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

<b>let </b><font color="Maroon" title="c1">O</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">G</font> be   <a href="group_9.html#NM2" title="GROUP_9:NM.2">GroupWithOperators</a> of <font color="Maroon" title="c1">O</font>;<br/><b>let </b><font color="Maroon" title="c3">s1</font>, <font color="Maroon" title="c4">s2</font> be   <a href="group_9.html#NM5" title="GROUP_9:NM.5">CompositionSeries</a> of <font color="Maroon" title="c2">G</font>;<br/><b>let </b><font color="Maroon" title="c5">i</font> be  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/>









<b>assume </b><a NAME="E1:181"/><i><font color="Green" title="E154">A1</font></i>: 
<font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">s1</font>
 ;<br/>

<b>assume </b><a NAME="E2:181"/><i><font color="Green" title="E155">A2</font></i>: 
<font color="Maroon" title="c5">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">s1</font>
 ;<br/>

<b>assume </b><a NAME="E3:181"/><i><font color="Green" title="E156">A3</font></i>: 
<font color="Maroon" title="c3">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>
 ;<br/>

<b>assume </b><a NAME="E4:181"/><i><font color="Green" title="E157">A4</font></i>: 
<font color="Maroon" title="c4">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> <font color="Maroon" title="c3">s1</font>,<font color="Maroon" title="c5">i</font>
 ;<br/>

<b>set </b><font color="Maroon" title="c6">f1</font> =  <a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c3">s1</font>;<br/>
<a NAME="E5:181"/><i><font color="Green" title="E158">A5</font></i>: 
<font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">s1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="group_9.html#E1:181"><i><font color="Green" title="E154">A1</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E6:181"/><b>then </b>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">i</font> &amp; <font color="Maroon" title="c5">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">s1</font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E7:181"/><b>then </b><i><font color="Green" title="E159">A6</font></i>: 
1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/>
<a NAME="E8:181"/>
<font color="Maroon" title="c5">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">s1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="group_9.html#E2:181"><i><font color="Green" title="E155">A2</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E9:181"/><b>then </b><i><font color="Green" title="E160">A7</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 &amp; <font color="Maroon" title="c5">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">s1</font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E10:181"/><b>then </b><i><font color="Green" title="E161">A8</font></i>: 
2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">s1</font>
 
<b>by </b><i><a class="txt" href="group_9.html#E7:181"><i><font color="Green" title="E159">A6</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c7">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E12:181"/><i><font color="Green" title="E162">A9</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> <font color="Maroon" title="c3">s1</font>,<font color="Maroon" title="c5">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">k</font> )
 <b>by </b><i><a class="txt" href="group_9.html#E1:181"><i><font color="Green" title="E154">A1</font></i></a>, <a class="ref" href="finseq_3.html#T113" title="FINSEQ_3:th.113">FINSEQ_3:113</a></i>;<br/>
<a NAME="E13:181"/><i><font color="Green" title="E163">A10</font></i>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">s1</font>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a>, <a class="txt" href="group_9.html#E10:181"><i><font color="Green" title="E161">A8</font></i></a></i>;<br/>
<a NAME="E14:181"/><b>then </b><i><font color="Green" title="E164">A11</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c3">s1</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="ref" href="group_9.html#D36" target="_self" title="GROUP_9:def.36">Def36</a></i>;<br/>
<a NAME="E15:181"/>
1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">s2</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="txt" href="group_9.html#E4:181"><i><font color="Green" title="E157">A4</font></i></a>, <a class="txt" href="group_9.html#E12:181"><i><font color="Green" title="E162">A9</font></i></a>, <a class="txt" href="group_9.html#E7:181"><i><font color="Green" title="E159">A6</font></i></a>, <a class="txt" href="group_9.html#E9:181"><i><font color="Green" title="E160">A7</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E16:181"/><b>then </b><i><font color="Green" title="E165">A12</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">s2</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="181_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:181_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">s2</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 )
 </span><b>by </b><i><a class="txt" href="group_9.html#E16:181"><i><font color="Green" title="E165">A12</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="181_1_1"><b>suppose </b></a><a NAME="E1:181_1_1"/><i><font color="Green" title="E166">A13</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:181_1_1"/><b>then </b><i><font color="Green" title="E167">A14</font></i>: 
 <a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c4">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="ref" href="group_9.html#D36" target="_self" title="GROUP_9:def.36">Def36</a></i>;<br/><a NAME="E3:181_1_1"/>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <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="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c3">s1</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="group_9.html#E14:181"><i><font color="Green" title="E164">A11</font></i></a>, <a class="txt" href="group_9.html#E1:181_1_1"><i><font color="Green" title="E166">A13</font></i></a>, <a class="txt" href="group_9.html#E4:181"><i><font color="Green" title="E157">A4</font></i></a>, <a class="txt" href="group_9.html#E12:181"><i><font color="Green" title="E162">A9</font></i></a></i>;<br/><a NAME="E4:181_1_1"/><b>then </b>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c3">s1</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><b>then consider </b><font color="Maroon" title="c8">k1</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E6:181_1_1"/><i><font color="Green" title="E168">A15</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">k1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> <span class="p2">(<span class="default"><a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c3">s1</font></span>)</span>,1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">k1</font> )
 <b>by </b><i><a class="ref" href="finseq_3.html#T113" title="FINSEQ_3:th.113">FINSEQ_3:113</a></i>;<br/><a NAME="E7:181_1_1"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">i</font> &amp; <font color="Maroon" title="c5">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 <b>by </b><i><a class="txt" href="group_9.html#E1:181_1_1"><i><font color="Green" title="E166">A13</font></i></a>, <a class="txt" href="group_9.html#E4:181"><i><font color="Green" title="E157">A4</font></i></a>, <a class="txt" href="group_9.html#E12:181"><i><font color="Green" title="E162">A9</font></i></a>, <a class="txt" href="group_9.html#E5:181"><i><font color="Green" title="E158">A5</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a>, <a class="txt" href="group_9.html#E9:181"><i><font color="Green" title="E160">A7</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E8:181_1_1"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> <span class="p2">(<span class="default"><a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c3">s1</font></span>)</span>,<font color="Maroon" title="c5">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a>, <a class="txt" href="group_9.html#E6:181_1_1"><i><font color="Green" title="E168">A15</font></i></a>, <a class="txt" href="group_9.html#E14:181"><i><font color="Green" title="E164">A11</font></i></a>, <a class="txt" href="group_9.html#E1:181_1_1"><i><font color="Green" title="E166">A13</font></i></a>, <a class="txt" href="group_9.html#E4:181"><i><font color="Green" title="E157">A4</font></i></a>, <a class="txt" href="group_9.html#E12:181"><i><font color="Green" title="E162">A9</font></i></a></i>;<br/><b>hence </b><a NAME="E9:181_1_1"/>
 <a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c4">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> <span class="p1">(<span class="default"><a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c3">s1</font></span>)</span>,<font color="Maroon" title="c5">i</font>
 <b>by </b><i><a class="txt" href="group_9.html#E2:181_1_1"><i><font color="Green" title="E167">A14</font></i></a>, <a class="ref" href="finseq_1.html#T25" title="FINSEQ_1:th.25">FINSEQ_1:25</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="181_1_2"><b>suppose </b></a><a NAME="E1:181_1_2"/><i><font color="Green" title="E166">A16</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">s2</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 ;<br/><div class="add"><a NAME="E2:181_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <b>by </b><i><a class="txt" href="group_9.html#E9:181"><i><font color="Green" title="E160">A7</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><a NAME="E3:181_1_2"/><b>then </b>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">i</font> &amp; <font color="Maroon" title="c5">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c3">s1</font></span>)</span> )
 <b>by </b><i><a class="txt" href="group_9.html#E14:181"><i><font color="Green" title="E164">A11</font></i></a>, <a class="txt" href="group_9.html#E5:181"><i><font color="Green" title="E158">A5</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E4:181_1_2"/><b>then </b>
<font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <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="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c3">s1</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E5:181_1_2"/><b>then </b><i><font color="Green" title="E167">A17</font></i>: 
<font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c3">s1</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><b>then consider </b><font color="Maroon" title="c8">k1</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E7:181_1_2"/><i><font color="Green" title="E168">A18</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">k1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> <span class="p2">(<span class="default"><a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c3">s1</font></span>)</span>,<font color="Maroon" title="c5">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">k1</font> )
 <b>by </b><i><a class="ref" href="finseq_3.html#T113" title="FINSEQ_3:th.113">FINSEQ_3:113</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="181_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c9">n</font> be  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/><b>assume </b><a NAME="E1:181_1_2_1"/><i><font color="Green" title="E169">A19</font></i>: 
<font color="Maroon" title="c9">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> <span class="p2">(<span class="default"><a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c3">s1</font></span>)</span>,<font color="Maroon" title="c5">i</font></span>)</span>
 ;<br/><a NAME="E2:181_1_2_1"/><b>then </b><i><font color="Green" title="E170">A20</font></i>: 
<font color="Maroon" title="c9">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <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="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> <span class="p3">(<span class="default"><a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c3">s1</font></span>)</span>,<font color="Maroon" title="c5">i</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><b>let </b><font color="Maroon" title="c10">H1</font> be    <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Maroon" title="c2">G</font>;<br/><b>let </b><font color="Maroon" title="c11">N1</font> be  <a href="group_9.html#V4" title="GROUP_9:attr.4">normal</a>  <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Maroon" title="c10">H1</font>;<br/><b>assume </b><a NAME="E3:181_1_2_1"/><i><font color="Green" title="E171">A21</font></i>: 
<font color="Maroon" title="c10">H1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">s2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font>
 ;<br/><b>assume </b><a NAME="E4:181_1_2_1"/><i><font color="Green" title="E172">A22</font></i>: 
<font color="Maroon" title="c11">N1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">s2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>
 ;<br/><a NAME="E5:181_1_2_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c3">s1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> <span class="p4">(<span class="default"><a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c3">s1</font></span>)</span>,<font color="Maroon" title="c5">i</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <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="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> <span class="p3">(<span class="default"><a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c3">s1</font></span>)</span>,<font color="Maroon" title="c5">i</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <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="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> <span class="p3">(<span class="default"><a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c3">s1</font></span>)</span>,<font color="Maroon" title="c5">i</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="group_9.html#E7:181_1_2"><i><font color="Green" title="E168">A18</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E6:181_1_2_1"/><b>then </b>
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <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="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> <span class="p3">(<span class="default"><a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c3">s1</font></span>)</span>,<font color="Maroon" title="c5">i</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <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="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c3">s1</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T7" title="FINSEQ_1:th.7">FINSEQ_1:7</a></i>;<br/><a NAME="E7:181_1_2_1"/><b>then </b>
<font color="Maroon" title="c9">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <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="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c3">s1</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="group_9.html#E2:181_1_2_1"><i><font color="Green" title="E170">A20</font></i></a></i>;<br/><a NAME="E8:181_1_2_1"/><b>then </b><i><font color="Green" title="E173">A23</font></i>: 
<font color="Maroon" title="c9">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c3">s1</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E9:181_1_2_1"/><i><font color="Green" title="E174">A24</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">n</font> &amp; <font color="Maroon" title="c9">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> <span class="p2">(<span class="default"><a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c3">s1</font></span>)</span>,<font color="Maroon" title="c5">i</font></span>)</span> )
 <b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a>, <a class="txt" href="group_9.html#E2:181_1_2_1"><i><font color="Green" title="E170">A20</font></i></a></i>;<br/><a NAME="E10:181_1_2_1"/><i><font color="Green" title="E175">A25</font></i>: 
<font color="Maroon" title="c9">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">k1</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a>, <a class="txt" href="group_9.html#E2:181_1_2_1"><i><font color="Green" title="E170">A20</font></i></a>, <a class="txt" href="group_9.html#E7:181_1_2"><i><font color="Green" title="E168">A18</font></i></a></i>;<br/><b>set </b><font color="Maroon" title="c12">n1</font> = <font color="Maroon" title="c9">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1;<br/><a NAME="E11:181_1_2_1"/><i><font color="Green" title="E176">A26</font></i>: 
<font color="Maroon" title="c9">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">k</font>
 <b>by </b><i><a class="txt" href="group_9.html#E7:181_1_2"><i><font color="Green" title="E168">A18</font></i></a>, <a class="txt" href="group_9.html#E12:181"><i><font color="Green" title="E162">A9</font></i></a>, <a class="txt" href="group_9.html#E14:181"><i><font color="Green" title="E164">A11</font></i></a>, <a class="txt" href="group_9.html#E10:181_1_2_1"><i><font color="Green" title="E175">A25</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E12:181_1_2_1"/>
0 <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c9">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c9">n</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E13:181_1_2_1"/><b>then </b><i><font color="Green" title="E177">A27</font></i>: 
<font color="Maroon" title="c9">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">k</font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a>, <a class="txt" href="group_9.html#E11:181_1_2_1"><i><font color="Green" title="E176">A26</font></i></a></i>;<br/><a NAME="E14:181_1_2_1"/>
1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="group_9.html#E9:181_1_2_1"><i><font color="Green" title="E174">A24</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E15:181_1_2_1"/><b>then </b>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E16:181_1_2_1"/><b>then </b>
<font color="Maroon" title="c9">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <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="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c3">s1</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="group_9.html#E12:181"><i><font color="Green" title="E162">A9</font></i></a>, <a class="txt" href="group_9.html#E14:181"><i><font color="Green" title="E164">A11</font></i></a>, <a class="txt" href="group_9.html#E11:181_1_2_1"><i><font color="Green" title="E176">A26</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E17:181_1_2_1"/><b>then </b><i><font color="Green" title="E178">A28</font></i>: 
<font color="Maroon" title="c9">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c3">s1</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c13">n1</font> = <font color="Maroon" title="c9">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 as   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="181_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:181_1_2_1_1"/>
( <font color="Maroon" title="c9">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">i</font> or <font color="Maroon" title="c9">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c5">i</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="181_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:181_1_2_1_1_1"/><i><font color="Green" title="E179">A29</font></i>: 
<font color="Maroon" title="c9">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">i</font>
 ;<br/><div class="add"><a NAME="E2:181_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E180">A30</font></i>: 
<font color="Maroon" title="c13">n1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">i</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="181_1_2_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:181_1_2_1_1_1_1"/>
( <font color="Maroon" title="c13">n1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">i</font> or <font color="Maroon" title="c13">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">i</font> )
 </span><b>by </b><i><a class="txt" href="group_9.html#E2:181_1_2_1_1_1"><i><font color="Green" title="E180">A30</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="181_1_2_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:181_1_2_1_1_1_1_1"/><i><font color="Green" title="E181">A31</font></i>: 
<font color="Maroon" title="c13">n1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">i</font>
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon" title="c14">n'</font> = <font color="Maroon" title="c9">n</font> as    <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>  <b>by </b><i><a class="ref" href="int_1.html#T16" title="INT_1:th.16">INT_1:16</a></i>;<br/><a NAME="E3:181_1_2_1_1_1_1_1"/>
( <font color="Maroon" title="c3">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">n'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">H1</font> &amp; <font color="Maroon" title="c3">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">n'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">N1</font> )
 <b>by </b><i><a class="txt" href="group_9.html#E3:181_1_2_1"><i><font color="Green" title="E171">A21</font></i></a>, <a class="txt" href="group_9.html#E4:181_1_2_1"><i><font color="Green" title="E172">A22</font></i></a>, <a class="txt" href="group_9.html#E1:181_1_2_1_1_1"><i><font color="Green" title="E179">A29</font></i></a>, <a class="txt" href="group_9.html#E4:181"><i><font color="Green" title="E157">A4</font></i></a>, <a class="txt" href="group_9.html#E12:181"><i><font color="Green" title="E162">A9</font></i></a>, <a class="txt" href="group_9.html#E1:181"><i><font color="Green" title="E154">A1</font></i></a>, <a class="ref" href="finseq_3.html#T119" title="FINSEQ_3:th.119">FINSEQ_3:119</a>, <a class="txt" href="group_9.html#E1:181_1_2_1_1_1_1_1"><i><font color="Green" title="E181">A31</font></i></a></i>;<br/><a NAME="E4:181_1_2_1_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">H1</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c11">N1</font>
 <b>by </b><i><a class="txt" href="group_9.html#E8:181_1_2_1"><i><font color="Green" title="E173">A23</font></i></a>, <a class="txt" href="group_9.html#E13:181"><i><font color="Green" title="E163">A10</font></i></a>, <a class="ref" href="group_9.html#D36" target="_self" title="GROUP_9:def.36">Def36</a></i>;<br/><b>hence </b><a NAME="E5:181_1_2_1_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> <span class="p2">(<span class="default"><a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c3">s1</font></span>)</span>,<font color="Maroon" title="c5">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">H1</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c11">N1</font>
 <b>by </b><i><a class="txt" href="group_9.html#E1:181"><i><font color="Green" title="E154">A1</font></i></a>, <a class="txt" href="group_9.html#E1:181_1_2_1"><i><font color="Green" title="E169">A19</font></i></a>, <a class="txt" href="group_9.html#E7:181_1_2"><i><font color="Green" title="E168">A18</font></i></a>, <a class="txt" href="group_9.html#E1:181_1_2_1_1_1"><i><font color="Green" title="E179">A29</font></i></a>, <a class="ref" href="finseq_3.html#T119" title="FINSEQ_3:th.119">FINSEQ_3:119</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="181_1_2_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:181_1_2_1_1_1_1_2"/>
<font color="Maroon" title="c13">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">i</font>
 ;<br/><div class="add"><a NAME="E2:181_1_2_1_1_1_1_2"/><b>then </b>
( <font color="Maroon" title="c3">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">H1</font> &amp; <font color="Maroon" title="c3">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">N1</font> )
 <b>by </b><i><a class="txt" href="group_9.html#E11:181_1_2_1"><i><font color="Green" title="E176">A26</font></i></a>, <a class="txt" href="group_9.html#E3:181_1_2_1"><i><font color="Green" title="E171">A21</font></i></a>, <a class="txt" href="group_9.html#E4:181_1_2_1"><i><font color="Green" title="E172">A22</font></i></a>, <a class="txt" href="group_9.html#E1:181_1_2_1_1_1"><i><font color="Green" title="E179">A29</font></i></a>, <a class="txt" href="group_9.html#E12:181"><i><font color="Green" title="E162">A9</font></i></a>, <a class="txt" href="group_9.html#E1:181"><i><font color="Green" title="E154">A1</font></i></a>, <a class="txt" href="group_9.html#E1:181_1_2_1"><i><font color="Green" title="E169">A19</font></i></a>, <a class="ref" href="finseq_3.html#T119" title="FINSEQ_3:th.119">FINSEQ_3:119</a>, <a class="txt" href="group_9.html#E4:181"><i><font color="Green" title="E157">A4</font></i></a>, <a class="txt" href="group_9.html#E3:181"><i><font color="Green" title="E156">A3</font></i></a>, <a class="ref" href="finseq_3.html#T120" title="FINSEQ_3:th.120">FINSEQ_3:120</a></i>;<br/><a NAME="E3:181_1_2_1_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">H1</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c11">N1</font>
 <b>by </b><i><a class="txt" href="group_9.html#E8:181_1_2_1"><i><font color="Green" title="E173">A23</font></i></a>, <a class="txt" href="group_9.html#E13:181"><i><font color="Green" title="E163">A10</font></i></a>, <a class="ref" href="group_9.html#D36" target="_self" title="GROUP_9:def.36">Def36</a></i>;<br/><b>hence </b><a NAME="E4:181_1_2_1_1_1_1_2"/>
<span class="p1">(<span class="default"><a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> <span class="p2">(<span class="default"><a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c3">s1</font></span>)</span>,<font color="Maroon" title="c5">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">H1</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c11">N1</font>
 <b>by </b><i><a class="txt" href="group_9.html#E1:181"><i><font color="Green" title="E154">A1</font></i></a>, <a class="txt" href="group_9.html#E1:181_1_2_1"><i><font color="Green" title="E169">A19</font></i></a>, <a class="txt" href="group_9.html#E7:181_1_2"><i><font color="Green" title="E168">A18</font></i></a>, <a class="txt" href="group_9.html#E1:181_1_2_1_1_1"><i><font color="Green" title="E179">A29</font></i></a>, <a class="ref" href="finseq_3.html#T119" title="FINSEQ_3:th.119">FINSEQ_3:119</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="181_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:181_1_2_1_1_2"/><i><font color="Green" title="E179">A32</font></i>: 
<font color="Maroon" title="c9">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c5">i</font>
 ;<br/><div class="add"><a NAME="E2:181_1_2_1_1_2"/><i><font color="Green" title="E180">A33</font></i>: 
0 <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c5">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c5">i</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E3:181_1_2_1_1_2"/>
<font color="Maroon" title="c9">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c5">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a>, <a class="txt" href="group_9.html#E1:181_1_2_1_1_2"><i><font color="Green" title="E179">A32</font></i></a></i>;<br/><a NAME="E4:181_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E181">A34</font></i>: 
<font color="Maroon" title="c13">n1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c5">i</font>
 <b>by </b><i><a class="txt" href="group_9.html#E2:181_1_2_1_1_2"><i><font color="Green" title="E180">A33</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c14">n1'</font> = <font color="Maroon" title="c13">n1</font> as    <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>  <b>by </b><i><a class="ref" href="int_1.html#T16" title="INT_1:th.16">INT_1:16</a></i>;<br/><a NAME="E6:181_1_2_1_1_2"/>
( <font color="Maroon" title="c3">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">n1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">H1</font> &amp; <font color="Maroon" title="c3">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">n1'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">N1</font> )
 <b>by </b><i><a class="txt" href="group_9.html#E4:181_1_2_1_1_2"><i><font color="Green" title="E181">A34</font></i></a>, <a class="txt" href="group_9.html#E11:181_1_2_1"><i><font color="Green" title="E176">A26</font></i></a>, <a class="txt" href="group_9.html#E3:181_1_2_1"><i><font color="Green" title="E171">A21</font></i></a>, <a class="txt" href="group_9.html#E4:181_1_2_1"><i><font color="Green" title="E172">A22</font></i></a>, <a class="txt" href="group_9.html#E1:181_1_2_1"><i><font color="Green" title="E169">A19</font></i></a>, <a class="txt" href="group_9.html#E13:181_1_2_1"><i><font color="Green" title="E177">A27</font></i></a>, <a class="txt" href="group_9.html#E1:181_1_2_1_1_2"><i><font color="Green" title="E179">A32</font></i></a>, <a class="txt" href="group_9.html#E1:181"><i><font color="Green" title="E154">A1</font></i></a>, <a class="txt" href="group_9.html#E12:181"><i><font color="Green" title="E162">A9</font></i></a>, <a class="txt" href="group_9.html#E4:181"><i><font color="Green" title="E157">A4</font></i></a>, <a class="ref" href="finseq_3.html#T120" title="FINSEQ_3:th.120">FINSEQ_3:120</a></i>;<br/><a NAME="E7:181_1_2_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">H1</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c11">N1</font>
 <b>by </b><i><a class="txt" href="group_9.html#E17:181_1_2_1"><i><font color="Green" title="E178">A28</font></i></a>, <a class="txt" href="group_9.html#E13:181"><i><font color="Green" title="E163">A10</font></i></a>, <a class="ref" href="group_9.html#D36" target="_self" title="GROUP_9:def.36">Def36</a></i>;<br/><b>hence </b><a NAME="E8:181_1_2_1_1_2"/>
<span class="p1">(<span class="default"><a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> <span class="p2">(<span class="default"><a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c3">s1</font></span>)</span>,<font color="Maroon" title="c5">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">H1</font> <a href="group_9.html#K10" title="GROUP_9:func.10">./.</a> <font color="Maroon" title="c11">N1</font>
 <b>by </b><i><a class="txt" href="group_9.html#E10:181_1_2_1"><i><font color="Green" title="E175">A25</font></i></a>, <a class="txt" href="group_9.html#E5:181_1_2"><i><font color="Green" title="E167">A17</font></i></a>, <a class="txt" href="group_9.html#E7:181_1_2"><i><font color="Green" title="E168">A18</font></i></a>, <a class="txt" href="group_9.html#E1:181_1_2_1_1_2"><i><font color="Green" title="E179">A32</font></i></a>, <a class="txt" href="group_9.html#E1:181_1_2_1"><i><font color="Green" title="E169">A19</font></i></a>, <a class="ref" href="finseq_3.html#T120" title="FINSEQ_3:th.120">FINSEQ_3:120</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E9:181_1_2"/>
 <a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c4">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> <span class="p1">(<span class="default"><a href="group_9.html#K20" title="GROUP_9:func.20">the_series_of_quotients_of</a> <font color="Maroon" title="c3">s1</font></span>)</span>,<font color="Maroon" title="c5">i</font>
 <b>by </b><i><a class="txt" href="group_9.html#E1:181_1_2"><i><font color="Green" title="E166">A16</font></i></a>, <a class="txt" href="group_9.html#E14:181"><i><font color="Green" title="E164">A11</font></i></a>, <a class="txt" href="group_9.html#E4:181"><i><font color="Green" title="E157">A4</font></i></a>, <a class="txt" href="group_9.html#E12:181"><i><font color="Green" title="E162">A9</font></i></a>, <a class="txt" href="group_9.html#E7:181_1_2"><i><font color="Green" title="E168">A18</font></i></a>, <a class="ref" href="group_9.html#D36" target="_self" title="GROUP_9:def.36">Def36</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
