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

<b>let </b><font color="Maroon" title="c1">I</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">A</font> be  <a href="pencil_1.html#V15" title="PENCIL_1:attr.15">PLS-yielding</a>  <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font>;<br/>



<b>assume </b><a NAME="E1:25"/><i><font color="Green" title="E24">A1</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font> holds  <font color="Maroon" title="c2">A</font> <a href="pencil_1.html#K2" title="PENCIL_1:func.2">.</a> <font color="Olive" title="b1">i</font> is <a href="pencil_1.html#V9" title="PENCIL_1:attr.9">connected</a>
 ;<br/>

<b>let </b><font color="Maroon" title="c3">B1</font>, <font color="Maroon" title="c4">B2</font> be    <a href="pencil_2.html#M1" title="PENCIL_2:mode.1">Segre-Coset</a> of <font color="Maroon" title="c2">A</font>;<br/>

<b>assume </b><a NAME="E2:25"/><i><font color="Green" title="E25">A2</font></i>: 
<font color="Maroon" title="c3">B1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c4">B2</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c5">b1</font>, <font color="Maroon" title="c6">b2</font> be  non <a href="pencil_1.html#V13" title="PENCIL_1:attr.13">trivial-yielding</a> <a href="pencil_1.html#V16" title="PENCIL_1:attr.16">Segre-like</a>  <a href="pboole.html#M4" title="PBOOLE:mode.4">ManySortedSubset</a> of  <a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">A</font>;<br/>

<b>assume </b><a NAME="E3:25"/><i><font color="Green" title="E26">A3</font></i>: 
( <font color="Maroon" title="c3">B1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c5">b1</font> &amp; <font color="Maroon" title="c4">B2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c6">b2</font> )
 ;<br/>

<b>thus </b><a NAME="E4:25"/>
(  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c5">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c6">b2</font> implies  ex <font color="Olive" title="b1">D</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="pencil_1.html#K6" title="PENCIL_1:func.6">Segre_Product</a> <font color="Maroon" title="c2">A</font></span>)</span> st <br/>( <font color="Olive" title="b1">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">B1</font> &amp; <font color="Olive" title="b1">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">D</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">B2</font> &amp; ( for <font color="Olive" title="b2">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b2">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="Olive" title="b1">D</font> holds <br/><font color="Olive" title="b1">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">i</font> is    <a href="pencil_2.html#M1" title="PENCIL_2:mode.1">Segre-Coset</a> of <font color="Maroon" title="c2">A</font> ) &amp; ( for <font color="Olive" title="b2">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">i</font> &amp; <font color="Olive" title="b2">i</font> <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="Olive" title="b1">D</font> holds <br/>for <font color="Olive" title="b3">Di</font>, <font color="Olive" title="b4">Di1</font> being   <a href="pencil_2.html#M1" title="PENCIL_2:mode.1">Segre-Coset</a> of <font color="Maroon" title="c2">A</font>  st <font color="Olive" title="b3">Di</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">i</font> &amp; <font color="Olive" title="b4">Di1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> holds <br/>( <font color="Olive" title="b3">Di</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b4">Di1</font> &amp; <font color="Olive" title="b3">Di</font> <a href="pencil_3.html#R1" title="PENCIL_3:pred.1">'||'</a> <font color="Olive" title="b4">Di1</font> ) ) ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:25_1"/><i><font color="Green" title="E27">A4</font></i>: 
 <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c5">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c6">b2</font>
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <b>means </b>for <font color="Olive" title="b1">c1</font>, <font color="Olive" title="b2">c2</font> being non <a href="pencil_1.html#V13" title="PENCIL_1:attr.13">trivial-yielding</a> <a href="pencil_1.html#V16" title="PENCIL_1:attr.16">Segre-like</a>  <a href="pboole.html#M4" title="PBOOLE:mode.4">ManySortedSubset</a> of  <a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">A</font>  st  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Olive" title="b1">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Olive" title="b2">c2</font> &amp;  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Olive" title="b1">c1</font> is    <a href="pencil_2.html#M1" title="PENCIL_2:mode.1">Segre-Coset</a> of <font color="Maroon" title="c2">A</font> &amp;  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Olive" title="b2">c2</font> is    <a href="pencil_2.html#M1" title="PENCIL_2:mode.1">Segre-Coset</a> of <font color="Maroon" title="c2">A</font> &amp;  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Olive" title="b1">c1</font> <a href="subset_1.html#R1" title="SUBSET_1:pred.1">misses</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Olive" title="b2">c2</font> &amp;  <a href="pencil_3.html#K1" title="PENCIL_3:func.1">diff</a> <font color="Olive" title="b1">c1</font>,<font color="Olive" title="b2">c2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 holds <br/> ex <font color="Olive" title="b3">D</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="pencil_1.html#K6" title="PENCIL_1:func.6">Segre_Product</a> <font color="Maroon" title="c2">A</font></span>)</span> st <br/>( <font color="Olive" title="b3">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Olive" title="b1">c1</font> &amp; <font color="Olive" title="b3">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b3">D</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Olive" title="b2">c2</font> &amp; ( for <font color="Olive" title="b4">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b4">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="Olive" title="b3">D</font> holds <br/><font color="Olive" title="b3">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">i</font> is    <a href="pencil_2.html#M1" title="PENCIL_2:mode.1">Segre-Coset</a> of <font color="Maroon" title="c2">A</font> ) &amp; ( for <font color="Olive" title="b4">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b4">i</font> &amp; <font color="Olive" title="b4">i</font> <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="Olive" title="b3">D</font> holds <br/>for <font color="Olive" title="b5">Di</font>, <font color="Olive" title="b6">Di1</font> being   <a href="pencil_2.html#M1" title="PENCIL_2:mode.1">Segre-Coset</a> of <font color="Maroon" title="c2">A</font>  st <font color="Olive" title="b5">Di</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">i</font> &amp; <font color="Olive" title="b6">Di1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> holds <br/>( <font color="Olive" title="b5">Di</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b6">Di1</font> &amp; <font color="Olive" title="b5">Di</font> <a href="pencil_3.html#R1" title="PENCIL_3:pred.1">'||'</a> <font color="Olive" title="b6">Di1</font> ) ) );<br/>
<a NAME="E2:25_1"/><i><font color="Green" title="E28">A5</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">c1</font>, <font color="Maroon" title="c8">c2</font> be  non <a href="pencil_1.html#V13" title="PENCIL_1:attr.13">trivial-yielding</a> <a href="pencil_1.html#V16" title="PENCIL_1:attr.16">Segre-like</a>  <a href="pboole.html#M4" title="PBOOLE:mode.4">ManySortedSubset</a> of  <a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">A</font>;<br/>

<b>assume </b><a NAME="E1:25_1_1"/><i><font color="Green" title="E29">A6</font></i>: 
(  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c7">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c8">c2</font> &amp;  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c7">c1</font> is    <a href="pencil_2.html#M1" title="PENCIL_2:mode.1">Segre-Coset</a> of <font color="Maroon" title="c2">A</font> &amp;  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c8">c2</font> is    <a href="pencil_2.html#M1" title="PENCIL_2:mode.1">Segre-Coset</a> of <font color="Maroon" title="c2">A</font> &amp;  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c7">c1</font> <a href="subset_1.html#R1" title="SUBSET_1:pred.1">misses</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c8">c2</font> &amp;  <a href="pencil_3.html#K1" title="PENCIL_3:func.1">diff</a> <font color="Maroon" title="c7">c1</font>,<font color="Maroon" title="c8">c2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 )
 ;<br/>

<a NAME="E2:25_1_1"/><b>then </b>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">i</font> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font> : <font color="Maroon" title="c7">c1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">c2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 
;<br/>
<a NAME="E3:25_1_1"/><b>then </b><i><font color="Green" title="E30">A7</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive" title="b1">i</font> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font> : <font color="Maroon" title="c7">c1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">c2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> </span>}</span>  <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="card_2.html#T59" title="CARD_2:th.59">CARD_2:59</a></i>;<br/>
<i><font color="Green" title="E31">A8</font></i>:  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c7">c1</font> = 
<font color="Maroon" title="c1">I</font>
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>
<br/>.= 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c8">c2</font>
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c9">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:25_1_1_2"/>
<font color="Maroon" title="c9">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c7">c1</font>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c10">j</font> = <font color="Maroon" title="c9">a</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font> <b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/><b>assume </b><a NAME="E3:25_1_1_2"/>
<font color="Maroon" title="c7">c1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">c2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">a</font>
 ;<br/><a NAME="E4:25_1_1_2"/><b>then </b>
<font color="Maroon" title="c10">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">i</font> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font> : <font color="Maroon" title="c7">c1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">c2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> </span>}</span> 
 ;<br/><b>hence </b><a NAME="E5:25_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="pencil_3.html#E3:25_1_1"><i><font color="Green" title="E30">A7</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E6:25_1_1"/><b>then </b>
<font color="Maroon" title="c7">c1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c8">c2</font>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E4:25_1_1"><i><font color="Green" title="E31">A8</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>;<br/>
<b>hence </b><a NAME="E7:25_1_1"/>
 ex <font color="Olive" title="b1">D</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="pencil_1.html#K6" title="PENCIL_1:func.6">Segre_Product</a> <font color="Maroon" title="c2">A</font></span>)</span> st <br/>( <font color="Olive" title="b1">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c7">c1</font> &amp; <font color="Olive" title="b1">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">D</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c8">c2</font> &amp; ( for <font color="Olive" title="b2">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b2">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="Olive" title="b1">D</font> holds <br/><font color="Olive" title="b1">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">i</font> is    <a href="pencil_2.html#M1" title="PENCIL_2:mode.1">Segre-Coset</a> of <font color="Maroon" title="c2">A</font> ) &amp; ( for <font color="Olive" title="b2">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">i</font> &amp; <font color="Olive" title="b2">i</font> <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="Olive" title="b1">D</font> holds <br/>for <font color="Olive" title="b3">Di</font>, <font color="Olive" title="b4">Di1</font> being   <a href="pencil_2.html#M1" title="PENCIL_2:mode.1">Segre-Coset</a> of <font color="Maroon" title="c2">A</font>  st <font color="Olive" title="b3">Di</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">i</font> &amp; <font color="Olive" title="b4">Di1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> holds <br/>( <font color="Olive" title="b3">Di</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b4">Di1</font> &amp; <font color="Olive" title="b3">Di</font> <a href="pencil_3.html#R1" title="PENCIL_3:pred.1">'||'</a> <font color="Olive" title="b4">Di1</font> ) ) )
 <b>by </b><i><a class="txt" href="pencil_3.html#E1:25_1_1"><i><font color="Green" title="E29">A6</font></i></a></i>;<br/>


</div><b>end;</b></div>
<div><i><font color="Green" title="E29">A9</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>assume </b><a NAME="E1:25_1_2"/><i><font color="Green" title="E30">A10</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">k</font>]
 ;<br/><b>thus </b><a NAME="E2:25_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">c1</font>, <font color="Maroon" title="c9">c2</font> be  non <a href="pencil_1.html#V13" title="PENCIL_1:attr.13">trivial-yielding</a> <a href="pencil_1.html#V16" title="PENCIL_1:attr.16">Segre-like</a>  <a href="pboole.html#M4" title="PBOOLE:mode.4">ManySortedSubset</a> of  <a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">A</font>;<br/>

<b>assume </b><a NAME="E1:25_1_2_1"/><i><font color="Green" title="E31">A11</font></i>: 
(  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c8">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c9">c2</font> &amp;  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c8">c1</font> is    <a href="pencil_2.html#M1" title="PENCIL_2:mode.1">Segre-Coset</a> of <font color="Maroon" title="c2">A</font> &amp;  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c9">c2</font> is    <a href="pencil_2.html#M1" title="PENCIL_2:mode.1">Segre-Coset</a> of <font color="Maroon" title="c2">A</font> &amp;  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c8">c1</font> <a href="subset_1.html#R1" title="SUBSET_1:pred.1">misses</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c9">c2</font> &amp;  <a href="pencil_3.html#K1" title="PENCIL_3:func.1">diff</a> <font color="Maroon" title="c8">c1</font>,<font color="Maroon" title="c9">c2</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 )
 ;<br/>

<a NAME="E2:25_1_2_1"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">i</font> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font> : <font color="Maroon" title="c8">c1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">c2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> </span>}</span>  <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
 
<b>by </b><i><a class="txt" href="pencil_3.html#E1:25_1_2_1"><i><font color="Green" title="E31">A11</font></i></a></i>;<br/>
<a NAME="E3:25_1_2_1"/><b>then </b>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">i</font> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font> : <font color="Maroon" title="c8">c1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">c2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> </span>}</span>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 
;<br/>
<a NAME="E4:25_1_2_1"/><b>then </b>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">i</font> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font> : <font color="Maroon" title="c8">c1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">c2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> </span>}</span>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="ref" href="card_1.html#D5" title="CARD_1:def.5">CARD_1:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c10">j0</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E6:25_1_2_1"/><i><font color="Green" title="E32">A12</font></i>: 
<font color="Maroon" title="c10">j0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">i</font> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font> : <font color="Maroon" title="c8">c1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">c2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c11">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font><b> such that </b><br/><a NAME="E8:25_1_2_1"/><i><font color="Green" title="E33">A13</font></i>: 
( <font color="Maroon" title="c10">j0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">j</font> &amp; <font color="Maroon" title="c8">c1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">c2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">j</font> )
 <b>by </b><i><a class="txt" href="pencil_3.html#E6:25_1_2_1"><i><font color="Green" title="E32">A12</font></i></a></i>;<br/>
<a NAME="E9:25_1_2_1"/><i><font color="Green" title="E34">A14</font></i>: 
( <font color="Maroon" title="c8">c1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c8">c1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="pencil_1.html#K2" title="PENCIL_1:func.2">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c8">c1</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c9">c2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c8">c1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="pencil_1.html#K2" title="PENCIL_1:func.2">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c8">c1</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="pencil_3.html#E1:25_1_2_1"><i><font color="Green" title="E31">A11</font></i></a>, <a class="ref" href="pencil_3.html#T10" target="_self" title="PENCIL_3:th.10">Th10</a></i>;<br/>
<a NAME="E10:25_1_2_1"/><b>then </b><i><font color="Green" title="E35">A15</font></i>: 
<font color="Maroon" title="c11">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c8">c1</font>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E8:25_1_2_1"><i><font color="Green" title="E33">A13</font></i></a></i>;<br/>
<a NAME="E11:25_1_2_1"/>
( not <font color="Maroon" title="c8">c1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">j</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; <font color="Maroon" title="c8">c1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">j</font> is <a href="realset1.html#V1" title="REALSET1:attr.1">trivial</a> )
 
<b>by </b><i><a class="txt" href="pencil_3.html#E10:25_1_2_1"><i><font color="Green" title="E35">A15</font></i></a>, <a class="ref" href="pencil_1.html#T12" title="PENCIL_1:th.12">PENCIL_1:12</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c12">p</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E13:25_1_2_1"/><i><font color="Green" title="E36">A16</font></i>: 
<font color="Maroon" title="c8">c1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c12">p</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="realset1.html#D4" title="REALSET1:def.4">REALSET1:def 4</a></i>;<br/>
<a NAME="E14:25_1_2_1"/>
<font color="Maroon" title="c8">c1</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a>  <a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">A</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D23" title="PBOOLE:def.23">PBOOLE:def 23</a></i>;<br/>
<a NAME="E15:25_1_2_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c12">p</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">j</font>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E13:25_1_2_1"><i><font color="Green" title="E36">A16</font></i></a>, <a class="ref" href="pboole.html#D5" title="PBOOLE:def.5">PBOOLE:def 5</a></i>;<br/>
<a NAME="E16:25_1_2_1"/><b>then </b>
<font color="Maroon" title="c12">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">j</font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E17:25_1_2_1"/><b>then </b>
<font color="Maroon" title="c12">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="pencil_1.html#K2" title="PENCIL_1:func.2">.</a> <font color="Maroon" title="c11">j</font></span>)</span>
 
<b>by </b><i><a class="ref" href="pencil_3.html#T7" target="_self" title="PENCIL_3:th.7">Th7</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c13">p</font> = <font color="Maroon" title="c12">p</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="pencil_1.html#K2" title="PENCIL_1:func.2">.</a> <font color="Maroon" title="c11">j</font></span>)</span> ;<br/>
<a NAME="E19:25_1_2_1"/>
( not <font color="Maroon" title="c9">c2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">j</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; <font color="Maroon" title="c9">c2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">j</font> is <a href="realset1.html#V1" title="REALSET1:attr.1">trivial</a> )
 
<b>by </b><i><a class="txt" href="pencil_3.html#E1:25_1_2_1"><i><font color="Green" title="E31">A11</font></i></a>, <a class="txt" href="pencil_3.html#E10:25_1_2_1"><i><font color="Green" title="E35">A15</font></i></a>, <a class="ref" href="pencil_1.html#T12" title="PENCIL_1:th.12">PENCIL_1:12</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c14">r</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E21:25_1_2_1"/><i><font color="Green" title="E37">A17</font></i>: 
<font color="Maroon" title="c9">c2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c14">r</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="realset1.html#D4" title="REALSET1:def.4">REALSET1:def 4</a></i>;<br/>
<a NAME="E22:25_1_2_1"/>
<font color="Maroon" title="c9">c2</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a>  <a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">A</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D23" title="PBOOLE:def.23">PBOOLE:def 23</a></i>;<br/>
<a NAME="E23:25_1_2_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c14">r</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">j</font>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E21:25_1_2_1"><i><font color="Green" title="E37">A17</font></i></a>, <a class="ref" href="pboole.html#D5" title="PBOOLE:def.5">PBOOLE:def 5</a></i>;<br/>
<a NAME="E24:25_1_2_1"/><b>then </b>
<font color="Maroon" title="c14">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">j</font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E25:25_1_2_1"/><b>then </b>
<font color="Maroon" title="c14">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="pencil_1.html#K2" title="PENCIL_1:func.2">.</a> <font color="Maroon" title="c11">j</font></span>)</span>
 
<b>by </b><i><a class="ref" href="pencil_3.html#T7" target="_self" title="PENCIL_3:th.7">Th7</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c15">r</font> = <font color="Maroon" title="c14">r</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="pencil_1.html#K2" title="PENCIL_1:func.2">.</a> <font color="Maroon" title="c11">j</font></span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon" title="c16">c3</font> = <font color="Maroon" title="c9">c2</font> <a href="polynom1.html#K2" title="POLYNOM1:func.2">+*</a> <font color="Maroon" title="c11">j</font>,<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c13">p</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> as  non <a href="pencil_1.html#V13" title="PENCIL_1:attr.13">trivial-yielding</a> <a href="pencil_1.html#V16" title="PENCIL_1:attr.16">Segre-like</a>  <a href="pboole.html#M4" title="PBOOLE:mode.4">ManySortedSubset</a> of  <a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">A</font> <b>by </b><i><a class="txt" href="pencil_3.html#E1:25_1_2_1"><i><font color="Green" title="E31">A11</font></i></a>, <a class="txt" href="pencil_3.html#E10:25_1_2_1"><i><font color="Green" title="E35">A15</font></i></a>, <a class="ref" href="pencil_3.html#T13" target="_self" title="PENCIL_3:th.13">Th13</a></i>;<br/>
<a NAME="E28:25_1_2_1"/><i><font color="Green" title="E38">A18</font></i>: 
 <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c16">c3</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="pencil_1.html#K6" title="PENCIL_1:func.6">Segre_Product</a> <font color="Maroon" title="c2">A</font></span>)</span>
 
<b>by </b><i><a class="ref" href="pencil_3.html#T14" target="_self" title="PENCIL_3:th.14">Th14</a></i>;<br/>
<a NAME="E29:25_1_2_1"/><i><font color="Green" title="E39">A19</font></i>: 
not <font color="Maroon" title="c2">A</font> <a href="pencil_1.html#K2" title="PENCIL_1:func.2">.</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c8">c1</font></span>)</span> is <a href="realset2.html#V3" title="REALSET2:attr.3">trivial</a>
 
<b>by </b><i><a class="ref" href="pencil_3.html#T2" target="_self" title="PENCIL_3:th.2">Th2</a></i>;<br/>
<a NAME="E30:25_1_2_1"/><i><font color="Green" title="E40">A20</font></i>: 
<font color="Maroon" title="c16">c3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c8">c1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="pencil_1.html#K2" title="PENCIL_1:func.2">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c8">c1</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E9:25_1_2_1"><i><font color="Green" title="E34">A14</font></i></a>, <a class="txt" href="pencil_3.html#E10:25_1_2_1"><i><font color="Green" title="E35">A15</font></i></a>, <a class="ref" href="funct_7.html#T34" title="FUNCT_7:th.34">FUNCT_7:34</a></i>;<br/>
<a NAME="E31:25_1_2_1"/><b>then </b>
<font color="Maroon" title="c16">c3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c8">c1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="pencil_1.html#K2" title="PENCIL_1:func.2">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c8">c1</font></span>)</span></span>)</span>
 
;<br/>
<a NAME="E32:25_1_2_1"/><b>then </b>
not <font color="Maroon" title="c16">c3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c8">c1</font></span>)</span> is <a href="realset1.html#V1" title="REALSET1:attr.1">trivial</a>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E29:25_1_2_1"><i><font color="Green" title="E39">A19</font></i></a>, <a class="ref" href="realset2.html#D5" title="REALSET2:def.5">REALSET2:def 5</a></i>;<br/>
<a NAME="E33:25_1_2_1"/><b>then </b><i><font color="Green" title="E41">A21</font></i>: 
 <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c16">c3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c8">c1</font>
 
<b>by </b><i><a class="ref" href="pencil_1.html#D21" title="PENCIL_1:def.21">PENCIL_1:def 21</a></i>;<br/>
<a NAME="E34:25_1_2_1"/><b>then </b>
<font color="Maroon" title="c16">c3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c16">c3</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="pencil_1.html#K2" title="PENCIL_1:func.2">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c16">c3</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E30:25_1_2_1"><i><font color="Green" title="E40">A20</font></i></a></i>;<br/>
<a NAME="E35:25_1_2_1"/><b>then </b><i><font color="Green" title="E42">A22</font></i>: 
 <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c16">c3</font> is    <a href="pencil_2.html#M1" title="PENCIL_2:mode.1">Segre-Coset</a> of <font color="Maroon" title="c2">A</font>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E28:25_1_2_1"><i><font color="Green" title="E38">A18</font></i></a>, <a class="ref" href="pencil_2.html#D2" title="PENCIL_2:def.2">PENCIL_2:def 2</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_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="E1:25_1_2_1_1"/>
(  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c8">c1</font> <a href="subset_1.html#R1" title="SUBSET_1:pred.1">misses</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c16">c3</font> or  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c8">c1</font> <a href="subset_1.html#R2" title="SUBSET_1:pred.2">meets</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c16">c3</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:25_1_2_1_1_1"/><i><font color="Green" title="E43">A23</font></i>: 
 <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c8">c1</font> <a href="subset_1.html#R1" title="SUBSET_1:pred.1">misses</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c16">c3</font>
 ;<br/><div class="add"><a NAME="E2:25_1_2_1_1_1"/>
 <a href="pencil_3.html#K1" title="PENCIL_3:func.1">diff</a> <font color="Maroon" title="c8">c1</font>,<font color="Maroon" title="c9">c2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pencil_3.html#K1" title="PENCIL_3:func.1">diff</a> <font color="Maroon" title="c8">c1</font>,<font color="Maroon" title="c16">c3</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="pencil_3.html#E8:25_1_2_1"><i><font color="Green" title="E33">A13</font></i></a>, <a class="txt" href="pencil_3.html#E13:25_1_2_1"><i><font color="Green" title="E36">A16</font></i></a>, <a class="ref" href="pencil_3.html#T19" target="_self" title="PENCIL_3:th.19">Th19</a></i>;<br/><a NAME="E3:25_1_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c7">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pencil_3.html#K1" title="PENCIL_3:func.1">diff</a> <font color="Maroon" title="c8">c1</font>,<font color="Maroon" title="c16">c3</font></span>)</span> <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
 <b>by </b><i><a class="txt" href="pencil_3.html#E1:25_1_2_1"><i><font color="Green" title="E31">A11</font></i></a></i>;<br/><a NAME="E4:25_1_2_1_1_1"/><b>then </b>
 <a href="pencil_3.html#K1" title="PENCIL_3:func.1">diff</a> <font color="Maroon" title="c8">c1</font>,<font color="Maroon" title="c16">c3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">k</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c17">E</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="pencil_1.html#K6" title="PENCIL_1:func.6">Segre_Product</a> <font color="Maroon" title="c2">A</font></span>)</span><b> such that </b><br/><a NAME="E6:25_1_2_1_1_1"/><i><font color="Green" title="E44">A24</font></i>: 
( <font color="Maroon" title="c17">E</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c8">c1</font> &amp; <font color="Maroon" title="c17">E</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">E</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c16">c3</font> &amp; ( for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">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="c17">E</font> holds <br/><font color="Maroon" title="c17">E</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> is    <a href="pencil_2.html#M1" title="PENCIL_2:mode.1">Segre-Coset</a> of <font color="Maroon" title="c2">A</font> ) &amp; ( for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <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="c17">E</font> holds <br/>for <font color="Olive" title="b2">Ei</font>, <font color="Olive" title="b3">Ei1</font> being   <a href="pencil_2.html#M1" title="PENCIL_2:mode.1">Segre-Coset</a> of <font color="Maroon" title="c2">A</font>  st <font color="Olive" title="b2">Ei</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">E</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b3">Ei1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">E</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> holds <br/>( <font color="Olive" title="b2">Ei</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b3">Ei1</font> &amp; <font color="Olive" title="b2">Ei</font> <a href="pencil_3.html#R1" title="PENCIL_3:pred.1">'||'</a> <font color="Olive" title="b3">Ei1</font> ) ) )
 <b>by </b><i><a class="txt" href="pencil_3.html#E1:25_1_2"><i><font color="Green" title="E30">A10</font></i></a>, <a class="txt" href="pencil_3.html#E1:25_1_2_1"><i><font color="Green" title="E31">A11</font></i></a>, <a class="txt" href="pencil_3.html#E33:25_1_2_1"><i><font color="Green" title="E41">A21</font></i></a>, <a class="txt" href="pencil_3.html#E35:25_1_2_1"><i><font color="Green" title="E42">A22</font></i></a>, <a class="txt" href="pencil_3.html#E1:25_1_2_1_1_1"><i><font color="Green" title="E43">A23</font></i></a></i>;<br/><a NAME="E7:25_1_2_1_1_1"/><i><font color="Green" title="E45">A25</font></i>: 
not <font color="Maroon" title="c13">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">c2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">j</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E8:25_1_2_1"><i><font color="Green" title="E33">A13</font></i></a>, <a class="txt" href="pencil_3.html#E13:25_1_2_1"><i><font color="Green" title="E36">A16</font></i></a>, <a class="txt" href="pencil_3.html#E21:25_1_2_1"><i><font color="Green" title="E37">A17</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>then consider </b><font color="Maroon" title="c18">F</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="pencil_1.html#K6" title="PENCIL_1:func.6">Segre_Product</a> <font color="Maroon" title="c2">A</font></span>)</span><b> such that </b><br/><a NAME="E9:25_1_2_1_1_1"/><i><font color="Green" title="E46">A26</font></i>: 
( <font color="Maroon" title="c18">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c16">c3</font> &amp; <font color="Maroon" title="c18">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c9">c2</font> &amp; ( for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">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="c18">F</font> holds <br/><font color="Maroon" title="c18">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> is    <a href="pencil_2.html#M1" title="PENCIL_2:mode.1">Segre-Coset</a> of <font color="Maroon" title="c2">A</font> ) &amp; ( for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <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="c18">F</font> holds <br/>for <font color="Olive" title="b2">Di</font>, <font color="Olive" title="b3">Di1</font> being   <a href="pencil_2.html#M1" title="PENCIL_2:mode.1">Segre-Coset</a> of <font color="Maroon" title="c2">A</font>  st <font color="Olive" title="b2">Di</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b3">Di1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> holds <br/>( <font color="Olive" title="b2">Di</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b3">Di1</font> &amp; <font color="Olive" title="b2">Di</font> <a href="pencil_3.html#R1" title="PENCIL_3:pred.1">'||'</a> <font color="Olive" title="b3">Di1</font> ) ) )
 <b>by </b><i><a class="txt" href="pencil_3.html#E1:25"><i><font color="Green" title="E24">A1</font></i></a>, <a class="txt" href="pencil_3.html#E1:25_1_2_1"><i><font color="Green" title="E31">A11</font></i></a>, <a class="txt" href="pencil_3.html#E35:25_1_2_1"><i><font color="Green" title="E42">A22</font></i></a>, <a class="ref" href="pencil_3.html#T22" target="_self" title="PENCIL_3:th.22">Th22</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c19">D</font> = <font color="Maroon" title="c17">E</font> <a href="graph_2.html#K4" title="GRAPH_2:func.4">^'</a> <font color="Maroon" title="c18">F</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="pencil_1.html#K6" title="PENCIL_1:func.6">Segre_Product</a> <font color="Maroon" title="c2">A</font></span>)</span> ;<br/><b>take </b>
<font color="Maroon" title="c19">D</font>
;<br/><a NAME="E11:25_1_2_1_1_1"/>
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="c17">E</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E6:25_1_2_1_1_1"><i><font color="Green" title="E44">A24</font></i></a>, <a class="ref" href="funct_1.html#D4" title="FUNCT_1:def.4">FUNCT_1:def 4</a></i>;<br/><a NAME="E12:25_1_2_1_1_1"/><b>then </b>
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="c17">E</font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><b>hence </b><a NAME="E13:25_1_2_1_1_1"/>
<font color="Maroon" title="c19">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c8">c1</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E6:25_1_2_1_1_1"><i><font color="Green" title="E44">A24</font></i></a>, <a class="ref" href="graph_2.html#T14" title="GRAPH_2:th.14">GRAPH_2:14</a></i>;<br/><a NAME="E14:25_1_2_1_1_1"/>
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="c18">F</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E9:25_1_2_1_1_1"><i><font color="Green" title="E46">A26</font></i></a>, <a class="ref" href="funct_1.html#D4" title="FUNCT_1:def.4">FUNCT_1:def 4</a></i>;<br/><a NAME="E15:25_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E47">A27</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="c18">F</font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_2_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:25_1_2_1_1_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><a NAME="E2:25_1_2_1_1_1_1"/><b>then </b>
 <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c16">c3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c9">c2</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E9:25_1_2_1_1_1"><i><font color="Green" title="E46">A26</font></i></a></i>;<br/><a NAME="E3:25_1_2_1_1_1_1"/><b>then </b><i><font color="Green" title="E48">A28</font></i>: 
<font color="Maroon" title="c16">c3</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c9">c2</font>
 <b>by </b><i><a class="ref" href="pua2mss1.html#T2" title="PUA2MSS1:th.2">PUA2MSS1:2</a></i>;<br/><a NAME="E4:25_1_2_1_1_1_1"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c9">c2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I</font>
 <b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/><a NAME="E5:25_1_2_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c16">c3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c13">p</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="ref" href="funct_7.html#T33" title="FUNCT_7:th.33">FUNCT_7:33</a></i>;<br/><b>hence </b><a NAME="E6:25_1_2_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="pencil_3.html#E7:25_1_2_1_1_1"><i><font color="Green" title="E45">A25</font></i></a>, <a class="txt" href="pencil_3.html#E3:25_1_2_1_1_1_1"><i><font color="Green" title="E48">A28</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div><a NAME="E17:25_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E48">A29</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">F</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 <b>by </b><i><a class="txt" href="pencil_3.html#E15:25_1_2_1_1_1"><i><font color="Green" title="E47">A27</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><b>hence </b><a NAME="E18:25_1_2_1_1_1"/>
<font color="Maroon" title="c19">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c19">D</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c9">c2</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E9:25_1_2_1_1_1"><i><font color="Green" title="E46">A26</font></i></a>, <a class="ref" href="graph_2.html#T16" title="GRAPH_2:th.16">GRAPH_2:16</a></i>;<br/><b>thus </b><a NAME="E19:25_1_2_1_1_1"/>
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">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="c19">D</font> holds <br/><font color="Maroon" title="c19">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> is    <a href="pencil_2.html#M1" title="PENCIL_2:mode.1">Segre-Coset</a> of <font color="Maroon" title="c2">A</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_2_1_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c20">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:25_1_2_1_1_1_2"/>
<font color="Maroon" title="c20">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="c19">D</font>
 ;<br/>

<a NAME="E2:25_1_2_1_1_1_2"/><b>then </b><i><font color="Green" title="E49">A30</font></i>: 
<font color="Maroon" title="c19">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c19">D</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/>
<a NAME="E3:25_1_2_1_1_1_2"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c19">D</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c17">E</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c18">F</font></span>)</span>
 
<b>by </b><i><a class="ref" href="graph_2.html#T17" title="GRAPH_2:th.17">GRAPH_2:17</a></i>;<br/>
<a NAME="E4:25_1_2_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c19">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c17">E</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c18">F</font></span>)</span>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E2:25_1_2_1_1_1_2"><i><font color="Green" title="E49">A30</font></i></a></i>;<br/>
<a NAME="E5:25_1_2_1_1_1_2"/><b>then </b><i><font color="Green" title="E50">A31</font></i>: 
( <font color="Maroon" title="c19">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c17">E</font> or <font color="Maroon" title="c19">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c18">F</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_2_1_1_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:25_1_2_1_1_1_2_1"/>
( <font color="Maroon" title="c19">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c17">E</font> or <font color="Maroon" title="c19">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c18">F</font> )
 </span><b>by </b><i><a class="txt" href="pencil_3.html#E5:25_1_2_1_1_1_2"><i><font color="Green" title="E50">A31</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_2_1_1_1_2_1_1"><b>suppose </b></a><a NAME="E1:25_1_2_1_1_1_2_1_1"/>
<font color="Maroon" title="c19">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c17">E</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c21">i0</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:25_1_2_1_1_1_2_1_1"/><i><font color="Green" title="E51">A32</font></i>: 
( <font color="Maroon" title="c21">i0</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="c17">E</font> &amp; <font color="Maroon" title="c19">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">E</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i0</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c22">i0</font> = <font color="Maroon" title="c21">i0</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="pencil_3.html#E3:25_1_2_1_1_1_2_1_1"><i><font color="Green" title="E51">A32</font></i></a></i>;<br/><a NAME="E5:25_1_2_1_1_1_2_1_1"/>
<font color="Maroon" title="c22">i0</font> is   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>
 ;<br/><b>hence </b><a NAME="E6:25_1_2_1_1_1_2_1_1"/>
<font color="Maroon" title="c19">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">i</font> is    <a href="pencil_2.html#M1" title="PENCIL_2:mode.1">Segre-Coset</a> of <font color="Maroon" title="c2">A</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E6:25_1_2_1_1_1"><i><font color="Green" title="E44">A24</font></i></a>, <a class="txt" href="pencil_3.html#E3:25_1_2_1_1_1_2_1_1"><i><font color="Green" title="E51">A32</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_2_1_1_1_2_1_2"><b>suppose </b></a><a NAME="E1:25_1_2_1_1_1_2_1_2"/>
<font color="Maroon" title="c19">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c18">F</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c21">i0</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:25_1_2_1_1_1_2_1_2"/><i><font color="Green" title="E51">A33</font></i>: 
( <font color="Maroon" title="c21">i0</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="c18">F</font> &amp; <font color="Maroon" title="c19">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">i0</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c22">i0</font> = <font color="Maroon" title="c21">i0</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="pencil_3.html#E3:25_1_2_1_1_1_2_1_2"><i><font color="Green" title="E51">A33</font></i></a></i>;<br/><a NAME="E5:25_1_2_1_1_1_2_1_2"/>
<font color="Maroon" title="c22">i0</font> is   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>
 ;<br/><b>hence </b><a NAME="E6:25_1_2_1_1_1_2_1_2"/>
<font color="Maroon" title="c19">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">i</font> is    <a href="pencil_2.html#M1" title="PENCIL_2:mode.1">Segre-Coset</a> of <font color="Maroon" title="c2">A</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E9:25_1_2_1_1_1"><i><font color="Green" title="E46">A26</font></i></a>, <a class="txt" href="pencil_3.html#E3:25_1_2_1_1_1_2_1_2"><i><font color="Green" title="E51">A33</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>thus </b><a NAME="E20:25_1_2_1_1_1"/>
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <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="c19">D</font> holds <br/>for <font color="Olive" title="b2">Di</font>, <font color="Olive" title="b3">Di1</font> being   <a href="pencil_2.html#M1" title="PENCIL_2:mode.1">Segre-Coset</a> of <font color="Maroon" title="c2">A</font>  st <font color="Olive" title="b2">Di</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b3">Di1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> holds <br/>( <font color="Olive" title="b2">Di</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b3">Di1</font> &amp; <font color="Olive" title="b2">Di</font> <a href="pencil_3.html#R1" title="PENCIL_3:pred.1">'||'</a> <font color="Olive" title="b3">Di1</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_2_1_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c20">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:25_1_2_1_1_1_3"/><i><font color="Green" title="E49">A34</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">i</font> &amp; <font color="Maroon" title="c20">i</font> <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="c19">D</font> )
 ;<br/>

<b>let </b><font color="Maroon" title="c21">Di</font>, <font color="Maroon" title="c22">Di1</font> be    <a href="pencil_2.html#M1" title="PENCIL_2:mode.1">Segre-Coset</a> of <font color="Maroon" title="c2">A</font>;<br/>

<b>assume </b><a NAME="E2:25_1_2_1_1_1_3"/><i><font color="Green" title="E50">A35</font></i>: 
( <font color="Maroon" title="c21">Di</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">i</font> &amp; <font color="Maroon" title="c22">Di1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> )
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c23">i</font> = <font color="Maroon" title="c20">i</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_2_1_1_1_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:25_1_2_1_1_1_3_1"/>
( <font color="Maroon" title="c23">i</font> <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="c17">E</font> or <font color="Maroon" title="c23">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">E</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_2_1_1_1_3_1_1"><b>suppose </b></a><a NAME="E1:25_1_2_1_1_1_3_1_1"/><i><font color="Green" title="E51">A36</font></i>: 
<font color="Maroon" title="c23">i</font> <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="c17">E</font>
 ;<br/><div class="add"><a NAME="E2:25_1_2_1_1_1_3_1_1"/><b>then </b><i><font color="Green" title="E52">A37</font></i>: 
<font color="Maroon" title="c21">Di</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">E</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E1:25_1_2_1_1_1_3"><i><font color="Green" title="E49">A34</font></i></a>, <a class="txt" href="pencil_3.html#E2:25_1_2_1_1_1_3"><i><font color="Green" title="E50">A35</font></i></a>, <a class="ref" href="graph_2.html#T14" title="GRAPH_2:th.14">GRAPH_2:14</a></i>;<br/><a NAME="E3:25_1_2_1_1_1_3_1_1"/>
<font color="Maroon" title="c23">i</font> <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>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">E</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E1:25_1_2_1_1_1_3_1_1"><i><font color="Green" title="E51">A36</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E4:25_1_2_1_1_1_3_1_1"/><b>then </b>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c23">i</font> <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>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">E</font> )
 <b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/><a NAME="E5:25_1_2_1_1_1_3_1_1"/><b>then </b>
<font color="Maroon" title="c22">Di1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">E</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="pencil_3.html#E2:25_1_2_1_1_1_3"><i><font color="Green" title="E50">A35</font></i></a>, <a class="ref" href="graph_2.html#T14" title="GRAPH_2:th.14">GRAPH_2:14</a></i>;<br/><b>hence </b><a NAME="E6:25_1_2_1_1_1_3_1_1"/>
( <font color="Maroon" title="c21">Di</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c22">Di1</font> &amp; <font color="Maroon" title="c21">Di</font> <a href="pencil_3.html#R1" title="PENCIL_3:pred.1">'||'</a> <font color="Maroon" title="c22">Di1</font> )
 <b>by </b><i><a class="txt" href="pencil_3.html#E6:25_1_2_1_1_1"><i><font color="Green" title="E44">A24</font></i></a>, <a class="txt" href="pencil_3.html#E1:25_1_2_1_1_1_3"><i><font color="Green" title="E49">A34</font></i></a>, <a class="txt" href="pencil_3.html#E1:25_1_2_1_1_1_3_1_1"><i><font color="Green" title="E51">A36</font></i></a>, <a class="txt" href="pencil_3.html#E2:25_1_2_1_1_1_3_1_1"><i><font color="Green" title="E52">A37</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_2_1_1_1_3_1_2"><b>suppose </b></a><a NAME="E1:25_1_2_1_1_1_3_1_2"/>
<font color="Maroon" title="c23">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">E</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c24">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E3:25_1_2_1_1_1_3_1_2"/><i><font color="Green" title="E51">A38</font></i>: 
<font color="Maroon" title="c23">i</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> <font color="Maroon" title="c17">E</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c24">k</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T10" title="NAT_1:th.10">NAT_1:10</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c25">k</font> = <font color="Maroon" title="c24">k</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><a NAME="E5:25_1_2_1_1_1_3_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">F</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 <b>by </b><i><a class="txt" href="pencil_3.html#E17:25_1_2_1_1_1"><i><font color="Green" title="E48">A29</font></i></a></i>;<br/><a NAME="E6:25_1_2_1_1_1_3_1_2"/><b>then </b>
<font color="Maroon" title="c18">F</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="ref" href="finseq_1.html#T25" title="FINSEQ_1:th.25">FINSEQ_1:25</a></i>;<br/><a NAME="E7:25_1_2_1_1_1_3_1_2"/><b>then </b><i><font color="Green" title="E52">A39</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c19">D</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <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> <font color="Maroon" title="c17">E</font></span>)</span> <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> <font color="Maroon" title="c18">F</font></span>)</span>
 <b>by </b><i><a class="ref" href="graph_2.html#T13" title="GRAPH_2:th.13">GRAPH_2:13</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_2_1_1_1_3_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:25_1_2_1_1_1_3_1_2_1"/>
( <font color="Maroon" title="c25">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 or <font color="Maroon" title="c25">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_2_1_1_1_3_1_2_1_1"><b>suppose </b></a><a NAME="E1:25_1_2_1_1_1_3_1_2_1_1"/>
<font color="Maroon" title="c25">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:25_1_2_1_1_1_3_1_2_1_1"/><b>then </b><i><font color="Green" title="E53">A40</font></i>: 
0 <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="c25">k</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E3:25_1_2_1_1_1_3_1_2_1_1"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">E</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c25">k</font> <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="c19">D</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E1:25_1_2_1_1_1_3"><i><font color="Green" title="E49">A34</font></i></a>, <a class="txt" href="pencil_3.html#E3:25_1_2_1_1_1_3_1_2"><i><font color="Green" title="E51">A38</font></i></a></i>;<br/><a NAME="E4:25_1_2_1_1_1_3_1_2_1_1"/><b>then </b>
<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> <font color="Maroon" title="c17">E</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c25">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&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="c19">D</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="E5:25_1_2_1_1_1_3_1_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">E</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&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="c17">E</font></span>)</span> <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> <font color="Maroon" title="c18">F</font></span>)</span>
 <b>by </b><i><a class="txt" href="pencil_3.html#E7:25_1_2_1_1_1_3_1_2"><i><font color="Green" title="E52">A39</font></i></a></i>;<br/><a NAME="E6:25_1_2_1_1_1_3_1_2_1_1"/><b>then </b><i><font color="Green" title="E54">A41</font></i>: 
<font color="Maroon" title="c25">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 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="c18">F</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="E7:25_1_2_1_1_1_3_1_2_1_1"/>
<font color="Maroon" title="c25">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c25">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E8:25_1_2_1_1_1_3_1_2_1_1"/><b>then </b>
<font color="Maroon" title="c25">k</font> <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="c18">F</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E6:25_1_2_1_1_1_3_1_2_1_1"><i><font color="Green" title="E54">A41</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E9:25_1_2_1_1_1_3_1_2_1_1"/><b>then </b><i><font color="Green" title="E55">A42</font></i>: 
<font color="Maroon" title="c21">Di</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="pencil_3.html#E2:25_1_2_1_1_1_3"><i><font color="Green" title="E50">A35</font></i></a>, <a class="txt" href="pencil_3.html#E3:25_1_2_1_1_1_3_1_2"><i><font color="Green" title="E51">A38</font></i></a>, <a class="txt" href="pencil_3.html#E2:25_1_2_1_1_1_3_1_2_1_1"><i><font color="Green" title="E53">A40</font></i></a>, <a class="ref" href="graph_2.html#T15" title="GRAPH_2:th.15">GRAPH_2:15</a></i>;<br/><a NAME="E10:25_1_2_1_1_1_3_1_2_1_1"/><i><font color="Green" title="E56">A43</font></i>: 
<font color="Maroon" title="c22">Di1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <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> <font color="Maroon" title="c17">E</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c25">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="pencil_3.html#E2:25_1_2_1_1_1_3"><i><font color="Green" title="E50">A35</font></i></a>, <a class="txt" href="pencil_3.html#E3:25_1_2_1_1_1_3_1_2"><i><font color="Green" title="E51">A38</font></i></a></i>;<br/><a NAME="E11:25_1_2_1_1_1_3_1_2_1_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c25">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/><a NAME="E12:25_1_2_1_1_1_3_1_2_1_1"/><b>then </b><i><font color="Green" title="E57">A44</font></i>: 
<font color="Maroon" title="c22">Di1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c25">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="pencil_3.html#E6:25_1_2_1_1_1_3_1_2_1_1"><i><font color="Green" title="E54">A41</font></i></a>, <a class="txt" href="pencil_3.html#E10:25_1_2_1_1_1_3_1_2_1_1"><i><font color="Green" title="E56">A43</font></i></a>, <a class="ref" href="graph_2.html#T15" title="GRAPH_2:th.15">GRAPH_2:15</a></i>;<br/><a NAME="E13:25_1_2_1_1_1_3_1_2_1_1"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c25">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c25">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 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="c18">F</font> )
 <b>by </b><i><a class="txt" href="pencil_3.html#E6:25_1_2_1_1_1_3_1_2_1_1"><i><font color="Green" title="E54">A41</font></i></a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/><b>hence </b><a NAME="E14:25_1_2_1_1_1_3_1_2_1_1"/>
( <font color="Maroon" title="c21">Di</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c22">Di1</font> &amp; <font color="Maroon" title="c21">Di</font> <a href="pencil_3.html#R1" title="PENCIL_3:pred.1">'||'</a> <font color="Maroon" title="c22">Di1</font> )
 <b>by </b><i><a class="txt" href="pencil_3.html#E9:25_1_2_1_1_1"><i><font color="Green" title="E46">A26</font></i></a>, <a class="txt" href="pencil_3.html#E9:25_1_2_1_1_1_3_1_2_1_1"><i><font color="Green" title="E55">A42</font></i></a>, <a class="txt" href="pencil_3.html#E12:25_1_2_1_1_1_3_1_2_1_1"><i><font color="Green" title="E57">A44</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_2_1_1_1_3_1_2_1_2"><b>suppose </b></a><a NAME="E1:25_1_2_1_1_1_3_1_2_1_2"/>
<font color="Maroon" title="c25">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:25_1_2_1_1_1_3_1_2_1_2"/><b>then </b><i><font color="Green" title="E53">A45</font></i>: 
<font color="Maroon" title="c23">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">E</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E3:25_1_2_1_1_1_3_1_2"><i><font color="Green" title="E51">A38</font></i></a></i>;<br/><a NAME="E3:25_1_2_1_1_1_3_1_2_1_2"/><b>then </b><i><font color="Green" title="E54">A46</font></i>: 
<font color="Maroon" title="c21">Di</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
 <b>by </b><i><a class="txt" href="pencil_3.html#E6:25_1_2_1_1_1"><i><font color="Green" title="E44">A24</font></i></a>, <a class="txt" href="pencil_3.html#E9:25_1_2_1_1_1"><i><font color="Green" title="E46">A26</font></i></a>, <a class="txt" href="pencil_3.html#E1:25_1_2_1_1_1_3"><i><font color="Green" title="E49">A34</font></i></a>, <a class="txt" href="pencil_3.html#E2:25_1_2_1_1_1_3"><i><font color="Green" title="E50">A35</font></i></a>, <a class="ref" href="graph_2.html#T14" title="GRAPH_2:th.14">GRAPH_2:14</a></i>;<br/><a NAME="E4:25_1_2_1_1_1_3_1_2_1_2"/>
<font color="Maroon" title="c22">Di1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="pencil_3.html#E17:25_1_2_1_1_1"><i><font color="Green" title="E48">A29</font></i></a>, <a class="txt" href="pencil_3.html#E2:25_1_2_1_1_1_3"><i><font color="Green" title="E50">A35</font></i></a>, <a class="txt" href="pencil_3.html#E2:25_1_2_1_1_1_3_1_2_1_2"><i><font color="Green" title="E53">A45</font></i></a>, <a class="ref" href="graph_2.html#T15" title="GRAPH_2:th.15">GRAPH_2:15</a></i>;<br/><b>hence </b><a NAME="E5:25_1_2_1_1_1_3_1_2_1_2"/>
( <font color="Maroon" title="c21">Di</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c22">Di1</font> &amp; <font color="Maroon" title="c21">Di</font> <a href="pencil_3.html#R1" title="PENCIL_3:pred.1">'||'</a> <font color="Maroon" title="c22">Di1</font> )
 <b>by </b><i><a class="txt" href="pencil_3.html#E9:25_1_2_1_1_1"><i><font color="Green" title="E46">A26</font></i></a>, <a class="txt" href="pencil_3.html#E17:25_1_2_1_1_1"><i><font color="Green" title="E48">A29</font></i></a>, <a class="txt" href="pencil_3.html#E3:25_1_2_1_1_1_3_1_2_1_2"><i><font color="Green" title="E54">A46</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></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="25_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:25_1_2_1_1_2"/>
 <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c8">c1</font> <a href="subset_1.html#R2" title="SUBSET_1:pred.2">meets</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c16">c3</font>
 ;<br/><div class="add"><a NAME="E2:25_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E43">A47</font></i>: 
<font color="Maroon" title="c8">c1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c16">c3</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E1:25_1_2_1"><i><font color="Green" title="E31">A11</font></i></a>, <a class="txt" href="pencil_3.html#E33:25_1_2_1"><i><font color="Green" title="E41">A21</font></i></a>, <a class="txt" href="pencil_3.html#E35:25_1_2_1"><i><font color="Green" title="E42">A22</font></i></a>, <a class="ref" href="pencil_3.html#T11" target="_self" title="PENCIL_3:th.11">Th11</a></i>;<br/><a NAME="E3:25_1_2_1_1_2"/>
not <font color="Maroon" title="c13">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">c2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">j</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E8:25_1_2_1"><i><font color="Green" title="E33">A13</font></i></a>, <a class="txt" href="pencil_3.html#E13:25_1_2_1"><i><font color="Green" title="E36">A16</font></i></a>, <a class="txt" href="pencil_3.html#E21:25_1_2_1"><i><font color="Green" title="E37">A17</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E4:25_1_2_1_1_2"/>
 ex <font color="Olive" title="b1">D</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="pencil_1.html#K6" title="PENCIL_1:func.6">Segre_Product</a> <font color="Maroon" title="c2">A</font></span>)</span> st <br/>( <font color="Olive" title="b1">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c8">c1</font> &amp; <font color="Olive" title="b1">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">D</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c9">c2</font> &amp; ( for <font color="Olive" title="b2">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b2">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="Olive" title="b1">D</font> holds <br/><font color="Olive" title="b1">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">i</font> is    <a href="pencil_2.html#M1" title="PENCIL_2:mode.1">Segre-Coset</a> of <font color="Maroon" title="c2">A</font> ) &amp; ( for <font color="Olive" title="b2">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">i</font> &amp; <font color="Olive" title="b2">i</font> <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="Olive" title="b1">D</font> holds <br/>for <font color="Olive" title="b3">Di</font>, <font color="Olive" title="b4">Di1</font> being   <a href="pencil_2.html#M1" title="PENCIL_2:mode.1">Segre-Coset</a> of <font color="Maroon" title="c2">A</font>  st <font color="Olive" title="b3">Di</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">i</font> &amp; <font color="Olive" title="b4">Di1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> holds <br/>( <font color="Olive" title="b3">Di</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b4">Di1</font> &amp; <font color="Olive" title="b3">Di</font> <a href="pencil_3.html#R1" title="PENCIL_3:pred.1">'||'</a> <font color="Olive" title="b4">Di1</font> ) ) )
 <b>by </b><i><a class="txt" href="pencil_3.html#E1:25"><i><font color="Green" title="E24">A1</font></i></a>, <a class="txt" href="pencil_3.html#E1:25_1_2_1"><i><font color="Green" title="E31">A11</font></i></a>, <a class="txt" href="pencil_3.html#E2:25_1_2_1_1_2"><i><font color="Green" title="E43">A47</font></i></a>, <a class="ref" href="pencil_3.html#T22" target="_self" title="PENCIL_3:th.22">Th22</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E4:25_1"/>
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S2" title="NAT_1:sch.2">NAT_1:sch 2</a>(<a class="txt" href="pencil_3.html#E2:25_1"><i><font color="Green" title="E28">A5</font></i></a>, <a class="txt" href="pencil_3.html#E3:25_1"><i><font color="Green" title="E29">A9</font></i></a>);<br/></i>
<a NAME="E5:25_1"/><b>then </b>
<font color="Maroon">S<sub>1</sub></font>[ <a href="pencil_3.html#K1" title="PENCIL_3:func.1">diff</a> <font color="Maroon" title="c5">b1</font>,<font color="Maroon" title="c6">b2</font>]
 
;<br/>
<b>then consider </b><font color="Maroon" title="c7">D</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="pencil_1.html#K6" title="PENCIL_1:func.6">Segre_Product</a> <font color="Maroon" title="c2">A</font></span>)</span><b> such that </b><br/><a NAME="E7:25_1"/><i><font color="Green" title="E30">A48</font></i>: 
( <font color="Maroon" title="c7">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c5">b1</font> &amp; <font color="Maroon" title="c7">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">D</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c6">b2</font> &amp; ( for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">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="c7">D</font> holds <br/><font color="Maroon" title="c7">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> is    <a href="pencil_2.html#M1" title="PENCIL_2:mode.1">Segre-Coset</a> of <font color="Maroon" title="c2">A</font> ) &amp; ( for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <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="c7">D</font> holds <br/>for <font color="Olive" title="b2">Di</font>, <font color="Olive" title="b3">Di1</font> being   <a href="pencil_2.html#M1" title="PENCIL_2:mode.1">Segre-Coset</a> of <font color="Maroon" title="c2">A</font>  st <font color="Olive" title="b2">Di</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b3">Di1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> holds <br/>( <font color="Olive" title="b2">Di</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b3">Di1</font> &amp; <font color="Olive" title="b2">Di</font> <a href="pencil_3.html#R1" title="PENCIL_3:pred.1">'||'</a> <font color="Olive" title="b3">Di1</font> ) ) )
 <b>by </b><i><a class="txt" href="pencil_3.html#E2:25"><i><font color="Green" title="E25">A2</font></i></a>, <a class="txt" href="pencil_3.html#E3:25"><i><font color="Green" title="E26">A3</font></i></a>, <a class="txt" href="pencil_3.html#E1:25_1"><i><font color="Green" title="E27">A4</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c7">D</font>
;<br/>

<b>thus </b><a NAME="E8:25_1"/>
( <font color="Maroon" title="c7">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">B1</font> &amp; <font color="Maroon" title="c7">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">D</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">B2</font> )
 <b>by </b><i><a class="txt" href="pencil_3.html#E3:25"><i><font color="Green" title="E26">A3</font></i></a>, <a class="txt" href="pencil_3.html#E7:25_1"><i><font color="Green" title="E30">A48</font></i></a></i>;<br/>

<b>thus </b><a NAME="E9:25_1"/>
( ( for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">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="c7">D</font> holds <br/><font color="Maroon" title="c7">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> is    <a href="pencil_2.html#M1" title="PENCIL_2:mode.1">Segre-Coset</a> of <font color="Maroon" title="c2">A</font> ) &amp; ( for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <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="c7">D</font> holds <br/>for <font color="Olive" title="b2">Di</font>, <font color="Olive" title="b3">Di1</font> being   <a href="pencil_2.html#M1" title="PENCIL_2:mode.1">Segre-Coset</a> of <font color="Maroon" title="c2">A</font>  st <font color="Olive" title="b2">Di</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b3">Di1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> holds <br/>( <font color="Olive" title="b2">Di</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b3">Di1</font> &amp; <font color="Olive" title="b2">Di</font> <a href="pencil_3.html#R1" title="PENCIL_3:pred.1">'||'</a> <font color="Olive" title="b3">Di1</font> ) ) )
 <b>by </b><i><a class="txt" href="pencil_3.html#E7:25_1"><i><font color="Green" title="E30">A48</font></i></a></i>;<br/>


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

<b>given </b><font color="Maroon" title="c7">D</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="pencil_1.html#K6" title="PENCIL_1:func.6">Segre_Product</a> <font color="Maroon" title="c2">A</font></span>)</span><b> such that </b><a NAME="E5:25"/><i><font color="Green" title="E27">A49</font></i>: 
( <font color="Maroon" title="c7">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">B1</font> &amp; <font color="Maroon" title="c7">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">D</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">B2</font> &amp; ( for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">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="c7">D</font> holds <br/><font color="Maroon" title="c7">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> is    <a href="pencil_2.html#M1" title="PENCIL_2:mode.1">Segre-Coset</a> of <font color="Maroon" title="c2">A</font> ) &amp; ( for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <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="c7">D</font> holds <br/>for <font color="Olive" title="b2">Di</font>, <font color="Olive" title="b3">Di1</font> being   <a href="pencil_2.html#M1" title="PENCIL_2:mode.1">Segre-Coset</a> of <font color="Maroon" title="c2">A</font>  st <font color="Olive" title="b2">Di</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b3">Di1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> holds <br/>( <font color="Olive" title="b2">Di</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b3">Di1</font> &amp; <font color="Olive" title="b2">Di</font> <a href="pencil_3.html#R1" title="PENCIL_3:pred.1">'||'</a> <font color="Olive" title="b3">Di1</font> ) ) )
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <b>means </b>for <font color="Olive" title="b1">bb</font> being non <a href="pencil_1.html#V13" title="PENCIL_1:attr.13">trivial-yielding</a> <a href="pencil_1.html#V16" title="PENCIL_1:attr.16">Segre-like</a>  <a href="pboole.html#M4" title="PBOOLE:mode.4">ManySortedSubset</a> of  <a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">A</font>  st  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Olive" title="b1">bb</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 holds <br/> <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c5">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Olive" title="b1">bb</font>;<br/>
<a NAME="E6:25"/><i><font color="Green" title="E28">A50</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">bb</font> be  non <a href="pencil_1.html#V13" title="PENCIL_1:attr.13">trivial-yielding</a> <a href="pencil_1.html#V16" title="PENCIL_1:attr.16">Segre-like</a>  <a href="pboole.html#M4" title="PBOOLE:mode.4">ManySortedSubset</a> of  <a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">A</font>;<br/>

<b>assume </b><a NAME="E1:25_2"/><i><font color="Green" title="E29">A51</font></i>: 
 <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c8">bb</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 0
 ;<br/>

<a NAME="E2:25_2"/>
not 0 <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="c7">D</font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T26" title="FINSEQ_3:th.26">FINSEQ_3:26</a></i>;<br/>
<b>hence </b><a NAME="E3:25_2"/>
 <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c5">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c8">bb</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E1:25_2"><i><font color="Green" title="E29">A51</font></i></a>, <a class="ref" href="funct_1.html#D4" title="FUNCT_1:def.4">FUNCT_1:def 4</a></i>;<br/>


</div><b>end;</b></div>
<div><i><font color="Green" title="E29">A52</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="25_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>assume </b><a NAME="E1:25_3"/><i><font color="Green" title="E30">A53</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">k</font>]
 ;<br/><b>thus </b><a NAME="E2:25_3"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">bb</font> be  non <a href="pencil_1.html#V13" title="PENCIL_1:attr.13">trivial-yielding</a> <a href="pencil_1.html#V16" title="PENCIL_1:attr.16">Segre-like</a>  <a href="pboole.html#M4" title="PBOOLE:mode.4">ManySortedSubset</a> of  <a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">A</font>;<br/>

<b>assume </b><a NAME="E1:25_3_1"/><i><font color="Green" title="E31">A54</font></i>: 
 <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c9">bb</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>
 ;<br/>

<a NAME="E2:25_3_1"/><i><font color="Green" title="E32">A55</font></i>: 
<font color="Maroon" title="c8">k</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="c7">D</font>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E1:25_3_1"><i><font color="Green" title="E31">A54</font></i></a>, <a class="ref" href="funct_1.html#D4" title="FUNCT_1:def.4">FUNCT_1:def 4</a></i>;<br/>
<a NAME="E3:25_3_1"/><b>then </b><i><font color="Green" title="E33">A56</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 &amp; <font color="Maroon" title="c8">k</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="c7">D</font> )
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:25_3_1_1"/>
( <font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 or 0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">k</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_3_1_1_1"><b>suppose </b></a><a NAME="E1:25_3_1_1_1"/>
<font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:25_3_1_1_1"/><b>then </b>
 <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c9">bb</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c5">b1</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E3:25"><i><font color="Green" title="E26">A3</font></i></a>, <a class="txt" href="pencil_3.html#E5:25"><i><font color="Green" title="E27">A49</font></i></a>, <a class="txt" href="pencil_3.html#E1:25_3_1"><i><font color="Green" title="E31">A54</font></i></a></i>;<br/><b>hence </b><a NAME="E3:25_3_1_1_1"/>
 <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c5">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c9">bb</font>
 <b>by </b><i><a class="ref" href="pua2mss1.html#T2" title="PUA2MSS1:th.2">PUA2MSS1:2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_3_1_1_2"><b>suppose </b></a><a NAME="E1:25_3_1_1_2"/>
0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">k</font>
 ;<br/><div class="add"><a NAME="E2:25_3_1_1_2"/><b>then </b>
0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">k</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="E3:25_3_1_1_2"/><b>then </b><i><font color="Green" title="E34">A57</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">k</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E4:25_3_1_1_2"/>
<font color="Maroon" title="c8">k</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="c7">D</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E3:25_3_1"><i><font color="Green" title="E33">A56</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E5:25_3_1_1_2"/><b>then </b>
<font color="Maroon" title="c8">k</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="c7">D</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E3:25_3_1_1_2"><i><font color="Green" title="E34">A57</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c10">B0</font> = <font color="Maroon" title="c7">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">k</font> as    <a href="pencil_2.html#M1" title="PENCIL_2:mode.1">Segre-Coset</a> of <font color="Maroon" title="c2">A</font> <b>by </b><i><a class="txt" href="pencil_3.html#E5:25"><i><font color="Green" title="E27">A49</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c11">B3</font> = <font color="Maroon" title="c7">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> as    <a href="pencil_2.html#M1" title="PENCIL_2:mode.1">Segre-Coset</a> of <font color="Maroon" title="c2">A</font> <b>by </b><i><a class="txt" href="pencil_3.html#E5:25"><i><font color="Green" title="E27">A49</font></i></a>, <a class="txt" href="pencil_3.html#E2:25_3_1"><i><font color="Green" title="E32">A55</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c12">b0</font> being  non <a href="pencil_1.html#V13" title="PENCIL_1:attr.13">trivial-yielding</a> <a href="pencil_1.html#V16" title="PENCIL_1:attr.16">Segre-like</a>  <a href="pboole.html#M4" title="PBOOLE:mode.4">ManySortedSubset</a> of  <a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">A</font><b> such that </b><br/><a NAME="E9:25_3_1_1_2"/><i><font color="Green" title="E35">A58</font></i>: 
( <font color="Maroon" title="c10">B0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c12">b0</font> &amp; <font color="Maroon" title="c12">b0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c12">b0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="pencil_1.html#K2" title="PENCIL_1:func.2">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c12">b0</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="pencil_2.html#D2" title="PENCIL_2:def.2">PENCIL_2:def 2</a></i>;<br/><a NAME="E10:25_3_1_1_2"/><i><font color="Green" title="E36">A59</font></i>: 
 <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c5">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c12">b0</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E1:25_3"><i><font color="Green" title="E30">A53</font></i></a>, <a class="txt" href="pencil_3.html#E9:25_3_1_1_2"><i><font color="Green" title="E35">A58</font></i></a></i>;<br/><a NAME="E11:25_3_1_1_2"/>
<font color="Maroon" title="c8">k</font> <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="c7">D</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E3:25_3_1"><i><font color="Green" title="E33">A56</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E12:25_3_1_1_2"/><b>then </b>
( <font color="Maroon" title="c10">B0</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c11">B3</font> &amp; <font color="Maroon" title="c10">B0</font> <a href="pencil_3.html#R1" title="PENCIL_3:pred.1">'||'</a> <font color="Maroon" title="c11">B3</font> )
 <b>by </b><i><a class="txt" href="pencil_3.html#E5:25"><i><font color="Green" title="E27">A49</font></i></a>, <a class="txt" href="pencil_3.html#E3:25_3_1_1_2"><i><font color="Green" title="E34">A57</font></i></a></i>;<br/><a NAME="E13:25_3_1_1_2"/><b>then </b>
 <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c12">b0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c9">bb</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E1:25_3_1"><i><font color="Green" title="E31">A54</font></i></a>, <a class="txt" href="pencil_3.html#E9:25_3_1_1_2"><i><font color="Green" title="E35">A58</font></i></a>, <a class="ref" href="pencil_3.html#T21" target="_self" title="PENCIL_3:th.21">Th21</a></i>;<br/><b>hence </b><a NAME="E14:25_3_1_1_2"/>
 <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c5">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c9">bb</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E10:25_3_1_1_2"><i><font color="Green" title="E36">A59</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E8:25"/>
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S2" title="NAT_1:sch.2">NAT_1:sch 2</a>(<a class="txt" href="pencil_3.html#E6:25"><i><font color="Green" title="E28">A50</font></i></a>, <a class="txt" href="pencil_3.html#E7:25"><i><font color="Green" title="E29">A52</font></i></a>);<br/></i>
<b>hence </b><a NAME="E9:25"/>
 <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c5">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c6">b2</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E3:25"><i><font color="Green" title="E26">A3</font></i></a>, <a class="txt" href="pencil_3.html#E5:25"><i><font color="Green" title="E27">A49</font></i></a></i>;<br/>


</div>
