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

<b>let </b><font color="Maroon">I</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">A</font> be  <a href="pencil_1.html#V15">PLS-yielding</a>  <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">I</font>;<br/>



<b>assume </b><a NAME="E1:25"/><i><font color="Green">E34</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> holds  <font color="Maroon">A</font> <a href="pencil_1.html#K2">.</a> <font color="Olive">i</font> is <a href="pencil_1.html#V9">connected</a>
 ;<br/>

<b>let </b><font color="Maroon">B1</font> be    <a href="pencil_2.html#M1">Segre-Coset</a> of <font color="Maroon">A</font>, <font color="Maroon">B2</font> be    <a href="pencil_2.html#M1">Segre-Coset</a> of <font color="Maroon">A</font>;<br/>

<b>assume </b><a NAME="E2:25"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">B1</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">B2</font>
 ;<br/>

<b>let </b><font color="Maroon">b1</font> be  non <a href="pencil_1.html#V13">trivial-yielding</a> <a href="pencil_1.html#V16">Segre-like</a>  <a href="pboole.html#M4">ManySortedSubset</a> of  <a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">A</font>, <font color="Maroon">b2</font> be  non <a href="pencil_1.html#V13">trivial-yielding</a> <a href="pencil_1.html#V16">Segre-like</a>  <a href="pboole.html#M4">ManySortedSubset</a> of  <a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">A</font>;<br/>

<b>assume </b><a NAME="E3:25"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">B1</font> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">B2</font> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">b2</font> )
 ;<br/>

<b>thus </b><a NAME="E4:25"/>
(  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font> <a href="hidden.html#R1">=</a>  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">b2</font> implies  ex <font color="Olive">D</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="zfmisc_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="pencil_1.html#K6">Segre_Product</a> <font color="Maroon">A</font></span>)</span> st <br/>( <font color="Olive">D</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">B1</font> &amp; <font color="Olive">D</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Olive">D</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">B2</font> &amp; ( for <font color="Olive">i</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Olive">D</font> holds <br/><font color="Olive">D</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> is    <a href="pencil_2.html#M1">Segre-Coset</a> of <font color="Maroon">A</font> ) &amp; ( for <font color="Olive">i</font> being  <a href="nat_1.html#NM1">Nat</a>  st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Olive">D</font> holds <br/>for <font color="Olive">Di</font>, <font color="Olive">Di1</font> being   <a href="pencil_2.html#M1">Segre-Coset</a> of <font color="Maroon">A</font>  st <font color="Olive">Di</font> <a href="hidden.html#R1">=</a> <font color="Olive">D</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> &amp; <font color="Olive">Di1</font> <a href="hidden.html#R1">=</a> <font color="Olive">D</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">i</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> holds <br/>( <font color="Olive">Di</font> <a href="xboole_0.html#R1">misses</a> <font color="Olive">Di1</font> &amp; <font color="Olive">Di</font> <a href="pencil_3.html#R1" target="_self">'||'</a> <font color="Olive">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">E47</font></i>: 
 <a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font> <a href="hidden.html#R1">=</a>  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">b2</font>
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] <b>means </b>for <font color="Olive">c1</font>, <font color="Olive">c2</font> being non <a href="pencil_1.html#V13">trivial-yielding</a> <a href="pencil_1.html#V16">Segre-like</a>  <a href="pboole.html#M4">ManySortedSubset</a> of  <a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">A</font>  st  <a href="pencil_1.html#K3">indx</a> <font color="Olive">c1</font> <a href="hidden.html#R1">=</a>  <a href="pencil_1.html#K3">indx</a> <font color="Olive">c2</font> &amp;  <a href="card_3.html#K4">product</a> <font color="Olive">c1</font> is    <a href="pencil_2.html#M1">Segre-Coset</a> of <font color="Maroon">A</font> &amp;  <a href="card_3.html#K4">product</a> <font color="Olive">c2</font> is    <a href="pencil_2.html#M1">Segre-Coset</a> of <font color="Maroon">A</font> &amp;  <a href="card_3.html#K4">product</a> <font color="Olive">c1</font> <a href="subset_1.html#R1">misses</a>  <a href="card_3.html#K4">product</a> <font color="Olive">c2</font> &amp;  <a href="pencil_3.html#K1" target="_self">diff</a> <font color="Olive">c1</font>,<font color="Olive">c2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> holds <br/> ex <font color="Olive">D</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="zfmisc_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="pencil_1.html#K6">Segre_Product</a> <font color="Maroon">A</font></span>)</span> st <br/>( <font color="Olive">D</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <font color="Olive">c1</font> &amp; <font color="Olive">D</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Olive">D</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <font color="Olive">c2</font> &amp; ( for <font color="Olive">i</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Olive">D</font> holds <br/><font color="Olive">D</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> is    <a href="pencil_2.html#M1">Segre-Coset</a> of <font color="Maroon">A</font> ) &amp; ( for <font color="Olive">i</font> being  <a href="nat_1.html#NM1">Nat</a>  st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Olive">D</font> holds <br/>for <font color="Olive">Di</font>, <font color="Olive">Di1</font> being   <a href="pencil_2.html#M1">Segre-Coset</a> of <font color="Maroon">A</font>  st <font color="Olive">Di</font> <a href="hidden.html#R1">=</a> <font color="Olive">D</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> &amp; <font color="Olive">Di1</font> <a href="hidden.html#R1">=</a> <font color="Olive">D</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">i</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> holds <br/>( <font color="Olive">Di</font> <a href="xboole_0.html#R1">misses</a> <font color="Olive">Di1</font> &amp; <font color="Olive">Di</font> <a href="pencil_3.html#R1" target="_self">'||'</a> <font color="Olive">Di1</font> ) ) );<br/>
<a NAME="E2:25_1"/><i><font color="Green">E48</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">c1</font> be  non <a href="pencil_1.html#V13">trivial-yielding</a> <a href="pencil_1.html#V16">Segre-like</a>  <a href="pboole.html#M4">ManySortedSubset</a> of  <a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">A</font>, <font color="Maroon">c2</font> be  non <a href="pencil_1.html#V13">trivial-yielding</a> <a href="pencil_1.html#V16">Segre-like</a>  <a href="pboole.html#M4">ManySortedSubset</a> of  <a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">A</font>;<br/>

<b>assume </b><a NAME="E1:25_1_1"/><i><font color="Green">E49</font></i>: 
(  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">c1</font> <a href="hidden.html#R1">=</a>  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">c2</font> &amp;  <a href="card_3.html#K4">product</a> <font color="Maroon">c1</font> is    <a href="pencil_2.html#M1">Segre-Coset</a> of <font color="Maroon">A</font> &amp;  <a href="card_3.html#K4">product</a> <font color="Maroon">c2</font> is    <a href="pencil_2.html#M1">Segre-Coset</a> of <font color="Maroon">A</font> &amp;  <a href="card_3.html#K4">product</a> <font color="Maroon">c1</font> <a href="subset_1.html#R1">misses</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">c2</font> &amp;  <a href="pencil_3.html#K1" target="_self">diff</a> <font color="Maroon">c1</font>,<font color="Maroon">c2</font> <a href="hidden.html#R1">=</a> 0 )
 ;<br/>

<a NAME="E2:25_1_1"/><b>then </b>
 <a href="card_1.html#K1">Card</a> <span class="p1">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> : <font color="Maroon">c1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> </span>}</span>  <a href="hidden.html#R1">=</a> 0
 
;<br/>
<a NAME="E3:25_1_1"/><b>then </b><i><font color="Green">E51</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> : <font color="Maroon">c1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> </span>}</span>  <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="card_2.html#T59">CARD_2:59</a></i>;<br/>
<i><font color="Green">E53</font></i>:  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c1</font> = 
<font color="Maroon">I</font>
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>
<br/>.= 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c2</font>
<b>by </b><i><a class="ref" href="pboole.html#D3">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">a</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:25_1_1_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c1</font>
 ;<br/><b>then reconsider </b><font color="Maroon">j</font> = <font color="Maroon">a</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><b>assume </b><a NAME="E3:25_1_1_2"/>
<font color="Maroon">c1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font>
 ;<br/><a NAME="E4:25_1_1_2"/><b>then </b>
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> : <font color="Maroon">c1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> </span>}</span> 
 ;<br/><b>hence </b><a NAME="E5:25_1_1_2"/>
contradiction
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<a NAME="E6:25_1_1"/><b>then </b>
<font color="Maroon">c1</font> <a href="pboole.html#R6">=</a> <font color="Maroon">c2</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>
<b>hence </b><a NAME="E7:25_1_1"/>
 ex <font color="Olive">D</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="zfmisc_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="pencil_1.html#K6">Segre_Product</a> <font color="Maroon">A</font></span>)</span> st <br/>( <font color="Olive">D</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">c1</font> &amp; <font color="Olive">D</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Olive">D</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">c2</font> &amp; ( for <font color="Olive">i</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Olive">D</font> holds <br/><font color="Olive">D</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> is    <a href="pencil_2.html#M1">Segre-Coset</a> of <font color="Maroon">A</font> ) &amp; ( for <font color="Olive">i</font> being  <a href="nat_1.html#NM1">Nat</a>  st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Olive">D</font> holds <br/>for <font color="Olive">Di</font>, <font color="Olive">Di1</font> being   <a href="pencil_2.html#M1">Segre-Coset</a> of <font color="Maroon">A</font>  st <font color="Olive">Di</font> <a href="hidden.html#R1">=</a> <font color="Olive">D</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> &amp; <font color="Olive">Di1</font> <a href="hidden.html#R1">=</a> <font color="Olive">D</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">i</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> holds <br/>( <font color="Olive">Di</font> <a href="xboole_0.html#R1">misses</a> <font color="Olive">Di1</font> &amp; <font color="Olive">Di</font> <a href="pencil_3.html#R1" target="_self">'||'</a> <font color="Olive">Di1</font> ) ) )
 <b>by </b><i><a class="ref" href="pencil_3.html#T1" target="_self">Th1</a></i>;<br/>


</div><b>end;</b></div>
<div><i><font color="Green">E54</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">k</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:25_1_2"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>]
 ;<br/><b>thus </b><a NAME="E2:25_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</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">c1</font> be  non <a href="pencil_1.html#V13">trivial-yielding</a> <a href="pencil_1.html#V16">Segre-like</a>  <a href="pboole.html#M4">ManySortedSubset</a> of  <a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">A</font>, <font color="Maroon">c2</font> be  non <a href="pencil_1.html#V13">trivial-yielding</a> <a href="pencil_1.html#V16">Segre-like</a>  <a href="pboole.html#M4">ManySortedSubset</a> of  <a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">A</font>;<br/>

<b>assume </b><a NAME="E1:25_1_2_1"/><i><font color="Green">E70</font></i>: 
(  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">c1</font> <a href="hidden.html#R1">=</a>  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">c2</font> &amp;  <a href="card_3.html#K4">product</a> <font color="Maroon">c1</font> is    <a href="pencil_2.html#M1">Segre-Coset</a> of <font color="Maroon">A</font> &amp;  <a href="card_3.html#K4">product</a> <font color="Maroon">c2</font> is    <a href="pencil_2.html#M1">Segre-Coset</a> of <font color="Maroon">A</font> &amp;  <a href="card_3.html#K4">product</a> <font color="Maroon">c1</font> <a href="subset_1.html#R1">misses</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">c2</font> &amp;  <a href="pencil_3.html#K1" target="_self">diff</a> <font color="Maroon">c1</font>,<font color="Maroon">c2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1 )
 ;<br/>

<a NAME="E2:25_1_2_1"/>
 <a href="card_1.html#K1">Card</a> <span class="p1">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> : <font color="Maroon">c1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> </span>}</span>  <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1
 
<b>by </b><i/>;<br/>
<a NAME="E3:25_1_2_1"/><b>then </b>
 <a href="card_1.html#K1">Card</a> <span class="p1">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> : <font color="Maroon">c1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> </span>}</span>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
;<br/>
<a NAME="E4:25_1_2_1"/><b>then </b>
<span class="p1">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> : <font color="Maroon">c1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> </span>}</span>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="card_2.html#T19">CARD_2:19</a></i>;<br/>
<b>then consider </b><font color="Maroon">j0</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:25_1_2_1"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">j0</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">i</font> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> : <font color="Maroon">c1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon">j</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font><b> such that </b><br/><a NAME="E8:25_1_2_1"/><i><font color="Green">E74</font></i>: 
( <font color="Maroon">j0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">c1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> )
 <b>by </b><i/>;<br/>
<a NAME="E9:25_1_2_1"/><i><font color="Green">E75</font></i>: 
( <font color="Maroon">c1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">c1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pencil_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">c1</font></span>)</span></span>)</span> &amp; <font color="Maroon">c2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">c1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pencil_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">c1</font></span>)</span></span>)</span> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E10:25_1_2_1"/><b>then </b><i><font color="Green">E102</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">c1</font>
 
<b>by </b><i/>;<br/>
<a NAME="E11:25_1_2_1"/>
( not <font color="Maroon">c1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Maroon">c1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> is <a href="realset1.html#V1">trivial</a> )
 
<b>by </b><i><a class="txt" href="pencil_3.html#E2:25"><i><font color="Green">E35</font></i></a>, <a class="ref" href="pencil_1.html#T12">PENCIL_1:12</a></i>;<br/>
<b>then consider </b><font color="Maroon">p</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E13:25_1_2_1"/><i><font color="Green">E103</font></i>: 
<font color="Maroon">c1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">p</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="realset1.html#D4">REALSET1:def 4</a></i>;<br/>
<a NAME="E14:25_1_2_1"/>
<font color="Maroon">c1</font> <a href="pboole.html#R2">c=</a>  <a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D23">PBOOLE:def 23</a></i>;<br/>
<a NAME="E15:25_1_2_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">p</font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 
<b>by </b><i>, <a class="ref" href="pboole.html#D5">PBOOLE:def 5</a></i>;<br/>
<a NAME="E16:25_1_2_1"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E17:25_1_2_1"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pencil_1.html#K2">.</a> <font color="Maroon">j</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">p</font> = <font color="Maroon">p</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pencil_1.html#K2">.</a> <font color="Maroon">j</font></span>)</span> ;<br/>
<a NAME="E19:25_1_2_1"/>
( not <font color="Maroon">c2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Maroon">c2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> is <a href="realset1.html#V1">trivial</a> )
 
<b>by </b><i>, <a class="txt" href="pencil_3.html#E2:25"><i><font color="Green">E35</font></i></a>, <a class="ref" href="pencil_1.html#T12">PENCIL_1:12</a></i>;<br/>
<b>then consider </b><font color="Maroon">r</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E21:25_1_2_1"/><i><font color="Green">E104</font></i>: 
<font color="Maroon">c2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">r</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="realset1.html#D4">REALSET1:def 4</a></i>;<br/>
<a NAME="E22:25_1_2_1"/>
<font color="Maroon">c2</font> <a href="pboole.html#R2">c=</a>  <a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D23">PBOOLE:def 23</a></i>;<br/>
<a NAME="E23:25_1_2_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">r</font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E3:25"><i><font color="Green">E37</font></i></a>, <a class="ref" href="pboole.html#D5">PBOOLE:def 5</a></i>;<br/>
<a NAME="E24:25_1_2_1"/><b>then </b>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E25:25_1_2_1"/><b>then </b>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pencil_1.html#K2">.</a> <font color="Maroon">j</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">r</font> = <font color="Maroon">r</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pencil_1.html#K2">.</a> <font color="Maroon">j</font></span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon">c3</font> = <font color="Maroon">c2</font> <a href="polynom1.html#K2">+*</a> <font color="Maroon">j</font>,<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span> as  non <a href="pencil_1.html#V13">trivial-yielding</a> <a href="pencil_1.html#V16">Segre-like</a>  <a href="pboole.html#M4">ManySortedSubset</a> of  <a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">A</font> <b>by </b><i>, <a class="txt" href="pencil_3.html#E2:25"><i><font color="Green">E35</font></i></a>, </i>;<br/>
<a NAME="E28:25_1_2_1"/><i><font color="Green">E105</font></i>: 
 <a href="card_3.html#K4">product</a> <font color="Maroon">c3</font> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="pencil_1.html#K6">Segre_Product</a> <font color="Maroon">A</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E29:25_1_2_1"/><i><font color="Green">E106</font></i>: 
not <font color="Maroon">A</font> <a href="pencil_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">c1</font></span>)</span> is <a href="realset2.html#V3">trivial</a>
 
<b>by </b><i/>;<br/>
<a NAME="E30:25_1_2_1"/><i><font color="Green">E107</font></i>: 
<font color="Maroon">c3</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">c1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pencil_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">c1</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E1:25"><i><font color="Green">E34</font></i></a>, <a class="txt" href="pencil_3.html#E2:25"><i><font color="Green">E35</font></i></a>, <a class="ref" href="funct_7.html#T34">FUNCT_7:34</a></i>;<br/>
<a NAME="E31:25_1_2_1"/><b>then </b>
<font color="Maroon">c3</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">c1</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pencil_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">c1</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D3">PRE_TOPC:def 3</a></i>;<br/>
<a NAME="E32:25_1_2_1"/><b>then </b>
not <font color="Maroon">c3</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">c1</font></span>)</span> is <a href="realset1.html#V1">trivial</a>
 
<b>by </b><i>, <a class="ref" href="realset2.html#D5">REALSET2:def 5</a></i>;<br/>
<a NAME="E33:25_1_2_1"/><b>then </b><i><font color="Green">E108</font></i>: 
 <a href="pencil_1.html#K3">indx</a> <font color="Maroon">c3</font> <a href="hidden.html#R1">=</a>  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">c1</font>
 
<b>by </b><i><a class="ref" href="pencil_1.html#D21">PENCIL_1:def 21</a></i>;<br/>
<a NAME="E34:25_1_2_1"/><b>then </b>
<font color="Maroon">c3</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">c3</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pencil_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">c3</font></span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E35:25_1_2_1"/><b>then </b><i><font color="Green">E109</font></i>: 
 <a href="card_3.html#K4">product</a> <font color="Maroon">c3</font> is    <a href="pencil_2.html#M1">Segre-Coset</a> of <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="pencil_3.html#T2" target="_self">Th2</a>, <a class="ref" href="pencil_2.html#D2">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">product</a> <font color="Maroon">c1</font> <a href="subset_1.html#R1">misses</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">c3</font> or  <a href="card_3.html#K4">product</a> <font color="Maroon">c1</font> <a href="subset_1.html#R2">meets</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">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">E110</font></i>: 
 <a href="card_3.html#K4">product</a> <font color="Maroon">c1</font> <a href="subset_1.html#R1">misses</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">c3</font>
 ;<br/><div class="add"><a NAME="E2:25_1_2_1_1_1"/>
 <a href="pencil_3.html#K1" target="_self">diff</a> <font color="Maroon">c1</font>,<font color="Maroon">c2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pencil_3.html#K1" target="_self">diff</a> <font color="Maroon">c1</font>,<font color="Maroon">c3</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1
 <b>by </b><i>, , </i>;<br/><a NAME="E3:25_1_2_1_1_1"/><b>then </b>
<font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pencil_3.html#K1" target="_self">diff</a> <font color="Maroon">c1</font>,<font color="Maroon">c3</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i/>;<br/><a NAME="E4:25_1_2_1_1_1"/><b>then </b>
 <a href="pencil_3.html#K1" target="_self">diff</a> <font color="Maroon">c1</font>,<font color="Maroon">c3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 ;<br/><b>then consider </b><font color="Maroon">E</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="zfmisc_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="pencil_1.html#K6">Segre_Product</a> <font color="Maroon">A</font></span>)</span><b> such that </b><br/><a NAME="E6:25_1_2_1_1_1"/><i><font color="Green">E111</font></i>: 
( <font color="Maroon">E</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">c1</font> &amp; <font color="Maroon">E</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">E</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">c3</font> &amp; ( for <font color="Olive">i</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">E</font> holds <br/><font color="Maroon">E</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> is    <a href="pencil_2.html#M1">Segre-Coset</a> of <font color="Maroon">A</font> ) &amp; ( for <font color="Olive">i</font> being  <a href="nat_1.html#NM1">Nat</a>  st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">E</font> holds <br/>for <font color="Olive">Ei</font>, <font color="Olive">Ei1</font> being   <a href="pencil_2.html#M1">Segre-Coset</a> of <font color="Maroon">A</font>  st <font color="Olive">Ei</font> <a href="hidden.html#R1">=</a> <font color="Maroon">E</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> &amp; <font color="Olive">Ei1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">E</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">i</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> holds <br/>( <font color="Olive">Ei</font> <a href="xboole_0.html#R1">misses</a> <font color="Olive">Ei1</font> &amp; <font color="Olive">Ei</font> <a href="pencil_3.html#R1" target="_self">'||'</a> <font color="Olive">Ei1</font> ) ) )
 <b>by </b><i>, , , <a class="ref" href="pencil_3.html#T3" target="_self">Th3</a>, </i>;<br/><a NAME="E7:25_1_2_1_1_1"/><i><font color="Green">E112</font></i>: 
not <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">c2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 <b>by </b><i>, , <a class="txt" href="pencil_3.html#E3:25"><i><font color="Green">E37</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>then consider </b><font color="Maroon">F</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="zfmisc_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="pencil_1.html#K6">Segre_Product</a> <font color="Maroon">A</font></span>)</span><b> such that </b><br/><a NAME="E9:25_1_2_1_1_1"/><i><font color="Green">E113</font></i>: 
( <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">c3</font> &amp; <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">c2</font> &amp; ( for <font color="Olive">i</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font> holds <br/><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> is    <a href="pencil_2.html#M1">Segre-Coset</a> of <font color="Maroon">A</font> ) &amp; ( for <font color="Olive">i</font> being  <a href="nat_1.html#NM1">Nat</a>  st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> holds <br/>for <font color="Olive">Di</font>, <font color="Olive">Di1</font> being   <a href="pencil_2.html#M1">Segre-Coset</a> of <font color="Maroon">A</font>  st <font color="Olive">Di</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> &amp; <font color="Olive">Di1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">i</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> holds <br/>( <font color="Olive">Di</font> <a href="xboole_0.html#R1">misses</a> <font color="Olive">Di1</font> &amp; <font color="Olive">Di</font> <a href="pencil_3.html#R1" target="_self">'||'</a> <font color="Olive">Di1</font> ) ) )
 <b>by </b><i>, , <a class="ref" href="pencil_3.html#T3" target="_self">Th3</a>, </i>;<br/><b>reconsider </b><font color="Maroon">D</font> = <font color="Maroon">E</font> <a href="graph_2.html#K4">^'</a> <font color="Maroon">F</font> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="zfmisc_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="pencil_1.html#K6">Segre_Product</a> <font color="Maroon">A</font></span>)</span> ;<br/><b>take </b>
<font color="Maroon">D</font>
;<br/><a NAME="E11:25_1_2_1_1_1"/>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">E</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D4">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">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">E</font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>hence </b><a NAME="E13:25_1_2_1_1_1"/>
<font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">c1</font>
 <b>by </b><i>, <a class="ref" href="graph_2.html#T14">GRAPH_2:14</a></i>;<br/><a NAME="E14:25_1_2_1_1_1"/>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D4">FUNCT_1:def 4</a></i>;<br/><a NAME="E15:25_1_2_1_1_1"/><b>then </b><i><font color="Green">E140</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">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">len</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><a NAME="E2:25_1_2_1_1_1_1"/><b>then </b>
 <a href="card_3.html#K4">product</a> <font color="Maroon">c3</font> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">c2</font>
 <b>by </b><i/>;<br/><a NAME="E3:25_1_2_1_1_1_1"/><b>then </b><i><font color="Green">E141</font></i>: 
<font color="Maroon">c3</font> <a href="pboole.html#R6">=</a> <font color="Maroon">c2</font>
 <b>by </b><i><a class="ref" href="pua2mss1.html#T2">PUA2MSS1:2</a></i>;<br/><a NAME="E4:25_1_2_1_1_1_1"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font>
 <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><a NAME="E5:25_1_2_1_1_1_1"/><b>then </b>
<font color="Maroon">c3</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="funct_7.html#T33">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#E2:25_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="tarski.html#D1">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">E143</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 <b>by </b><i><a class="txt" href="pencil_3.html#E1:25_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><b>hence </b><a NAME="E18:25_1_2_1_1_1"/>
<font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">c2</font>
 <b>by </b><i>, <a class="ref" href="graph_2.html#T16">GRAPH_2:16</a></i>;<br/><b>thus </b><a NAME="E19:25_1_2_1_1_1"/>
for <font color="Olive">i</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D</font> holds <br/><font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> is    <a href="pencil_2.html#M1">Segre-Coset</a> of <font color="Maroon">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">i</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:25_1_2_1_1_1_2"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D</font>
 ;<br/>

<a NAME="E2:25_1_2_1_1_1_2"/><b>then </b><i><font color="Green">E144</font></i>: 
<font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">D</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/>
<a NAME="E3:25_1_2_1_1_1_2"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">D</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">E</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">F</font></span>)</span>
 
<b>by </b><i><a class="ref" href="graph_2.html#T17">GRAPH_2:17</a></i>;<br/>
<a NAME="E4:25_1_2_1_1_1_2"/><b>then </b>
<font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">E</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">F</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E5:25_1_2_1_1_1_2"/><b>then </b><i><font color="Green">E145</font></i>: 
( <font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">E</font> or <font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">F</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">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">D</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">E</font> or <font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">F</font> )
 </span><b>by </b><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">D</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">E</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">i0</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:25_1_2_1_1_1_2_1_1"/><i><font color="Green">E146</font></i>: 
( <font color="Maroon">i0</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">E</font> &amp; <font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">E</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i0</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon">i0</font> = <font color="Maroon">i0</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/><a NAME="E5:25_1_2_1_1_1_2_1_1"/>
<font color="Maroon">i0</font> is   <a href="nat_1.html#NM1">Nat</a>
 ;<br/><b>hence </b><a NAME="E6:25_1_2_1_1_1_2_1_1"/>
<font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> is    <a href="pencil_2.html#M1">Segre-Coset</a> of <font color="Maroon">A</font>
 <b>by </b><i>, </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">D</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">F</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">i0</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:25_1_2_1_1_1_2_1_2"/><i><font color="Green">E147</font></i>: 
( <font color="Maroon">i0</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font> &amp; <font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i0</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon">i0</font> = <font color="Maroon">i0</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/><a NAME="E5:25_1_2_1_1_1_2_1_2"/>
<font color="Maroon">i0</font> is   <a href="nat_1.html#NM1">Nat</a>
 ;<br/><b>hence </b><a NAME="E6:25_1_2_1_1_1_2_1_2"/>
<font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> is    <a href="pencil_2.html#M1">Segre-Coset</a> of <font color="Maroon">A</font>
 <b>by </b><i>, </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">i</font> being  <a href="nat_1.html#NM1">Nat</a>  st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font> holds <br/>for <font color="Olive">Di</font>, <font color="Olive">Di1</font> being   <a href="pencil_2.html#M1">Segre-Coset</a> of <font color="Maroon">A</font>  st <font color="Olive">Di</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> &amp; <font color="Olive">Di1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">i</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> holds <br/>( <font color="Olive">Di</font> <a href="xboole_0.html#R1">misses</a> <font color="Olive">Di1</font> &amp; <font color="Olive">Di</font> <a href="pencil_3.html#R1" target="_self">'||'</a> <font color="Olive">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">i</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:25_1_2_1_1_1_3"/><i><font color="Green">E148</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font> )
 ;<br/>

<b>let </b><font color="Maroon">Di</font> be    <a href="pencil_2.html#M1">Segre-Coset</a> of <font color="Maroon">A</font>, <font color="Maroon">Di1</font> be    <a href="pencil_2.html#M1">Segre-Coset</a> of <font color="Maroon">A</font>;<br/>

<b>assume </b><a NAME="E2:25_1_2_1_1_1_3"/><i><font color="Green">E149</font></i>: 
( <font color="Maroon">Di</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">Di1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> )
 ;<br/>

<b>reconsider </b><font color="Maroon">i</font> = <font color="Maroon">i</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">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">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">E</font> or <font color="Maroon">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">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">E150</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">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">E151</font></i>: 
<font color="Maroon">Di</font> <a href="hidden.html#R1">=</a> <font color="Maroon">E</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, , <a class="ref" href="graph_2.html#T14">GRAPH_2:14</a></i>;<br/><a NAME="E3:25_1_2_1_1_1_3_1_1"/>
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">E</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</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">&lt;=</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">E</font> )
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E5:25_1_2_1_1_1_3_1_1"/><b>then </b>
<font color="Maroon">Di1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">E</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, <a class="ref" href="graph_2.html#T14">GRAPH_2:14</a></i>;<br/><b>hence </b><a NAME="E6:25_1_2_1_1_1_3_1_1"/>
( <font color="Maroon">Di</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">Di1</font> &amp; <font color="Maroon">Di</font> <a href="pencil_3.html#R1" target="_self">'||'</a> <font color="Maroon">Di1</font> )
 <b>by </b><i>, , , </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">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">E</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">k</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:25_1_2_1_1_1_3_1_2"/><i><font color="Green">E152</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">E</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/><b>reconsider </b><font color="Maroon">k</font> = <font color="Maroon">k</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><a NAME="E5:25_1_2_1_1_1_3_1_2"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="txt" href="pencil_3.html#E2:25_1"><i><font color="Green">E48</font></i></a></i>;<br/><a NAME="E6:25_1_2_1_1_1_3_1_2"/><b>then </b>
<font color="Maroon">F</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="finseq_1.html#T25">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">E153</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">E</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span>
 <b>by </b><i><a class="ref" href="graph_2.html#T13">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">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 or <font color="Maroon">k</font> <a href="hidden.html#R1">=</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">k</font> <a href="xxreal_0.html#NR4" target="_self">&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">E155</font></i>: 
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</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">len</a> <font color="Maroon">E</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font>
 <b>by </b><i>, </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">len</a> <font color="Maroon">E</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">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">len</a> <font color="Maroon">E</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">E</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E6:25_1_2_1_1_1_3_1_2_1_1"/><b>then </b><i><font color="Green">E156</font></i>: 
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E7:25_1_2_1_1_1_3_1_2_1_1"/>
<font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E8:25_1_2_1_1_1_3_1_2_1_1"/><b>then </b>
<font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E3:25_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="xxreal_0.html#T2">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">E157</font></i>: 
<font color="Maroon">Di</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, , , <a class="ref" href="graph_2.html#T15">GRAPH_2:15</a></i>;<br/><a NAME="E10:25_1_2_1_1_1_3_1_2_1_1"/><i><font color="Green">E158</font></i>: 
<font color="Maroon">Di1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">E</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E11:25_1_2_1_1_1_3_1_2_1_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E12:25_1_2_1_1_1_3_1_2_1_1"/><b>then </b><i><font color="Green">E160</font></i>: 
<font color="Maroon">Di1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="pencil_3.html#E3:25_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="pencil_3.html#T4" target="_self">Th4</a>, <a class="ref" href="graph_2.html#T15">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">&lt;=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> )
 <b>by </b><i><a class="txt" href="pencil_3.html#E3:25_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><b>hence </b><a NAME="E14:25_1_2_1_1_1_3_1_2_1_1"/>
( <font color="Maroon">Di</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">Di1</font> &amp; <font color="Maroon">Di</font> <a href="pencil_3.html#R1" target="_self">'||'</a> <font color="Maroon">Di1</font> )
 <b>by </b><i>, <a class="txt" href="pencil_3.html#E1:25_1_2"><i><font color="Green">E55</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">k</font> <a href="hidden.html#R1">=</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">E161</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">E</font>
 <b>by </b><i/>;<br/><a NAME="E3:25_1_2_1_1_1_3_1_2_1_2"/><b>then </b><i><font color="Green">E162</font></i>: 
<font color="Maroon">Di</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> 1
 <b>by </b><i>, , , , <a class="ref" href="graph_2.html#T14">GRAPH_2:14</a></i>;<br/><a NAME="E4:25_1_2_1_1_1_3_1_2_1_2"/>
<font color="Maroon">Di1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="pencil_3.html#E2:25_1"><i><font color="Green">E48</font></i></a>, , , <a class="ref" href="graph_2.html#T15">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">Di</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">Di1</font> &amp; <font color="Maroon">Di</font> <a href="pencil_3.html#R1" target="_self">'||'</a> <font color="Maroon">Di1</font> )
 <b>by </b><i>, <a class="txt" href="pencil_3.html#E2:25_1"><i><font color="Green">E48</font></i></a>, <a class="txt" href="pencil_3.html#E3:25_1"><i><font color="Green">E54</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">product</a> <font color="Maroon">c1</font> <a href="subset_1.html#R2">meets</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">c3</font>
 ;<br/><div class="add"><a NAME="E2:25_1_2_1_1_2"/><b>then </b><i><font color="Green">E163</font></i>: 
<font color="Maroon">c1</font> <a href="pboole.html#R6">=</a> <font color="Maroon">c3</font>
 <b>by </b><i>, , <a class="ref" href="pencil_3.html#T3" target="_self">Th3</a>, </i>;<br/><a NAME="E3:25_1_2_1_1_2"/>
not <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">c2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 <b>by </b><i>, , <a class="txt" href="pencil_3.html#E3:25"><i><font color="Green">E37</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E4:25_1_2_1_1_2"/>
 ex <font color="Olive">D</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="zfmisc_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="pencil_1.html#K6">Segre_Product</a> <font color="Maroon">A</font></span>)</span> st <br/>( <font color="Olive">D</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">c1</font> &amp; <font color="Olive">D</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Olive">D</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">c2</font> &amp; ( for <font color="Olive">i</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Olive">D</font> holds <br/><font color="Olive">D</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> is    <a href="pencil_2.html#M1">Segre-Coset</a> of <font color="Maroon">A</font> ) &amp; ( for <font color="Olive">i</font> being  <a href="nat_1.html#NM1">Nat</a>  st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Olive">D</font> holds <br/>for <font color="Olive">Di</font>, <font color="Olive">Di1</font> being   <a href="pencil_2.html#M1">Segre-Coset</a> of <font color="Maroon">A</font>  st <font color="Olive">Di</font> <a href="hidden.html#R1">=</a> <font color="Olive">D</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> &amp; <font color="Olive">Di1</font> <a href="hidden.html#R1">=</a> <font color="Olive">D</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">i</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> holds <br/>( <font color="Olive">Di</font> <a href="xboole_0.html#R1">misses</a> <font color="Olive">Di1</font> &amp; <font color="Olive">Di</font> <a href="pencil_3.html#R1" target="_self">'||'</a> <font color="Olive">Di1</font> ) ) )
 <b>by </b><i>, , , </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">n</font> being  <a href="nat_1.html#NM1">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S2">NAT_1:sch 2</a>(, <a class="ref" href="pencil_3.html#T1" target="_self">Th1</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" target="_self">diff</a> <font color="Maroon">b1</font>,<font color="Maroon">b2</font>]
 
;<br/>
<b>then consider </b><font color="Maroon">D</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="zfmisc_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="pencil_1.html#K6">Segre_Product</a> <font color="Maroon">A</font></span>)</span><b> such that </b><br/><a NAME="E7:25_1"/><i><font color="Green">E164</font></i>: 
( <font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">b1</font> &amp; <font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">b2</font> &amp; ( for <font color="Olive">i</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D</font> holds <br/><font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> is    <a href="pencil_2.html#M1">Segre-Coset</a> of <font color="Maroon">A</font> ) &amp; ( for <font color="Olive">i</font> being  <a href="nat_1.html#NM1">Nat</a>  st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font> holds <br/>for <font color="Olive">Di</font>, <font color="Olive">Di1</font> being   <a href="pencil_2.html#M1">Segre-Coset</a> of <font color="Maroon">A</font>  st <font color="Olive">Di</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> &amp; <font color="Olive">Di1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">i</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> holds <br/>( <font color="Olive">Di</font> <a href="xboole_0.html#R1">misses</a> <font color="Olive">Di1</font> &amp; <font color="Olive">Di</font> <a href="pencil_3.html#R1" target="_self">'||'</a> <font color="Olive">Di1</font> ) ) )
 <b>by </b><i>, , </i>;<br/>
<b>take </b>
<font color="Maroon">D</font>
;<br/>

<b>thus </b><a NAME="E8:25_1"/>
( <font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">B1</font> &amp; <font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">B2</font> )
 <b>by </b><i>, </i>;<br/>

<b>thus </b><a NAME="E9:25_1"/>
( ( for <font color="Olive">i</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D</font> holds <br/><font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> is    <a href="pencil_2.html#M1">Segre-Coset</a> of <font color="Maroon">A</font> ) &amp; ( for <font color="Olive">i</font> being  <a href="nat_1.html#NM1">Nat</a>  st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font> holds <br/>for <font color="Olive">Di</font>, <font color="Olive">Di1</font> being   <a href="pencil_2.html#M1">Segre-Coset</a> of <font color="Maroon">A</font>  st <font color="Olive">Di</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> &amp; <font color="Olive">Di1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">i</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> holds <br/>( <font color="Olive">Di</font> <a href="xboole_0.html#R1">misses</a> <font color="Olive">Di1</font> &amp; <font color="Olive">Di</font> <a href="pencil_3.html#R1" target="_self">'||'</a> <font color="Olive">Di1</font> ) ) )
 <b>by </b><i/>;<br/>


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

<b>given </b><font color="Maroon">D</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="zfmisc_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="pencil_1.html#K6">Segre_Product</a> <font color="Maroon">A</font></span>)</span><b> such that </b><a NAME="E5:25"/><i><font color="Green">E165</font></i>: 
( <font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">B1</font> &amp; <font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">B2</font> &amp; ( for <font color="Olive">i</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D</font> holds <br/><font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> is    <a href="pencil_2.html#M1">Segre-Coset</a> of <font color="Maroon">A</font> ) &amp; ( for <font color="Olive">i</font> being  <a href="nat_1.html#NM1">Nat</a>  st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font> holds <br/>for <font color="Olive">Di</font>, <font color="Olive">Di1</font> being   <a href="pencil_2.html#M1">Segre-Coset</a> of <font color="Maroon">A</font>  st <font color="Olive">Di</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> &amp; <font color="Olive">Di1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">i</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> holds <br/>( <font color="Olive">Di</font> <a href="xboole_0.html#R1">misses</a> <font color="Olive">Di1</font> &amp; <font color="Olive">Di</font> <a href="pencil_3.html#R1" target="_self">'||'</a> <font color="Olive">Di1</font> ) ) )
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] <b>means </b>for <font color="Olive">bb</font> being non <a href="pencil_1.html#V13">trivial-yielding</a> <a href="pencil_1.html#V16">Segre-like</a>  <a href="pboole.html#M4">ManySortedSubset</a> of  <a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">A</font>  st  <a href="card_3.html#K4">product</a> <font color="Olive">bb</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> holds <br/> <a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font> <a href="hidden.html#R1">=</a>  <a href="pencil_1.html#K3">indx</a> <font color="Olive">bb</font>;<br/>
<a NAME="E6:25"/><i><font color="Green">E166</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">bb</font> be  non <a href="pencil_1.html#V13">trivial-yielding</a> <a href="pencil_1.html#V16">Segre-like</a>  <a href="pboole.html#M4">ManySortedSubset</a> of  <a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">A</font>;<br/>

<b>assume </b><a NAME="E1:25_2"/><i><font color="Green">E167</font></i>: 
 <a href="card_3.html#K4">product</a> <font color="Maroon">bb</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> 0
 ;<br/>

<a NAME="E2:25_2"/>
not 0 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D</font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T26">FINSEQ_3:26</a></i>;<br/>
<b>hence </b><a NAME="E3:25_2"/>
 <a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font> <a href="hidden.html#R1">=</a>  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">bb</font>
 <b>by </b><i><a class="ref" href="pencil_3.html#T1" target="_self">Th1</a>, <a class="ref" href="funct_1.html#D4">FUNCT_1:def 4</a></i>;<br/>


</div><b>end;</b></div>
<div><i><font color="Green">E168</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">k</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:25_3"/><i><font color="Green">E169</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>]
 ;<br/><b>thus </b><a NAME="E2:25_3"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</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">bb</font> be  non <a href="pencil_1.html#V13">trivial-yielding</a> <a href="pencil_1.html#V16">Segre-like</a>  <a href="pboole.html#M4">ManySortedSubset</a> of  <a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">A</font>;<br/>

<b>assume </b><a NAME="E1:25_3_1"/><i><font color="Green">E170</font></i>: 
 <a href="card_3.html#K4">product</a> <font color="Maroon">bb</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>
 ;<br/>

<a NAME="E2:25_3_1"/><i><font color="Green">E171</font></i>: 
<font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#D4">FUNCT_1:def 4</a></i>;<br/>
<a NAME="E3:25_3_1"/><b>then </b><i><font color="Green">E172</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1 &amp; <font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font> )
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27">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">k</font> <a href="hidden.html#R1">=</a> 0 or 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">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">k</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:25_3_1_1_1"/><b>then </b>
 <a href="card_3.html#K4">product</a> <font color="Maroon">bb</font> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">b1</font>
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E3:25_3_1_1_1"/>
 <a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font> <a href="hidden.html#R1">=</a>  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">bb</font>
 <b>by </b><i><a class="ref" href="pua2mss1.html#T2">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" target="_self">&lt;</a> <font color="Maroon">k</font>
 ;<br/><div class="add"><a NAME="E2:25_3_1_1_2"/><b>then </b>
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E3:25_3_1_1_2"/><b>then </b><i><font color="Green">E173</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E4:25_3_1_1_2"/>
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E5:25_3_1_1_2"/><b>then </b>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">D</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E1:25"><i><font color="Green">E34</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>then reconsider </b><font color="Maroon">B0</font> = <font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> as    <a href="pencil_2.html#M1">Segre-Coset</a> of <font color="Maroon">A</font> <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">B3</font> = <font color="Maroon">D</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> as    <a href="pencil_2.html#M1">Segre-Coset</a> of <font color="Maroon">A</font> <b>by </b><i>, </i>;<br/><b>consider </b><font color="Maroon">b0</font> being  non <a href="pencil_1.html#V13">trivial-yielding</a> <a href="pencil_1.html#V16">Segre-like</a>  <a href="pboole.html#M4">ManySortedSubset</a> of  <a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">A</font><b> such that </b><br/><a NAME="E9:25_3_1_1_2"/><i><font color="Green">E175</font></i>: 
( <font color="Maroon">B0</font> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">b0</font> &amp; <font color="Maroon">b0</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b0</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="pencil_1.html#K2">.</a> <span class="p2">(<span class="default"><a href="pencil_1.html#K3">indx</a> <font color="Maroon">b0</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="pencil_2.html#D2">PENCIL_2:def 2</a></i>;<br/><a NAME="E10:25_3_1_1_2"/><i><font color="Green">E176</font></i>: 
 <a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font> <a href="hidden.html#R1">=</a>  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">b0</font>
 <b>by </b><i>, <a class="txt" href="pencil_3.html#E2:25"><i><font color="Green">E35</font></i></a></i>;<br/><a NAME="E11:25_3_1_1_2"/>
<font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">D</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E12:25_3_1_1_2"/><b>then </b>
( <font color="Maroon">B0</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">B3</font> &amp; <font color="Maroon">B0</font> <a href="pencil_3.html#R1" target="_self">'||'</a> <font color="Maroon">B3</font> )
 <b>by </b><i>, <a class="txt" href="pencil_3.html#E1:25"><i><font color="Green">E34</font></i></a></i>;<br/><a NAME="E13:25_3_1_1_2"/><b>then </b>
 <a href="pencil_1.html#K3">indx</a> <font color="Maroon">b0</font> <a href="hidden.html#R1">=</a>  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">bb</font>
 <b>by </b><i>, <a class="txt" href="pencil_3.html#E2:25"><i><font color="Green">E35</font></i></a>, </i>;<br/><b>hence </b><a NAME="E14:25_3_1_1_2"/>
 <a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font> <a href="hidden.html#R1">=</a>  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">bb</font>
 <b>by </b><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">n</font> being  <a href="nat_1.html#NM1">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S2">NAT_1:sch 2</a>(, <a class="ref" href="pencil_3.html#T1" target="_self">Th1</a>);<br/></i>
<b>hence </b><a NAME="E9:25"/>
 <a href="pencil_1.html#K3">indx</a> <font color="Maroon">b1</font> <a href="hidden.html#R1">=</a>  <a href="pencil_1.html#K3">indx</a> <font color="Maroon">b2</font>
 <b>by </b><i>, </i>;<br/>


</div>
