon">p2</font> <a href="polynom1.html#R3">divides</a> <font color="Maroon">b2</font>
 <b>and </b><br/><a NAME="E5:18_3_2"/><i><font color="Green">E267</font></i>: 
<font color="Maroon">p</font> <a href="pboole.html#R6">=</a> <font color="Maroon">p1</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">p2</font>
 <b>by </b><i/>;<br/>
<a NAME="E6:18_3_2"/>
<font color="Maroon">p1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T</font>
 
<b>by </b><i>, <a class="ref" href="polynom6.html#T1" target="_self">Th1</a></i>;<br/>
<b>then consider </b><font color="Maroon">i</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E8:18_3_2"/><i><font color="Green">E268</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">b1</font></span>)</span>
 <b>and </b><br/><a NAME="E9:18_3_2"/><i><font color="Green">E269</font></i>: 
<font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">b1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E10:18_3_2"/>
<font color="Maroon">p2</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">b2</font></span>)</span>
 
<b>by </b><i>, , , </i>;<br/>
<b>then consider </b><font color="Maroon">j</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E12:18_3_2"/><i><font color="Green">E270</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">b2</font></span>)</span>
 <b>and </b><br/><a NAME="E13:18_3_2"/><i><font color="Green">E271</font></i>: 
<font color="Maroon">p2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">b2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">i</font> = <font color="Maroon">i</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">a1'</font> being    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o1</font>, <font color="Maroon">Fk</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="polynom1.html#K14">Bags</a> <span class="p1">(<span class="default"><font color="Maroon">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font></span>)</span><b> such that </b><br/><a NAME="E16:18_3_2"/><i><font color="Green">E272</font></i>: 
<font color="Maroon">Fk</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="matrlin.html#K1">/.</a> <font color="Maroon">i</font>
 <b>and </b><br/><a NAME="E17:18_3_2"/><i><font color="Green">E273</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">b1</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a1'</font>
 <b>and </b><br/><a NAME="E18:18_3_2"/><i><font color="Green">E274</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">Fk</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">b2</font></span>)</span>
 <b>and </b><br/><a NAME="E19:18_3_2"/><i><font color="Green">E275</font></i>: 
for <font color="Olive">m</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">Fk</font> holds <br/> ex <font color="Olive">a1''</font> being   <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o2</font> st <br/>( <span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">b2</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">m</font> <a href="hidden.html#R1">=</a> <font color="Olive">a1''</font> &amp; <font color="Maroon">Fk</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a1'</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Olive">a1''</font> )
 <b>by </b><i>, </i>;<br/>
<a NAME="E20:18_3_2"/><i><font color="Green">E276</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">b2</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="polynom6.html#E2:18"><i><font color="Green">E36</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E21:18_3_2"/>
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">b2</font></span>)</span></span>)</span>
 
<b>by </b><i>, , <a class="txt" href="polynom6.html#E6:18"><i><font color="Green">E38</font></i></a>, <a class="txt" href="polynom6.html#E10:18"><i><font color="Green">E40</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<a NAME="E22:18_3_2"/><b>then </b><i><font color="Green">E277</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
 
<b>by </b><i><a class="txt" href="polynom6.html#E14:18"><i><font color="Green">E42</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">j</font> = <font color="Maroon">j</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="txt" href="polynom6.html#E2:18"><i><font color="Green">E36</font></i></a></i>;<br/>
<a NAME="E24:18_3_2"/><i><font color="Green">E278</font></i>: 
( <span class="p1">(<span class="default"><a href="wsierp_1.html#K7">Sum</a> <span class="p2">(<span class="default"><a href="polynom1.html#K5">Card</a> <span class="p3">(<span class="default"><font color="Maroon">G</font> <a href="finseq_1.html#K17">|</a> <span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15">FlattenSeq</a> <font color="Maroon">G</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15">FlattenSeq</a> <font color="Maroon">G</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="wsierp_1.html#K7">Sum</a> <span class="p3">(<span class="default"><a href="polynom1.html#K5">Card</a> <span class="p4">(<span class="default"><font color="Maroon">G</font> <a href="finseq_1.html#K17">|</a> <span class="p5">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font></span>)</span> )
 
<b>by </b><i>, , <a class="txt" href="polynom6.html#E15:18"><i><font color="Green">E43</font></i></a>, <a class="ref" href="polynom1.html#T33">POLYNOM1:33</a></i>;<br/>
<a NAME="E25:18_3_2"/><i><font color="Green">E279</font></i>: 
<font color="Maroon">G</font> <a href="matrlin.html#K1">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 
<b>by </b><i>, , <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<b>then consider </b><font color="Maroon">a1''</font> being    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o2</font><b> such that </b><br/><a NAME="E27:18_3_2"/><i><font color="Green">E280</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">b2</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a1''</font>
 <b>and </b><br/><a NAME="E28:18_3_2"/><i><font color="Green">E281</font></i>: 
<font color="Maroon">Fk</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a1'</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">a1''</font>
 <b>by </b><i><a class="txt" href="polynom6.html#E6:18"><i><font color="Green">E38</font></i></a>, <a class="txt" href="polynom6.html#E11:18"><i><font color="Green">E41</font></i></a>, <a class="txt" href="polynom6.html#E15:18"><i><font color="Green">E43</font></i></a></i>;<br/>
<a NAME="E29:18_3_2"/><i><font color="Green">E282</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a1'</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">a1''</font>
 
<b>by </b><i><a class="txt" href="polynom6.html#E6:18"><i><font color="Green">E38</font></i></a>, <a class="txt" href="polynom6.html#E15:18"><i><font color="Green">E43</font></i></a>, <a class="ref" href="polynom6.html#T2" target="_self">Th2</a>, , <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<a NAME="E30:18_3_2"/><i><font color="Green">E283</font></i>: 
<font color="Maroon">a1'</font> <a href="pboole.html#R6">=</a> <font color="Maroon">p1</font>
 
<b>by </b><i>, <a class="txt" href="polynom6.html#E1:18"><i><font color="Green">E35</font></i></a>, <a class="txt" href="polynom6.html#E8:18"><i><font color="Green">E39</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<a NAME="E31:18_3_2"/>
<font color="Maroon">a1''</font> <a href="pboole.html#R6">=</a> <font color="Maroon">p2</font>
 
<b>by </b><i><a class="txt" href="polynom6.html#E2:18"><i><font color="Green">E36</font></i></a>, <a class="txt" href="polynom6.html#E5:18"><i><font color="Green">E37</font></i></a>, , <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<b>hence </b><a NAME="E32:18_3_2"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 <b>by </b><i>, <a class="txt" href="polynom6.html#E22:18"><i><font color="Green">E44</font></i></a>, , , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div>


</div><b>end;</b></div>
<b>hence </b><a NAME="E26:18"/>
 <a href="polynom1.html#K20">divisors</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">b2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="dtconstr.html#K15">FlattenSeq</a> <font color="Maroon">G</font>
 <b>by </b><i>, <a class="ref" href="polynom1.html#D18">POLYNOM1:def 18</a></i>;<br/>


</div>
