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

<span class="kw">let </span><font color="Maroon" title="c1">fp</font> be    <a href="trees_4.html#M1" title="TREES_4:mode.1">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">a</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><br/>  for <font color="Olive" title="b2">b</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c1">fp</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c1">fp</font> &amp; <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">b</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">fp</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 holds <br/><font color="Maroon" title="c1">fp</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b2">b</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a>  <a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Product</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> (<font color="Maroon" title="c1">fp</font>,<font color="Olive" title="b1">a</font>)</span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">a</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">b</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c1">fp</font> &amp; <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c1">fp</font> &amp; <font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">b</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">fp</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 holds <br/><font color="Maroon" title="c1">fp</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">b</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a>  <a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Product</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> (<font color="Maroon" title="c1">fp</font>,<font color="Maroon" title="c2">a</font>)</span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c3">b</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c1">fp</font> &amp; <font color="Maroon" title="c3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c1">fp</font> &amp; <font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">b</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">fp</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 implies <font color="Maroon" title="c1">fp</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c3">b</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a>  <a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Product</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> (<font color="Maroon" title="c1">fp</font>,<font color="Maroon" title="c2">a</font>)</span>)</span> )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:48"/><span class="lab"><font color="Green" title="E32">A1</font></span>: 
<font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c1">fp</font>
 <span class="kw">and </span><br/><a NAME="E2:48"/><span class="lab"><font color="Green" title="E33">A2</font></span>: 
<font color="Maroon" title="c3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c1">fp</font>
 <span class="kw">and </span><br/><a NAME="E3:48"/><span class="lab"><font color="Green" title="E34">A3</font></span>: 
<font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">b</font>
 <span class="kw">and </span><br/><a NAME="E4:48"/><span class="lab"><font color="Green" title="E35">A4</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">fp</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">fp</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c3">b</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a>  <a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Product</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> (<font color="Maroon" title="c1">fp</font>,<font color="Maroon" title="c2">a</font>)</span>)</span></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c4">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E6:48"/><span class="lab"><font color="Green" title="E36">A5</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">fp</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:48"><span class="lab"><font color="Green" title="E35">A4</font></span></a>, <a class="ref" href="nat_1.html#T6" title="NAT_1:th.6">NAT_1:6</a></span>;<br/>
<a NAME="E7:48"/><span class="lab"><font color="Green" title="E37">A6</font></span>: 
<font color="Maroon" title="c2">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48"><span class="lab"><font color="Green" title="E32">A1</font></span></a>, <a class="txt" href="#E6:48"><span class="lab"><font color="Green" title="E36">A5</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<a NAME="E8:48"/><span class="lab"><font color="Green" title="E38">A7</font></span>: 
<font color="Maroon" title="c2">a</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48"><span class="lab"><font color="Green" title="E32">A1</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<a NAME="E9:48"/><span class="lab"><font color="Green" title="E39">A8</font></span>: 
<font color="Maroon" title="c3">b</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48"><span class="lab"><font color="Green" title="E33">A2</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<a NAME="E10:48"/><span class="lab"><font color="Green" title="E40">A9</font></span>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> (<font color="Maroon" title="c1">fp</font>,<font color="Maroon" title="c2">a</font>)</span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48"><span class="lab"><font color="Green" title="E32">A1</font></span></a>, <a class="txt" href="#E6:48"><span class="lab"><font color="Green" title="E36">A5</font></span></a>, <a class="ref" href="wsierp_1.html#D1" title="WSIERP_1:def.1">WSIERP_1:def 1</a></span>;<br/>
<a NAME="E11:48"/><span class="lab"><font color="Green" title="E41">A10</font></span>: 
<font color="Maroon" title="c3">b</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48"><span class="lab"><font color="Green" title="E33">A2</font></span></a>, <a class="txt" href="#E6:48"><span class="lab"><font color="Green" title="E36">A5</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:48_1"/>
( <font color="Maroon" title="c2">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font> or <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1 )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:48"><span class="lab"><font color="Green" title="E37">A6</font></span></a>, <a class="ref" href="nat_1.html#T8" title="NAT_1:th.8">NAT_1:8</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1"><span class="kw">suppose </span></a><a NAME="E1:48_1_1"/><span class="lab"><font color="Green" title="E42">A11</font></span>: 
<font color="Maroon" title="c2">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">fp</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c3">b</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a>  <a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Product</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> (<font color="Maroon" title="c1">fp</font>,<font color="Maroon" title="c2">a</font>)</span>)</span></span><br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:48_1_1_1"/>
( <font color="Maroon" title="c3">b</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">a</font> or <font color="Maroon" title="c2">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">b</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:48"><span class="lab"><font color="Green" title="E34">A3</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:48_1_1_1_1"/><span class="lab"><font color="Green" title="E43">A12</font></span>: 
<font color="Maroon" title="c3">b</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">a</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">fp</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c3">b</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a>  <a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Product</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> (<font color="Maroon" title="c1">fp</font>,<font color="Maroon" title="c2">a</font>)</span>)</span></span><br/><div class="add"><a NAME="E2:48_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">b</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48_1_1"><span class="lab"><font color="Green" title="E42">A11</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E3:48_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A13</font></span>: 
<font color="Maroon" title="c3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> (<font color="Maroon" title="c1">fp</font>,<font color="Maroon" title="c2">a</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:48"><span class="lab"><font color="Green" title="E39">A8</font></span></a>, <a class="txt" href="#E10:48"><span class="lab"><font color="Green" title="E40">A9</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E4:48_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> (<font color="Maroon" title="c1">fp</font>,<font color="Maroon" title="c2">a</font>)</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">fp</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c3">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48"><span class="lab"><font color="Green" title="E32">A1</font></span></a>, <a class="txt" href="#E1:48_1_1_1_1"><span class="lab"><font color="Green" title="E43">A12</font></span></a>, <a class="ref" href="wsierp_1.html#D1" title="WSIERP_1:def.1">WSIERP_1:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E5:48_1_1_1_1"/>
<font color="Maroon" title="c1">fp</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c3">b</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a>  <a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Product</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> (<font color="Maroon" title="c1">fp</font>,<font color="Maroon" title="c2">a</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:48_1_1_1_1"><span class="lab"><font color="Green" title="E44">A13</font></span></a>, <a class="ref" href="int_4.html#T32" target="_self" title="INT_4:th.32">Th32</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:48_1_1_1_2"/><span class="lab"><font color="Green" title="E43">A14</font></span>: 
<font color="Maroon" title="c2">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">fp</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c3">b</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a>  <a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Product</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> (<font color="Maroon" title="c1">fp</font>,<font color="Maroon" title="c2">a</font>)</span>)</span></span><br/><div class="add"><a NAME="E2:48_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">b</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">a</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E3:48_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">b</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T19" title="XREAL_1:th.19">XREAL_1:19</a></span>;<br/><a NAME="E4:48_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">b</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:48"><span class="lab"><font color="Green" title="E39">A8</font></span></a>, <a class="ref" href="xreal_1.html#T233" title="XREAL_1:th.233">XREAL_1:233</a></span>;<br/><a NAME="E5:48_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A15</font></span>: 
<span class="p1">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> (<font color="Maroon" title="c1">fp</font>,<font color="Maroon" title="c2">a</font>)</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">fp</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48"><span class="lab"><font color="Green" title="E32">A1</font></span></a>, <a class="ref" href="wsierp_1.html#D1" title="WSIERP_1:def.1">WSIERP_1:def 1</a></span>;<br/><a NAME="E6:48_1_1_1_2"/>
<font color="Maroon" title="c3">b</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:48"><span class="lab"><font color="Green" title="E38">A7</font></span></a>, <a class="txt" href="#E1:48_1_1_1_2"><span class="lab"><font color="Green" title="E43">A14</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E7:48_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">b</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T50" title="XREAL_1:th.50">XREAL_1:50</a></span>;<br/><a NAME="E8:48_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">b</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:48"><span class="lab"><font color="Green" title="E39">A8</font></span></a>, <a class="ref" href="xreal_1.html#T233" title="XREAL_1:th.233">XREAL_1:233</a></span>;<br/><a NAME="E9:48_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A16</font></span>: 
<font color="Maroon" title="c3">b</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="int_4.html#E12"><span class="lab"><font color="Green" title="E12">Lm7</font></span></a></span>;<br/><a NAME="E10:48_1_1_1_2"/>
<font color="Maroon" title="c3">b</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:48"><span class="lab"><font color="Green" title="E41">A10</font></span></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/><a NAME="E11:48_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">b</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:48"><span class="lab"><font color="Green" title="E39">A8</font></span></a>, <a class="ref" href="xreal_1.html#T233" title="XREAL_1:th.233">XREAL_1:233</a></span>;<br/><a NAME="E12:48_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">b</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> (<font color="Maroon" title="c1">fp</font>,<font color="Maroon" title="c2">a</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:48"><span class="lab"><font color="Green" title="E40">A9</font></span></a>, <a class="txt" href="#E9:48_1_1_1_2"><span class="lab"><font color="Green" title="E45">A16</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E13:48_1_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> (<font color="Maroon" title="c1">fp</font>,<font color="Maroon" title="c2">a</font>)</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a>  <a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Product</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> (<font color="Maroon" title="c1">fp</font>,<font color="Maroon" title="c2">a</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_4.html#T32" target="_self" title="INT_4:th.32">Th32</a></span>;<br/><span class="kw">hence </span><a NAME="E14:48_1_1_1_2"/>
<font color="Maroon" title="c1">fp</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c3">b</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a>  <a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Product</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> (<font color="Maroon" title="c1">fp</font>,<font color="Maroon" title="c2">a</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:48"><span class="lab"><font color="Green" title="E39">A8</font></span></a>, <a class="txt" href="#E5:48_1_1_1_2"><span class="lab"><font color="Green" title="E44">A15</font></span></a>, <a class="ref" href="xreal_1.html#T235" title="XREAL_1:th.235">XREAL_1:235</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_2"><span class="kw">suppose </span></a><a NAME="E1:48_1_2"/><span class="lab"><font color="Green" title="E42">A17</font></span>: 
<font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">fp</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c3">b</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a>  <a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Product</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> (<font color="Maroon" title="c1">fp</font>,<font color="Maroon" title="c2">a</font>)</span>)</span></span><br/><div class="add"><a NAME="E2:48_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">b</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">n</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:48"><span class="lab"><font color="Green" title="E34">A3</font></span></a>, <a class="txt" href="#E11:48"><span class="lab"><font color="Green" title="E41">A10</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E3:48_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">b</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E4:48_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A18</font></span>: 
<font color="Maroon" title="c3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> (<font color="Maroon" title="c1">fp</font>,<font color="Maroon" title="c2">a</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:48"><span class="lab"><font color="Green" title="E39">A8</font></span></a>, <a class="txt" href="#E10:48"><span class="lab"><font color="Green" title="E40">A9</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E5:48_1_2"/>
<font color="Maroon" title="c3">b</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:48"><span class="lab"><font color="Green" title="E34">A3</font></span></a>, <a class="txt" href="#E11:48"><span class="lab"><font color="Green" title="E41">A10</font></span></a>, <a class="txt" href="#E1:48_1_2"><span class="lab"><font color="Green" title="E42">A17</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E6:48_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> (<font color="Maroon" title="c1">fp</font>,<font color="Maroon" title="c2">a</font>)</span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">fp</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c3">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48"><span class="lab"><font color="Green" title="E32">A1</font></span></a>, <a class="ref" href="wsierp_1.html#D1" title="WSIERP_1:def.1">WSIERP_1:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E7:48_1_2"/>
<font color="Maroon" title="c1">fp</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c3">b</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a>  <a href="wsierp_1.html#K4" title="WSIERP_1:func.4">Product</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K6" title="WSIERP_1:func.6">Del</a> (<font color="Maroon" title="c1">fp</font>,<font color="Maroon" title="c2">a</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:48_1_2"><span class="lab"><font color="Green" title="E43">A18</font></span></a>, <a class="ref" href="int_4.html#T32" target="_self" title="INT_4:th.32">Th32</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div>
