<?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="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>let </b><font color="Maroon" title="c3">X</font>, <font color="Maroon" title="c4">Y</font> be   <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>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:15"/><i><font color="Green" title="E13">A1</font></i>: 
( not <font color="Maroon" title="c3">X</font> is <a href="realset1.html#V1" title="REALSET1:attr.1">trivial</a> &amp; <font color="Maroon" title="c3">X</font> is <a href="pencil_1.html#V1" title="PENCIL_1:attr.1">closed_under_lines</a> &amp; <font color="Maroon" title="c3">X</font> is <a href="pencil_1.html#V2" title="PENCIL_1:attr.2">strong</a> )
 <b>and </b><br/><a NAME="E2:15"/><i><font color="Green" title="E14">A2</font></i>: 
( not <font color="Maroon" title="c4">Y</font> is <a href="realset1.html#V1" title="REALSET1:attr.1">trivial</a> &amp; <font color="Maroon" title="c4">Y</font> is <a href="pencil_1.html#V1" title="PENCIL_1:attr.1">closed_under_lines</a> &amp; <font color="Maroon" title="c4">Y</font> is <a href="pencil_1.html#V2" title="PENCIL_1:attr.2">strong</a> )
 <b>and </b><br/><a NAME="E3:15"/><i><font color="Green" title="E15">A3</font></i>: 
<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c4">Y</font> <a href="pencil_2.html#R1" title="PENCIL_2:pred.1">are_joinable</a> 
 ;<br/>

<b>let </b><font color="Maroon" title="c5">X1</font>, <font color="Maroon" title="c6">Y1</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="E4:15"/><i><font color="Green" title="E16">A4</font></i>: 
( <font color="Maroon" title="c3">X</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">X1</font> &amp; <font color="Maroon" title="c4">Y</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">Y1</font> )
 ;<br/>

<b>consider </b><font color="Maroon" title="c7">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="E6:15"/><i><font color="Green" title="E17">A5</font></i>: 
( <font color="Maroon" title="c3">X</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="c7">B0</font> &amp; ( for <font color="Olive" title="b1">C</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</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="c7">B0</font></span>)</span></span>)</span>  st <font color="Olive" title="b1">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">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="c7">B0</font></span>)</span> holds <br/>( <font color="Olive" title="b1">C</font> is <a href="pencil_1.html#V2" title="PENCIL_1:attr.2">strong</a> &amp; <font color="Olive" title="b1">C</font> is <a href="pencil_1.html#V1" title="PENCIL_1:attr.1">closed_under_lines</a> ) ) )
 <b>by </b><i><a class="txt" href="pencil_2.html#E1:15"><i><font color="Green" title="E13">A1</font></i></a>, <a class="ref" href="pencil_1.html#T29" title="PENCIL_1:th.29">PENCIL_1:29</a></i>;<br/>
<a NAME="E7:15"/><i><font color="Green" title="E18">A6</font></i>: 
<font color="Maroon" title="c7">B0</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c5">X1</font>
 
<b>by </b><i><a class="txt" href="pencil_2.html#E4:15"><i><font color="Green" title="E16">A4</font></i></a>, <a class="txt" href="pencil_2.html#E6:15"><i><font color="Green" title="E17">A5</font></i></a>, <a class="ref" href="pua2mss1.html#T2" title="PUA2MSS1:th.2">PUA2MSS1:2</a></i>;<br/>
<b>set </b><font color="Maroon" title="c8">B</font> =  <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>consider </b><font color="Maroon" title="c9">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="E9:15"/><i><font color="Green" title="E19">A7</font></i>: 
( <font color="Maroon" title="c3">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 &amp; <font color="Maroon" title="c4">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">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="c9">f</font></span>)</span> &amp; ( for <font color="Olive" title="b1">W</font> being  <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>  st <font color="Olive" title="b1">W</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="c9">f</font> holds <br/>( <font color="Olive" title="b1">W</font> is <a href="pencil_1.html#V1" title="PENCIL_1:attr.1">closed_under_lines</a> &amp; <font color="Olive" title="b1">W</font> is <a href="pencil_1.html#V2" title="PENCIL_1:attr.2">strong</a> ) ) &amp; ( for <font color="Olive" title="b1">i</font> being   <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>   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="c9">f</font> holds <br/>2 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> ) )
 <b>by </b><i><a class="txt" href="pencil_2.html#E3:15"><i><font color="Green" title="E15">A3</font></i></a>, <a class="ref" href="pencil_2.html#D3" target="_self" title="PENCIL_2:def.3">Def3</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <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>means </b>for <font color="Olive" title="b1">H</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 <font color="Maroon" title="c9">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="Olive" title="b1">H</font> holds <br/>(  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c5">X1</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">H</font> &amp; ( for <font color="Olive" title="b2">i</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">i</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="c5">X1</font> holds <br/><font color="Maroon" title="c5">X1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">i</font> ) );<br/>
<a NAME="E10:15"/><i><font color="Green" title="E20">A8</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[1]
 
<b>by </b><i><a class="txt" href="pencil_2.html#E4:15"><i><font color="Green" title="E16">A4</font></i></a>, <a class="txt" href="pencil_2.html#E9:15"><i><font color="Green" title="E19">A7</font></i></a>, <a class="ref" href="pua2mss1.html#T2" title="PUA2MSS1:th.2">PUA2MSS1:2</a></i>;<br/>
<a NAME="E11:15"/><i><font color="Green" title="E21">A9</font></i>: 
for <font color="Olive" title="b1">j</font> being   <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>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</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="c9">f</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">j</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">j</font> be    <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> ;<br/>

<b>assume </b><a NAME="E1:15_1"/><i><font color="Green" title="E22">A10</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">j</font> &amp; <font color="Maroon" title="c10">j</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="c9">f</font> )
 ;<br/>

<a NAME="E2:15_1"/><b>then </b><i><font color="Green" title="E23">A11</font></i>: 
<font color="Maroon" title="c10">j</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="c9">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/>
<a NAME="E3:15_1"/><i><font color="Green" title="E24">A12</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">j</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="E4:15_1"/>
<font color="Maroon" title="c10">j</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="c9">f</font>
 
<b>by </b><i><a class="txt" href="pencil_2.html#E1:15_1"><i><font color="Green" title="E22">A10</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E5:15_1"/><b>then </b>
<font color="Maroon" title="c10">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c9">f</font>
 
<b>by </b><i><a class="txt" href="pencil_2.html#E3:15_1"><i><font color="Green" title="E24">A12</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E6:15_1"/><b>then </b><i><font color="Green" title="E25">A13</font></i>: 
( <font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">j</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="c9">f</font> &amp; <font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <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="c9">f</font> )
 
<b>by </b><i><a class="txt" href="pencil_2.html#E2:15_1"><i><font color="Green" title="E23">A11</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c11">fj</font> = <font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">j</font>, <font color="Maroon" title="c12">fj1</font> = <font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> as   <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> ;<br/>
<a NAME="E8:15_1"/>
<font color="Maroon" title="c11">fj</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c12">fj1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">fj</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E9:15_1"/><b>then </b><i><font color="Green" title="E26">A14</font></i>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">fj</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c12">fj1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c11">fj</font>
 
<b>by </b><i><a class="ref" href="card_1.html#T27" title="CARD_1:th.27">CARD_1:27</a></i>;<br/>
<b>assume </b><a NAME="E10:15_1"/><i><font color="Green" title="E27">A15</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c10">j</font>]
 ;<br/>

<a NAME="E11:15_1"/>
2 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">fj</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c12">fj1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="pencil_2.html#E9:15"><i><font color="Green" title="E19">A7</font></i></a>, <a class="txt" href="pencil_2.html#E1:15_1"><i><font color="Green" title="E22">A10</font></i></a></i>;<br/>
<a NAME="E12:15_1"/><b>then </b>
2 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c11">fj</font>
 
<b>by </b><i><a class="txt" href="pencil_2.html#E9:15_1"><i><font color="Green" title="E26">A14</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E13:15_1"/><b>then </b>
( not <font color="Maroon" title="c11">fj</font> is <a href="realset1.html#V1" title="REALSET1:attr.1">trivial</a> &amp; <font color="Maroon" title="c11">fj</font> is <a href="pencil_1.html#V1" title="PENCIL_1:attr.1">closed_under_lines</a> &amp; <font color="Maroon" title="c11">fj</font> is <a href="pencil_1.html#V2" title="PENCIL_1:attr.2">strong</a> )
 
<b>by </b><i><a class="txt" href="pencil_2.html#E9:15"><i><font color="Green" title="E19">A7</font></i></a>, <a class="txt" href="pencil_2.html#E6:15_1"><i><font color="Green" title="E25">A13</font></i></a>, <a class="ref" href="pencil_1.html#T4" title="PENCIL_1:th.4">PENCIL_1:4</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c13">B1</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="E15:15_1"/><i><font color="Green" title="E28">A16</font></i>: 
( <font color="Maroon" title="c11">fj</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="c13">B1</font> &amp; ( for <font color="Olive" title="b1">C</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</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="c13">B1</font></span>)</span></span>)</span>  st <font color="Olive" title="b1">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">B1</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="c13">B1</font></span>)</span> holds <br/>( <font color="Olive" title="b1">C</font> is <a href="pencil_1.html#V2" title="PENCIL_1:attr.2">strong</a> &amp; <font color="Olive" title="b1">C</font> is <a href="pencil_1.html#V1" title="PENCIL_1:attr.1">closed_under_lines</a> ) ) )
 <b>by </b><i><a class="ref" href="pencil_1.html#T29" title="PENCIL_1:th.29">PENCIL_1:29</a></i>;<br/>
<a NAME="E16:15_1"/><i><font color="Green" title="E29">A17</font></i>: 
(  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c7">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="c13">B1</font> &amp; ( for <font color="Olive" title="b1">i</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">i</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="c7">B0</font> holds <br/><font color="Maroon" title="c7">B0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">B1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> ) )
 
<b>by </b><i><a class="txt" href="pencil_2.html#E7:15"><i><font color="Green" title="E18">A6</font></i></a>, <a class="txt" href="pencil_2.html#E10:15_1"><i><font color="Green" title="E27">A15</font></i></a>, <a class="txt" href="pencil_2.html#E15:15_1"><i><font color="Green" title="E28">A16</font></i></a></i>;<br/>
<a NAME="E17:15_1"/><i><font color="Green" title="E30">A18</font></i>: 
2 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">fj</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c12">fj1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="pencil_2.html#E9:15"><i><font color="Green" title="E19">A7</font></i></a>, <a class="txt" href="pencil_2.html#E1:15_1"><i><font color="Green" title="E22">A10</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c14">H</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="E1:15_1_1"/><i><font color="Green" title="E31">A19</font></i>: 
<font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c14">H</font>
 ;<br/><b>hence </b><a NAME="E2:15_1_1"/>
 <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c5">X1</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="c14">H</font>
 <b>by </b><i><a class="txt" href="pencil_2.html#E7:15"><i><font color="Green" title="E18">A6</font></i></a>, <a class="txt" href="pencil_2.html#E15:15_1"><i><font color="Green" title="E28">A16</font></i></a>, <a class="txt" href="pencil_2.html#E16:15_1"><i><font color="Green" title="E29">A17</font></i></a>, <a class="txt" href="pencil_2.html#E17:15_1"><i><font color="Green" title="E30">A18</font></i></a>, <a class="ref" href="pencil_1.html#T26" title="PENCIL_1:th.26">PENCIL_1:26</a></i>;<br/><b>thus </b><a NAME="E3:15_1_1"/>
for <font color="Olive" title="b1">i</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">i</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="c5">X1</font> holds <br/><font color="Maroon" title="c5">X1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c15">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:15_1_1_1"/><i><font color="Green" title="E32">A20</font></i>: 
<font color="Maroon" title="c15">i</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="c5">X1</font>
 ;<br/>

<a NAME="E2:15_1_1_1"/><b>then </b><i><font color="Green" title="E33">A21</font></i>: 
<font color="Maroon" title="c15">i</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="c13">B1</font>
 
<b>by </b><i><a class="txt" href="pencil_2.html#E10:15_1"><i><font color="Green" title="E27">A15</font></i></a>, <a class="txt" href="pencil_2.html#E15:15_1"><i><font color="Green" title="E28">A16</font></i></a></i>;<br/>
<b>thus </b><font color="Maroon" title="c5">X1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font> = 
<font color="Maroon" title="c13">B1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font>
<b>by </b><i><a class="txt" href="pencil_2.html#E10:15_1"><i><font color="Green" title="E27">A15</font></i></a>, <a class="txt" href="pencil_2.html#E15:15_1"><i><font color="Green" title="E28">A16</font></i></a>, <a class="txt" href="pencil_2.html#E1:15_1_1_1"><i><font color="Green" title="E32">A20</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c14">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font>
<b>by </b><i><a class="txt" href="pencil_2.html#E15:15_1"><i><font color="Green" title="E28">A16</font></i></a>, <a class="txt" href="pencil_2.html#E17:15_1"><i><font color="Green" title="E30">A18</font></i></a>, <a class="txt" href="pencil_2.html#E1:15_1_1"><i><font color="Green" title="E31">A19</font></i></a>, <a class="txt" href="pencil_2.html#E2:15_1_1_1"><i><font color="Green" title="E33">A21</font></i></a>, <a class="ref" href="pencil_1.html#T26" title="PENCIL_1:th.26">PENCIL_1:26</a></i>
;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E19:15_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c10">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E12:15"/><i><font color="Green" title="E22">A22</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  <a href="numbers.html#K5" title="NUMBERS:func.5">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#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="c9">f</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font>]
 
<b>from </b><i><a class="ref" href="polynom2.html#S4" title="POLYNOM2:sch.4">POLYNOM2:sch 4</a>(<a class="txt" href="pencil_2.html#E10:15"><i><font color="Green" title="E20">A8</font></i></a>, <a class="txt" href="pencil_2.html#E11:15"><i><font color="Green" title="E21">A9</font></i></a>);<br/></i>
<a NAME="E13:15"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">f</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="c9">f</font>
 
<b>by </b><i><a class="txt" href="pencil_2.html#E2:15"><i><font color="Green" title="E14">A2</font></i></a>, <a class="txt" href="pencil_2.html#E9:15"><i><font color="Green" title="E19">A7</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="E14:15"/><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="c9">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/>
<b>hence </b><a NAME="E15:15"/>
(  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c5">X1</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">Y1</font> &amp; ( for <font color="Olive" title="b1">i</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">i</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="c5">X1</font> holds <br/><font color="Maroon" title="c5">X1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">Y1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> ) )
 <b>by </b><i><a class="txt" href="pencil_2.html#E4:15"><i><font color="Green" title="E16">A4</font></i></a>, <a class="txt" href="pencil_2.html#E9:15"><i><font color="Green" title="E19">A7</font></i></a>, <a class="txt" href="pencil_2.html#E12:15"><i><font color="Green" title="E22">A22</font></i></a></i>;<br/>


</div>
