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



<b>assume </b><a NAME="E1:26"/><i><font color="Green" title="E25">A1</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font> holds  <font color="Maroon" title="c2">A</font> <a href="pencil_1.html#K2" title="PENCIL_1:func.2">.</a> <font color="Olive" title="b1">i</font> is <a href="pencil_1.html#V10" title="PENCIL_1:attr.10">strongly_connected</a>
 ;<br/>

<div><i><font color="Green" title="E26">A2</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font>;<br/><a NAME="E1:26_1"/>
<font color="Maroon" title="c2">A</font> <a href="pencil_1.html#K2" title="PENCIL_1:func.2">.</a> <font color="Maroon" title="c3">i</font> is <a href="pencil_1.html#V10" title="PENCIL_1:attr.10">strongly_connected</a>
 <b>by </b><i><a class="txt" href="pencil_3.html#E1:26"><i><font color="Green" title="E25">A1</font></i></a></i>;<br/><b>hence </b><a NAME="E2:26_1"/>
<font color="Maroon" title="c2">A</font> <a href="pencil_1.html#K2" title="PENCIL_1:func.2">.</a> <font color="Maroon" title="c3">i</font> is <a href="pencil_1.html#V9" title="PENCIL_1:attr.9">connected</a>
 <b>by </b><i><a class="ref" href="pencil_3.html#T4" target="_self" title="PENCIL_3:th.4">Th4</a></i>;<br/></div><b>end;</b></div>
<b>let </b><font color="Maroon" title="c3">f</font> be   <a href="pencil_2.html#NM2" title="PENCIL_2:NM.2">Collineation</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>let </b><font color="Maroon" title="c4">B1</font>, <font color="Maroon" title="c5">B2</font> be    <a href="pencil_2.html#M1" title="PENCIL_2:mode.1">Segre-Coset</a> of <font color="Maroon" title="c2">A</font>;<br/><b>let </b><font color="Maroon" title="c6">b1</font>, <font color="Maroon" title="c7">b2</font>, <font color="Maroon" title="c8">b3</font>, <font color="Maroon" title="c9">b4</font> be  non <a href="pencil_1.html#V13" title="PENCIL_1:attr.13">trivial-yielding</a> <a href="pencil_1.html#V16" title="PENCIL_1:attr.16">Segre-like</a>  <a href="pboole.html#M4" title="PBOOLE:mode.4">ManySortedSubset</a> of  <a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">A</font>;<br/>





<b>assume </b><a NAME="E3:26"/><i><font color="Green" title="E27">A3</font></i>: 
( <font color="Maroon" title="c4">B1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c6">b1</font> &amp; <font color="Maroon" title="c5">B2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c7">b2</font> &amp; <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c4">B1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c8">b3</font> &amp; <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c5">B2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c9">b4</font> )
 ;<br/>

<b>assume </b><a NAME="E4:26"/><i><font color="Green" title="E28">A4</font></i>: 
 <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c6">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c7">b2</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:26_2"/>
( <font color="Maroon" title="c4">B1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c5">B2</font> or <font color="Maroon" title="c4">B1</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c5">B2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_2_1"><b>suppose </b></a><a NAME="E1:26_2_1"/><i><font color="Green" title="E29">A5</font></i>: 
<font color="Maroon" title="c4">B1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c5">B2</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c10">D</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="pencil_1.html#K6" title="PENCIL_1:func.6">Segre_Product</a> <font color="Maroon" title="c2">A</font></span>)</span><b> such that </b><br/><a NAME="E3:26_2_1"/><i><font color="Green" title="E30">A6</font></i>: 
( <font color="Maroon" title="c10">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">B1</font> &amp; <font color="Maroon" title="c10">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">D</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">B2</font> &amp; ( for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c10">D</font> holds <br/><font color="Maroon" title="c10">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> is    <a href="pencil_2.html#M1" title="PENCIL_2:mode.1">Segre-Coset</a> of <font color="Maroon" title="c2">A</font> ) &amp; ( for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">D</font> holds <br/>for <font color="Olive" title="b2">Di</font>, <font color="Olive" title="b3">Di1</font> being   <a href="pencil_2.html#M1" title="PENCIL_2:mode.1">Segre-Coset</a> of <font color="Maroon" title="c2">A</font>  st <font color="Olive" title="b2">Di</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b3">Di1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> holds <br/>( <font color="Olive" title="b2">Di</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b3">Di1</font> &amp; <font color="Olive" title="b2">Di</font> <a href="pencil_3.html#R1" title="PENCIL_3:pred.1">'||'</a> <font color="Olive" title="b3">Di1</font> ) ) )
 <b>by </b><i><a class="txt" href="pencil_3.html#E2:26"><i><font color="Green" title="E26">A2</font></i></a>, <a class="txt" href="pencil_3.html#E3:26"><i><font color="Green" title="E27">A3</font></i></a>, <a class="txt" href="pencil_3.html#E4:26"><i><font color="Green" title="E28">A4</font></i></a>, <a class="ref" href="pencil_3.html#T23" target="_self" title="PENCIL_3:th.23">Th23</a></i>;<br/><b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</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> = <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1</span>)</span>;<br/><b>consider </b><font color="Maroon" title="c11">E</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="pencil_1.html#K6" title="PENCIL_1:func.6">Segre_Product</a> <font color="Maroon" title="c2">A</font></span>)</span><b> such that </b><br/><a NAME="E5:26_2_1"/><i><font color="Green" title="E31">A7</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">D</font> &amp; ( for <font color="Olive" title="b1">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">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="c11">E</font> holds <br/><font color="Maroon" title="c11">E</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">j</font>) ) )
 <b>from </b><i><a class="ref" href="finseq_2.html#S1" title="FINSEQ_2:sch.1">FINSEQ_2:sch 1</a>();<br/></i><a NAME="E6:26_2_1"/><i><font color="Green" title="E32">A8</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">D</font></span>)</span>
 <b>by </b><i><a class="txt" href="pencil_3.html#E5:26_2_1"><i><font color="Green" title="E31">A7</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E7:26_2_1"/>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c10">D</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E3:26"><i><font color="Green" title="E27">A3</font></i></a>, <a class="txt" href="pencil_3.html#E3:26_2_1"><i><font color="Green" title="E30">A6</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="E8:26_2_1"/><b>then </b>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">D</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E9:26_2_1"/><b>then </b><i><font color="Green" title="E33">A9</font></i>: 
<font color="Maroon" title="c11">E</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c4">B1</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E3:26_2_1"><i><font color="Green" title="E30">A6</font></i></a>, <a class="txt" href="pencil_3.html#E5:26_2_1"><i><font color="Green" title="E31">A7</font></i></a>, <a class="txt" href="pencil_3.html#E6:26_2_1"><i><font color="Green" title="E32">A8</font></i></a></i>;<br/><a NAME="E10:26_2_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">E</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="c10">D</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E3:26"><i><font color="Green" title="E27">A3</font></i></a>, <a class="txt" href="pencil_3.html#E3:26_2_1"><i><font color="Green" title="E30">A6</font></i></a>, <a class="txt" href="pencil_3.html#E5:26_2_1"><i><font color="Green" title="E31">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="E11:26_2_1"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">E</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">D</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E12:26_2_1"/><b>then </b><i><font color="Green" title="E34">A10</font></i>: 
<font color="Maroon" title="c11">E</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">E</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c5">B2</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E3:26_2_1"><i><font color="Green" title="E30">A6</font></i></a>, <a class="txt" href="pencil_3.html#E5:26_2_1"><i><font color="Green" title="E31">A7</font></i></a>, <a class="txt" href="pencil_3.html#E6:26_2_1"><i><font color="Green" title="E32">A8</font></i></a></i>;<br/><a NAME="E13:26_2_1"/><i><font color="Green" title="E35">A11</font></i>: 
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">E</font> holds <br/><font color="Maroon" title="c11">E</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> is    <a href="pencil_2.html#M1" title="PENCIL_2:mode.1">Segre-Coset</a> of <font color="Maroon" title="c2">A</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c12">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:26_2_1_1"/>
<font color="Maroon" title="c12">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">E</font>
 ;<br/>

<a NAME="E2:26_2_1_1"/><b>then </b><i><font color="Green" title="E36">A12</font></i>: 
<font color="Maroon" title="c12">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">D</font></span>)</span>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E5:26_2_1"><i><font color="Green" title="E31">A7</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:26_2_1_1"/><b>then </b>
<font color="Maroon" title="c12">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c10">D</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c13">di</font> = <font color="Maroon" title="c10">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font> as    <a href="pencil_2.html#M1" title="PENCIL_2:mode.1">Segre-Coset</a> of <font color="Maroon" title="c2">A</font> <b>by </b><i><a class="txt" href="pencil_3.html#E3:26_2_1"><i><font color="Green" title="E30">A6</font></i></a></i>;<br/>
<a NAME="E5:26_2_1_1"/>
<font color="Maroon" title="c11">E</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c13">di</font>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E5:26_2_1"><i><font color="Green" title="E31">A7</font></i></a>, <a class="txt" href="pencil_3.html#E2:26_2_1_1"><i><font color="Green" title="E36">A12</font></i></a>, <a class="txt" href="pencil_3.html#E6:26_2_1"><i><font color="Green" title="E32">A8</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:26_2_1_1"/>
<font color="Maroon" title="c11">E</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font> is    <a href="pencil_2.html#M1" title="PENCIL_2:mode.1">Segre-Coset</a> of <font color="Maroon" title="c2">A</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E1:26"><i><font color="Green" title="E25">A1</font></i></a>, <a class="ref" href="pencil_2.html#T24" title="PENCIL_2:th.24">PENCIL_2:24</a></i>;<br/>


</div><b>end;</b></div><a NAME="E14:26_2_1"/><i><font color="Green" title="E36">A13</font></i>: 
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">E</font> holds <br/>for <font color="Olive" title="b2">Ei</font>, <font color="Olive" title="b3">Ei1</font> being   <a href="pencil_2.html#M1" title="PENCIL_2:mode.1">Segre-Coset</a> of <font color="Maroon" title="c2">A</font>  st <font color="Olive" title="b2">Ei</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">E</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b3">Ei1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">E</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> holds <br/>( <font color="Olive" title="b2">Ei</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b3">Ei1</font> &amp; <font color="Olive" title="b2">Ei</font> <a href="pencil_3.html#R1" title="PENCIL_3:pred.1">'||'</a> <font color="Olive" title="b3">Ei1</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_2_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c12">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:26_2_1_2"/><i><font color="Green" title="E37">A14</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">i</font> &amp; <font color="Maroon" title="c12">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="c11">E</font> )
 ;<br/>

<b>let </b><font color="Maroon" title="c13">Ei</font>, <font color="Maroon" title="c14">Ei1</font> be    <a href="pencil_2.html#M1" title="PENCIL_2:mode.1">Segre-Coset</a> of <font color="Maroon" title="c2">A</font>;<br/>

<b>assume </b><a NAME="E2:26_2_1_2"/><i><font color="Green" title="E38">A15</font></i>: 
( <font color="Maroon" title="c13">Ei</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">E</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font> &amp; <font color="Maroon" title="c14">Ei1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">E</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> )
 ;<br/>

<a NAME="E3:26_2_1_2"/>
<font color="Maroon" title="c12">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">D</font></span>)</span>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E5:26_2_1"><i><font color="Green" title="E31">A7</font></i></a>, <a class="txt" href="pencil_3.html#E1:26_2_1_2"><i><font color="Green" title="E37">A14</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E4:26_2_1_2"/><b>then </b><i><font color="Green" title="E39">A16</font></i>: 
<font color="Maroon" title="c13">Ei</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font></span>)</span>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E5:26_2_1"><i><font color="Green" title="E31">A7</font></i></a>, <a class="txt" href="pencil_3.html#E2:26_2_1_2"><i><font color="Green" title="E38">A15</font></i></a>, <a class="txt" href="pencil_3.html#E6:26_2_1"><i><font color="Green" title="E32">A8</font></i></a></i>;<br/>
<a NAME="E5:26_2_1_2"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c12">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">E</font> )
 
<b>by </b><i><a class="txt" href="pencil_3.html#E1:26_2_1_2"><i><font color="Green" title="E37">A14</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E6:26_2_1_2"/><b>then </b>
<font color="Maroon" title="c12">i</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#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">D</font></span>)</span>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E5:26_2_1"><i><font color="Green" title="E31">A7</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E7:26_2_1_2"/><b>then </b><i><font color="Green" title="E40">A17</font></i>: 
<font color="Maroon" title="c14">Ei1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E5:26_2_1"><i><font color="Green" title="E31">A7</font></i></a>, <a class="txt" href="pencil_3.html#E2:26_2_1_2"><i><font color="Green" title="E38">A15</font></i></a>, <a class="txt" href="pencil_3.html#E6:26_2_1"><i><font color="Green" title="E32">A8</font></i></a></i>;<br/>
<a NAME="E8:26_2_1_2"/>
<font color="Maroon" title="c12">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c10">D</font>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E5:26_2_1"><i><font color="Green" title="E31">A7</font></i></a>, <a class="txt" href="pencil_3.html#E1:26_2_1_2"><i><font color="Green" title="E37">A14</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c15">Di</font> = <font color="Maroon" title="c10">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font> as    <a href="pencil_2.html#M1" title="PENCIL_2:mode.1">Segre-Coset</a> of <font color="Maroon" title="c2">A</font> <b>by </b><i><a class="txt" href="pencil_3.html#E3:26_2_1"><i><font color="Green" title="E30">A6</font></i></a></i>;<br/>
<a NAME="E10:26_2_1_2"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c12">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">E</font> )
 
<b>by </b><i><a class="txt" href="pencil_3.html#E1:26_2_1_2"><i><font color="Green" title="E37">A14</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E11:26_2_1_2"/><b>then </b>
<font color="Maroon" title="c12">i</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="c10">D</font>
 
<b>by </b><i><a class="txt" href="pencil_3.html#E5:26_2_1"><i><font color="Green" title="E31">A7</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c16">Di1</font> = <font color="Maroon" title="c10">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> as    <a href="pencil_2.html#M1" title="PENCIL_2:mode.1">Segre-Coset</a> of <font color="Maroon" title="c2">A</font> <b>by </b><i><a class="txt" href="pencil_3.html#E3:26_2_1"><i><font color="Green" title="E30">A6</font></i></a></i>;<br/>
<a NAME="E13:26_2_1_2"/><i><font color="Green" title="E41">A18</font></i>: 
( <font color="Maroon" title="c15">Di</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c16">Di1</font> &amp; <font color="Maroon" title="c15">Di</font> <a href="pencil_3.html#R1" title="PENCIL_3:pred.1">'||'</a> <font color="Maroon" title="c16">Di1</font> )
 
<b>by </b><i><a class="txt" href="pencil_3.html#E3:26_2_1"><i><font color="Green" title="E30">A6</font></i></a>, <a class="txt" href="pencil_3.html#E5:26_2_1"><i><font color="Green" title="E31">A7</font></i></a>, <a class="txt" href="pencil_3.html#E1:26_2_1_2"><i><font color="Green" title="E37">A14</font></i></a></i>;<br/>
<a NAME="E14:26_2_1_2"/>
<font color="Maroon" title="c3">f</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a> <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of 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>,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>by </b><i><a class="ref" href="pencil_2.html#D4" title="PENCIL_2:def.4">PENCIL_2:def 4</a></i>;<br/>
<b>hence </b><a NAME="E15:26_2_1_2"/>
<font color="Maroon" title="c13">Ei</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c14">Ei1</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E4:26_2_1_2"><i><font color="Green" title="E39">A16</font></i></a>, <a class="txt" href="pencil_3.html#E7:26_2_1_2"><i><font color="Green" title="E40">A17</font></i></a>, <a class="txt" href="pencil_3.html#E13:26_2_1_2"><i><font color="Green" title="E41">A18</font></i></a>, <a class="ref" href="funct_1.html#T125" title="FUNCT_1:th.125">FUNCT_1:125</a></i>;<br/>

<b>thus </b><a NAME="E16:26_2_1_2"/>
<font color="Maroon" title="c13">Ei</font> <a href="pencil_3.html#R1" title="PENCIL_3:pred.1">'||'</a> <font color="Maroon" title="c14">Ei1</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E4:26_2_1_2"><i><font color="Green" title="E39">A16</font></i></a>, <a class="txt" href="pencil_3.html#E7:26_2_1_2"><i><font color="Green" title="E40">A17</font></i></a>, <a class="txt" href="pencil_3.html#E13:26_2_1_2"><i><font color="Green" title="E41">A18</font></i></a>, <a class="ref" href="pencil_3.html#T20" target="_self" title="PENCIL_3:th.20">Th20</a></i>;<br/>


</div><b>end;</b></div><b>reconsider </b><font color="Maroon" title="c12">fB1</font> = <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c4">B1</font>, <font color="Maroon" title="c13">fB2</font> = <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c5">B2</font> as    <a href="pencil_2.html#M1" title="PENCIL_2:mode.1">Segre-Coset</a> of <font color="Maroon" title="c2">A</font> <b>by </b><i><a class="txt" href="pencil_3.html#E1:26"><i><font color="Green" title="E25">A1</font></i></a>, <a class="ref" href="pencil_2.html#T24" title="PENCIL_2:th.24">PENCIL_2:24</a></i>;<br/><a NAME="E16:26_2_1"/>
<font color="Maroon" title="c3">f</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a> <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of 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>,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>by </b><i><a class="ref" href="pencil_2.html#D4" title="PENCIL_2:def.4">PENCIL_2:def 4</a></i>;<br/><a NAME="E17:26_2_1"/><b>then </b>
<font color="Maroon" title="c12">fB1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c13">fB2</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E1:26_2_1"><i><font color="Green" title="E29">A5</font></i></a>, <a class="ref" href="funct_1.html#T125" title="FUNCT_1:th.125">FUNCT_1:125</a></i>;<br/><b>hence </b><a NAME="E18:26_2_1"/>
 <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c8">b3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c9">b4</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E2:26"><i><font color="Green" title="E26">A2</font></i></a>, <a class="txt" href="pencil_3.html#E3:26"><i><font color="Green" title="E27">A3</font></i></a>, <a class="txt" href="pencil_3.html#E9:26_2_1"><i><font color="Green" title="E33">A9</font></i></a>, <a class="txt" href="pencil_3.html#E12:26_2_1"><i><font color="Green" title="E34">A10</font></i></a>, <a class="txt" href="pencil_3.html#E13:26_2_1"><i><font color="Green" title="E35">A11</font></i></a>, <a class="txt" href="pencil_3.html#E14:26_2_1"><i><font color="Green" title="E36">A13</font></i></a>, <a class="ref" href="pencil_3.html#T23" target="_self" title="PENCIL_3:th.23">Th23</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_2_2"><b>suppose </b></a><a NAME="E1:26_2_2"/>
<font color="Maroon" title="c4">B1</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c5">B2</font>
 ;<br/><div class="add"><a NAME="E2:26_2_2"/><b>then </b>
<font color="Maroon" title="c4">B1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">B2</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E3:26"><i><font color="Green" title="E27">A3</font></i></a>, <a class="txt" href="pencil_3.html#E4:26"><i><font color="Green" title="E28">A4</font></i></a>, <a class="ref" href="pencil_3.html#T11" target="_self" title="PENCIL_3:th.11">Th11</a></i>;<br/><b>hence </b><a NAME="E3:26_2_2"/>
 <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c8">b3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pencil_1.html#K3" title="PENCIL_1:func.3">indx</a> <font color="Maroon" title="c9">b4</font>
 <b>by </b><i><a class="txt" href="pencil_3.html#E3:26"><i><font color="Green" title="E27">A3</font></i></a>, <a class="ref" href="pua2mss1.html#T2" title="PUA2MSS1:th.2">PUA2MSS1:2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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